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"""
.. todo::
WRITEME
"""
import logging
import warnings
import theano
import theano.tensor as T
from theano.compat.six.moves import input, zip as izip
import numpy
np = numpy
from theano.compat import six
from ..commons import EPS, global_rng
from functools import partial
WRAPPER_ASSIGNMENTS = ('__module__', '__name__')
WRAPPER_CONCATENATIONS = ('__doc__',)
WRAPPER_UPDATES = ('__dict__',)
logger = logging.getLogger(__name__)
#Groundhog related imports
import numpy
import random
import string
import copy as pycopy
import theano
import theano.tensor as TT
import inspect
import re
import os
def ensure_dir_exists(directory):
"""
If the dir does not exist recreate it
"""
if not os.path.exists(directory):
os.makedirs(directory)
def overrides(method):
"""Decorator to indicate that the decorated method overrides a method
in superclass.
The decorator code is executed while loading class. Using this method should
have minimal runtime performance implications.
This is based on my idea about how to do this and fwc:s highly improved
algorithm for the implementation
fwc:s algorithm : http://stackoverflow.com/a/14631397/308189
answer : http://stackoverflow.com/a/8313042/308189
How to use:
from overrides import overrides
class SuperClass(object):
def method(self):
return 2
class SubClass(SuperClass):
@overrides
def method(self):
return 1
:raises AssertionError if no match in super classes for the method name
:return method with possibly added (if the method doesn't have one)
docstring from super class
@NOTE: This is based on pip overrides package.
"""
stack = inspect.stack()
base_classes = [s.strip() for s in re.search(r'class.+\((.+)\)\s*:', \
stack[2][4][0]).group(1).split(',')]
if not base_classes:
raise ValueError("overrides decorator: unable to determine base class"
"for method %s" % method.__name__)
# replace each class name in base_classes with the actual class type
derived_class_locals = stack[2][0].f_locals
for i, base_class in enumerate(base_classes):
if '.' not in base_class:
base_classes[i] = derived_class_locals[base_class]
else:
components = base_class.split('.')
# obj is either a module or a class
obj = derived_class_locals[components[0]]
for c in components[1:]:
assert(inspect.ismodule(obj) or inspect.isclass(obj))
obj = getattr(obj, c)
base_classes[i] = obj
for super_class in base_classes:
if hasattr(super_class, method.__name__):
if not method.__doc__:
method.__doc__ = getattr(super_class, method.__name__).__doc__
return method
raise AssertionError('No super class method found for "%s"' % method.__name__)
def concatenate(tensor_list, axis=0):
"""
Alternative implementation of `theano.tensor.concatenate`.
This function does exactly the same thing, but contrary to Theano's own
implementation, the gradient is implemented on the GPU.
Backpropagating through `theano.tensor.concatenate` yields slowdowns
because the inverse operation (splitting) needs to be done on the CPU.
This implementation does not have that problem.
:usage:
>>> x, y = theano.tensor.matrices('x', 'y')
>>> c = concatenate([x, y], axis=1)
:parameters:
- tensor_list : list
list of Theano tensor expressions that should be concatenated.
- axis : int
the tensors will be joined along this axis.
:returns:
- out : tensor
the concatenated tensor expression.
"""
concat_size = sum(tt.shape[axis] for tt in tensor_list)
output_shape = ()
for k in range(axis):
output_shape += (tensor_list[0].shape[k],)
output_shape += (concat_size,)
for k in range(axis + 1, tensor_list[0].ndim):
output_shape += (tensor_list[0].shape[k],)
out = TT.zeros(output_shape)
offset = 0
for tt in tensor_list:
indices = ()
for k in range(axis):
indices += (slice(None),)
indices += (slice(offset, offset + tt.shape[axis]),)
for k in range(axis + 1, tensor_list[0].ndim):
indices += (slice(None),)
out = TT.set_subtensor(out[indices], tt)
offset += tt.shape[axis]
return out
def kl_divergence(p, p_hat):
term1 = p * T.log(p)
term2 = p * T.log(p_hat)
term3 = (1-p) * T.log(1 - p + EPS)
term4 = (1-p) * T.log(1 - p_hat + EPS)
return term1 - term2 + term3 - term4
def sparsity_penalty(h, sparsity_level=0.05, sparse_reg=1e-4):
if h.ndim == 2:
sparsity_level = T.extra_ops.repeat(sparsity_level, h.shape[1])
else:
sparsity_level = T.extra_ops.repeat(sparsity_level, h.shape[0])
sparsity_penalty = 0
avg_act = h.mean(axis=0)
kl_div = self.kl_divergence(sparsity_level, avg_act)
sparsity_penalty = sparse_reg * kl_div.sum()
# Implement KL divergence here.
return sparsity_penalty
def get_key_byname_from_dict(dict_, name):
keys = dict_.keys()
keyval = None
for key in keys:
if key.name == name:
keyval = key
break
return keyval
def print_time(secs):
if secs < 120.:
return '%6.3f sec' % secs
elif secs <= 60 * 60:
return '%6.3f min' % (secs / 60.)
else:
return '%6.3f h ' % (secs / 3600.)
def print_mem(context=None):
if theano.sandbox.cuda.cuda_enabled:
rvals = theano.sandbox.cuda.cuda_ndarray.cuda_ndarray.mem_info()
# Avaliable memory in Mb
available = float(rvals[0]) / 1024. / 1024.
# Total memory in Mb
total = float(rvals[1]) / 1024. / 1024.
if context == None:
print ('Used %.3f Mb Free %.3f Mb, total %.3f Mb' %
(total - available, available, total))
else:
info = str(context)
print (('GPU status : Used %.3f Mb Free %.3f Mb,'
'total %.3f Mb [context %s]') %
(total - available, available, total, info))
def safe_grad(cost, params, known_grads=None):
from collections import OrderedDict
grad_list = T.grad(cost, params, known_grads=known_grads, add_names=True)
grads = OrderedDict({param: grad for param, grad in safe_izip(params, grad_list)})
return grads
def const(value):
return TT.constant(numpy.asarray(value, dtype=theano.config.floatX))
def as_floatX(variable):
"""
This code is taken from pylearn2:
Casts a given variable into dtype config.floatX
numpy ndarrays will remain numpy ndarrays
python floats will become 0-D ndarrays
all other types will be treated as theano tensors
"""
if isinstance(variable, float):
return numpy.cast[theano.config.floatX](variable)
if isinstance(variable, numpy.ndarray):
return numpy.cast[theano.config.floatX](variable)
return theano.tensor.cast(variable, theano.config.floatX)
def copy(x):
new_x = pycopy.copy(x)
new_x.params = [x for x in new_x.params]
new_x.params_grad_scale = [x for x in new_x.params_grad_scale ]
new_x.noise_params = [x for x in new_x.noise_params ]
new_x.noise_params_shape_fn = [x for x in new_x.noise_params_shape_fn]
new_x.updates = [x for x in new_x.updates ]
new_x.additional_gradients = [x for x in new_x.additional_gradients ]
new_x.inputs = [x for x in new_x.inputs ]
new_x.schedules = [x for x in new_x.schedules ]
new_x.properties = [x for x in new_x.properties ]
return new_x
def softmax(x):
if x.ndim == 2:
e = TT.exp(x)
return e / TT.sum(e, axis=1).dimshuffle(0, 'x')
else:
e = TT.exp(x)
return e/ TT.sum(e)
def id_generator(size=5, chars=string.ascii_uppercase + string.digits):
return ''.join(random.choice(chars) for i in xrange(size))
def constant_shape(shape):
return lambda *args, **kwargs : shape
def binVec2Int(binVec):
add = lambda x,y: x+y
return reduce(add,
[int(x) * 2 ** y
for x, y in zip(
list(binVec),range(len(binVec) - 1, -1,
-1))])
def Int2binVec(val, nbits=10):
strVal = '{0:b}'.format(val)
value = numpy.zeros((nbits,), dtype=theano.config.floatX)
if theano.config.floatX == 'float32':
value[:len(strVal)] = [numpy.float32(x) for x in strVal[::-1]]
else:
value[:len(strVal)] = [numpy.float64(x) for x in strVal[::-1]]
return value
def dot(inp, matrix):
"""
Decide the right type of dot product depending on the input
arguments
"""
if 'int' in inp.dtype and inp.ndim >= 2:
return matrix[inp.flatten()]
elif 'int' in inp.dtype:
return matrix[inp]
elif 'float' in inp.dtype and inp.ndim == 3:
shape0 = inp.shape[0]
shape1 = inp.shape[1]
shape2 = inp.shape[2]
return TT.dot(inp.reshape((shape0*shape1, shape2)), matrix)
elif 'float' in inp.dtype and inp.ndim == 2:
return TT.dot(inp, matrix)
else:
return TT.dot(inp, matrix)
def dbg_hook(hook, x):
if not isinstance(x, TT.TensorVariable):
x.out = theano.printing.Print(global_fn=hook)(x.out)
return x
else:
return theano.printing.Print(global_fn=hook)(x)
def make_name(variable, anon="anonymous_variable"):
"""
If variable has a name, returns that name. Otherwise, returns anon.
Parameters
----------
variable : tensor_like
WRITEME
anon : str, optional
WRITEME
Returns
-------
WRITEME
"""
if hasattr(variable, 'name') and variable.name is not None:
return variable.name
return anon
def sharedX(value, name=None, borrow=False, dtype=None, broadcastable=None):
"""
Transform value into a shared variable of type floatX
Parameters
----------
value : WRITEME
name : WRITEME
borrow : WRITEME
dtype : str, optional
data type. Default value is theano.config.floatX
Returns
-------
WRITEME
"""
if dtype is None:
dtype = theano.config.floatX
return theano.shared(numpy.cast[dtype](value),
name=name,
borrow=borrow,
broadcastable=broadcastable)
def as_floatX(variable):
"""
Casts a given variable into dtype `config.floatX`. Numpy ndarrays will
remain numpy ndarrays, python floats will become 0-D ndarrays and
all other types will be treated as theano tensors
Parameters
----------
variable : WRITEME
Returns
-------
WRITEME
"""
if isinstance(variable, float):
return np.cast[theano.config.floatX](variable)
if isinstance(variable, np.ndarray):
return np.cast[theano.config.floatX](variable)
return theano.tensor.cast(variable, theano.config.floatX)
def constantX(value):
"""
Returns a constant of value `value` with floatX dtype
Parameters
----------
variable : WRITEME
Returns
-------
WRITEME
"""
return theano.tensor.constant(np.asarray(value,
dtype=theano.config.floatX))
def subdict(d, keys):
"""
Create a subdictionary of d with the keys in keys
Parameters
----------
d : WRITEME
keys : WRITEME
Returns
-------
WRITEME
"""
result = {}
for key in keys:
if key in d:
result[key] = d[key]
return result
def safe_update(dict_to, dict_from):
"""
Like dict_to.update(dict_from), except don't overwrite any keys.
Parameters
----------
dict_to : WRITEME
dict_from : WRITEME
Returns
-------
WRITEME
"""
for key, val in six.iteritems(dict_from):
if key in dict_to:
raise KeyError(key)
dict_to[key] = val
return dict_to
class CallbackOp(theano.gof.Op):
"""
A Theano Op that implements the identity transform but also does an
arbitrary (user-specified) side effect.
Parameters
----------
callback : WRITEME
"""
view_map = {0: [0]}
def __init__(self, callback):
self.callback = callback
def make_node(self, xin):
"""
.. todo::
WRITEME
"""
xout = xin.type.make_variable()
return theano.gof.Apply(op=self, inputs=[xin], outputs=[xout])
def perform(self, node, inputs, output_storage):
"""
.. todo::
WRITEME
"""
xin, = inputs
xout, = output_storage
xout[0] = xin
self.callback(xin)
def grad(self, inputs, output_gradients):
"""
.. todo::
WRITEME
"""
return output_gradients
def R_op(self, inputs, eval_points):
"""
.. todo::
WRITEME
"""
return [x for x in eval_points]
def __eq__(self, other):
"""
.. todo::
WRITEME
"""
return type(self) == type(other) and self.callback == other.callback
def hash(self):
"""
.. todo::
WRITEME
"""
return hash(self.callback)
def __hash__(self):
"""
.. todo::
WRITEME
"""
return self.hash()
def get_dataless_dataset(model):
"""
Loads the dataset that model was trained on, without loading data.
This is useful if you just need the dataset's metadata, like for
formatting views of the model's weights.
Parameters
----------
model : Model
Returns
-------
dataset : Dataset
The data-less dataset as described above.
"""
global yaml_parse
global control
if yaml_parse is None:
from pylearn2.config import yaml_parse
if control is None:
from pylearn2.datasets import control
control.push_load_data(False)
try:
rval = yaml_parse.load(model.dataset_yaml_src)
finally:
control.pop_load_data()
return rval
def safe_zip(*args):
"""Like zip, but ensures arguments are of same length"""
base = len(args[0])
for i, arg in enumerate(args[1:]):
if len(arg) != base:
raise ValueError("Argument 0 has length %d but argument %d has "
"length %d" % (base, i+1, len(arg)))
return zip(*args)
def safe_izip(*args):
"""Like izip, but ensures arguments are of same length"""
assert all([len(arg) == len(args[0]) for arg in args])
return izip(*args)
def gpu_mem_free():
"""
Memory free on the GPU
Returns
-------
megs_free : float
Number of megabytes of memory free on the GPU used by Theano
"""
global cuda
if cuda is None:
from theano.sandbox import cuda
return cuda.mem_info()[0]/1024./1024
class _ElemwiseNoGradient(theano.tensor.Elemwise):
"""
A Theano Op that applies an elementwise transformation and reports
having no gradient.
"""
def connection_pattern(self, node):
"""
Report being disconnected to all inputs in order to have no gradient
at all.
Parameters
----------
node : WRITEME
"""
return [[False]]
def grad(self, inputs, output_gradients):
"""
Report being disconnected to all inputs in order to have no gradient
at all.
Parameters
----------
inputs : WRITEME
output_gradients : WRITEME
"""
return [theano.gradient.DisconnectedType()()]
# Call this on a theano variable to make a copy of that variable
# No gradient passes through the copying operation
# This is equivalent to making my_copy = var.copy() and passing
# my_copy in as part of consider_constant to tensor.grad
# However, this version doesn't require as much long range
# communication between parts of the code
block_gradient = _ElemwiseNoGradient(theano.scalar.identity)
def is_block_gradient(op):
"""
Parameters
----------
op : object
Returns
-------
is_block_gradient : bool
True if op is a gradient-blocking op, False otherwise
"""
return isinstance(op, _ElemwiseNoGradient)
def safe_union(a, b):
"""
Does the logic of a union operation without the non-deterministic ordering
of python sets.
Parameters
----------
a : list
b : list
Returns
-------
c : list
A list containing one copy of each element that appears in at
least one of `a` or `b`.
"""
if not isinstance(a, list):
raise TypeError("Expected first argument to be a list, but got " +
str(type(a)))
assert isinstance(b, list)
c = []
for x in a + b:
if x not in c:
c.append(x)
return c
# This was moved to theano, but I include a link to avoid breaking
# old imports
from theano.printing import hex_digest as _hex_digest
def hex_digest(*args, **kwargs):
warnings.warn("hex_digest has been moved into Theano. "
"pylearn2.utils.hex_digest will be removed on or after "
"2014-08-26")
def function(*args, **kwargs):
"""
A wrapper around theano.function that disables the on_unused_input error.
Almost no part of pylearn2 can assume that an unused input is an error, so
the default from theano is inappropriate for this project.
"""
return theano.function(*args, on_unused_input='ignore', **kwargs)
def grad(*args, **kwargs):
"""
A wrapper around theano.gradient.grad that disable the disconnected_inputs
error. Almost no part of pylearn2 can assume that a disconnected input
is an error.
"""
return theano.gradient.grad(*args, disconnected_inputs='ignore', **kwargs)
# Groups of Python types that are often used together in `isinstance`
if six.PY3:
py_integer_types = (int, np.integer)
py_number_types = (int, float, complex, np.number)
else:
py_integer_types = (int, long, np.integer) # noqa
py_number_types = (int, long, float, complex, np.number) # noqa
py_float_types = (float, np.floating)
py_complex_types = (complex, np.complex)
def get_choice(choice_to_explanation):
"""
.. todo::
WRITEME
Parameters
----------
choice_to_explanation : dict
Dictionary mapping possible user responses to strings describing
what that response will cause the script to do
Returns
-------
WRITEME
"""
d = choice_to_explanation
for key in d:
logger.info('\t{0}: {1}'.format(key, d[key]))
prompt = '/'.join(d.keys())+'? '
first = True
choice = ''
while first or choice not in d.keys():
if not first:
warnings.warn('unrecognized choice')
first = False
choice = input(prompt)
return choice
def float32_floatX(f):
"""
This function changes floatX to float32 for the call to f.
Useful in GPU tests.
Parameters
----------
f : WRITEME
Returns
-------
WRITEME
"""
def new_f(*args, **kwargs):
"""
.. todo::
WRITEME
"""
old_floatX = theano.config.floatX
theano.config.floatX = 'float32'
try:
f(*args, **kwargs)
finally:
theano.config.floatX = old_floatX
# If we don't do that, tests function won't be run.
new_f.func_name = f.func_name
return new_f
|
State Before: Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
β’ MeasurableSet (f '' s) State After: Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
β’ MeasurableSet (range fun x => f βx) Tactic: rw [image_eq_range] State Before: Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
β’ MeasurableSet (range fun x => f βx) State After: Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ MeasurableSet (range fun x => f βx) Tactic: haveI : PolishSpace s := IsClosed.polishSpace hs State Before: Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ MeasurableSet (range fun x => f βx) State After: case f_cont
Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ Continuous fun x => f βx
case f_inj
Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ Injective fun x => f βx Tactic: apply measurableSet_range_of_continuous_injective State Before: case f_cont
Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ Continuous fun x => f βx State After: no goals Tactic: rwa [continuousOn_iff_continuous_restrict] at f_cont State Before: case f_inj
Ξ± : Type ?u.320629
instββ΅ : TopologicalSpace Ξ±
ΞΉ : Type ?u.320635
Ξ³ : Type u_2
tΞ³ : TopologicalSpace Ξ³
instββ΄ : PolishSpace Ξ³
Ξ² : Type u_1
instβΒ³ : TopologicalSpace Ξ²
instβΒ² : T2Space Ξ²
instβΒΉ : MeasurableSpace Ξ²
instβ : BorelSpace Ξ²
s : Set Ξ³
hs : IsClosed s
f : Ξ³ β Ξ²
f_cont : ContinuousOn f s
f_inj : InjOn f s
this : PolishSpace βs
β’ Injective fun x => f βx State After: no goals Tactic: rwa [injOn_iff_injective] at f_inj |
namespace hidden
class inhabited (Ξ± : Type _) := (default : Ξ±)
#reduce default Prop
#print default
instance Prop_inhabited : inhabited Prop :=
β¨trueβ©
instance bool_inhabited : inhabited bool :=
β¨ttβ©
instance nat_inhabited : inhabited nat :=
β¨0β©
instance unit_inhabited : inhabited unit :=
β¨()β©
def default (Ξ± : Type) [s : inhabited Ξ±] : Ξ± :=
@inhabited.default Ξ± s
instance prod_inhabited
{Ξ± Ξ² : Type} [inhabited Ξ±] [inhabited Ξ²] : inhabited (prod Ξ± Ξ²) :=
β¨(default Ξ±, default Ξ²)β©
#reduce default (nat Γ bool)
instance inhabited_fun (Ξ± : Type) {Ξ² : Type} [inhabited Ξ²] :
inhabited (Ξ± β Ξ²) :=
β¨(Ξ» a : Ξ±, default Ξ²)β©
#check default (nat β nat Γ bool)
#reduce default (nat β nat Γ bool)
universes u v
instance prod_has_add {Ξ± : Type u} {Ξ² : Type v}
[has_add Ξ±] [has_add Ξ²] :
has_add (Ξ± Γ Ξ²) :=
β¨Ξ» β¨aβ, bββ© β¨aβ, bββ©, β¨aβ + aβ, bβ + bββ©β©
#check (1, 2) + (3, 4) -- β Γ β
#reduce (1, 2) + (3, 4) -- (4, 6)
end hidden
universe u
def list.add {Ξ± : Type u} [has_add Ξ±] :
list Ξ± β list Ξ± β list Ξ±
| a [] := a
| [] b := b
| (a::as) (b::bs) := (a + b)::(list.add as bs)
instance {Ξ± : Type u} [has_add Ξ±] : has_add (list Ξ±) :=
β¨list.addβ©
#reduce [1,2,3] + [4,5,6,7,8]
set_option pp.implicit true
-- set_option trace.class_instances true
def step (a b x : β) : β :=
if x < a β¨ x > b then 0 else 1
#print definition step
def inhabited.set (Ξ± : Type*) : inhabited (set Ξ±) :=
by unfold set; apply_instance
#print inhabited.set
instance bool_to_Prop : has_coe bool Prop :=
β¨Ξ» b, b = ttβ©
#reduce if tt then 3 else 5
#reduce if ff then 3 else 5
#print notation
def list.to_set {Ξ± : Type u} : list Ξ± β set Ξ±
| [] := β
| (h::t) := {h} βͺ list.to_set t
instance list_to_set_coe (Ξ± : Type u) :
has_coe (list Ξ±) (set Ξ±) :=
β¨list.to_setβ©
def s : set nat := {1, 2}
def l : list nat := [3, 4]
#check s βͺ l -- set nat
structure Semigroup : Type (u+1) :=
(carrier : Type u)
(mul : carrier β carrier β carrier)
(mul_assoc : β a b c : carrier,
mul (mul a b) c = mul a (mul b c))
instance Semigroup_has_mul (S : Semigroup) :
has_mul (S.carrier) :=
β¨S.mulβ©
#check Semigroup.carrier
instance Semigroup_to_sort : has_coe_to_sort Semigroup :=
{S := Type u, coe := Ξ» S, S.carrier}
example (S : Semigroup) (a b c : S) :
(a * b) * c = a * (b * c) :=
Semigroup.mul_assoc _ a b c
structure morphism (S1 S2 : Semigroup) :=
(mor : S1 β S2)
(resp_mul : β a b : S1, mor (a * b) = (mor a) * (mor b))
#check @morphism.mor
instance morphism_to_fun (S1 S2 : Semigroup) :
has_coe_to_fun (morphism S1 S2) :=
{ F := Ξ» _, S1 β S2,
coe := Ξ» m, m.mor }
lemma resp_mul {S1 S2 : Semigroup}
(f : morphism S1 S2) (a b : S1) :
f (a * b) = f a * f b :=
f.resp_mul a b
theorem semi_assoc (S1 S2 : Semigroup) (f : morphism S1 S2) (a : S1) :
f (a * a * a) = f a * f a * f a :=
calc
f (a * a * a) = f (a * a) * f a : by rw [resp_mul f]
... = f a * f a * f a : by rw [resp_mul f]
#check semi_assoc
set_option pp.coercions false
#check semi_assoc |
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
L M N K : ModuleCat R
f : L βΆ M
g : M βΆ N
wβ : f β« g = 0
h k : _root_.homology f g wβ βΆ K
w :
β (x : { x // x β LinearMap.ker g }),
βh (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x)) =
βk (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x))
β’ h = k
[PROOFSTEP]
refine'
cokernel_funext fun n =>
_
-- porting note: as `equiv_rw` was not ported, it was replaced by `Equiv.surjective`
-- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `β`.
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
L M N K : ModuleCat R
f : L βΆ M
g : M βΆ N
wβ : f β« g = 0
h k : _root_.homology f g wβ βΆ K
w :
β (x : { x // x β LinearMap.ker g }),
βh (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x)) =
βk (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x))
n : (CategoryTheory.forget (ModuleCat R)).obj (Subobject.underlying.obj (kernelSubobject g))
β’ βh (β(cokernel.Ο (imageToKernel f g wβ)) n) = βk (β(cokernel.Ο (imageToKernel f g wβ)) n)
[PROOFSTEP]
obtain β¨n, rflβ© := (kernelSubobjectIso g βͺβ« ModuleCat.kernelIsoKer g).toLinearEquiv.toEquiv.symm.surjective n
[GOAL]
case intro
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
L M N K : ModuleCat R
f : L βΆ M
g : M βΆ N
wβ : f β« g = 0
h k : _root_.homology f g wβ βΆ K
w :
β (x : { x // x β LinearMap.ker g }),
βh (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x)) =
βk (β(cokernel.Ο (imageToKernel f g wβ)) (βtoKernelSubobject x))
n : β(of R { x // x β LinearMap.ker g })
β’ βh
(β(cokernel.Ο (imageToKernel f g wβ))
(β(LinearEquiv.toEquiv (Iso.toLinearEquiv (kernelSubobjectIso g βͺβ« kernelIsoKer g))).symm n)) =
βk
(β(cokernel.Ο (imageToKernel f g wβ))
(β(LinearEquiv.toEquiv (Iso.toLinearEquiv (kernelSubobjectIso g βͺβ« kernelIsoKer g))).symm n))
[PROOFSTEP]
exact w n
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
Cβ D C : HomologicalComplex (ModuleCat R) c
i : ΞΉ
x y : β(Subobject.underlying.obj (cycles C i))
w : β(Subobject.arrow (cycles C i)) x = β(Subobject.arrow (cycles C i)) y
β’ x = y
[PROOFSTEP]
apply_fun (C.cycles i).arrow using (ModuleCat.mono_iff_injective _).mp (cycles C i).arrow_mono
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
Cβ D C : HomologicalComplex (ModuleCat R) c
i : ΞΉ
x y : β(Subobject.underlying.obj (cycles C i))
w : β(Subobject.arrow (cycles C i)) x = β(Subobject.arrow (cycles C i)) y
β’ β(Subobject.arrow (cycles C i)) x = β(Subobject.arrow (cycles C i)) y
[PROOFSTEP]
exact w
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f : C βΆ D
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(Hom.f f i) βx β LinearMap.ker (dFrom D i)
[PROOFSTEP]
rw [LinearMap.mem_ker, Hom.comm_from_apply, x.2, map_zero]
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f : C βΆ D
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(cyclesMap f i) (toCycles x) =
toCycles { val := β(Hom.f f i) βx, property := (_ : β(Hom.f f i) βx β LinearMap.ker (dFrom D i)) }
[PROOFSTEP]
ext
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f : C βΆ D
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(Subobject.arrow (cycles D i)) (β(cyclesMap f i) (toCycles x)) =
β(Subobject.arrow (cycles D i))
(toCycles { val := β(Hom.f f i) βx, property := (_ : β(Hom.f f i) βx β LinearMap.ker (dFrom D i)) })
[PROOFSTEP]
rw [cyclesMap_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow]
[GOAL]
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
β’ (homologyFunctor (ModuleCat R) c i).map f = (homologyFunctor (ModuleCat R) c i).map g
[PROOFSTEP]
apply homology_ext
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
β’ β (x : { x // x β LinearMap.ker (dFrom C i) }),
β((homologyFunctor (ModuleCat R) c i).map f)
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x)) =
β((homologyFunctor (ModuleCat R) c i).map g)
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x))
[PROOFSTEP]
intro x
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β((homologyFunctor (ModuleCat R) c i).map f)
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x)) =
β((homologyFunctor (ModuleCat R) c i).map g)
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x))
[PROOFSTEP]
simp only [homologyFunctor_map]
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(homology.map (_ : dTo C i β« dFrom C i = 0) (_ : dTo D i β« dFrom D i = 0) (Hom.sqTo f i) (Hom.sqFrom f i)
(_ : (Hom.sqTo f i).right = (Hom.sqTo f i).right))
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x)) =
β(homology.map (_ : dTo C i β« dFrom C i = 0) (_ : dTo D i β« dFrom D i = 0) (Hom.sqTo g i) (Hom.sqFrom g i)
(_ : (Hom.sqTo g i).right = (Hom.sqTo g i).right))
(β(cokernel.Ο (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i β« dFrom C i = 0))) (βtoKernelSubobject x))
[PROOFSTEP]
erw [homology.Ο_map_apply, homology.Ο_map_apply]
-- To check that two elements are equal mod boundaries, it suffices to exhibit a boundary:
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(homology.Ο (dTo D i) (dFrom D i) (_ : dTo D i β« dFrom D i = 0))
(β(kernelSubobjectMap (Hom.sqFrom f i)) (βtoKernelSubobject x)) =
β(homology.Ο (dTo D i) (dFrom D i) (_ : dTo D i β« dFrom D i = 0))
(β(kernelSubobjectMap (Hom.sqFrom g i)) (βtoKernelSubobject x))
[PROOFSTEP]
refine'
cokernel_Ο_imageSubobject_ext _ _ ((toPrev i h.hom) x.1)
_
-- Moreover, to check that two cycles are equal, it suffices to check their underlying elements:
[GOAL]
case w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(kernelSubobjectMap (Hom.sqFrom f i)) (βtoKernelSubobject x) =
β(kernelSubobjectMap (Hom.sqFrom g i)) (βtoKernelSubobject x) +
β(imageToKernel (dTo D i) (dFrom D i) (_ : dTo D i β« dFrom D i = 0))
(β(factorThruImageSubobject (dTo D i)) (β(β(toPrev i) h.hom) βx))
[PROOFSTEP]
ext
[GOAL]
case w.w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(Subobject.arrow (cycles D i)) (β(kernelSubobjectMap (Hom.sqFrom f i)) (βtoKernelSubobject x)) =
β(Subobject.arrow (cycles D i))
(β(kernelSubobjectMap (Hom.sqFrom g i)) (βtoKernelSubobject x) +
β(imageToKernel (dTo D i) (dFrom D i) (_ : dTo D i β« dFrom D i = 0))
(β(factorThruImageSubobject (dTo D i)) (β(β(toPrev i) h.hom) βx)))
[PROOFSTEP]
erw [map_add, CategoryTheory.Limits.kernelSubobjectMap_arrow_apply,
CategoryTheory.Limits.kernelSubobjectMap_arrow_apply, ModuleCat.toKernelSubobject_arrow, imageToKernel_arrow_apply,
imageSubobject_arrow_comp_apply]
[GOAL]
case w.w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ β(Hom.sqFrom f i).left βx = β(Hom.sqFrom g i).left βx + β(dTo D i) (β(β(toPrev i) h.hom) βx)
[PROOFSTEP]
rw [Hom.sqFrom_left, Hom.sqFrom_left, h.comm i, LinearMap.add_apply, LinearMap.add_apply, prevD_eq_toPrev_dTo,
dNext_eq_dFrom_fromNext, comp_apply, comp_apply, x.2, map_zero]
[GOAL]
case w.w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ 0 + β(dTo D i) (β(β(toPrev i) h.hom) βx) + β(Hom.f g i) βx = β(Hom.f g i) βx + β(dTo D i) (β(β(toPrev i) h.hom) βx)
[PROOFSTEP]
dsimp
[GOAL]
case w.w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ 0 + β(dTo D i) (β(β(toPrev i) h.hom) βx) + β(Hom.f g i) βx = β(Hom.f g i) βx + β(dTo D i) (β(β(toPrev i) h.hom) βx)
[PROOFSTEP]
abel
[GOAL]
case w.w
R : Type v
instβ : Ring R
ΞΉ : Type u_1
c : ComplexShape ΞΉ
C D : HomologicalComplex (ModuleCat R) c
f g : C βΆ D
h : Homotopy f g
i : ΞΉ
x : { x // x β LinearMap.ker (dFrom C i) }
β’ 0 + β(dTo D i) (β(β(toPrev i) h.hom) βx) + β(Hom.f g i) βx = β(Hom.f g i) βx + β(dTo D i) (β(β(toPrev i) h.hom) βx)
[PROOFSTEP]
abel
|
REBOL [
Title: "Whatever"
Date: 2-Feb-2000
File: %whatever.reb
Author: "Whatever"
Version: 1.2.3
]
dict: [
"name" john
"surname" doe
"country" Spain
]
loop 200000 [
print [select dict "surname"]
] |
(* Title: HOL/TLA/Stfun.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
*)
section \<open>States and state functions for TLA as an "intensional" logic\<close>
theory Stfun
imports Intensional
begin
typedecl state
instance state :: world ..
type_synonym 'a stfun = "state \<Rightarrow> 'a"
type_synonym stpred = "bool stfun"
consts
(* Formalizing type "state" would require formulas to be tagged with
their underlying state space and would result in a system that is
much harder to use. (Unlike Hoare logic or Unity, TLA has quantification
over state variables, and therefore one usually works with different
state spaces within a single specification.) Instead, "state" is just
an anonymous type whose only purpose is to provide "Skolem" constants.
Moreover, we do not define a type of state variables separate from that
of arbitrary state functions, again in order to simplify the definition
of flexible quantification later on. Nevertheless, we need to distinguish
state variables, mainly to define the enabledness of actions. The user
identifies (tuples of) "base" state variables in a specification via the
"meta predicate" basevars, which is defined here.
*)
stvars :: "'a stfun \<Rightarrow> bool"
syntax
"_PRED" :: "lift \<Rightarrow> 'a" ("PRED _")
"_stvars" :: "lift \<Rightarrow> bool" ("basevars _")
translations
"PRED P" => "(P::state \<Rightarrow> _)"
"_stvars" == "CONST stvars"
(* Base variables may be assigned arbitrary (type-correct) values.
Note that vs may be a tuple of variables. The correct identification
of base variables is up to the user who must take care not to
introduce an inconsistency. For example, "basevars (x,x)" would
definitely be inconsistent.
*)
overloading stvars \<equiv> stvars
begin
definition stvars :: "(state \<Rightarrow> 'a) \<Rightarrow> bool"
where basevars_def: "stvars vs == range vs = UNIV"
end
lemma basevars: "\<And>vs. basevars vs \<Longrightarrow> \<exists>u. vs u = c"
apply (unfold basevars_def)
apply (rule_tac b = c and f = vs in rangeE)
apply auto
done
lemma base_pair1: "\<And>x y. basevars (x,y) \<Longrightarrow> basevars x"
apply (simp (no_asm) add: basevars_def)
apply (rule equalityI)
apply (rule subset_UNIV)
apply (rule subsetI)
apply (drule_tac c = "(xa, _) " in basevars)
apply auto
done
lemma base_pair2: "\<And>x y. basevars (x,y) \<Longrightarrow> basevars y"
apply (simp (no_asm) add: basevars_def)
apply (rule equalityI)
apply (rule subset_UNIV)
apply (rule subsetI)
apply (drule_tac c = "(_, xa) " in basevars)
apply auto
done
lemma base_pair: "\<And>x y. basevars (x,y) \<Longrightarrow> basevars x & basevars y"
apply (rule conjI)
apply (erule base_pair1)
apply (erule base_pair2)
done
(* Since the unit type has just one value, any state function can be
regarded as "base". The following axiom can sometimes be useful
because it gives a trivial solution for "basevars" premises.
*)
lemma unit_base: "basevars (v::unit stfun)"
apply (unfold basevars_def)
apply auto
done
lemma baseE: "\<lbrakk> basevars v; \<And>x. v x = c \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
apply (erule basevars [THEN exE])
apply blast
done
(* -------------------------------------------------------------------------------
The following shows that there should not be duplicates in a "stvars" tuple:
*)
lemma "\<And>v. basevars (v::bool stfun, v) \<Longrightarrow> False"
apply (erule baseE)
apply (subgoal_tac "(LIFT (v,v)) x = (True, False)")
prefer 2
apply assumption
apply simp
done
end
|
The left-hand side of the equation $a(x - y) = ax - ay$ is equal to the right-hand side. |
Our Intake Unit is available to individuals and families in order to assess individual service needs, and secure appropriate supports, whether directly through Arc/Morris or through a referral to another entity.
Additionally, the Intake Unit can provide information on an array of topics germane to intellectual and related developmental disabilities.
The Intake Unit of The Arc/Morris can be reached by calling 973.326.9750 x211 or you can fill out the form (PDF) and return to Intake Unit, The Arc/Morris, Executive Drive, PO Box 123, Morris Plains, NJ 07950. |
\newcommand{\shcmd}[2]{
\subsubsection{\mach{\##1} - #2}
\index{#1@\mach{\##1}}
\index{\##1@\mach{\##1}}
}
\section{The SwapForth shell} \index{shell}
The SwapForth shell is a Python program that runs on the host PC.
It has a number of advantages over raw UART access:
\begin{itemize}
\item command-line editing
\item command history
\item word completion on TAB
\item local file \mach{include}
\item \mach{\^{}C} for interrupt
\end{itemize}
\subsection{Invocation}
The shell is a Python program. To run it, go to the appropriate directory and type:
\begin{framed}
\begin{Verbatim}
python shell.py -h /dev/ttyUSB0
\end{Verbatim}
\end{framed}
\subsection{Command reference}
\shcmd{bye}{quit SwapForth shell}
\shcmd{flash}{copy the target state to a local file}
\shcmd{include}{send local source file}
\shcmd{noverbose}{turn off include echo}
\shcmd{time}{measure execution time}
\section{Tethered Mode}
J1b SwapForth supports
\term{tethered mode},
which makes the UART protocol easier to use for host programs.
The SwapForth shell uses
tethered mode.
To enter tethered mode, write one to the variable \wordidx{tth}:
\begin{framed}
\begin{Verbatim}
1 tth !
\end{Verbatim}
\end{framed}
In tethered mode, \word{accept} transmits byte value 30 (hex \mach{1e}, ASCII code RS).
This allows the listening program to know that the target machine is ready to accept a line of input.
In addition, \word{accept} does not echo characters as they are typed.
|
(* Title: HOL/Auth/n_flash_lemma_on_inv__60.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_flash Protocol Case Study*}
theory n_flash_lemma_on_inv__60 imports n_flash_base
begin
section{*All lemmas on causal relation between inv__60 and some rule r*}
lemma n_NI_Local_Get_Put_HeadVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put_Head N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_Get_Put_Head N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Remote_Get_Nak_HomeVsinv__60:
assumes a1: "(\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Get_Nak_Home dst)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain dst where a1:"dst\<le>N\<and>r=n_NI_Remote_Get_Nak_Home dst" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(dst=p__Inv4)\<or>(dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(dst=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(dst~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Remote_Get_Put_HomeVsinv__60:
assumes a1: "(\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Get_Put_Home dst)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain dst where a1:"dst\<le>N\<and>r=n_NI_Remote_Get_Put_Home dst" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(dst=p__Inv4)\<or>(dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(dst=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(dst~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_1Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_1 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_1 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_2Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_2 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_2 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_3Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_3 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_3 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_4Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_4 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_4 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_5Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_5 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_5 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_6Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_6 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_6 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_7__part__0Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__0 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__0 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_7__part__1Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__1 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7__part__1 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_7_NODE_Get__part__0Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__0 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__0 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_7_NODE_Get__part__1Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__1 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_7_NODE_Get__part__1 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_8_HomeVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeShrSet'')) (Const true)))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_8_Home_NODE_GetVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home_NODE_Get N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_8_Home_NODE_Get N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeShrSet'')) (Const true)))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_8Vsinv__60:
assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8 N src pp)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8 N src pp" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_8_NODE_GetVsinv__60:
assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8_NODE_Get N src pp)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_8_NODE_Get N src pp" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_9__part__0Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__0 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__0 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_9__part__1Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__1 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_9__part__1 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_10_HomeVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_10_Home N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_10_Home N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeShrSet'')) (Const true)))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_10Vsinv__60:
assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_10 N src pp)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_Local_GetX_PutX_10 N src pp" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)"
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 b1 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4)))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadVld'')) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_GetX_PutX_11Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_PutX_11 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Local_GetX_PutX_11 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Remote_GetX_Nak_HomeVsinv__60:
assumes a1: "(\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_GetX_Nak_Home dst)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain dst where a1:"dst\<le>N\<and>r=n_NI_Remote_GetX_Nak_Home dst" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(dst=p__Inv4)\<or>(dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(dst=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(dst~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Remote_GetX_PutX_HomeVsinv__60:
assumes a1: "(\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_GetX_PutX_Home dst)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain dst where a1:"dst\<le>N\<and>r=n_NI_Remote_GetX_PutX_Home dst" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(dst=p__Inv4)\<or>(dst~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(dst=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_GetX)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''InvSet'') p__Inv4)) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(dst~=p__Inv4)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_GetX)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''InvSet'') p__Inv4)) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_InvAck_exists_HomeVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_InvAck_exists_Home src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_InvAck_exists_Home src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_InvAck_existsVsinv__60:
assumes a1: "(\<exists> src pp. src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_InvAck_exists src pp)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src pp where a1:"src\<le>N\<and>pp\<le>N\<and>src~=pp\<and>r=n_NI_InvAck_exists src pp" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4\<and>pp~=p__Inv4)\<or>(src~=p__Inv4\<and>pp=p__Inv4)\<or>(src~=p__Inv4\<and>pp~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4\<and>pp~=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4\<and>pp~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_InvAck_1Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_InvAck_1 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_InvAck_1 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_InvAck_2Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_InvAck_2 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_InvAck_2 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_InvAck_3Vsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_InvAck_3 N src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_InvAck_3 N src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_ReplaceVsinv__60:
assumes a1: "(\<exists> src. src\<le>N\<and>r=n_NI_Replace src)" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain src where a1:"src\<le>N\<and>r=n_NI_Replace src" apply fastforce done
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "(src=p__Inv4)\<or>(src~=p__Inv4)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(src=p__Inv4)"
have "((formEval (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) s))" by auto
moreover {
assume c1: "((formEval (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) s))"
have "?P1 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) s))"
have "?P2 s"
proof(cut_tac a1 a2 b1 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
assume b1: "(src~=p__Inv4)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_PI_Local_Get_GetVsinv__60:
assumes a1: "(r=n_PI_Local_Get_Get )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_PI_Local_GetX_GetX__part__0Vsinv__60:
assumes a1: "(r=n_PI_Local_GetX_GetX__part__0 )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_PI_Local_GetX_GetX__part__1Vsinv__60:
assumes a1: "(r=n_PI_Local_GetX_GetX__part__1 )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_PI_Local_GetX_PutX_HeadVld__part__0Vsinv__60:
assumes a1: "(r=n_PI_Local_GetX_PutX_HeadVld__part__0 N )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_PI_Local_GetX_PutX_HeadVld__part__1Vsinv__60:
assumes a1: "(r=n_PI_Local_GetX_PutX_HeadVld__part__1 N )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))\<or>((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))) (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''Dirty'')) (Const false)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HeadPtr'')) (Const (index p__Inv4))))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''ShrVld'')) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))) (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''Dir'') ''HomeHeadPtr'')) (Const false)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Nak_HomeVsinv__60:
assumes a1: "(r=n_NI_Nak_Home )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_NI_Local_PutVsinv__60:
assumes a1: "(r=n_NI_Local_Put )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_NI_Local_PutXAcksDoneVsinv__60:
assumes a1: "(r=n_NI_Local_PutXAcksDone )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "?P1 s"
proof(cut_tac a1 a2 , auto) qed
then show "invHoldForRule s f r (invariants N)" by auto
qed
lemma n_NI_ShWbVsinv__60:
assumes a1: "(r=n_NI_ShWb N )" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a2 obtain p__Inv4 where a2:"p__Inv4\<le>N\<and>f=inv__60 p__Inv4" apply fastforce done
have "((formEval (andForm (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc''))) (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) s))\<or>((formEval (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) s))\<or>((formEval (andForm (neg (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc'')))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))\<or>((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))" by auto
moreover {
assume c1: "((formEval (andForm (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc''))) (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Cmd'')) (Const SHWB_ShWb)) (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)) s))"
have "?P3 s"
apply (cut_tac a1 a2 c1, simp, rule_tac x="(neg (andForm (eqn (IVar (Field (Field (Ident ''Sta'') ''HomeUniMsg'') ''Cmd'')) (Const UNI_PutX)) (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (Const (index p__Inv4)) (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''Proc'')))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume c1: "((formEval (andForm (neg (eqn (IVar (Field (Field (Ident ''Sta'') ''ShWbMsg'') ''HomeProc'')) (Const false))) (neg (eqn (IVar (Para (Field (Field (Ident ''Sta'') ''Dir'') ''ShrSet'') p__Inv4)) (Const true)))) s))"
have "?P1 s"
proof(cut_tac a1 a2 c1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_NI_Local_Get_Get__part__1Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Get__part__1 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Remote_GetVsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_Get src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Local_GetX_PutX__part__0Vsinv__60:
assumes a1: "r=n_PI_Local_GetX_PutX__part__0 " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_WbVsinv__60:
assumes a1: "r=n_NI_Wb " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_StoreVsinv__60:
assumes a1: "\<exists> src data. src\<le>N\<and>data\<le>N\<and>r=n_Store src data" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_GetX_GetX__part__1Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_GetX__part__1 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Remote_ReplaceVsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_Replace src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_Store_HomeVsinv__60:
assumes a1: "\<exists> data. data\<le>N\<and>r=n_Store_Home data" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Local_ReplaceVsinv__60:
assumes a1: "r=n_PI_Local_Replace " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_GetX_Nak__part__1Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__1 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_Nak__part__1Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__1 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_Get__part__0Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Get__part__0 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_GetX_Nak__part__2Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__2 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Remote_PutXVsinv__60:
assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_PI_Remote_PutX dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_InvVsinv__60:
assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Inv dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Local_PutXVsinv__60:
assumes a1: "r=n_PI_Local_PutX " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_Nak__part__2Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__2 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_GetX_GetX__part__0Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_GetX__part__0 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Local_Get_PutVsinv__60:
assumes a1: "r=n_PI_Local_Get_Put " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_GetX_NakVsinv__60:
assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_GetX_Nak src dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_NakVsinv__60:
assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Nak dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Remote_GetXVsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_PI_Remote_GetX src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_PI_Local_GetX_PutX__part__1Vsinv__60:
assumes a1: "r=n_PI_Local_GetX_PutX__part__1 " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_PutXVsinv__60:
assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_PutX dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_PutVsinv__60:
assumes a1: "\<exists> dst. dst\<le>N\<and>r=n_NI_Remote_Put dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_PutVsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_GetX_Nak__part__0Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_GetX_Nak__part__0 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Replace_HomeVsinv__60:
assumes a1: "r=n_NI_Replace_Home " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_GetX_PutXVsinv__60:
assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_GetX_PutX src dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_Get_NakVsinv__60:
assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_Get_Nak src dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Nak_ClearVsinv__60:
assumes a1: "r=n_NI_Nak_Clear " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_Put_DirtyVsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Put_Dirty src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Local_Get_Nak__part__0Vsinv__60:
assumes a1: "\<exists> src. src\<le>N\<and>r=n_NI_Local_Get_Nak__part__0 src" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_Remote_Get_PutVsinv__60:
assumes a1: "\<exists> src dst. src\<le>N\<and>dst\<le>N\<and>src~=dst\<and>r=n_NI_Remote_Get_Put src dst" and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
lemma n_NI_FAckVsinv__60:
assumes a1: "r=n_NI_FAck " and
a2: "(\<exists> p__Inv4. p__Inv4\<le>N\<and>f=inv__60 p__Inv4)"
shows "invHoldForRule s f r (invariants N)"
apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
end
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- List Zipper-related properties
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.List.Zipper.Properties where
open import Data.List.Base as List using (List ; [] ; _β·_)
open import Data.List.Properties
open import Data.List.Zipper
open import Data.Maybe.Base using (Maybe; just; nothing)
open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
open import Relation.Binary.PropositionalEquality
open β‘-Reasoning
open import Function
-- Invariant: Zipper represents a given list
------------------------------------------------------------------------
module _ {a} {A : Set a} where
-- Stability under moving left or right
toList-left-identity : (zp : Zipper A) β All ((_β‘_ on toList) zp) (left zp)
toList-left-identity (mkZipper [] val) = nothing
toList-left-identity (mkZipper (x β· ctx) val) = just $β² begin
List.reverse (x β· ctx) List.++ val
β‘β¨ cong (List._++ val) (unfold-reverse x ctx) β©
(List.reverse ctx List.++ List.[ x ]) List.++ val
β‘β¨ ++-assoc (List.reverse ctx) List.[ x ] val β©
toList (mkZipper ctx (x β· val))
β
toList-right-identity : (zp : Zipper A) β All ((_β‘_ on toList) zp) (right zp)
toList-right-identity (mkZipper ctx []) = nothing
toList-right-identity (mkZipper ctx (x β· val)) = just $β² begin
List.reverse ctx List.++ x β· val
β‘β¨ sym (++-assoc (List.reverse ctx) List.[ x ] val) β©
(List.reverse ctx List.++ List.[ x ]) List.++ val
β‘β¨ cong (List._++ val) (sym (unfold-reverse x ctx)) β©
List.reverse (x β· ctx) List.++ val
β
-- Applying reverse does correspond to reversing the represented list
toList-reverse-commute : (zp : Zipper A) β toList (reverse zp) β‘ List.reverse (toList zp)
toList-reverse-commute (mkZipper ctx val) = begin
List.reverse val List.++ ctx
β‘β¨ cong (List.reverse val List.++_) (sym (reverse-involutive ctx)) β©
List.reverse val List.++ List.reverse (List.reverse ctx)
β‘β¨ sym (reverse-++-commute (List.reverse ctx) val) β©
List.reverse (List.reverse ctx List.++ val)
β
-- Properties of the insertion functions
------------------------------------------------------------------------
-- _Λ‘++_ properties
toList-Λ‘++-commute : β xs (zp : Zipper A) β toList (xs Λ‘++ zp) β‘ xs List.++ toList zp
toList-Λ‘++-commute xs (mkZipper ctx val) = begin
List.reverse (ctx List.++ List.reverse xs) List.++ val
β‘β¨ cong (List._++ _) (reverse-++-commute ctx (List.reverse xs)) β©
(List.reverse (List.reverse xs) List.++ List.reverse ctx) List.++ val
β‘β¨ ++-assoc (List.reverse (List.reverse xs)) (List.reverse ctx) val β©
List.reverse (List.reverse xs) List.++ List.reverse ctx List.++ val
β‘β¨ cong (List._++ _) (reverse-involutive xs) β©
xs List.++ List.reverse ctx List.++ val
β
Λ‘++-assoc : β xs ys (zp : Zipper A) β xs Λ‘++ (ys Λ‘++ zp) β‘ (xs List.++ ys) Λ‘++ zp
Λ‘++-assoc xs ys (mkZipper ctx val) = cong (Ξ» ctx β mkZipper ctx val) $ begin
(ctx List.++ List.reverse ys) List.++ List.reverse xs
β‘β¨ ++-assoc ctx _ _ β©
ctx List.++ List.reverse ys List.++ List.reverse xs
β‘β¨ cong (ctx List.++_) (sym (reverse-++-commute xs ys)) β©
ctx List.++ List.reverse (xs List.++ ys)
β
-- _Κ³++_ properties
Κ³++-assoc : β xs ys (zp : Zipper A) β xs Κ³++ (ys Κ³++ zp) β‘ (ys List.++ xs) Κ³++ zp
Κ³++-assoc xs ys (mkZipper ctx val) = cong (Ξ» ctx β mkZipper ctx val) $ begin
List.reverse xs List.++ List.reverse ys List.++ ctx
β‘β¨ sym (++-assoc (List.reverse xs) (List.reverse ys) ctx) β©
(List.reverse xs List.++ List.reverse ys) List.++ ctx
β‘β¨ cong (List._++ ctx) (sym (reverse-++-commute ys xs)) β©
List.reverse (ys List.++ xs) List.++ ctx
β
-- _++Λ‘_ properties
++Λ‘-assoc : β xs ys (zp : Zipper A) β zp ++Λ‘ xs ++Λ‘ ys β‘ zp ++Λ‘ (ys List.++ xs)
++Λ‘-assoc xs ys (mkZipper ctx val) = cong (mkZipper ctx) $ sym $ ++-assoc ys xs val
-- _++Κ³_ properties
toList-++Κ³-commute : β (zp : Zipper A) xs β toList (zp ++Κ³ xs) β‘ toList zp List.++ xs
toList-++Κ³-commute (mkZipper ctx val) xs = begin
List.reverse ctx List.++ val List.++ xs
β‘β¨ sym (++-assoc (List.reverse ctx) val xs) β©
(List.reverse ctx List.++ val) List.++ xs
β
++Κ³-assoc : β xs ys (zp : Zipper A) β zp ++Κ³ xs ++Κ³ ys β‘ zp ++Κ³ (xs List.++ ys)
++Κ³-assoc xs ys (mkZipper ctx val) = cong (mkZipper ctx) $ ++-assoc val xs ys
-- List-like operations indeed correspond to their counterparts
------------------------------------------------------------------------
module _ {a b} {A : Set a} {B : Set b} where
toList-map-commute : β (f : A β B) zp β toList (map f zp) β‘ List.map f (toList zp)
toList-map-commute f zp@(mkZipper ctx val) = begin
List.reverse (List.map f ctx) List.++ List.map f val
β‘β¨ cong (List._++ _) (sym (reverse-map-commute f ctx)) β©
List.map f (List.reverse ctx) List.++ List.map f val
β‘β¨ sym (map-++-commute f (List.reverse ctx) val) β©
List.map f (List.reverse ctx List.++ val)
β
toList-foldr-commute : β (c : A β B β B) n zp β foldr c n zp β‘ List.foldr c n (toList zp)
toList-foldr-commute c n (mkZipper ctx val) = begin
List.foldl (flip c) (List.foldr c n val) ctx
β‘β¨ sym (reverse-foldr c (List.foldr c n val) ctx) β©
List.foldr c (List.foldr c n val) (List.reverse ctx)
β‘β¨ sym (foldr-++ c n (List.reverse ctx) val) β©
List.foldr c n (List.reverse ctx List.++ val)
β
|
Require Import Blech.Defaults.
Require Import Coq.Setoids.Setoid.
Require Import Coq.Classes.SetoidClass.
Require Import Blech.Proset.
Import ProsetNotations.
Reserved Notation "C 'α΅α΅'" (at level 1).
#[program]
Definition Op (P: Proset): Proset := {|
T := P;
preorder x y := y β x ;
|}.
Next Obligation.
Proof.
exists.
- intro.
reflexivity.
- intros ? ? ? p q.
rewrite q, p.
reflexivity.
Defined.
Module OpNotations.
Notation "C 'α΅α΅'" := (Op C) : proset_scope.
End OpNotations.
|
lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)" |
State Before: L : Language
L' : Language
M : Type w
N : Type ?u.66568
P : Type ?u.66571
instβΒ² : Structure L M
instβΒΉ : Structure L N
instβ : Structure L P
Ξ± : Type u'
Ξ² : Type v'
n lβ : β
Ο Ο : BoundedFormula L Ξ± lβ
ΞΈ : BoundedFormula L Ξ± (Nat.succ lβ)
vβ : Ξ± β M
xsβ : Fin lβ β M
l : List (BoundedFormula L Ξ± n)
v : Ξ± β M
xs : Fin n β M
β’ Realize (List.foldr (fun x x_1 => x β x_1) β₯ l) v xs β β Ο, Ο β l β§ Realize Ο v xs State After: case nil
L : Language
L' : Language
M : Type w
N : Type ?u.66568
P : Type ?u.66571
instβΒ² : Structure L M
instβΒΉ : Structure L N
instβ : Structure L P
Ξ± : Type u'
Ξ² : Type v'
n l : β
Ο Ο : BoundedFormula L Ξ± l
ΞΈ : BoundedFormula L Ξ± (Nat.succ l)
vβ : Ξ± β M
xsβ : Fin l β M
v : Ξ± β M
xs : Fin n β M
β’ Realize (List.foldr (fun x x_1 => x β x_1) β₯ []) v xs β β Ο, Ο β [] β§ Realize Ο v xs
case cons
L : Language
L' : Language
M : Type w
N : Type ?u.66568
P : Type ?u.66571
instβΒ² : Structure L M
instβΒΉ : Structure L N
instβ : Structure L P
Ξ± : Type u'
Ξ² : Type v'
n lβ : β
Οβ Ο : BoundedFormula L Ξ± lβ
ΞΈ : BoundedFormula L Ξ± (Nat.succ lβ)
vβ : Ξ± β M
xsβ : Fin lβ β M
v : Ξ± β M
xs : Fin n β M
Ο : BoundedFormula L Ξ± n
l : List (BoundedFormula L Ξ± n)
ih : Realize (List.foldr (fun x x_1 => x β x_1) β₯ l) v xs β β Ο, Ο β l β§ Realize Ο v xs
β’ Realize (List.foldr (fun x x_1 => x β x_1) β₯ (Ο :: l)) v xs β β Ο_1, Ο_1 β Ο :: l β§ Realize Ο_1 v xs Tactic: induction' l with Ο l ih State Before: case nil
L : Language
L' : Language
M : Type w
N : Type ?u.66568
P : Type ?u.66571
instβΒ² : Structure L M
instβΒΉ : Structure L N
instβ : Structure L P
Ξ± : Type u'
Ξ² : Type v'
n l : β
Ο Ο : BoundedFormula L Ξ± l
ΞΈ : BoundedFormula L Ξ± (Nat.succ l)
vβ : Ξ± β M
xsβ : Fin l β M
v : Ξ± β M
xs : Fin n β M
β’ Realize (List.foldr (fun x x_1 => x β x_1) β₯ []) v xs β β Ο, Ο β [] β§ Realize Ο v xs State After: no goals Tactic: simp State Before: case cons
L : Language
L' : Language
M : Type w
N : Type ?u.66568
P : Type ?u.66571
instβΒ² : Structure L M
instβΒΉ : Structure L N
instβ : Structure L P
Ξ± : Type u'
Ξ² : Type v'
n lβ : β
Οβ Ο : BoundedFormula L Ξ± lβ
ΞΈ : BoundedFormula L Ξ± (Nat.succ lβ)
vβ : Ξ± β M
xsβ : Fin lβ β M
v : Ξ± β M
xs : Fin n β M
Ο : BoundedFormula L Ξ± n
l : List (BoundedFormula L Ξ± n)
ih : Realize (List.foldr (fun x x_1 => x β x_1) β₯ l) v xs β β Ο, Ο β l β§ Realize Ο v xs
β’ Realize (List.foldr (fun x x_1 => x β x_1) β₯ (Ο :: l)) v xs β β Ο_1, Ο_1 β Ο :: l β§ Realize Ο_1 v xs State After: no goals Tactic: simp_rw [List.foldr_cons, realize_sup, ih, exists_prop, List.mem_cons, or_and_right, exists_or,
exists_eq_left] |
(* Title: HOL/Auth/n_mutualExFsm_lemma_inv__2_on_rules.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_mutualExFsm Protocol Case Study*}
theory n_mutualExFsm_lemma_inv__2_on_rules imports n_mutualExFsm_lemma_on_inv__2
begin
section{*All lemmas on causal relation between inv__2*}
lemma lemma_inv__2_on_rules:
assumes b1: "r \<in> rules N" and b2: "(\<exists> p__Inv0. p__Inv0\<le>N\<and>f=inv__2 p__Inv0)"
shows "invHoldForRule s f r (invariants N)"
proof -
have c1: "(\<exists> i. i\<le>N\<and>r=n_fsm i)"
apply (cut_tac b1, auto) done
moreover {
assume d1: "(\<exists> i. i\<le>N\<and>r=n_fsm i)"
have "invHoldForRule s f r (invariants N)"
apply (cut_tac b2 d1, metis n_fsmVsinv__2) done
}
ultimately show "invHoldForRule s f r (invariants N)"
by satx
qed
end
|
[STATEMENT]
lemma not_strict_par2:
assumes "A B Par C D" and
"Col A B X" and
"Col C D X"
shows "Col A B D"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. Col A B D
[PROOF STEP]
using Par_cases assms(1) assms(2) assms(3) not_col_permutation_4 not_strict_par1
[PROOF STATE]
proof (prove)
using this:
?A ?B Par ?C ?D \<or> ?B ?A Par ?C ?D \<or> ?A ?B Par ?D ?C \<or> ?B ?A Par ?D ?C \<or> ?C ?D Par ?A ?B \<or> ?C ?D Par ?B ?A \<or> ?D ?C Par ?A ?B \<or> ?D ?C Par ?B ?A \<Longrightarrow> ?A ?B Par ?C ?D
A B Par C D
Col A B X
Col C D X
\<not> Col ?A ?B ?C \<Longrightarrow> \<not> Col ?B ?A ?C
\<lbrakk>?A ?B Par ?C ?D; Col ?A ?B ?X; Col ?C ?D ?X\<rbrakk> \<Longrightarrow> Col ?A ?B ?C
goal (1 subgoal):
1. Col A B D
[PROOF STEP]
by blast |
#ifndef COMMAND_UTILS_INCLUDE
#define COMMAND_UTILS_INCLUDE
#include <limits>
#include <string>
#include <vector>
#include <gsl/string_span>
#include "tmpFile.h"
namespace execHelper {
namespace test {
namespace baseUtils {
using ConfigFile = TmpFile;
const gsl::czstring<> EXEC_HELPER_BINARY = "exec-helper";
const gsl::czstring<> COMMAND_KEY = "commands";
const gsl::czstring<> COMMAND_LINE_COMMAND_KEY = "command-line-command";
const gsl::czstring<> COMMAND_LINE_COMMAND_LINE_KEY = "command-line";
using ReturnCode = int32_t;
static const ReturnCode SUCCESS = EXIT_SUCCESS;
static const ReturnCode RUNTIME_ERROR = std::numeric_limits<ReturnCode>::max();
} // namespace baseUtils
} // namespace test
} // namespace execHelper
#endif /* COMMAND_UTILS_INCLUDE */
|
```python
!pip install cirq
```
Collecting cirq
Downloading cirq-0.14.0-py3-none-any.whl (7.8 kB)
Collecting cirq-web==0.14.0
Downloading cirq_web-0.14.0-py3-none-any.whl (593 kB)
Collecting cirq-pasqal==0.14.0
Downloading cirq_pasqal-0.14.0-py3-none-any.whl (30 kB)
Collecting cirq-ionq==0.14.0
Downloading cirq_ionq-0.14.0-py3-none-any.whl (48 kB)
Collecting cirq-rigetti==0.14.0
Downloading cirq_rigetti-0.14.0-py3-none-any.whl (56 kB)
Collecting cirq-core==0.14.0
Downloading cirq_core-0.14.0-py3-none-any.whl (1.8 MB)
Collecting cirq-aqt==0.14.0
Downloading cirq_aqt-0.14.0-py3-none-any.whl (19 kB)
Collecting cirq-google==0.14.0
Downloading cirq_google-0.14.0-py3-none-any.whl (541 kB)
Requirement already satisfied: requests~=2.18 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-aqt==0.14.0->cirq) (2.25.1)
Requirement already satisfied: sortedcontainers~=2.0 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (2.3.0)
Requirement already satisfied: typing-extensions in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (3.7.4.3)
Requirement already satisfied: scipy in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (1.6.2)
Requirement already satisfied: matplotlib~=3.0 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (3.3.4)
Requirement already satisfied: networkx~=2.4 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (2.5)
Requirement already satisfied: sympy<1.10 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (1.8)
Requirement already satisfied: pandas in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (1.2.4)
Requirement already satisfied: tqdm in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (4.59.0)
Collecting duet~=0.2.0
Downloading duet-0.2.5-py3-none-any.whl (28 kB)
Requirement already satisfied: numpy~=1.16 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-core==0.14.0->cirq) (1.20.1)
Collecting google-api-core[grpc]<2.0.0dev,>=1.14.0
Downloading google_api_core-1.31.5-py2.py3-none-any.whl (93 kB)
Requirement already satisfied: protobuf>=3.13.0 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-google==0.14.0->cirq) (3.19.4)
Collecting httpx~=0.15.5
Downloading httpx-0.15.5-py3-none-any.whl (65 kB)
Requirement already satisfied: attrs~=20.3.0 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-rigetti==0.14.0->cirq) (20.3.0)
Collecting rfc3986~=1.5.0
Downloading rfc3986-1.5.0-py2.py3-none-any.whl (31 kB)
Requirement already satisfied: toml~=0.10.2 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-rigetti==0.14.0->cirq) (0.10.2)
Collecting certifi~=2021.5.30
Downloading certifi-2021.5.30-py2.py3-none-any.whl (145 kB)
Requirement already satisfied: python-dateutil~=2.8.1 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-rigetti==0.14.0->cirq) (2.8.1)
Collecting iso8601~=0.1.14
Downloading iso8601-0.1.16-py2.py3-none-any.whl (10 kB)
Collecting six~=1.16.0
Using cached six-1.16.0-py2.py3-none-any.whl (11 kB)
Collecting httpcore~=0.11.1
Downloading httpcore-0.11.1-py3-none-any.whl (52 kB)
Collecting pyjwt~=1.7.1
Using cached PyJWT-1.7.1-py2.py3-none-any.whl (18 kB)
Requirement already satisfied: idna~=2.10 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-rigetti==0.14.0->cirq) (2.10)
Collecting retrying~=1.3.3
Downloading retrying-1.3.3.tar.gz (10 kB)
Requirement already satisfied: sniffio~=1.2.0 in c:\users\ameyb\anaconda3\lib\site-packages (from cirq-rigetti==0.14.0->cirq) (1.2.0)
Collecting rfc3339~=6.2
Downloading rfc3339-6.2-py3-none-any.whl (5.5 kB)
Collecting pydantic~=1.8.2
Downloading pydantic-1.8.2-cp38-cp38-win_amd64.whl (2.0 MB)
Collecting pyquil~=3.0.0
Downloading pyquil-3.0.1-py3-none-any.whl (220 kB)
Collecting qcs-api-client~=0.8.0
Downloading qcs_api_client-0.8.0-py3-none-any.whl (97 kB)
Collecting h11~=0.9.0
Downloading h11-0.9.0-py2.py3-none-any.whl (53 kB)
Collecting googleapis-common-protos<2.0dev,>=1.6.0
Downloading googleapis_common_protos-1.56.0-py2.py3-none-any.whl (241 kB)
Collecting google-auth<2.0dev,>=1.25.0
Downloading google_auth-1.35.0-py2.py3-none-any.whl (152 kB)
Requirement already satisfied: pytz in c:\users\ameyb\anaconda3\lib\site-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (2021.1)
Requirement already satisfied: packaging>=14.3 in c:\users\ameyb\anaconda3\lib\site-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (20.9)
Requirement already satisfied: setuptools>=40.3.0 in c:\users\ameyb\anaconda3\lib\site-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (52.0.0.post20210125)
Requirement already satisfied: grpcio<2.0dev,>=1.29.0 in c:\users\ameyb\anaconda3\lib\site-packages (from google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (1.44.0)
Requirement already satisfied: pyasn1-modules>=0.2.1 in c:\users\ameyb\anaconda3\lib\site-packages (from google-auth<2.0dev,>=1.25.0->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (0.2.8)
Collecting cachetools<5.0,>=2.0.0
Downloading cachetools-4.2.4-py3-none-any.whl (10 kB)
Requirement already satisfied: rsa<5,>=3.1.4 in c:\users\ameyb\anaconda3\lib\site-packages (from google-auth<2.0dev,>=1.25.0->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (4.8)
Requirement already satisfied: cycler>=0.10 in c:\users\ameyb\anaconda3\lib\site-packages (from matplotlib~=3.0->cirq-core==0.14.0->cirq) (0.10.0)
Requirement already satisfied: pyparsing!=2.0.4,!=2.1.2,!=2.1.6,>=2.0.3 in c:\users\ameyb\anaconda3\lib\site-packages (from matplotlib~=3.0->cirq-core==0.14.0->cirq) (2.4.7)
Requirement already satisfied: kiwisolver>=1.0.1 in c:\users\ameyb\anaconda3\lib\site-packages (from matplotlib~=3.0->cirq-core==0.14.0->cirq) (1.3.1)
Requirement already satisfied: pillow>=6.2.0 in c:\users\ameyb\anaconda3\lib\site-packages (from matplotlib~=3.0->cirq-core==0.14.0->cirq) (8.2.0)
Requirement already satisfied: decorator>=4.3.0 in c:\users\ameyb\anaconda3\lib\site-packages (from networkx~=2.4->cirq-core==0.14.0->cirq) (5.0.6)
Requirement already satisfied: pyasn1<0.5.0,>=0.4.6 in c:\users\ameyb\anaconda3\lib\site-packages (from pyasn1-modules>=0.2.1->google-auth<2.0dev,>=1.25.0->google-api-core[grpc]<2.0.0dev,>=1.14.0->cirq-google==0.14.0->cirq) (0.4.8)
Collecting lark<0.12.0,>=0.11.1
Downloading lark-0.11.3.tar.gz (229 kB)
Collecting rpcq<4.0.0,>=3.6.0
Downloading rpcq-3.9.2.tar.gz (43 kB)
Collecting retry<0.10.0,>=0.9.2
Downloading retry-0.9.2-py2.py3-none-any.whl (8.0 kB)
Requirement already satisfied: urllib3<1.27,>=1.21.1 in c:\users\ameyb\anaconda3\lib\site-packages (from requests~=2.18->cirq-aqt==0.14.0->cirq) (1.26.4)
Requirement already satisfied: chardet<5,>=3.0.2 in c:\users\ameyb\anaconda3\lib\site-packages (from requests~=2.18->cirq-aqt==0.14.0->cirq) (4.0.0)
Requirement already satisfied: py<2.0.0,>=1.4.26 in c:\users\ameyb\anaconda3\lib\site-packages (from retry<0.10.0,>=0.9.2->pyquil~=3.0.0->cirq-rigetti==0.14.0->cirq) (1.10.0)
Collecting msgpack<1.0,>=0.6
Downloading msgpack-0.6.2.tar.gz (119 kB)
Collecting python-rapidjson
Downloading python_rapidjson-1.6-cp38-cp38-win_amd64.whl (143 kB)
Requirement already satisfied: pyzmq>=17 in c:\users\ameyb\anaconda3\lib\site-packages (from rpcq<4.0.0,>=3.6.0->pyquil~=3.0.0->cirq-rigetti==0.14.0->cirq) (20.0.0)
Collecting ruamel.yaml
Downloading ruamel.yaml-0.17.21-py3-none-any.whl (109 kB)
Requirement already satisfied: mpmath>=0.19 in c:\users\ameyb\anaconda3\lib\site-packages (from sympy<1.10->cirq-core==0.14.0->cirq) (1.2.1)
Collecting ruamel.yaml.clib>=0.2.6
Downloading ruamel.yaml.clib-0.2.6-cp38-cp38-win_amd64.whl (117 kB)
Building wheels for collected packages: lark, retrying, rpcq, msgpack
Building wheel for lark (setup.py): started
Building wheel for lark (setup.py): finished with status 'done'
Created wheel for lark: filename=lark-0.11.3-py2.py3-none-any.whl size=99635 sha256=155e7e2100b52b4d75f8b42f117bb6698394e89cd998645896af53ed1a02d1ba
Stored in directory: c:\users\ameyb\appdata\local\pip\cache\wheels\34\cb\6c\4df359c2a3f0a1af4cccae6392bee423bb5aff530103de3538
Building wheel for retrying (setup.py): started
Building wheel for retrying (setup.py): finished with status 'done'
Created wheel for retrying: filename=retrying-1.3.3-py3-none-any.whl size=11429 sha256=e25e954dd607b67304a1195b8285ae5e67180926f92f23e8530da1a489c1c5bd
Stored in directory: c:\users\ameyb\appdata\local\pip\cache\wheels\c4\a7\48\0a434133f6d56e878ca511c0e6c38326907c0792f67b476e56
Building wheel for rpcq (setup.py): started
Building wheel for rpcq (setup.py): finished with status 'done'
Created wheel for rpcq: filename=rpcq-3.9.2-py3-none-any.whl size=45865 sha256=74d9bce012f3f24fc0fe440ee93502f2efd71bd8bc10324efde8e15d89af21f1
Stored in directory: c:\users\ameyb\appdata\local\pip\cache\wheels\20\fd\8d\4d4a9f389a9c92210dbee8ca8bbd725a6204f64a8ca8cad841
Building wheel for msgpack (setup.py): started
Building wheel for msgpack (setup.py): finished with status 'done'
Created wheel for msgpack: filename=msgpack-0.6.2-cp38-cp38-win_amd64.whl size=73428 sha256=c9451975d58a7a983fc5b1ef7e660a774c871f8f4f2d6373645cdba003b70589
Stored in directory: c:\users\ameyb\appdata\local\pip\cache\wheels\5d\f2\04\0d19c10080b996bef17c908a6243e6e65d8da1a4094a3f604d
Successfully built lark retrying rpcq msgpack
Installing collected packages: rfc3986, h11, six, ruamel.yaml.clib, httpcore, certifi, cachetools, ruamel.yaml, rfc3339, retrying, python-rapidjson, pyjwt, pydantic, msgpack, iso8601, httpx, googleapis-common-protos, google-auth, rpcq, retry, qcs-api-client, lark, google-api-core, duet, pyquil, cirq-core, cirq-web, cirq-rigetti, cirq-pasqal, cirq-ionq, cirq-google, cirq-aqt, cirq
Attempting uninstall: six
Found existing installation: six 1.15.0
Uninstalling six-1.15.0:
Successfully uninstalled six-1.15.0
Attempting uninstall: certifi
Found existing installation: certifi 2020.12.5
Uninstalling certifi-2020.12.5:
Successfully uninstalled certifi-2020.12.5
Attempting uninstall: cachetools
Found existing installation: cachetools 5.0.0
Uninstalling cachetools-5.0.0:
Successfully uninstalled cachetools-5.0.0
Attempting uninstall: msgpack
Found existing installation: msgpack 1.0.2
Uninstalling msgpack-1.0.2:
Successfully uninstalled msgpack-1.0.2
Attempting uninstall: google-auth
Found existing installation: google-auth 2.6.2
Uninstalling google-auth-2.6.2:
Successfully uninstalled google-auth-2.6.2
Successfully installed cachetools-4.2.4 certifi-2021.5.30 cirq-0.14.0 cirq-aqt-0.14.0 cirq-core-0.14.0 cirq-google-0.14.0 cirq-ionq-0.14.0 cirq-pasqal-0.14.0 cirq-rigetti-0.14.0 cirq-web-0.14.0 duet-0.2.5 google-api-core-1.31.5 google-auth-1.35.0 googleapis-common-protos-1.56.0 h11-0.9.0 httpcore-0.11.1 httpx-0.15.5 iso8601-0.1.16 lark-0.11.3 msgpack-0.6.2 pydantic-1.8.2 pyjwt-1.7.1 pyquil-3.0.1 python-rapidjson-1.6 qcs-api-client-0.8.0 retry-0.9.2 retrying-1.3.3 rfc3339-6.2 rfc3986-1.5.0 rpcq-3.9.2 ruamel.yaml-0.17.21 ruamel.yaml.clib-0.2.6 six-1.16.0
ERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.
conda-repo-cli 1.0.4 requires pathlib, which is not installed.
```python
import tensorflow as tf
import numpy as np
import cirq
%matplotlib inline
import matplotlib.pyplot as plt
from cirq.contrib.svg import SVGCircuit
```
## Part 1
```python
#qubits = cirq.GridQubit.rect(1,5)
circuit = cirq.Circuit()
circuit.append((cirq.H(i)) for i in cirq.LineQubit.range(5))
for i in cirq.LineQubit.range(4):
circuit.append(cirq.CNOT(control=i , target = i+1))
circuit.append(cirq.SWAP(cirq.LineQubit(0),cirq.LineQubit(4)))
circuit.append(cirq.rx(np.pi/2).on(cirq.LineQubit(np.random.randint(5))))
SVGCircuit(circuit)
```
## Part 2
```python
circuit = cirq.Circuit()
qubit=cirq.LineQubit(0)
angle = np.random.random(5)/20
for i in range(5):
circuit.append(cirq.rx(angle[i]).on(qubit))
circuit.append(cirq.measure(qubit,key='result'))
SVGCircuit(circuit)
```
```python
simulator = cirq.Simulator()
samples = simulator.run(circuit, repetitions=1024)
hist_data = samples.measurements['result'].reshape(1024,)
print('The probability of measuring a qubit in |0> state : ', samples.histogram(key='result')[0]/1024)
print(samples.histogram(key='result'))
plt.hist(hist_data)
```
```python
```
|
State Before: Ξ± : Type u
Ξ² : Type v
ΞΉ : Type u_2
R : Type u_1
instβΒ³ : CompleteLinearOrder R
instβΒ² : TopologicalSpace R
instβΒΉ : OrderTopology R
x : R
as : ΞΉ β R
x_le : β (i : ΞΉ), x β€ as i
F : Filter ΞΉ
instβ : NeBot F
as_lim : Tendsto as F (π x)
β’ (β¨
(i : ΞΉ), as i) = x State After: Ξ± : Type u
Ξ² : Type v
ΞΉ : Type u_2
R : Type u_1
instβΒ³ : CompleteLinearOrder R
instβΒ² : TopologicalSpace R
instβΒΉ : OrderTopology R
x : R
as : ΞΉ β R
x_le : β (i : ΞΉ), x β€ as i
F : Filter ΞΉ
instβ : NeBot F
as_lim : Tendsto as F (π x)
β’ β (w : R), x < w β β i, as i < w Tactic: refine' iInf_eq_of_forall_ge_of_forall_gt_exists_lt (fun i β¦ x_le i) _ State Before: Ξ± : Type u
Ξ² : Type v
ΞΉ : Type u_2
R : Type u_1
instβΒ³ : CompleteLinearOrder R
instβΒ² : TopologicalSpace R
instβΒΉ : OrderTopology R
x : R
as : ΞΉ β R
x_le : β (i : ΞΉ), x β€ as i
F : Filter ΞΉ
instβ : NeBot F
as_lim : Tendsto as F (π x)
β’ β (w : R), x < w β β i, as i < w State After: no goals Tactic: apply fun w x_lt_w β¦ βΉFilter.NeBot FβΊ.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim) |
module LC.Confluence where
open import LC.Base
open import LC.Subst
open import LC.Reduction
open import Data.Product
open import Relation.Binary.Construct.Closure.ReflexiveTransitive
Ξ²βconfluent : β {M N O : Term} β (M Ξ²β N) β (M Ξ²β O) β β (Ξ» P β (N Ξ²β* P) Γ (O Ξ²β* P))
Ξ²βconfluent (Ξ²-Ζ-β {M} {N}) Ξ²-Ζ-β = M [ N ] , Ξ΅ , Ξ΅
Ξ²βconfluent (Ξ²-Ζ-β {M} {N}) (Ξ²-β-l {N = _} (Ξ²-Ζ {N = O} MβO)) = (O [ N ]) , cong-[]-l MβO , return Ξ²-Ζ-β
Ξ²βconfluent (Ξ²-Ζ-β {M} {N}) (Ξ²-β-r {N = O} NβO) = M [ O ] , cong-[]-r M NβO , return Ξ²-Ζ-β
Ξ²βconfluent (Ξ²-Ζ MβN) (Ξ²-Ζ MβO) with Ξ²βconfluent MβN MβO
... | P , NβP , OβP = Ζ P , cong-Ζ NβP , cong-Ζ OβP
Ξ²βconfluent (Ξ²-β-l {L} (Ξ²-Ζ {N = N} MβN)) Ξ²-Ζ-β = N [ L ] , return Ξ²-Ζ-β , cong-[]-l MβN
Ξ²βconfluent (Ξ²-β-l {L} MβN) (Ξ²-β-l MβO) with Ξ²βconfluent MβN MβO
... | P , NβP , OβP = P β L , cong-β-l NβP , cong-β-l OβP
Ξ²βconfluent (Ξ²-β-l {N = N} MβN) (Ξ²-β-r {N = O} LβO) = N β O , cong-β-r (return LβO) , cong-β-l (return MβN)
Ξ²βconfluent (Ξ²-β-r {N = N} MβN) (Ξ²-Ζ-β {O}) = O [ N ] , return Ξ²-Ζ-β , cong-[]-r O MβN
Ξ²βconfluent (Ξ²-β-r {N = N} MβN) (Ξ²-β-l {N = O} LβO) = O β N , cong-β-l (return LβO) , cong-β-r (return MβN)
Ξ²βconfluent (Ξ²-β-r {L} {M} {N} MβN) (Ξ²-β-r {N = O} MβO) with Ξ²βconfluent MβN MβO
... | P , NβP , OβP = L β P , cong-β-r NβP , cong-β-r OβP
-- Ξ²β*-confluent : β {M N O} β (M Ξ²β* N) β (M Ξ²β* O) β β (Ξ» P β (N Ξ²β* P) Γ (O Ξ²β* P))
-- Ξ²β*-confluent {O = O} Ξ΅ MβO = O , MβO , Ξ΅
-- Ξ²β*-confluent {N = N} MβN Ξ΅ = N , Ξ΅ , MβN
-- Ξ²β*-confluent {M} {N} {O} (_β
_ {j = L} MβL LβN) (_β
_ {j = K} MβK KβO) with Ξ²βconfluent MβL MβK
-- ... | M' , LβM' , KβM' = {! !} |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import group_theory.order_of_element
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `is_complement S T` where `S` and `T` are subsets of `G` states that every `g : G` can be
written uniquely as a product `s * t` for `s β S`, `t β T`.
- `left_transversals T` where `T` is a subset of `G` is the set of all left-complements of `T`,
i.e. the set of all `S : set G` that contain exactly one element of each left coset of `T`.
- `right_transversals S` where `S` is a subset of `G` is the set of all right-complements of `S`,
i.e. the set of all `T : set G` that contain exactly one element of each right coset of `S`.
## Main results
- `is_complement_of_coprime` : Subgroups of coprime order are complements.
-/
open_locale big_operators
namespace subgroup
variables {G : Type*} [group G] (H K : subgroup G) (S T : set G)
/-- `S` and `T` are complements if `(*) : S Γ T β G` is a bijection.
This notion generalizes left transversals, right transversals, and complementary subgroups. -/
@[to_additive "`S` and `T` are complements if `(*) : S Γ T β G` is a bijection"]
def is_complement : Prop := function.bijective (Ξ» x : S Γ T, x.1.1 * x.2.1)
/-- `H` and `K` are complements if `(*) : H Γ K β G` is a bijection -/
@[to_additive "`H` and `K` are complements if `(*) : H Γ K β G` is a bijection"]
abbreviation is_complement' := is_complement (H : set G) (K : set G)
/-- The set of left-complements of `T : set G` -/
@[to_additive "The set of left-complements of `T : set G`"]
def left_transversals : set (set G) := {S : set G | is_complement S T}
/-- The set of right-complements of `S : set G` -/
@[to_additive "The set of right-complements of `S : set G`"]
def right_transversals : set (set G) := {T : set G | is_complement S T}
variables {H K S T}
@[to_additive] lemma is_complement'_def :
is_complement' H K β is_complement (H : set G) (K : set G) := iff.rfl
@[to_additive] lemma is_complement_iff_exists_unique :
is_complement S T β β g : G, β! x : S Γ T, x.1.1 * x.2.1 = g :=
function.bijective_iff_exists_unique _
@[to_additive] lemma is_complement.exists_unique (h : is_complement S T) (g : G) :
β! x : S Γ T, x.1.1 * x.2.1 = g :=
is_complement_iff_exists_unique.mp h g
@[to_additive] lemma is_complement'.symm (h : is_complement' H K) : is_complement' K H :=
begin
let Ο : H Γ K β K Γ H := equiv.mk (Ξ» x, β¨x.2β»ΒΉ, x.1β»ΒΉβ©) (Ξ» x, β¨x.2β»ΒΉ, x.1β»ΒΉβ©)
(Ξ» x, prod.ext (inv_inv _) (inv_inv _)) (Ξ» x, prod.ext (inv_inv _) (inv_inv _)),
let Ο : G β G := equiv.mk (Ξ» g : G, gβ»ΒΉ) (Ξ» g : G, gβ»ΒΉ) inv_inv inv_inv,
suffices : Ο β (Ξ» x : H Γ K, x.1.1 * x.2.1) = (Ξ» x : K Γ H, x.1.1 * x.2.1) β Ο,
{ rwa [is_complement'_def, is_complement, βequiv.bijective_comp, βthis, equiv.comp_bijective] },
exact funext (Ξ» x, mul_inv_rev _ _),
end
@[to_additive] lemma is_complement'_comm : is_complement' H K β is_complement' K H :=
β¨is_complement'.symm, is_complement'.symmβ©
@[to_additive] lemma is_complement_top_singleton {g : G} : is_complement (β€ : set G) {g} :=
β¨Ξ» β¨x, _, rflβ© β¨y, _, rflβ© h, prod.ext (subtype.ext (mul_right_cancel h)) rfl,
Ξ» x, β¨β¨β¨x * gβ»ΒΉ, β¨β©β©, g, rflβ©, inv_mul_cancel_right x gβ©β©
@[to_additive] lemma is_complement_singleton_top {g : G} : is_complement ({g} : set G) β€ :=
β¨Ξ» β¨β¨_, rflβ©, xβ© β¨β¨_, rflβ©, yβ© h, prod.ext rfl (subtype.ext (mul_left_cancel h)),
Ξ» x, β¨β¨β¨g, rflβ©, gβ»ΒΉ * x, β¨β©β©, mul_inv_cancel_left g xβ©β©
@[to_additive] lemma is_complement_singleton_left {g : G} : is_complement {g} S β S = β€ :=
begin
refine β¨Ξ» h, top_le_iff.mp (Ξ» x hx, _), Ξ» h, (congr_arg _ h).mpr is_complement_singleton_topβ©,
obtain β¨β¨β¨z, rfl : z = gβ©, y, _β©, hyβ© := h.2 (g * x),
rwa β mul_left_cancel hy,
end
@[to_additive] lemma is_complement_singleton_right {g : G} : is_complement S {g} β S = β€ :=
begin
refine β¨Ξ» h, top_le_iff.mp (Ξ» x hx, _), Ξ» h, (congr_arg _ h).mpr is_complement_top_singletonβ©,
obtain β¨y, hyβ© := h.2 (x * g),
conv_rhs at hy { rw β (show y.2.1 = g, from y.2.2) },
rw β mul_right_cancel hy,
exact y.1.2,
end
@[to_additive] lemma is_complement_top_left : is_complement β€ S β β g : G, S = {g} :=
begin
refine β¨Ξ» h, set.exists_eq_singleton_iff_nonempty_unique_mem.mpr β¨_, Ξ» a b ha hb, _β©, _β©,
{ obtain β¨a, haβ© := h.2 1,
exact β¨a.2.1, a.2.2β© },
{ have : (β¨β¨_, mem_top aβ»ΒΉβ©, β¨a, haβ©β© : (β€ : set G) Γ S) = β¨β¨_, mem_top bβ»ΒΉβ©, β¨b, hbβ©β© :=
h.1 ((inv_mul_self a).trans (inv_mul_self b).symm),
exact subtype.ext_iff.mp ((prod.ext_iff.mp this).2) },
{ rintro β¨g, rflβ©,
exact is_complement_top_singleton },
end
@[to_additive] lemma is_complement_top_right : is_complement S β€ β β g : G, S = {g} :=
begin
refine β¨Ξ» h, set.exists_eq_singleton_iff_nonempty_unique_mem.mpr β¨_, Ξ» a b ha hb, _β©, _β©,
{ obtain β¨a, haβ© := h.2 1,
exact β¨a.1.1, a.1.2β© },
{ have : (β¨β¨a, haβ©, β¨_, mem_top aβ»ΒΉβ©β© : S Γ (β€ : set G)) = β¨β¨b, hbβ©, β¨_, mem_top bβ»ΒΉβ©β© :=
h.1 ((mul_inv_self a).trans (mul_inv_self b).symm),
exact subtype.ext_iff.mp ((prod.ext_iff.mp this).1) },
{ rintro β¨g, rflβ©,
exact is_complement_singleton_top },
end
@[to_additive] lemma is_complement'_top_bot : is_complement' (β€ : subgroup G) β₯ :=
is_complement_top_singleton
@[to_additive] lemma is_complement'_bot_top : is_complement' (β₯ : subgroup G) β€ :=
is_complement_singleton_top
@[simp, to_additive] lemma is_complement'_bot_left : is_complement' β₯ H β H = β€ :=
is_complement_singleton_left.trans coe_eq_univ
@[simp, to_additive] lemma is_complement'_bot_right : is_complement' H β₯ β H = β€ :=
is_complement_singleton_right.trans coe_eq_univ
@[simp, to_additive] lemma is_complement'_top_left : is_complement' β€ H β H = β₯ :=
is_complement_top_left.trans coe_eq_singleton
@[simp, to_additive] lemma is_complement'_top_right : is_complement' H β€ β H = β₯ :=
is_complement_top_right.trans coe_eq_singleton
@[to_additive] lemma mem_left_transversals_iff_exists_unique_inv_mul_mem :
S β left_transversals T β β g : G, β! s : S, (s : G)β»ΒΉ * g β T :=
begin
rw [left_transversals, set.mem_set_of_eq, is_complement_iff_exists_unique],
refine β¨Ξ» h g, _, Ξ» h g, _β©,
{ obtain β¨x, h1, h2β© := h g,
exact β¨x.1, (congr_arg (β T) (eq_inv_mul_of_mul_eq h1)).mp x.2.2, Ξ» y hy,
(prod.ext_iff.mp (h2 β¨y, yβ»ΒΉ * g, hyβ© (mul_inv_cancel_left y g))).1β© },
{ obtain β¨x, h1, h2β© := h g,
refine β¨β¨x, xβ»ΒΉ * g, h1β©, mul_inv_cancel_left x g, Ξ» y hy, _β©,
have := h2 y.1 ((congr_arg (β T) (eq_inv_mul_of_mul_eq hy)).mp y.2.2),
exact prod.ext this (subtype.ext (eq_inv_mul_of_mul_eq ((congr_arg _ this).mp hy))) },
end
@[to_additive] lemma mem_right_transversals_iff_exists_unique_mul_inv_mem :
S β right_transversals T β β g : G, β! s : S, g * (s : G)β»ΒΉ β T :=
begin
rw [right_transversals, set.mem_set_of_eq, is_complement_iff_exists_unique],
refine β¨Ξ» h g, _, Ξ» h g, _β©,
{ obtain β¨x, h1, h2β© := h g,
exact β¨x.2, (congr_arg (β T) (eq_mul_inv_of_mul_eq h1)).mp x.1.2, Ξ» y hy,
(prod.ext_iff.mp (h2 β¨β¨g * yβ»ΒΉ, hyβ©, yβ© (inv_mul_cancel_right g y))).2β© },
{ obtain β¨x, h1, h2β© := h g,
refine β¨β¨β¨g * xβ»ΒΉ, h1β©, xβ©, inv_mul_cancel_right g x, Ξ» y hy, _β©,
have := h2 y.2 ((congr_arg (β T) (eq_mul_inv_of_mul_eq hy)).mp y.1.2),
exact prod.ext (subtype.ext (eq_mul_inv_of_mul_eq ((congr_arg _ this).mp hy))) this },
end
@[to_additive] lemma mem_left_transversals_iff_exists_unique_quotient_mk'_eq :
S β left_transversals (H : set G) β
β q : quotient (quotient_group.left_rel H), β! s : S, quotient.mk' s.1 = q :=
begin
have key : β g h, quotient.mk' g = quotient.mk' h β gβ»ΒΉ * h β H :=
@quotient.eq' G (quotient_group.left_rel H),
simp_rw [mem_left_transversals_iff_exists_unique_inv_mul_mem, set_like.mem_coe, βkey],
exact β¨Ξ» h q, quotient.induction_on' q h, Ξ» h g, h (quotient.mk' g)β©,
end
@[to_additive]
@[to_additive] lemma mem_left_transversals_iff_bijective : S β left_transversals (H : set G) β
function.bijective (S.restrict (quotient.mk' : G β quotient (quotient_group.left_rel H))) :=
mem_left_transversals_iff_exists_unique_quotient_mk'_eq.trans
(function.bijective_iff_exists_unique (S.restrict quotient.mk')).symm
@[to_additive] lemma mem_right_transversals_iff_bijective : S β right_transversals (H : set G) β
function.bijective (set.restrict (quotient.mk' : G β quotient (quotient_group.right_rel H)) S) :=
mem_right_transversals_iff_exists_unique_quotient_mk'_eq.trans
(function.bijective_iff_exists_unique (S.restrict quotient.mk')).symm
@[to_additive] instance : inhabited (left_transversals (H : set G)) :=
β¨β¨set.range quotient.out', mem_left_transversals_iff_bijective.mpr β¨by
{ rintros β¨_, qβ, rflβ© β¨_, qβ, rflβ© hg,
rw (qβ.out_eq'.symm.trans hg).trans qβ.out_eq' }, Ξ» q, β¨β¨q.out', q, rflβ©, quotient.out_eq' qβ©β©β©β©
@[to_additive] instance : inhabited (right_transversals (H : set G)) :=
β¨β¨set.range quotient.out', mem_right_transversals_iff_bijective.mpr β¨by
{ rintros β¨_, qβ, rflβ© β¨_, qβ, rflβ© hg,
rw (qβ.out_eq'.symm.trans hg).trans qβ.out_eq' }, Ξ» q, β¨β¨q.out', q, rflβ©, quotient.out_eq' qβ©β©β©β©
lemma is_complement'.is_compl (h : is_complement' H K) : is_compl H K :=
begin
refine β¨Ξ» g β¨p, qβ©, let x : H Γ K := β¨β¨g, pβ©, 1β©, y : H Γ K := β¨1, g, qβ© in subtype.ext_iff.mp
(prod.ext_iff.mp (show x = y, from h.1 ((mul_one g).trans (one_mul g).symm))).1, Ξ» g _, _β©,
obtain β¨β¨h, kβ©, rflβ© := h.2 g,
exact subgroup.mul_mem_sup h.2 k.2,
end
lemma is_complement'.sup_eq_top (h : subgroup.is_complement' H K) : H β K = β€ :=
h.is_compl.sup_eq_top
lemma is_complement'.disjoint (h : is_complement' H K) : disjoint H K :=
h.is_compl.disjoint
lemma is_complement.card_mul [fintype G] [fintype S] [fintype T] (h : is_complement S T) :
fintype.card S * fintype.card T = fintype.card G :=
(fintype.card_prod _ _).symm.trans (fintype.card_of_bijective h)
lemma is_complement'.card_mul [fintype G] [fintype H] [fintype K] (h : is_complement' H K) :
fintype.card H * fintype.card K = fintype.card G :=
h.card_mul
lemma is_complement'_of_card_mul_and_disjoint [fintype G] [fintype H] [fintype K]
(h1 : fintype.card H * fintype.card K = fintype.card G) (h2 : disjoint H K) :
is_complement' H K :=
begin
refine (fintype.bijective_iff_injective_and_card _).mpr
β¨Ξ» x y h, _, (fintype.card_prod H K).trans h1β©,
rw [βeq_inv_mul_iff_mul_eq, βmul_assoc, βmul_inv_eq_iff_eq_mul] at h,
change β(x.2 * y.2β»ΒΉ) = β(x.1β»ΒΉ * y.1) at h,
rw [prod.ext_iff, β@inv_mul_eq_one H _ x.1 y.1, β@mul_inv_eq_one K _ x.2 y.2, subtype.ext_iff,
subtype.ext_iff, coe_one, coe_one, h, and_self, βmem_bot, βh2.eq_bot, mem_inf],
exact β¨subtype.mem ((x.1)β»ΒΉ * (y.1)), (congr_arg (β K) h).mp (subtype.mem (x.2 * (y.2)β»ΒΉ))β©,
end
lemma is_complement'_iff_card_mul_and_disjoint [fintype G] [fintype H] [fintype K] :
is_complement' H K β
fintype.card H * fintype.card K = fintype.card G β§ disjoint H K :=
β¨Ξ» h, β¨h.card_mul, h.disjointβ©, Ξ» h, is_complement'_of_card_mul_and_disjoint h.1 h.2β©
lemma is_complement'_of_coprime [fintype G] [fintype H] [fintype K]
(h1 : fintype.card H * fintype.card K = fintype.card G)
(h2 : nat.coprime (fintype.card H) (fintype.card K)) :
is_complement' H K :=
is_complement'_of_card_mul_and_disjoint h1 (disjoint_iff.mpr (inf_eq_bot_of_coprime h2))
end subgroup
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.int.basic
import Mathlib.category_theory.graded_object
import Mathlib.category_theory.differential_object
import Mathlib.PostPort
universes v u u_1
namespace Mathlib
/-!
# Chain complexes
We define a chain complex in `V` as a differential `β€`-graded object in `V`.
This is fancy language for the obvious definition,
and it seems we can use it straightforwardly:
```
example (C : chain_complex V) : C.X 5 βΆ C.X 6 := C.d 5
```
-/
/--
A `homological_complex V b` for `b : Ξ²` is a (co)chain complex graded by `Ξ²`,
with differential in grading `b`.
(We use the somewhat cumbersome `homological_complex` to avoid the name conflict with `β`.)
-/
def homological_complex (V : Type u) [category_theory.category V] [category_theory.limits.has_zero_morphisms V] {Ξ² : Type} [add_comm_group Ξ²] (b : Ξ²) :=
category_theory.differential_object (category_theory.graded_object_with_shift b V)
/--
A chain complex in `V` is "just" a differential `β€`-graded object in `V`,
with differential graded `-1`.
-/
def chain_complex (V : Type u) [category_theory.category V] [category_theory.limits.has_zero_morphisms V] :=
homological_complex V (-1)
/--
A cochain complex in `V` is "just" a differential `β€`-graded object in `V`,
with differential graded `+1`.
-/
def cochain_complex (V : Type u) [category_theory.category V] [category_theory.limits.has_zero_morphisms V] :=
homological_complex V 1
-- The chain groups of a chain complex `C` are accessed as `C.X i`,
-- and the differentials as `C.d i : C.X i βΆ C.X (i-1)`.
namespace homological_complex
@[simp] theorem d_squared {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] {Ξ² : Type} [add_comm_group Ξ²] {b : Ξ²} (C : homological_complex V b) (i : Ξ²) : category_theory.differential_object.d C i β« category_theory.differential_object.d C (i + b) = 0 := sorry
/--
A convenience lemma for morphisms of cochain complexes,
picking out one component of the commutation relation.
-/
-- I haven't been able to get this to work with projection notation: `f.comm_at i`
@[simp] theorem comm_at {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] {Ξ² : Type} [add_comm_group Ξ²] {b : Ξ²} {C : homological_complex V b} {D : homological_complex V b} (f : C βΆ D) (i : Ξ²) : category_theory.differential_object.d C i β« category_theory.differential_object.hom.f f (i + b) =
category_theory.differential_object.hom.f f i β« category_theory.differential_object.d D i := sorry
@[simp] theorem comm {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] {Ξ² : Type} [add_comm_group Ξ²] {b : Ξ²} {C : homological_complex V b} {D : homological_complex V b} (f : C βΆ D) : category_theory.differential_object.d C β«
category_theory.functor.map
(category_theory.equivalence.functor (category_theory.shift (category_theory.graded_object_with_shift b V) ^ 1))
(category_theory.differential_object.hom.f f) =
category_theory.differential_object.hom.f f β« category_theory.differential_object.d D :=
category_theory.differential_object.hom.comm (category_theory.graded_object_with_shift b V)
(category_theory.graded_object.category_of_graded_objects Ξ²) (category_theory.graded_object.has_zero_morphisms Ξ²)
(category_theory.graded_object.has_shift b) C D f
/-- The forgetful functor from cochain complexes to graded objects, forgetting the differential. -/
def forget (V : Type u) [category_theory.category V] [category_theory.limits.has_zero_morphisms V] {Ξ² : Type} [add_comm_group Ξ²] {b : Ξ²} : homological_complex V b β₯€ category_theory.graded_object Ξ² V :=
category_theory.differential_object.forget (category_theory.graded_object_with_shift b V)
protected instance inhabited {Ξ² : Type} [add_comm_group Ξ²] {b : Ξ²} : Inhabited (homological_complex (category_theory.discrete PUnit) b) :=
{ default := 0 }
|
Formal statement is: lemma lim_null_mult_right_bounded: fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra" assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F" shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F" Informal statement is: If $f$ converges to $0$ and $g$ is bounded, then $f \cdot g$ converges to $0$. |
import numpy as np
from numpy.lib.stride_tricks import as_strided as ast
from scipy.misc import lena
import cv2
import Image
from get_params import get_params
from matplotlib import pyplot as plt
def slide(boxes,height,width,stpx,stpy,sx,sy):
for i in range(0,height,stpy):
for j in range(0,width,stpx):
boxes.append([i, j, min(i+sy,height), min(j+sx,width)])
return boxes
def get_boxes(params):
height = params['height']
width = params['width']
sizes_x = [32,64,128,256,512]
sizes_y = [32,64,128,256,512]
boxes = []
for sx in sizes_x:
for sy in sizes_y:
stpx = sx/2
stpy = sy/2
boxes = slide(boxes,height,width,stpx,stpy,sx,sy)
return boxes
if __name__ == "__main__":
params = get_params()
boxes = get_boxes(params)
print np.shape(boxes)
|
# # Wrapper blocks
abstract type WrapperBlock <: AbstractBlock end
wrapped(w::WrapperBlock) = w.block
wrapped(b::Block) = b
function setwrapped(w::WrapperBlock, b)
return Setfield.@set w.block = b
end
mockblock(w::WrapperBlock) = mockblock(wrapped(w))
checkblock(w::WrapperBlock, data) = checkblock(wrapped(w), data)
# If not overwritten, encodings are applied to the wrapped block
"""
abstract type PropagateWrapper
Defines the default propagation behavior of a `WrapperBlock` when
an encoding is applied to it.
Propagation refers to what happens when an encoding is applied to
a `WrapperBlock`. If no `encode` method is defined for a wrapper block
`wrapper`, `encode` is instead called on the wrapped block.
Propagating the wrapper block means that the block resulting from
encoding the wrapped block is rewrapped in `wrapper.`.
```
wrapper = Wrapper(block)
# propagate
encodedblock(enc, wrapper) = Wrapper(encodedblock(enc, wrapped(wrapper)))
# don't propagate
encodedblock(enc, wrapper) = encodedblock(enc, wrapped(wrapper))
```
The following wrapping behaviors exist:
- `PropagateAlways`: Always propagate. This is the default behavior.
- `PropagateNever`: Never propagate
- `PropagateSameBlock`: Only propagate if the wrapped block is unchanged
"""
abstract type PropagateWrapper end
struct PropagateAlways <: PropagateWrapper end
struct PropagateSameBlock <: PropagateWrapper end
struct PropagateNever <: PropagateWrapper end
propagatewrapper(::WrapperBlock) = PropagateAlways()
encodedblock(enc::Encoding, wrapper::WrapperBlock) =
encodedblock(enc, wrapper, propagatewrapper(wrapper))
function encodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateAlways)
inner = encodedblock(enc, wrapped(wrapper))
return isnothing(inner) ? nothing : setwrapped(wrapper, inner)
end
function encodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateNever)
return encodedblock(enc, wrapped(wrapper))
end
function encodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateSameBlock)
inner = encodedblock(enc, wrapped(wrapper))
inner == wrapped(block) && return setwrapped(wrapper, inner)
return inner
end
decodedblock(enc::Encoding, wrapper::WrapperBlock) =
decodedblock(enc, wrapper, propagatewrapper(wrapper))
function decodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateAlways)
inner = decodedblock(enc, wrapped(wrapper))
return isnothing(inner) ? nothing : setwrapped(wrapper, inner)
end
function decodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateNever)
return decodedblock(enc, wrapped(wrapper))
end
function decodedblock(enc::Encoding, wrapper::WrapperBlock, ::PropagateSameBlock)
inner = decodedblock(enc, wrapped(wrapper))
inner == wrapped(block) && return setwrapped(wrapper, inner)
return inner
end
# Encoding and decoding, if not overwritten for specific wrapper, are fowarded
# to wrapped block.
function encode(enc::Encoding, ctx, wrapper::WrapperBlock, data; kwargs...)
return encode(enc, ctx, wrapped(wrapper), data; kwargs...)
end
function decode(enc::Encoding, ctx, wrapper::WrapperBlock, data; kwargs...)
return decode(enc, ctx, wrapped(wrapper), data; kwargs...)
end
# Training interface
blockbackbone(wrapper::WrapperBlock) = blockbackbone(wrapped(wrapper))
blockmodel(wrapper::WrapperBlock, out, args...) = blockmodel(wrapped(wrapper), out, args...)
blockmodel(in::Block, out::WrapperBlock, args...) = blockmodel(in, wrapped(out), args...)
blocklossfn(wrapper::WrapperBlock, out) = blocklossfn(wrapped(wrapper), out)
blocklossfn(in::Block, out::WrapperBlock) = blocklossfn(in, wrapped(out))
# ## Named
"""
Named(name, block)
Wrapper `Block` to attach a name to a block. Can be used in conjunction
with [`Only`](#) to apply encodings to specific blocks only.
"""
struct Named{Name,B<:AbstractBlock} <: WrapperBlock
block::B
end
Named(name::Symbol, block::B) where {B<:AbstractBlock} = Named{name,B}(block)
# the name is preserved through encodings and decodings
function encodedblock(enc::Encoding, named::Named{Name}) where {Name}
outblock = encodedblock(enc, wrapped(named))
return isnothing(outblock) ? nothing : Named(Name, outblock)
end
function decodedblock(enc::Encoding, named::Named{Name}) where {Name}
outblock = decodedblock(enc, wrapped(named))
return isnothing(outblock) ? nothing : Named(Name, outblock)
end
# ## Many
"""
Many(block) <: WrapperBlock
`Many` indicates that you can variable number of data instances for
`block`. Consider a bounding box detection task where there may be any
number of targets in an image and this number varies for different
samples. The blocks `(Image{2}(), BoundingBox{2}()` imply that there is exactly
one bounding box for every image, which is not the case. Instead you
would want to use `(Image{2}(), Many(BoundingBox{2}())`.
"""
struct Many{B<:AbstractBlock} <: WrapperBlock
block::B
end
FastAI.checkblock(many::Many, datas) =
all(checkblock(wrapped(many), data) for data in datas)
FastAI.mockblock(many::Many) = [mockblock(wrapped(many)), mockblock(wrapped(many))]
function FastAI.encode(enc::Encoding, ctx, many::Many, datas)
return map(datas) do data
encode(enc, ctx, wrapped(many), data)
end
end
function FastAI.decode(enc::Encoding, ctx, many::Many, datas)
return map(datas) do data
decode(enc, ctx, wrapped(many), data)
end
end
# # Wrapper encodings
"""
Only(fn, encoding)
Only(BlockType, encoding)
Only(name, encoding)
Wrapper that applies the wrapped `encoding` to a `block` if
`fn(block) === true`. Instead of a function you can also pass in
a type of block `BlockType` or the `name` of a `Named` block.
"""
struct Only{E<:Encoding} <: StatefulEncoding
fn::Any
encoding::E
end
function Only(name::Symbol, encoding::Encoding) where {E}
return Only(Named{name}, encoding)
end
function Only(B::Type{<:AbstractBlock}, encoding::Encoding) where {E}
return Only(block -> block isa B, encoding)
end
encodedblock(only::Only, block::Block) =
only.fn(block) ? encodedblock(only.encoding, block) : nothing
encodedblock(only::Only, block::WrapperBlock) =
only.fn(block) ? encodedblock(only.encoding, block) : nothing
encodedblock(only::Only, block::Named) =
only.fn(block) ? encodedblock(only.encoding, block) : nothing
function decodedblock(only::Only, block::Block)
inblock = decodedblock(only.encoding, block)
only.fn(inblock) || return nothing
return inblock
end
function decodedblock(only::Only, block::WrapperBlock)
inblock = decodedblock(only.encoding, block)
only.fn(inblock) || return nothing
return inblock
end
function decodedblock(only::Only, block::Named)
inblock = decodedblock(only.encoding, block)
only.fn(inblock) || return nothing
return inblock
end
encodestate(only::Only, args...) = encodestate(only.encoding, args...)
decodestate(only::Only, args...) = decodestate(only.encoding, args...)
function encode(only::Only, ctx, block::Block, data; kwargs...)
_encode(only, ctx, block, data; kwargs...)
end
function encode(only::Only, ctx, block::WrapperBlock, data; kwargs...)
_encode(only, ctx, block, data; kwargs...)
end
function _encode(only, ctx, block, data; kwargs...)
return only.fn(block) ? encode(only.encoding, ctx, block, data; kwargs...) : data
end
function decode(only::Only, ctx, block::Block, data; kwargs...)
_decode(only, ctx, block, data; kwargs...)
end
function decode(only::Only, ctx, block::WrapperBlock, data; kwargs...)
_decode(only, ctx, block, data; kwargs...)
end
function _decode(only, ctx, block, data; kwargs...)
return only.fn(decodedblock(only.encoding, block)) ?
decode(only.encoding, ctx, block, data; kwargs...) : data
end
InlineTest.@testset "Only" begin
encx = Only(:x, OneHot())
inblock = Label(1:100)
inblocknamed = Named(:x, inblock)
data = mockblock(inblock)
encdata = encode(OneHot(), Training(), inblock, data)
@test encodedblock(encx, inblock) === nothing
@test encodedblock(encx, inblocknamed) isa Named{:x}
@test encodedblock(Only(Named, OneHot()), inblock) === nothing
@test encodedblock(Only(Named, OneHot()), inblocknamed) isa Named{:x}
outblock = encodedblock(OneHot(), inblock)
outblocknamed = encodedblock(OneHot(), inblocknamed)
@test decodedblock(encx, outblock) === nothing
@test decodedblock(encx, outblocknamed) isa Named{:x}
@test decodedblock(Only(Named, OneHot()), outblock) === nothing
@test decodedblock(Only(Named, OneHot()), outblocknamed) isa Named{:x}
@test encode(encx, Training(), inblock, data) == data
@test encode(encx, Training(), inblocknamed, data) != data
@test decode(encx, Training(), outblock, encdata) == encdata
@test decode(encx, Training(), outblocknamed, encdata) != encdata
end
|
lemma has_vector_derivative_polynomial_function: fixes p :: "real \<Rightarrow> 'a::euclidean_space" assumes "polynomial_function p" obtains p' where "polynomial_function p'" "\<And>x. (p has_vector_derivative (p' x)) (at x)" |
[GOAL]
Ξ±β : Type u
Ξ² : Type v
instβΒ² : Monoid Ξ±β
instβΒΉ : Monoid Ξ²
Ξ± : Type u_1
instβ : CommMonoid Ξ±
a b : Ξ±
xβ : IsConj a b
c : Ξ±Λ£
hc : SemiconjBy (βc) a b
β’ a = b
[PROOFSTEP]
rw [SemiconjBy, mul_comm, β Units.mul_inv_eq_iff_eq_mul, mul_assoc, c.mul_inv, mul_one] at hc
[GOAL]
Ξ±β : Type u
Ξ² : Type v
instβΒ² : Monoid Ξ±β
instβΒΉ : Monoid Ξ²
Ξ± : Type u_1
instβ : CommMonoid Ξ±
a b : Ξ±
xβ : IsConj a b
c : Ξ±Λ£
hc : a = b
β’ a = b
[PROOFSTEP]
exact hc
[GOAL]
Ξ±β : Type u
Ξ² : Type v
instβΒ² : Monoid Ξ±β
instβΒΉ : Monoid Ξ²
Ξ± : Type u_1
instβ : CommMonoid Ξ±
a b : Ξ±
h : a = b
β’ IsConj a b
[PROOFSTEP]
rw [h]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβΒΉ : Monoid Ξ±
instβ : Monoid Ξ²
f : Ξ± β* Ξ²
a b : Ξ±
c : Ξ±Λ£
hc : SemiconjBy (βc) a b
β’ SemiconjBy (β(β(Units.map f) c)) (βf a) (βf b)
[PROOFSTEP]
rw [Units.coe_map, SemiconjBy, β f.map_mul, hc.eq, f.map_mul]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : CancelMonoid Ξ±
a : Ξ±
h : a = 1
β’ IsConj 1 a
[PROOFSTEP]
rw [h]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
i : β
a b : Ξ±
β’ (a * b * aβ»ΒΉ) ^ i = a * b ^ i * aβ»ΒΉ
[PROOFSTEP]
induction' i with i hi
[GOAL]
case zero
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
β’ (a * b * aβ»ΒΉ) ^ Nat.zero = a * b ^ Nat.zero * aβ»ΒΉ
[PROOFSTEP]
simp
[GOAL]
case succ
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
i : β
hi : (a * b * aβ»ΒΉ) ^ i = a * b ^ i * aβ»ΒΉ
β’ (a * b * aβ»ΒΉ) ^ Nat.succ i = a * b ^ Nat.succ i * aβ»ΒΉ
[PROOFSTEP]
simp [pow_succ, hi]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
i : β€
a b : Ξ±
β’ (a * b * aβ»ΒΉ) ^ i = a * b ^ i * aβ»ΒΉ
[PROOFSTEP]
induction' i
[GOAL]
case ofNat
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
aβ : β
β’ (a * b * aβ»ΒΉ) ^ Int.ofNat aβ = a * b ^ Int.ofNat aβ * aβ»ΒΉ
[PROOFSTEP]
change (a * b * aβ»ΒΉ) ^ (_ : β€) = a * b ^ (_ : β€) * aβ»ΒΉ
[GOAL]
case ofNat
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
aβ : β
β’ (a * b * aβ»ΒΉ) ^ Int.ofNat aβ = a * b ^ Int.ofNat aβ * aβ»ΒΉ
[PROOFSTEP]
simp [zpow_ofNat]
[GOAL]
case negSucc
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
aβ : β
β’ (a * b * aβ»ΒΉ) ^ Int.negSucc aβ = a * b ^ Int.negSucc aβ * aβ»ΒΉ
[PROOFSTEP]
simp [zpow_negSucc, conj_pow]
[GOAL]
case negSucc
Ξ± : Type u
Ξ² : Type v
instβ : Group Ξ±
a b : Ξ±
aβ : β
β’ a * ((b ^ (aβ + 1))β»ΒΉ * aβ»ΒΉ) = a * (b ^ (aβ + 1))β»ΒΉ * aβ»ΒΉ
[PROOFSTEP]
rw [mul_assoc]
-- Porting note: Added `change`, `zpow_ofNat`, and `rw`.
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : GroupWithZero Ξ±
a b : Ξ±
xβ : IsConj a b
c : Ξ±Λ£
hc : SemiconjBy (βc) a b
β’ βc β 0 β§ βc * a * (βc)β»ΒΉ = b
[PROOFSTEP]
rw [β Units.val_inv_eq_inv_val, Units.mul_inv_eq_iff_eq_mul]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : GroupWithZero Ξ±
a b : Ξ±
xβ : IsConj a b
c : Ξ±Λ£
hc : SemiconjBy (βc) a b
β’ βc β 0 β§ βc * a = b * βc
[PROOFSTEP]
exact β¨c.ne_zero, hcβ©
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : GroupWithZero Ξ±
a b : Ξ±
xβ : β c, c β 0 β§ c * a * cβ»ΒΉ = b
c : Ξ±
c0 : c β 0
hc : c * a * cβ»ΒΉ = b
β’ SemiconjBy (β(Units.mk0 c c0)) a b
[PROOFSTEP]
rw [SemiconjBy, β Units.mul_inv_eq_iff_eq_mul, Units.val_inv_eq_inv_val, Units.val_mk0]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : GroupWithZero Ξ±
a b : Ξ±
xβ : β c, c β 0 β§ c * a * cβ»ΒΉ = b
c : Ξ±
c0 : c β 0
hc : c * a * cβ»ΒΉ = b
β’ c * a * cβ»ΒΉ = b
[PROOFSTEP]
exact hc
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβΒΉ : Monoid Ξ±
instβ : Monoid Ξ²
f : Ξ± β* Ξ²
hf : Function.Surjective βf
β’ Function.Surjective (map f)
[PROOFSTEP]
intro b
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβΒΉ : Monoid Ξ±
instβ : Monoid Ξ²
f : Ξ± β* Ξ²
hf : Function.Surjective βf
b : ConjClasses Ξ²
β’ β a, map f a = b
[PROOFSTEP]
obtain β¨b, rflβ© := ConjClasses.mk_surjective b
[GOAL]
case intro
Ξ± : Type u
Ξ² : Type v
instβΒΉ : Monoid Ξ±
instβ : Monoid Ξ²
f : Ξ± β* Ξ²
hf : Function.Surjective βf
b : Ξ²
β’ β a, map f a = ConjClasses.mk b
[PROOFSTEP]
obtain β¨a, rflβ© := hf b
[GOAL]
case intro.intro
Ξ± : Type u
Ξ² : Type v
instβΒΉ : Monoid Ξ±
instβ : Monoid Ξ²
f : Ξ± β* Ξ²
hf : Function.Surjective βf
a : Ξ±
β’ β a_1, map f a_1 = ConjClasses.mk (βf a)
[PROOFSTEP]
exact β¨ConjClasses.mk a, rflβ©
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : CommMonoid Ξ±
β’ Function.RightInverse (Quotient.lift id (_ : β (a b : Ξ±), IsConj a b β a = b)) ConjClasses.mk
[PROOFSTEP]
rw [Function.RightInverse, Function.LeftInverse, forall_isConj]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : CommMonoid Ξ±
β’ β (a : Ξ±),
ConjClasses.mk (Quotient.lift id (_ : β (a b : Ξ±), IsConj a b β a = b) (ConjClasses.mk a)) = ConjClasses.mk a
[PROOFSTEP]
intro x
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : CommMonoid Ξ±
x : Ξ±
β’ ConjClasses.mk (Quotient.lift id (_ : β (a b : Ξ±), IsConj a b β a = b) (ConjClasses.mk x)) = ConjClasses.mk x
[PROOFSTEP]
rw [β quotient_mk_eq_mk, β quotient_mk_eq_mk, Quotient.lift_mk, id.def]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a b : Ξ±
h : conjugatesOf a = conjugatesOf b
β’ IsConj a b
[PROOFSTEP]
have ha := @mem_conjugatesOf_self _ _ b
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a b : Ξ±
h : conjugatesOf a = conjugatesOf b
ha : b β conjugatesOf b
β’ IsConj a b
[PROOFSTEP]
rwa [β h] at ha
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a : Ξ±
b : ConjClasses Ξ±
β’ a β carrier b β ConjClasses.mk a = b
[PROOFSTEP]
revert b
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a : Ξ±
β’ β {b : ConjClasses Ξ±}, a β carrier b β ConjClasses.mk a = b
[PROOFSTEP]
rw [forall_isConj]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a : Ξ±
β’ β (a_1 : Ξ±), a β carrier (ConjClasses.mk a_1) β ConjClasses.mk a = ConjClasses.mk a_1
[PROOFSTEP]
intro b
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a b : Ξ±
β’ a β carrier (ConjClasses.mk b) β ConjClasses.mk a = ConjClasses.mk b
[PROOFSTEP]
rw [carrier, eq_comm, mk_eq_mk_iff_isConj, β quotient_mk_eq_mk, Quotient.lift_mk]
[GOAL]
Ξ± : Type u
Ξ² : Type v
instβ : Monoid Ξ±
a b : Ξ±
β’ a β conjugatesOf b β IsConj b a
[PROOFSTEP]
rfl
|
(*
* Copyright (C) 2014, National ICT Australia Limited. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are
* met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* * Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* * The name of National ICT Australia Limited nor the names of its
* contributors may be used to endorse or promote products derived from
* this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
* IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
* TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
* PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*)
(* Author: Gerwin Klein and Thomas Sewell
Type class for enumerations.
*)
header "Enumerations"
theory Enumeration
imports "~~/src/HOL/Main"
begin
abbreviation
"enum \<equiv> enum_class.enum"
abbreviation
"enum_all \<equiv> enum_class.enum_all"
abbreviation
"enum_ex \<equiv> enum_class.enum_ex"
primrec
the_index :: "'a list \<Rightarrow> 'a \<Rightarrow> nat"
where
"the_index (x # xs) y = (if x = y then 0 else Suc (the_index xs y))"
lemma the_index_bounded:
"x \<in> set xs \<Longrightarrow> the_index xs x < length xs"
by (induct xs, clarsimp+)
lemma nth_the_index:
"x \<in> set xs \<Longrightarrow> xs ! the_index xs x = x"
by (induct xs, clarsimp+)
lemma distinct_the_index_is_index[simp]:
"\<lbrakk> distinct xs ; n < length xs \<rbrakk> \<Longrightarrow> the_index xs (xs ! n) = n"
apply (subst nth_eq_iff_index_eq[symmetric])
apply assumption
apply (rule the_index_bounded)
apply simp_all
apply (rule nth_the_index)
apply simp
done
lemma the_index_last_distinct:
"distinct xs \<and> xs \<noteq> [] \<Longrightarrow> the_index xs (last xs) = length xs - 1"
apply safe
apply (subgoal_tac "xs ! (length xs - 1) = last xs")
apply (subgoal_tac "xs ! the_index xs (last xs) = last xs")
apply (subst nth_eq_iff_index_eq[symmetric])
apply assumption
apply (rule the_index_bounded)
apply simp_all
apply (rule nth_the_index)
apply simp
apply (induct xs, auto)
done
context enum begin
(* These two are added for historical reasons.
* We had an enum class first, and these are the two
* assumptions we had, which were added to the simp set.
*)
lemmas enum_surj[simp] = enum_UNIV
declare enum_distinct[simp]
lemma enum_nonempty[simp]: "(enum :: 'a list) \<noteq> []"
apply (rule classical, simp)
apply (subgoal_tac "\<exists>X. X \<in> set (enum :: 'a list)")
apply simp
apply (subst enum_surj)
apply simp
done
definition
maxBound :: 'a where
"maxBound \<equiv> last enum"
definition
minBound :: 'a where
"minBound \<equiv> hd enum"
definition
toEnum :: "nat \<Rightarrow> 'a" where
"toEnum n \<equiv> if n < length (enum :: 'a list) then enum ! n else the None"
definition
fromEnum :: "'a \<Rightarrow> nat" where
"fromEnum x \<equiv> the_index enum x"
lemma maxBound_is_length:
"fromEnum maxBound = length (enum :: 'a list) - 1"
apply (simp add: maxBound_def fromEnum_def)
apply (subst the_index_last_distinct)
apply simp
apply simp
done
lemma maxBound_less_length:
"(x \<le> fromEnum maxBound) = (x < length (enum :: 'a list))"
apply (simp only: maxBound_is_length)
apply (case_tac "length (enum :: 'a list)")
apply simp
apply simp
apply arith
done
lemma maxBound_is_bound [simp]:
"fromEnum x \<le> fromEnum maxBound"
apply (simp only: maxBound_less_length)
apply (simp add: fromEnum_def)
apply (rule the_index_bounded)
by simp
lemma to_from_enum [simp]:
fixes x :: 'a
shows "toEnum (fromEnum x) = x"
proof -
have "x \<in> set enum" by simp
thus ?thesis
by (simp add: toEnum_def fromEnum_def nth_the_index the_index_bounded)
qed
lemma from_to_enum [simp]:
"x \<le> fromEnum maxBound \<Longrightarrow> fromEnum (toEnum x) = x"
apply (simp only: maxBound_less_length)
apply (simp add: toEnum_def fromEnum_def)
done
lemma map_enum:
fixes x :: 'a
shows "map f enum ! fromEnum x = f x"
proof -
have "fromEnum x \<le> fromEnum (maxBound :: 'a)"
by (rule maxBound_is_bound)
hence "fromEnum x < length (enum::'a list)"
by (simp add: maxBound_less_length)
hence "map f enum ! fromEnum x = f (enum ! fromEnum x)" by simp
also
have "x \<in> set enum" by simp
hence "enum ! fromEnum x = x"
by (simp add: fromEnum_def nth_the_index)
finally
show ?thesis .
qed
definition
assocs :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) list" where
"assocs f \<equiv> map (\<lambda>x. (x, f x)) enum"
end
(* For historical naming reasons. *)
lemmas enum_bool = enum_bool_def
lemma fromEnumTrue [simp]: "fromEnum True = 1"
by (simp add: fromEnum_def enum_bool)
lemma fromEnumFalse [simp]: "fromEnum False = 0"
by (simp add: fromEnum_def enum_bool)
class enum_alt =
fixes enum_alt :: "nat \<Rightarrow> 'a option"
class enumeration_alt = enum_alt +
assumes enum_alt_one_bound:
"enum_alt x = (None :: 'a option) \<Longrightarrow> enum_alt (Suc x) = (None :: 'a option)"
assumes enum_alt_surj: "range enum_alt \<union> {None} = UNIV"
assumes enum_alt_inj:
"(enum_alt x :: 'a option) = enum_alt y \<Longrightarrow> (x = y) \<or> (enum_alt x = (None :: 'a option))"
begin
lemma enum_alt_inj_2:
"\<lbrakk> enum_alt x = (enum_alt y :: 'a option);
enum_alt x \<noteq> (None :: 'a option) \<rbrakk>
\<Longrightarrow> x = y"
apply (subgoal_tac "(x = y) \<or> (enum_alt x = (None :: 'a option))")
apply clarsimp
apply (rule enum_alt_inj)
apply simp
done
lemma enum_alt_surj_2:
"\<exists>x. enum_alt x = Some y"
apply (subgoal_tac "Some y \<in> range enum_alt")
apply (erule rangeE)
apply (rule exI)
apply simp
apply (subgoal_tac "Some y \<in> range enum_alt \<union> {None}")
apply simp
apply (subst enum_alt_surj)
apply simp
done
end
definition
alt_from_ord :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option" where
"alt_from_ord L \<equiv> \<lambda>n. if (n < length L) then Some (L ! n) else None"
lemma handy_enum_lemma1: "((if P then Some A else None) = None) = (\<not> P)"
apply simp
done
lemma handy_enum_lemma2: "Some x \<notin> empty ` S"
apply safe
done
lemma handy_enum_lemma3: "((if P then Some A else None) = Some B) = (P \<and> (A = B))"
apply simp
done
class enumeration_both = enum_alt + enum +
assumes enum_alt_rel: "enum_alt = alt_from_ord enum"
instance enumeration_both < enumeration_alt
apply (intro_classes)
apply (simp_all add: enum_alt_rel alt_from_ord_def)
apply (simp add: handy_enum_lemma1)
apply (safe, simp_all)
apply (simp add: handy_enum_lemma2)
apply (rule rev_image_eqI, simp_all)
defer
apply (subst nth_the_index, simp_all)
apply (simp add: handy_enum_lemma3)
apply (subst nth_eq_iff_index_eq[symmetric], simp_all)
apply safe
apply (rule the_index_bounded)
apply simp
done
instantiation bool :: enumeration_both
begin
definition
enum_alt_bool: "enum_alt \<equiv> alt_from_ord [False, True]"
instance
by (intro_classes, simp add: enum_bool_def enum_alt_bool)
end
definition
toEnumAlt :: "nat \<Rightarrow> ('a :: enum_alt)" where
"toEnumAlt n \<equiv> the (enum_alt n)"
definition
fromEnumAlt :: "('a :: enum_alt) \<Rightarrow> nat" where
"fromEnumAlt x \<equiv> THE n. enum_alt n = Some x"
definition
upto_enum :: "('a :: enumeration_alt) \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_.e._])") where
"upto_enum n m \<equiv> map toEnumAlt [fromEnumAlt n ..< Suc (fromEnumAlt m)]"
lemma fromEnum_alt_red[simp]:
"fromEnumAlt = (fromEnum :: ('a :: enumeration_both) \<Rightarrow> nat)"
apply (rule ext)
apply (simp add: fromEnumAlt_def fromEnum_def)
apply (simp add: enum_alt_rel alt_from_ord_def)
apply (rule theI2)
apply safe
apply (rule nth_the_index, simp)
apply (rule the_index_bounded, simp)
apply simp_all
done
lemma toEnum_alt_red[simp]:
"toEnumAlt = (toEnum :: nat \<Rightarrow> ('a :: enumeration_both))"
apply (rule ext)
apply (unfold toEnum_def toEnumAlt_def)
apply (simp add: enum_alt_rel alt_from_ord_def)
done
lemma upto_enum_red:
"[(n :: ('a :: enumeration_both)) .e. m] = map toEnum [fromEnum n ..< Suc (fromEnum m)]"
apply (unfold upto_enum_def)
apply simp
done
instantiation nat :: enumeration_alt
begin
definition
enum_alt_nat: "enum_alt \<equiv> Some"
instance
apply (intro_classes)
apply (simp_all add: enum_alt_nat)
apply (safe, simp_all)
apply (case_tac x, simp_all)
done
end
lemma toEnumAlt_nat[simp]: "toEnumAlt = id"
apply (rule ext)
apply (simp add: toEnumAlt_def enum_alt_nat)
done
lemma fromEnumAlt_nat[simp]: "fromEnumAlt = id"
apply (rule ext)
apply (simp add: fromEnumAlt_def enum_alt_nat)
done
lemma upto_enum_nat[simp]: "[n .e. m] = [n ..< Suc m]"
apply (subst upto_enum_def)
apply simp
done
definition
zipE1 :: "('a :: enum_alt) \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
"zipE1 x L \<equiv> zip (map toEnumAlt [(fromEnumAlt x) ..< (fromEnumAlt x) + length L]) L"
definition
zipE2 :: "('a :: enum_alt) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
"zipE2 x xn L \<equiv> zip (map (\<lambda>n. toEnumAlt ((fromEnumAlt x) + ((fromEnumAlt xn) - (fromEnumAlt x)) * n)) [0 ..< length L]) L"
definition
zipE3 :: "'a list \<Rightarrow> ('b :: enum_alt) \<Rightarrow> ('a \<times> 'b) list" where
"zipE3 L x \<equiv> zip L (map toEnumAlt [(fromEnumAlt x) ..< (fromEnumAlt x) + length L])"
definition
zipE4 :: "'a list \<Rightarrow> ('b :: enum_alt) \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) list" where
"zipE4 L x xn \<equiv> zip L (map (\<lambda>n. toEnumAlt ((fromEnumAlt x) + ((fromEnumAlt xn) - (fromEnumAlt x)) * n)) [0 ..< length L])"
lemma handy_lemma: "a = Some b \<Longrightarrow> the a = b"
by (simp)
lemma to_from_enum_alt[simp]:
"toEnumAlt (fromEnumAlt x) = (x :: ('a :: enumeration_alt))"
apply (simp add: fromEnumAlt_def toEnumAlt_def)
apply (rule handy_lemma)
apply (rule theI')
apply safe
apply (rule enum_alt_surj_2)
apply (rule enum_alt_inj_2)
apply auto
done
end
|
using JuLIP.Testing, ASE, PyCall
@info("These tests to compare JuLIP vs ASE implementations of some potentials")
h3("Compare JuLIP vs ASE: EAM")
# JuLIP's EMT implementation
at = set_pbc!( bulk(:Cu, cubic=true) * (2,2,2), (true,false,false) )
rattle!(at, 0.02)
emt = EMT(at)
@info("Test JuLIP vs ASE EMT implementation")
pyemt = ASE.Models.EMTCalculator()
print(" energy: ")
println(@test abs(energy(emt, at) - energy(pyemt, at)) < 1e-10)
print(" forces: ")
println(@test norm(forces(pyemt, at) - forces(emt, at), Inf) < 1e-10)
# ------------------------------------------------------------------------
h3("Compare JuLIP EAM Implementation against ASE EAM Implementation")
@pyimport ase.calculators.eam as eam
pot_file = joinpath(dirname(pathof(JuLIP)), "..", "data", "w_eam4.fs")
@info("Generate the ASE potential")
eam4_ase = eam.EAM(potential=pot_file) |> ASECalculator
@info("Generate low-, med-, high-accuracy JuLIP potential")
eam4_jl1 = EAM(pot_file)
eam4_jl2 = EAM(pot_file; s = 1e-4)
eam4_jl3 = EAM(pot_file; s = 1e-6)
at1 = rattle!(bulk(:W, cubic=true) * 3, 0.1)
at2 = deleteat!(bulk(:W, cubic=true) * 3, 1)
at1_ase = ASEAtoms(at1)
at2_ase = ASEAtoms(at2)
for (i, (at, at_ase)) in enumerate(zip([at1, at2], [at1_ase, at2_ase]))
@info("Test $i")
err_low = (energy(eam4_ase, at_ase) - energy(eam4_jl1, at)) / length(at)
err_med = (energy(eam4_ase, at_ase) - energy(eam4_jl2, at)) / length(at)
err_hi = (energy(eam4_ase, at_ase) - energy(eam4_jl3, at)) / length(at)
print(" Low Accuracy energy error:", err_low, "; ",
(@test abs(err_low) < 0.02))
print(" Low Accuracy energy error:", err_med, "; ",
(@test abs(err_med) < 0.006))
print(" Low Accuracy energy error:", err_hi, "; ",
(@test abs(err_hi) < 0.0004))
end
# ------------------------------------------------------------------------
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
Extends `tendsto` to relations and partial functions.
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.order.filter.basic
import Mathlib.PostPort
universes u v w
namespace Mathlib
namespace filter
/-
Relations.
-/
def rmap {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ±) : filter Ξ² :=
mk (set_of fun (s : set Ξ²) => rel.core r s β f) sorry sorry sorry
theorem rmap_sets {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ±) : sets (rmap r f) = rel.core r β»ΒΉ' sets f :=
rfl
@[simp] theorem mem_rmap {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (l : filter Ξ±) (s : set Ξ²) : s β rmap r l β rel.core r s β l :=
iff.rfl
@[simp] theorem rmap_rmap {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) (l : filter Ξ±) : rmap s (rmap r l) = rmap (rel.comp r s) l := sorry
@[simp] theorem rmap_compose {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) : rmap s β rmap r = rmap (rel.comp r s) :=
funext (rmap_rmap r s)
def rtendsto {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) :=
rmap r lβ β€ lβ
theorem rtendsto_def {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) : rtendsto r lβ lβ β β (s : set Ξ²), s β lβ β rel.core r s β lβ :=
iff.rfl
def rcomap {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ²) : filter Ξ± :=
mk (rel.image (fun (s : set Ξ²) (t : set Ξ±) => rel.core r s β t) (sets f)) sorry sorry sorry
theorem rcomap_sets {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ²) : sets (rcomap r f) = rel.image (fun (s : set Ξ²) (t : set Ξ±) => rel.core r s β t) (sets f) :=
rfl
@[simp] theorem rcomap_rcomap {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) (l : filter Ξ³) : rcomap r (rcomap s l) = rcomap (rel.comp r s) l := sorry
@[simp] theorem rcomap_compose {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) : rcomap r β rcomap s = rcomap (rel.comp r s) :=
funext (rcomap_rcomap r s)
theorem rtendsto_iff_le_comap {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) : rtendsto r lβ lβ β lβ β€ rcomap r lβ := sorry
-- Interestingly, there does not seem to be a way to express this relation using a forward map.
-- Given a filter `f` on `Ξ±`, we want a filter `f'` on `Ξ²` such that `r.preimage s β f` if
-- and only if `s β f'`. But the intersection of two sets satsifying the lhs may be empty.
def rcomap' {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ²) : filter Ξ± :=
mk (rel.image (fun (s : set Ξ²) (t : set Ξ±) => rel.preimage r s β t) (sets f)) sorry sorry sorry
@[simp] theorem mem_rcomap' {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (l : filter Ξ²) (s : set Ξ±) : s β rcomap' r l β β (t : set Ξ²), β (H : t β l), rel.preimage r t β s :=
iff.rfl
theorem rcomap'_sets {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (f : filter Ξ²) : sets (rcomap' r f) = rel.image (fun (s : set Ξ²) (t : set Ξ±) => rel.preimage r s β t) (sets f) :=
rfl
@[simp] theorem rcomap'_rcomap' {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) (l : filter Ξ³) : rcomap' r (rcomap' s l) = rcomap' (rel.comp r s) l := sorry
@[simp] theorem rcomap'_compose {Ξ± : Type u} {Ξ² : Type v} {Ξ³ : Type w} (r : rel Ξ± Ξ²) (s : rel Ξ² Ξ³) : rcomap' r β rcomap' s = rcomap' (rel.comp r s) :=
funext (rcomap'_rcomap' r s)
def rtendsto' {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) :=
lβ β€ rcomap' r lβ
theorem rtendsto'_def {Ξ± : Type u} {Ξ² : Type v} (r : rel Ξ± Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) : rtendsto' r lβ lβ β β (s : set Ξ²), s β lβ β rel.preimage r s β lβ := sorry
theorem tendsto_iff_rtendsto {Ξ± : Type u} {Ξ² : Type v} (lβ : filter Ξ±) (lβ : filter Ξ²) (f : Ξ± β Ξ²) : tendsto f lβ lβ β rtendsto (function.graph f) lβ lβ := sorry
theorem tendsto_iff_rtendsto' {Ξ± : Type u} {Ξ² : Type v} (lβ : filter Ξ±) (lβ : filter Ξ²) (f : Ξ± β Ξ²) : tendsto f lβ lβ β rtendsto' (function.graph f) lβ lβ := sorry
/-
Partial functions.
-/
def pmap {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (l : filter Ξ±) : filter Ξ² :=
rmap (pfun.graph' f) l
@[simp] theorem mem_pmap {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (l : filter Ξ±) (s : set Ξ²) : s β pmap f l β pfun.core f s β l :=
iff.rfl
def ptendsto {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) :=
pmap f lβ β€ lβ
theorem ptendsto_def {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) : ptendsto f lβ lβ β β (s : set Ξ²), s β lβ β pfun.core f s β lβ :=
iff.rfl
theorem ptendsto_iff_rtendsto {Ξ± : Type u} {Ξ² : Type v} (lβ : filter Ξ±) (lβ : filter Ξ²) (f : Ξ± β. Ξ²) : ptendsto f lβ lβ β rtendsto (pfun.graph' f) lβ lβ :=
iff.rfl
theorem pmap_res {Ξ± : Type u} {Ξ² : Type v} (l : filter Ξ±) (s : set Ξ±) (f : Ξ± β Ξ²) : pmap (pfun.res f s) l = map f (l β principal s) := sorry
theorem tendsto_iff_ptendsto {Ξ± : Type u} {Ξ² : Type v} (lβ : filter Ξ±) (lβ : filter Ξ²) (s : set Ξ±) (f : Ξ± β Ξ²) : tendsto f (lβ β principal s) lβ β ptendsto (pfun.res f s) lβ lβ := sorry
theorem tendsto_iff_ptendsto_univ {Ξ± : Type u} {Ξ² : Type v} (lβ : filter Ξ±) (lβ : filter Ξ²) (f : Ξ± β Ξ²) : tendsto f lβ lβ β ptendsto (pfun.res f set.univ) lβ lβ := sorry
def pcomap' {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (l : filter Ξ²) : filter Ξ± :=
rcomap' (pfun.graph' f) l
def ptendsto' {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) :=
lβ β€ rcomap' (pfun.graph' f) lβ
theorem ptendsto'_def {Ξ± : Type u} {Ξ² : Type v} (f : Ξ± β. Ξ²) (lβ : filter Ξ±) (lβ : filter Ξ²) : ptendsto' f lβ lβ β β (s : set Ξ²), s β lβ β pfun.preimage f s β lβ :=
rtendsto'_def (pfun.graph' f) lβ lβ
theorem ptendsto_of_ptendsto' {Ξ± : Type u} {Ξ² : Type v} {f : Ξ± β. Ξ²} {lβ : filter Ξ±} {lβ : filter Ξ²} : ptendsto' f lβ lβ β ptendsto f lβ lβ := sorry
theorem ptendsto'_of_ptendsto {Ξ± : Type u} {Ξ² : Type v} {f : Ξ± β. Ξ²} {lβ : filter Ξ±} {lβ : filter Ξ²} (h : pfun.dom f β lβ) : ptendsto f lβ lβ β ptendsto' f lβ lβ := sorry
|
clear;close all;
im1=imread('page.png');
im2=imread('tshape.png');
bwim1=adaptivethreshold(im1,11,0.03,0);
bwim2=adaptivethreshold(im2,15,0.02,0);
subplot(2,2,1);
imshow(im1);
subplot(2,2,2);
imshow(bwim1);
subplot(2,2,3);
imshow(im2);
subplot(2,2,4);
imshow(bwim2); |
[STATEMENT]
lemma minus_plus_minus:
assumes "s adds t" and "u adds v"
shows "(t - s) + (v - u) = (t + v) - (s + u)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. t - s + (v - u) = t + v - (s + u)
[PROOF STEP]
using add_commute assms(1) assms(2) diff_diff_add minus_plus
[PROOF STATE]
proof (prove)
using this:
?a + ?b = ?b + ?a
s adds t
u adds v
?a - ?b - ?c = ?a - (?b + ?c)
?s adds ?t \<Longrightarrow> ?t - ?s + ?u = ?t + ?u - ?s
goal (1 subgoal):
1. t - s + (v - u) = t + v - (s + u)
[PROOF STEP]
by auto |
Require Import Coq.Lists.List.
Require Import Coq.ZArith.ZArith.
Require Import PL.RTClosure.
Import ListNotations.
Local Open Scope Z.
Require Import PL.Imp.
Require Import FunctionalExtensionality.
(** Splay tree is a kind of self-balanced binary search tree. You may learn this
data structure from online resources like:
<<
https://people.eecs.berkeley.edu/~jrs/61b/lec/36
>>
In this task, you should prove the functional correctness of the splay
operation, the key operation of splay trees. We provide a step-wise
description of splay. *)
Definition Key: Type := Z.
Definition Value: Type := Z.
Record Node := {
key_of_node : Key;
value_of_node : Value
}.
Inductive tree : Type :=
| E : tree
| T : tree -> Node -> tree -> tree.
Definition optionZ_lt (ok1 ok2: option Key): Prop :=
match ok1, ok2 with
| Some k1, Some k2 => k1 < k2
| _, _ => True
end.
Definition optionZ_le (ok1 ok2: option Key): Prop :=
match ok1, ok2 with
| Some k1, Some k2 => k1 <= k2
| _, _ => True
end.
Inductive SearchTree : option Key -> tree -> option Key -> Prop :=
| ST_E : forall lo hi, optionZ_lt lo hi -> SearchTree lo E hi
| ST_T: forall lo l n r hi,
SearchTree lo l (Some (key_of_node n)) ->
SearchTree (Some (key_of_node n)) r hi ->
SearchTree lo (T l n r) hi.
Definition relate_map := Key -> option Value .
Definition relate_default: relate_map := fun x => None.
Definition relate_single (k: Key) (v: Value): relate_map :=
fun x =>
if Z.eq_dec x k then Some v else None.
Definition combine (m1 m2: relate_map): relate_map :=
fun x =>
match m1 x, m2 x with
| None, Some v => Some v
| Some v, None => Some v
| _ ,_ => None
end.
Inductive Abs : tree -> relate_map -> Prop :=
| Abs_E : Abs E relate_default
| Abs_T: forall l n r lm rm,
Abs l lm ->
Abs r rm ->
Abs
(T l n r)
(combine lm
(combine (relate_single (key_of_node n) (value_of_node n)) rm)).
Inductive LeftOrRight :=
| L: LeftOrRight
| R: LeftOrRight.
Definition half_tree: Type := (LeftOrRight * Node * tree)%type.
Definition partial_tree: Type := list half_tree.
Inductive SearchTree_half_in: (*inner border of partial tree*)
option Key -> partial_tree -> option Key -> Prop :=
| ST_in_nil:
forall lo hi, optionZ_lt lo hi -> SearchTree_half_in lo nil hi
| ST_in_cons_L:
forall lo hi h l n,
SearchTree_half_in lo h hi ->
SearchTree lo l (Some (key_of_node n)) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) hi
| ST_in_cons_R:
forall lo hi h r n,
SearchTree_half_in lo h hi ->
SearchTree (Some (key_of_node n)) r hi ->
SearchTree_half_in lo ((R, n, r) :: h) (Some (key_of_node n)).
Fact example:
forall n1 n2,
key_of_node n1 = 11 ->
value_of_node n1 = 11 ->
key_of_node n2 = 9 ->
value_of_node n2 = 9 ->
SearchTree_half_in (Some (key_of_node n1)) [(L, n1,(T E n2 E))] (Some 10).
Proof.
intros.
assert (SearchTree_half_in (Some 8) [] (Some 10)).
{ constructor. simpl. lia. }
assert (SearchTree (Some 8) (T E n2 E) (Some (key_of_node n1))).
{ constructor. constructor. rewrite H1. constructor.
constructor. rewrite H, H1. constructor. }
pose proof ST_in_cons_L _ _ _ _ _ H3 H4.
exact H5.
Qed.
Inductive Abs_half : partial_tree -> relate_map -> Prop :=
| Abs_half_nil : Abs_half nil relate_default
| Abs_half_cons: forall LR n t h m1 m2,
Abs t m1 ->
Abs_half h m2 ->
Abs_half
((LR, n, t) :: h)
(combine m1
(combine (relate_single (key_of_node n) (value_of_node n)) m2)).
Inductive SearchTree_half_out: (*outer border of partial tree*)
option Key -> partial_tree -> option Key -> Prop :=
| ST_out_nil:
forall lo hi, optionZ_lt lo hi -> SearchTree_half_out lo nil hi
| ST_out_cons_L:
forall lo hi h l n,
SearchTree_half_out lo h hi ->
SearchTree lo l (Some (key_of_node n)) ->
optionZ_lt (Some (key_of_node n)) hi ->
SearchTree_half_out lo ((L, n, l) :: h) hi
| ST_out_cons_R:
forall lo hi h r n,
SearchTree_half_out lo h hi ->
SearchTree (Some (key_of_node n)) r hi ->
optionZ_lt lo (Some (key_of_node n)) ->
SearchTree_half_out lo ((R, n, r) :: h) hi.
Inductive splay_step: partial_tree * tree -> partial_tree * tree -> Prop :=
| Splay_LL: forall h a b c d n1 n2 n3,
splay_step
((R, n2, c) :: (R, n3, d) :: h, T a n1 b)
(h, T a n1 (T b n2 (T c n3 d)))
| Splay_RR: forall h a b c d n1 n2 n3,
splay_step
((L, n2, b) :: (L, n1, a) :: h, T c n3 d)
(h, T (T (T a n1 b) n2 c) n3 d)
| Splay_RL: forall h a b c d n1 n2 n3, (* right child of left child *)
splay_step
((L, n1, a) :: (R, n3, d) :: h, T b n2 c)
(h, T (T a n1 b) n2 (T c n3 d))
| Splay_LR: forall h a b c d n1 n2 n3, (* left child of right child *)
splay_step
((R, n3, d) :: (L, n1, a) :: h, T b n2 c)
(h, T (T a n1 b) n2 (T c n3 d))
| Splay_L: forall x y z n1 n2,
splay_step ((R, n2, z) :: nil, T x n1 y) (nil, T x n1 (T y n2 z))
| Splay_R: forall x y z n1 n2,
splay_step ((L, n1, x) :: nil, T y n2 z) (nil, T (T x n1 y) n2 z)
.
Definition splay (h: partial_tree) (t t': tree): Prop :=
clos_refl_trans splay_step (h, t) (nil, t').
Definition preserves: Prop :=
forall HI LO hi lo h t t',
optionZ_lt (Some LO) (Some lo) ->
optionZ_lt (Some hi) (Some HI) ->
SearchTree_half_in (Some lo) h (Some hi) ->
SearchTree_half_out (Some LO) h (Some HI) ->
SearchTree (Some lo) t (Some hi)->
splay h t t' ->
SearchTree (Some LO) t' (Some HI).
Definition correct: Prop :=
forall h t t' m1 m2 lo hi LO HI,
Abs_half h m1 ->
Abs t m2 ->
splay h t t' ->
SearchTree (Some lo) t (Some hi)->(* new *)
SearchTree_half_in (Some lo) h (Some hi)->(* new *)
SearchTree_half_out (Some LO) h (Some HI)->(* new *)
optionZ_lt (Some LO) (Some lo) ->
optionZ_lt (Some hi) (Some HI) ->
Abs t' (combine m1 m2).
Definition splay' :partial_tree * tree -> partial_tree * tree -> Prop :=
clos_refl_trans splay_step .
Lemma splay'_splay :
forall h t t',
splay' (h,t) (nil,t') -> splay h t t'.
Proof.
intros.
unfold splay.
unfold splay' in H.
exact H.
Qed.
Lemma splay_splay':
forall h t t',
splay h t t' -> splay' (h,t) (nil,t').
Proof.
intros. unfold splay'. unfold splay in H. exact H.
Qed.
(* =============================================================*)
(* =====================Proof of preserves =====================*)
(* =============================================================*)
Lemma lt_le:
forall a b,
optionZ_lt a b -> optionZ_le a b.
Proof.
intros. destruct a;destruct b;simpl in *;try tauto. lia.
Qed.
Lemma lt_le':
forall a b,
optionZ_lt (Some a) (Some b) -> optionZ_le (Some a) (Some (b-1)).
Proof.
intros. simpl in *. lia.
Qed.
Lemma lt_le'':
forall a b,
optionZ_lt (Some a) (Some b) -> optionZ_le (Some (a+1)) (Some b).
Proof.
intros.
simpl in *.
lia.
Qed.
Lemma lt_le''':
forall a b,
optionZ_lt (Some a) (Some (b+1)) -> optionZ_le (Some a) (Some b).
Proof.
intros. simpl in *. lia.
Qed.
Lemma optionZ_lt_cong: forall n lo hi,
optionZ_lt (Some (n)) hi->
optionZ_lt lo (Some (n))->
optionZ_lt lo hi.
Proof.
intros. induction hi; simpl in H;simpl; induction lo; simpl in H0; simpl; try exact I; unfold Key in *. lia. Qed.
Lemma optionZ_le_cong: forall n lo hi,
optionZ_le (Some (n)) hi->
optionZ_le lo (Some (n))->
optionZ_le lo hi.
Proof.
intros. induction hi; simpl in H;simpl; induction lo; simpl in H0; simpl; try exact I; unfold Key in *. lia. Qed.
Lemma optionZ_let_cong: forall n lo hi,
optionZ_le (Some (n)) hi->
optionZ_lt lo (Some (n))->
optionZ_lt lo hi.
Proof.
intros. induction hi; simpl in H;simpl; induction lo; simpl in H0; simpl; try exact I; unfold Key in *. lia. Qed.
Lemma optionZ_lte_cong: forall n lo hi,
optionZ_lt (Some (n)) hi->
optionZ_le lo (Some (n))->
optionZ_lt lo hi.
Proof.
intros. induction hi; simpl in H;simpl; induction lo; simpl in H0; simpl; try exact I; unfold Key in *. lia. Qed.
Lemma optionZ_lt_SearchTree: forall l lo hi,
SearchTree lo l hi
-> optionZ_lt lo hi.
Proof.
intros. induction H. tauto. pose proof optionZ_lt_cong _ _ _ IHSearchTree2 IHSearchTree1. tauto.
Qed.
Lemma looser_SearchTree_l:
forall lo' lo hi t,
optionZ_lt lo (Some lo') ->
SearchTree (Some lo') t (Some hi) ->
SearchTree lo t (Some hi).
Proof.
intros. revert H. revert lo. revert H0. revert lo'. revert hi.
induction t;subst.
2:{ intros. inversion H0; subst. constructor.
specialize (IHt1 (key_of_node n) lo' H6 lo H).
exact IHt1. exact H7. }
intros. constructor.
pose proof optionZ_lt_SearchTree _ _ _ H0.
pose proof optionZ_lt_cong _ _ _ H1 H.
tauto.
Qed.
Lemma looser_SearchTree_r:
forall hi' lo hi t,
optionZ_lt (Some hi') hi ->
SearchTree (Some lo) t (Some hi') ->
SearchTree (Some lo) t hi.
Proof.
intros. revert H. revert hi. revert H0. revert lo. revert hi'.
induction t;subst.
2:{ intros. inversion H0; subst.
constructor. exact H6.
specialize (IHt2 hi' (key_of_node n) H7 hi H).
exact IHt2. }
intros. constructor.
pose proof optionZ_lt_SearchTree _ _ _ H0.
pose proof optionZ_lt_cong _ _ _ H H1.
tauto.
Qed.
Lemma looser_SearchTree:
forall lo' hi' lo hi t,
optionZ_lt lo (Some lo') ->
optionZ_lt (Some hi') hi ->
SearchTree (Some lo') t (Some hi') ->
SearchTree lo t hi.
Proof.
intros.
inversion H1. subst. constructor. pose proof optionZ_lt_cong _ _ _ H0 H2. pose proof optionZ_lt_cong _ _ _ H3 H. exact H4.
subst. constructor. pose proof looser_SearchTree_l _ _ _ _ H H2. tauto. pose proof looser_SearchTree_r _ _ _ _ H0 H3. tauto. Qed.
Lemma looser_SearchTree_l_e:
forall lo' lo hi t,
optionZ_le lo (Some lo') ->
SearchTree (Some lo') t (Some hi) ->
SearchTree lo t (Some hi).
Proof.
intros. revert H. revert lo. revert H0. revert lo'. revert hi.
induction t;subst.
2:{ intros. inversion H0; subst. constructor.
specialize (IHt1 (key_of_node n) lo' H6 lo H).
exact IHt1. exact H7. }
intros. constructor.
pose proof optionZ_lt_SearchTree _ _ _ H0.
pose proof optionZ_lte_cong _ _ _ H1 H.
tauto.
Qed.
Lemma looser_SearchTree_r_e:
forall hi' lo hi t,
optionZ_le (Some hi') hi ->
SearchTree (Some lo) t (Some hi') ->
SearchTree (Some lo) t hi.
Proof.
intros. revert H. revert hi. revert H0. revert lo. revert hi'.
induction t;subst.
2:{ intros. inversion H0; subst.
constructor. exact H6.
specialize (IHt2 hi' (key_of_node n) H7 hi H).
exact IHt2. }
intros. constructor.
pose proof optionZ_lt_SearchTree _ _ _ H0.
pose proof optionZ_let_cong _ _ _ H H1.
tauto.
Qed.
Lemma looser_SearchTree_le:
forall lo' hi' lo hi t,
optionZ_le lo (Some lo') ->
optionZ_le (Some hi') hi ->
SearchTree (Some lo') t (Some hi') ->
SearchTree lo t hi.
Proof.
intros.
inversion H1; subst. constructor. pose proof optionZ_let_cong _ _ _ H0 H2. pose proof optionZ_lte_cong _ _ _ H3 H. exact H4.
subst. constructor. pose proof looser_SearchTree_l_e _ _ _ _ H H2. tauto. pose proof looser_SearchTree_r_e _ _ _ _ H0 H3. tauto. Qed.
Fixpoint supremum (t: tree): option Key:=
match t with
| E => None
| T _ n E => Some (key_of_node n)
| T l n r => supremum r
end.
Lemma sup_fact':
forall t,
t <> E ->
exists v, supremum t = Some v.
Proof.
intros.
induction t.
+ tauto.
+ destruct t2.
{ exists (key_of_node n). simpl. reflexivity. }
assert (T t2_1 n0 t2_2 <> E).
{ pose proof classic (T t2_1 n0 t2_2 = E).
destruct H0;[inversion H0|tauto]. }
specialize (IHt2 H0).
destruct IHt2.
exists x.
simpl in *.
exact H1.
Qed.
Lemma sup_fact :
forall l n r, exists v, supremum (T l n r) = Some v.
Proof.
intros.
assert ((T l n r) <> E).
{ pose proof classic ((T l n r) = E). destruct H;[inversion H|tauto]. }
apply sup_fact' in H. tauto.
Qed.
Lemma sup_property:
forall lo hi t sup,
SearchTree lo t hi ->
supremum t = sup ->
optionZ_lt sup hi.
Proof.
intros.
revert sup lo hi H H0.
induction t;intros.
+ subst. simpl. tauto.
+ destruct t2.
{ simpl in H0. subst.
inversion H;subst.
inversion H6;subst.
exact H0.
}
inversion H. subst lo0 l n1 r hi0.
specialize (IHt2 sup _ _ H7).
simpl in *.
specialize (IHt2 H0).
exact IHt2.
Qed.
Lemma SearchTree_sup:
forall lo t hi sup,
SearchTree lo t hi ->
supremum t = (Some sup) ->
SearchTree lo t (Some (sup+1)).
Proof.
intros.
revert lo hi sup H H0 .
induction t;intros.
+ discriminate H0.
+ inversion H. subst lo0 l n0 r hi0.
constructor;[tauto|].
destruct t2.
{ simpl in H0. injection H0. intros. rewrite H1.
constructor. simpl. lia. }
specialize (IHt2 _ _ sup H7).
simpl in *.
specialize (IHt2 H0).
exact IHt2.
Qed.
Fixpoint infimum (t: tree): option Key:=
match t with
| E => None
| T E n _ => Some (key_of_node n)
| T l n r => infimum l
end.
Lemma inf_fact':
forall t,
t <> E ->
exists v, infimum t = Some v.
Proof.
intros.
induction t.
+ tauto.
+ destruct t1.
{ exists (key_of_node n). simpl. reflexivity. }
assert (T t1_1 n0 t1_2 <> E).
{ pose proof classic (T t1_1 n0 t1_2 = E).
destruct H0;[inversion H0|tauto]. }
specialize (IHt1 H0).
destruct IHt1.
exists x.
simpl in *.
exact H1.
Qed.
Lemma inf_fact :
forall l n r, exists v, infimum (T l n r) = Some v.
Proof.
intros.
assert ((T l n r) <> E).
{ pose proof classic ((T l n r) = E). destruct H;[inversion H|tauto]. }
apply inf_fact' in H. tauto.
Qed.
Lemma inf_property:
forall lo hi t inf,
SearchTree lo t hi ->
infimum t = inf ->
optionZ_lt lo inf.
Proof.
intros.
revert inf lo hi H H0.
induction t;intros.
+ subst. destruct lo;simpl;tauto.
+ destruct t1.
{ simpl in H0. subst.
inversion H;subst.
inversion H5;subst.
exact H0.
}
inversion H. subst lo0 l n1 r hi0.
specialize (IHt1 inf _ _ H6).
simpl in *.
specialize (IHt1 H0).
exact IHt1.
Qed.
Lemma SearchTree_inf:
forall lo t hi inf,
SearchTree lo t hi ->
infimum t = (Some inf) ->
SearchTree (Some (inf-1)) t hi.
Proof.
intros.
revert lo hi inf H H0 .
induction t;intros.
+ discriminate H0.
+ inversion H. subst lo0 l n0 r hi0.
constructor;[|tauto].
destruct t1.
{ simpl in H0. injection H0. intros. rewrite H1.
constructor. simpl. lia. }
specialize (IHt1 _ _ inf H6).
simpl in *.
specialize (IHt1 H0).
exact IHt1.
Qed.
Inductive R_in: partial_tree -> half_tree -> Prop :=
| R_in_base: forall n r h, R_in ((R, n, r)::h) (R, n, r)
| R_in_forward: forall n n' l r h, R_in h (R, n, r) -> R_in ((L, n', l)::h) (R, n, r).
Inductive all_L: partial_tree ->Prop :=
| AL_nil: all_L nil
| AL_forward: forall h n l, all_L h -> all_L ((L, n, l)::h).
Inductive L_in: partial_tree -> half_tree -> Prop :=
| L_in_base: forall n l h, L_in ((L, n, l)::h) (L, n, l)
| L_in_forward: forall n n' l r h, L_in h (L, n, l) -> L_in ((R, n', r)::h) (L, n, l).
Inductive all_R: partial_tree ->Prop :=
| AR_nil: all_R nil
| AR_forward: forall h n r, all_R h -> all_R ((R, n, r)::h).
Lemma all_L_or_R_in: forall h,
all_L h \/ exists n r, R_in h (R, n, r).
Proof.
intros.
induction h.
+ left. constructor.
+ destruct IHh.
- destruct a. destruct p. destruct l.
-- left. constructor. tauto.
-- right. exists n,t. constructor.
- right.
destruct H as [n [r ?]].
destruct a. destruct p. destruct l.
-- exists n, r. constructor; tauto.
-- exists n0, t. constructor.
Qed.
Lemma not_all_L_R_in: forall h,
~ all_L h <-> exists n r, R_in h (R, n, r) .
Proof.
intros.
unfold iff;split;intros.
+ pose proof all_L_or_R_in h.
destruct H0;tauto.
+ pose proof classic (all_L h).
destruct H0;[|tauto].
destruct H as [n [r ?]].
induction H;intros.
- inversion H0.
- inversion H0;subst. tauto.
Qed.
Lemma all_R_or_L_in: forall h,
all_R h \/ exists n l, L_in h (L, n, l).
Proof.
intros.
induction h.
+ left. constructor.
+ destruct IHh.
- destruct a. destruct p. destruct l.
-- right. exists n,t. constructor.
-- left. constructor. tauto.
- right.
destruct H as [n [l ?]].
destruct a. destruct p. destruct l0.
-- exists n0, t. constructor.
-- exists n, l. constructor; tauto.
Qed.
Lemma not_all_R_L_in: forall h,
~ all_R h <-> exists n l, L_in h (L, n, l) .
Proof.
intros.
unfold iff;split;intros.
+ pose proof all_R_or_L_in h.
destruct H0;tauto.
+ pose proof classic (all_R h).
destruct H0;[|tauto].
destruct H as [n [l ?]].
induction H;intros.
- inversion H0.
- inversion H0;subst. tauto.
Qed.
Lemma r_none_all_L:
forall n l h,
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) None ->
all_L ((L, n, l) :: h).
Proof.
intros.
pose proof classic (all_L ((L, n, l) :: h)).
pose proof not_all_L_R_in ((L, n, l) :: h).
destruct H0;[tauto|].
assert (exists (n0 : Node) (r : tree), R_in ((L, n, l) :: h) (R, n0, r)) by tauto.
clear H0 H1.
destruct H2 as [n0 [r ?]].
inversion H0;subst.
remember None as hi.
remember (R, n0, r) as ht.
revert n l H H0.
induction H2;intros;subst.
+ inversion H;subst.
inversion H6;subst.
+ inversion H;subst.
inversion H0;subst.
inversion H7;subst.
specialize (IHR_in Heqht _ _ H7 H4).
constructor. exact IHR_in.
Qed.
Lemma r_none_tighter:
forall n l h hi,
optionZ_le (Some (key_of_node n)) hi ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) None ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) hi.
Proof.
intros.
pose proof r_none_all_L _ _ _ H0.
inversion H1;subst.
revert n l H H0 H1.
induction H3;intros.
+ inversion H0;subst.
clear H2.
assert (SearchTree_half_in lo [] hi).
{ constructor. apply optionZ_lt_SearchTree in H8.
pose proof optionZ_let_cong _ _ _ H H8. exact H2. }
Print SearchTree_half_in.
pose proof ST_in_cons_L _ _ _ _ _ H2 H8.
exact H3.
+ inversion H0;subst.
inversion H8;subst.
inversion H1;subst.
pose proof optionZ_lt_SearchTree _ _ _ H9.
pose proof optionZ_let_cong _ _ _ H H4.
apply lt_le in H6.
specialize (IHall_L _ _ H6 H8 H5).
Print SearchTree_half_in.
pose proof ST_in_cons_L _ _ _ _ _ IHall_L H9.
exact H7.
Qed.
Lemma l_none_all_R:
forall n r h,
SearchTree_half_in None ((R, n, r) :: h) (Some (key_of_node n)) ->
all_R ((R, n, r) :: h).
Proof.
intros.
pose proof classic (all_R ((R, n, r) :: h)).
pose proof not_all_R_L_in ((R, n, r) :: h).
destruct H0;[tauto|].
assert (exists (n0 : Node) (l : tree), L_in ((R, n, r) :: h) (L, n0, l)) by tauto.
clear H0 H1.
destruct H2 as [n0 [l ?]].
inversion H0;subst.
remember None as lo.
remember (L, n0, l) as ht.
revert n r H H0.
induction H2;intros;subst.
+ inversion H;subst.
inversion H4;subst.
+ inversion H;subst.
inversion H0;subst.
inversion H5;subst.
specialize (IHL_in Heqht _ _ H5 H3).
constructor. exact IHL_in.
Qed.
Lemma l_none_tighter:
forall n r h lo,
optionZ_le lo (Some (key_of_node n)) ->
SearchTree_half_in None ((R, n, r) :: h) (Some (key_of_node n)) ->
SearchTree_half_in lo ((R, n, r) :: h) (Some (key_of_node n)).
Proof.
intros.
pose proof l_none_all_R _ _ _ H0.
inversion H1;subst.
revert n r H H0 H1.
induction H3;intros.
+ inversion H0;subst.
assert (SearchTree_half_in lo [] hi).
{ constructor. apply optionZ_lt_SearchTree in H7.
pose proof optionZ_lte_cong _ _ _ H7 H. exact H2. }
Print SearchTree_half_in.
pose proof ST_in_cons_R _ _ _ _ _ H2 H7.
exact H3.
+ inversion H0;subst.
inversion H6;subst.
inversion H1;subst.
pose proof optionZ_lt_SearchTree _ _ _ H8.
pose proof optionZ_lte_cong _ _ _ H2 H.
apply lt_le in H5.
specialize (IHall_R _ _ H5 H6 H4).
Print SearchTree_half_in.
pose proof ST_in_cons_R _ _ _ _ _ IHall_R H8.
exact H7.
Qed.
Lemma all_L_r_some_tighter:
forall n l h hi k,
all_L ((L, n, l) :: h) ->
optionZ_le (Some (key_of_node n)) hi ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) (Some k) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) hi.
Proof.
intros.
inversion H;subst.
revert n l H0 H1 H.
induction H3;intros.
+ inversion H1;subst.
assert (SearchTree_half_in lo [] hi).
{ constructor. apply optionZ_lt_SearchTree in H8.
pose proof optionZ_let_cong _ _ _ H0 H8. exact H3. }
Print SearchTree_half_in.
pose proof ST_in_cons_L _ _ _ _ _ H3 H8.
exact H4.
+ inversion H1;subst.
inversion H8;subst.
inversion H;subst.
pose proof optionZ_lt_SearchTree _ _ _ H9.
pose proof optionZ_let_cong _ _ _ H0 H4.
apply lt_le in H6.
specialize (IHall_L _ _ H6 H8 H5).
Print SearchTree_half_in.
pose proof ST_in_cons_L _ _ _ _ _ IHall_L H9.
exact H7.
Qed.
Lemma all_R_l_some_tighter:
forall n r h lo k,
all_R ((R, n, r) :: h) ->
optionZ_le lo (Some (key_of_node n)) ->
SearchTree_half_in (Some k) ((R, n, r) :: h) (Some (key_of_node n)) ->
SearchTree_half_in lo ((R, n, r) :: h) (Some (key_of_node n)).
Proof.
intros.
inversion H;subst.
revert n r H0 H1 H.
induction H3;intros.
+ inversion H1;subst.
assert (SearchTree_half_in lo [] hi).
{ constructor. apply optionZ_lt_SearchTree in H7.
pose proof optionZ_lte_cong _ _ _ H7 H0. exact H2. }
Print SearchTree_half_in.
pose proof ST_in_cons_R _ _ _ _ _ H2 H7.
exact H3.
+ inversion H1;subst.
inversion H6;subst.
inversion H;subst.
pose proof optionZ_lt_SearchTree _ _ _ H8.
pose proof optionZ_lte_cong _ _ _ H2 H0.
apply lt_le in H5.
specialize (IHall_R _ _ H5 H6 H4).
Print SearchTree_half_in.
pose proof ST_in_cons_R _ _ _ _ _ IHall_R H8.
exact H7.
Qed.
Lemma R_in_r_bound:
forall n l n0 r0 h LO HI hi,
R_in ((L, n, l)::h) (R, n0, r0) ->
SearchTree_half_out (Some LO) ((L, n, l) :: h) (Some HI) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) hi ->
hi = (Some (key_of_node n0)) /\ optionZ_le hi (Some HI).
Proof.
intros.
inversion H;subst.
clear H.
remember (R, n0, r0) as h_t.
revert n l H0 H1.
induction H3;intros.
+ inversion H1;subst.
inversion H6;subst.
injection Heqh_t.
intros;subst.
inversion H0;subst.
inversion H8;subst.
apply optionZ_lt_SearchTree, lt_le in H15.
split;[reflexivity|exact H15].
+ inversion H0;subst.
inversion H1;subst.
inversion H10;subst.
specialize (IHR_in Heqh_t _ _ H6 H10).
exact IHR_in.
Qed.
Lemma L_in_l_bound:
forall n r n0 l0 h LO HI lo,
L_in ((R, n, r)::h) (L, n0, l0) ->
SearchTree_half_out (Some LO) ((R, n, r) :: h) (Some HI) ->
SearchTree_half_in lo ((R, n, r) :: h) (Some (key_of_node n)) ->
lo = (Some (key_of_node n0)) /\ optionZ_le (Some LO) lo.
Proof.
intros.
inversion H;subst.
clear H.
remember (L, n0, l0) as h_t.
revert n r H0 H1.
induction H3;intros.
+ inversion H1;subst.
inversion H4;subst.
injection Heqh_t.
intros;subst.
inversion H0;subst.
inversion H7;subst.
apply optionZ_lt_SearchTree, lt_le in H14.
split;[reflexivity|exact H14].
+ inversion H0;subst.
inversion H1;subst.
inversion H5;subst.
specialize (IHL_in Heqh_t _ _ H6 H5).
exact IHL_in.
Qed.
Lemma all_L_r_bound:
forall lt nt rt LO HI n l h k,
all_L ((L, n, l) :: h) ->
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some HI) ->
SearchTree_half_out (Some LO) ((L, n, l) :: h) (Some HI) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) (Some k) ->
exists hi',
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some hi') /\
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) (Some hi') /\
optionZ_le (Some hi') (Some HI).
Proof.
intros.
pose proof sup_fact lt nt rt. destruct H3 as [sup ?].
exists (sup +1).
pose proof SearchTree_sup _ _ _ _ H0 H3.
split;[exact H4|split].
{ inversion H4;subst.
apply optionZ_lt_SearchTree in H10. apply optionZ_lt_SearchTree in H11.
pose proof optionZ_lt_cong _ _ _ H11 H10. apply lt_le in H5.
pose proof all_L_r_some_tighter _ _ _ _ _ H H5 H2. exact H6. }
pose proof sup_property _ _ _ _ H0 H3.
apply lt_le'' in H5. exact H5.
Qed.
Lemma all_R_l_bound:
forall lt nt rt LO HI n r h k,
all_R ((R, n, r) :: h) ->
SearchTree (Some LO) (T lt nt rt) (Some (key_of_node n)) ->
SearchTree_half_out (Some LO) ((R, n, r) :: h) (Some HI) ->
SearchTree_half_in (Some k) ((R, n, r) :: h) (Some (key_of_node n)) ->
exists lo',
SearchTree (Some lo') (T lt nt rt) (Some (key_of_node n)) /\
SearchTree_half_in (Some lo') ((R, n, r) :: h) (Some (key_of_node n)) /\
optionZ_le (Some LO) (Some lo').
Proof.
intros.
pose proof inf_fact lt nt rt. destruct H3 as [inf ?].
exists (inf - 1).
pose proof SearchTree_inf _ _ _ _ H0 H3.
split;[exact H4|split].
{ inversion H4;subst.
apply optionZ_lt_SearchTree in H10. apply optionZ_lt_SearchTree in H11.
pose proof optionZ_lt_cong _ _ _ H11 H10. apply lt_le in H5.
pose proof all_R_l_some_tighter _ _ _ _ _ H H5 H2. exact H6. }
pose proof inf_property _ _ _ _ H0 H3.
apply lt_le' in H5. exact H5.
Qed.
Lemma r_bound_None:
forall lt nt rt LO HI n l h,
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some HI) ->
SearchTree_half_out (Some LO) ((L, n, l) :: h) (Some HI) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) None ->
exists hi',
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some hi') /\
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) (Some hi') /\
optionZ_le (Some hi') (Some HI).
Proof.
intros.
pose proof sup_fact lt nt rt. destruct H2 as [sup ?].
exists (sup + 1).
pose proof SearchTree_sup _ _ _ _ H H2.
split;[exact H3|split].
{ inversion H3;subst.
apply optionZ_lt_SearchTree in H9.
apply optionZ_lt_SearchTree in H10.
pose proof optionZ_lt_cong _ _ _ H10 H9.
apply lt_le in H4. pose proof r_none_tighter _ _ _ _ H4 H1. exact H5. }
Check sup_property.
pose proof sup_property _ _ _ _ H H2.
apply lt_le'' in H4. exact H4.
Qed.
Lemma l_bound_None:
forall lt nt rt LO HI n r h,
SearchTree (Some LO) (T lt nt rt) (Some (key_of_node n)) ->
SearchTree_half_out (Some LO) ((R, n, r) :: h) (Some HI) ->
SearchTree_half_in None ((R, n, r) :: h) (Some (key_of_node n)) ->
exists lo',
SearchTree (Some lo') (T lt nt rt) (Some (key_of_node n)) /\
SearchTree_half_in (Some lo') ((R, n, r) :: h) (Some (key_of_node n)) /\
optionZ_le (Some LO) (Some lo').
Proof.
intros.
pose proof inf_fact lt nt rt. destruct H2 as [inf ?].
exists (inf - 1).
pose proof SearchTree_inf _ _ _ _ H H2.
split;[exact H3|split].
{ inversion H3;subst.
apply optionZ_lt_SearchTree in H9.
apply optionZ_lt_SearchTree in H10.
pose proof optionZ_lt_cong _ _ _ H10 H9.
apply lt_le in H4. pose proof l_none_tighter _ _ _ _ H4 H1. exact H5. }
Check inf_property.
pose proof inf_property _ _ _ _ H H2.
apply lt_le' in H4. exact H4.
Qed.
Lemma inner_border_tighter_L:
forall n l h hi LO HI lt nt rt,
SearchTree (Some (key_of_node n)) (T lt nt rt) hi ->
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some HI) ->
SearchTree_half_out (Some LO) ((L, n, l) :: h) (Some HI) ->
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) hi ->
exists hi',
SearchTree (Some (key_of_node n)) (T lt nt rt) (Some hi') /\
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) (Some hi') /\
optionZ_le (Some hi') (Some HI).
Proof.
intros.
destruct hi.
2:{ pose proof r_bound_None _ _ _ _ _ _ _ _ H0 H1 H2. exact H3. }
pose proof all_L_or_R_in ((L, n, l)::h).
destruct H3;[|destruct H3 as [n0 [r ?]]].
2:{ pose proof R_in_r_bound _ _ _ _ _ _ _ _ H3 H1 H2.
destruct H4;injection H4;intros;subst.
exists (key_of_node n0). tauto. }
pose proof all_L_r_bound _ _ _ _ _ _ _ _ _ H3 H0 H1 H2. exact H4.
Qed.
Lemma inner_border_tighter_R:
forall n r h lo LO HI lt nt rt,
SearchTree lo (T lt nt rt) (Some (key_of_node n)) ->
SearchTree (Some LO) (T lt nt rt) (Some (key_of_node n)) ->
SearchTree_half_out (Some LO) ((R, n, r) :: h) (Some HI) ->
SearchTree_half_in lo ((R, n, r) :: h) (Some (key_of_node n)) ->
exists lo',
SearchTree (Some lo') (T lt nt rt) (Some (key_of_node n)) /\
SearchTree_half_in (Some lo') ((R, n, r) :: h) (Some (key_of_node n)) /\
optionZ_le (Some LO) (Some lo').
Proof.
intros.
destruct lo.
2:{ pose proof l_bound_None _ _ _ _ _ _ _ _ H0 H1 H2. exact H3. }
pose proof all_R_or_L_in ((R, n, r)::h).
destruct H3;[|destruct H3 as [n0 [l ?]]].
2:{ pose proof L_in_l_bound _ _ _ _ _ _ _ _ H3 H1 H2.
destruct H4;injection H4;intros;subst.
exists (key_of_node n0). tauto. }
pose proof all_R_l_bound _ _ _ _ _ _ _ _ _ H3 H0 H1 H2. exact H4.
Qed.
Lemma step_preserves:
forall h h' t t' lo hi LO HI,
optionZ_le (Some LO) (Some lo) ->
optionZ_le (Some hi) (Some HI) ->
SearchTree_half_in (Some lo) h (Some hi) ->
SearchTree_half_out (Some LO) h (Some HI) ->
SearchTree (Some lo) t (Some hi) ->
splay_step (h,t) (h',t') ->
exists lo' hi',
(optionZ_le (Some LO) (Some lo')) /\
(optionZ_le (Some hi') (Some HI)) /\
(SearchTree (Some lo') t' (Some hi')) /\
(SearchTree_half_in (Some lo') h' (Some hi')) /\
(SearchTree_half_out (Some LO) h' (Some HI)).
Proof.
intros.
inversion H4;subst.
+ inversion H1;subst.
inversion H8;subst.
rename H3 into H_Tn1, H11 into H_c, H13 into H_d.
rename H12 into H_h'.
inversion H2;subst.
inversion H9;subst.
exists lo.
inversion H_h';subst.
3:{ exists (key_of_node n).
inversion H10;subst.
apply optionZ_lt_SearchTree , lt_le in H19.
split;[exact H|split;[exact H19|split]].
{ inversion H_Tn1;subst.
constructor;try tauto;constructor;try tauto;constructor;try tauto. }
split;tauto. }
- exists HI.
split;[tauto|split;[simpl;lia|]].
split;[|split;[|exact H10]].
{ inversion H_Tn1;subst.
constructor;try tauto;constructor;try tauto;constructor;try tauto. }
constructor.
apply optionZ_lt_SearchTree in H_Tn1.
apply optionZ_lt_SearchTree in H11.
pose proof optionZ_lt_cong _ _ _ H11 H_Tn1.
exact H5.
- inversion H_Tn1;subst.
assert (SearchTree (Some (key_of_node n)) (T a n1 (T b n2 (T c n3 d))) hi).
{ constructor;try tauto;constructor;try tauto;constructor;try tauto. }
assert (SearchTree (Some (key_of_node n)) (T a n1 (T b n2 (T c n3 d))) (Some HI)).
{ constructor;try tauto;constructor;try tauto;constructor;try tauto. }
pose proof inner_border_tighter_L _ _ _ _ _ _ _ _ _ H3 H7 H10 H_h'.
destruct H13 as [hi' ?].
exists hi'. tauto.
+ inversion H1;subst. inversion H10;subst.
inversion H2;subst. inversion H9;subst.
inversion H3;subst.
inversion H12;subst.
2:{ exists (key_of_node n), hi.
inversion H14;subst. apply optionZ_lt_SearchTree, lt_le in H25.
assert (SearchTree (Some (key_of_node n)) (T (T (T a n1 b) n2 c) n3 d) (Some hi)).
{ constructor;[constructor;[constructor;tauto|tauto]|tauto]. }
tauto. }
- exists LO, hi.
assert (optionZ_le (Some LO) (Some LO)) by (simpl;lia).
assert (SearchTree (Some LO) (T (T (T a n1 b) n2 c) n3 d) (Some hi)).
{ constructor;[constructor;[constructor;tauto|tauto]|tauto]. }
apply optionZ_lt_SearchTree in H21. apply optionZ_lt_SearchTree in H20.
apply optionZ_lt_SearchTree in H15.
pose proof optionZ_lt_cong _ _ _ H21 H20.
pose proof optionZ_lt_cong _ _ _ H8 H15.
pose proof ST_in_nil _ _ H17.
tauto.
- assert (SearchTree lo (T (T (T a n1 b) n2 c) n3 d) (Some (key_of_node n))).
{ constructor;[constructor;[constructor;tauto|tauto]|tauto]. }
assert (SearchTree (Some LO) (T (T (T a n1 b) n2 c) n3 d) (Some (key_of_node n))).
{ constructor;[constructor;[constructor;tauto|tauto]|tauto]. }
pose proof inner_border_tighter_R _ _ _ _ _ _ _ _ _ H5 H7 H14 H12.
destruct H8 as [lo' ?].
exists lo' ,(key_of_node n). tauto.
+ inversion H1;subst. inversion H10;subst.
inversion H2;subst. inversion H12;subst.
inversion H8;subst.
- exists LO, HI.
inversion H3;subst.
assert (optionZ_le (Some LO) (Some LO)) by (simpl;lia).
assert (optionZ_le (Some HI) (Some HI)) by (simpl;lia).
assert (SearchTree (Some LO) (T (T a n1 b) n2 (T c n3 d)) (Some HI)).
{ constructor;constructor;tauto. }
inversion H14;subst.
assert (SearchTree_half_in (Some LO) [] (Some HI)) by (constructor;simpl;exact H17).
tauto.
- exists (key_of_node n).
inversion H14;subst.
apply optionZ_lt_SearchTree, lt_le in H23.
inversion H3;subst.
assert (SearchTree (Some (key_of_node n)) (T (T a n1 b) n2 (T c n3 d)) hi0).
{ constructor;constructor;tauto. }
assert (SearchTree (Some (key_of_node n)) (T (T a n1 b) n2 (T c n3 d)) (Some HI)).
{ constructor;constructor;tauto. }
pose proof inner_border_tighter_L _ _ _ _ _ _ _ _ _ H7 H9 H14 H8.
destruct H17 as [hi' ?].
exists hi'. tauto.
- inversion H14;subst.
apply optionZ_lt_SearchTree, lt_le in H23.
inversion H3;subst.
assert (SearchTree lo0 (T (T a n1 b) n2 (T c n3 d)) (Some (key_of_node n))).
{ constructor;constructor;tauto. }
assert (SearchTree (Some LO) (T (T a n1 b) n2 (T c n3 d)) (Some (key_of_node n))).
{ constructor;constructor;tauto. }
pose proof inner_border_tighter_R _ _ _ _ _ _ _ _ _ H7 H9 H14 H8.
destruct H17 as [lo' ?].
exists lo', (key_of_node n). tauto.
+ inversion H1;subst. inversion H8;subst.
inversion H2;subst. inversion H10;subst.
inversion H12;subst.
- exists LO, HI.
inversion H3;subst.
assert (optionZ_le (Some LO) (Some LO)) by (simpl;lia).
assert (optionZ_le (Some HI) (Some HI)) by (simpl;lia).
assert (SearchTree (Some LO) (T (T a n1 b) n2 (T c n3 d)) (Some HI)).
{ constructor;constructor;tauto. }
inversion H14;subst.
assert (SearchTree_half_in (Some LO) [] (Some HI)) by (constructor;simpl;exact H17).
tauto.
- exists (key_of_node n).
inversion H14;subst.
apply optionZ_lt_SearchTree, lt_le in H23.
inversion H3;subst.
assert (SearchTree (Some (key_of_node n)) (T (T a n1 b) n2 (T c n3 d)) hi0).
{ constructor;constructor;tauto. }
assert (SearchTree (Some (key_of_node n)) (T (T a n1 b) n2 (T c n3 d)) (Some HI)).
{ constructor;constructor;tauto. }
pose proof inner_border_tighter_L _ _ _ _ _ _ _ _ _ H7 H9 H14 H12.
destruct H17 as [hi' ?].
exists hi'. tauto.
- inversion H14;subst.
apply optionZ_lt_SearchTree, lt_le in H23.
inversion H3;subst.
assert (SearchTree lo0 (T (T a n1 b) n2 (T c n3 d)) (Some (key_of_node n))).
{ constructor;constructor;tauto. }
assert (SearchTree (Some LO) (T (T a n1 b) n2 (T c n3 d)) (Some (key_of_node n))).
{ constructor;constructor;tauto. }
pose proof inner_border_tighter_R _ _ _ _ _ _ _ _ _ H7 H9 H14 H12.
destruct H17 as [lo' ?].
exists lo', (key_of_node n). tauto.
+ inversion H1;subst.
inversion H2;subst.
inversion H3;subst.
exists lo, HI.
assert (optionZ_le (Some HI) (Some HI)) by (simpl;lia).
assert (SearchTree (Some lo) (T x n1 (T y n2 z)) (Some HI)).
{ constructor;[tauto|constructor;tauto]. }
pose proof optionZ_lt_SearchTree _ _ _ H6.
assert (SearchTree_half_in (Some lo) [] (Some HI)) by (constructor;exact H7).
tauto.
+ inversion H1;subst.
inversion H2;subst.
inversion H3;subst.
exists LO, hi.
assert (optionZ_le (Some LO) (Some LO)) by (simpl;lia).
assert (SearchTree (Some LO) (T (T x n1 y) n2 z) (Some hi)).
{ constructor;[constructor;tauto|tauto]. }
pose proof optionZ_lt_SearchTree _ _ _ H6.
assert (SearchTree_half_in (Some LO) [] (Some hi)) by (constructor;exact H7).
tauto.
Qed.
Lemma preserves_le:
forall HI LO hi lo h t t',
optionZ_le (Some LO) (Some lo) ->
optionZ_le (Some hi) (Some HI) ->
SearchTree_half_in (Some lo) h (Some hi) ->
SearchTree_half_out (Some LO) h (Some HI) ->
SearchTree (Some lo) t (Some hi)->
splay h t t' ->
SearchTree (Some LO) t' (Some HI).
Proof.
intros.
apply splay_splay' in H4.
revert H H0 H1 H2 H3.
revert lo hi LO HI.
induction_1n H4; intros.
2:{ rename p into h'.
pose proof step_preserves _ _ _ _ _ _ _ _ H1 H2 H3 H4 H5 H.
clear H1 H2 H3 H4 H5 H.
destruct H6 as [lo' [hi' [? [? [? [? ?]]]]]].
specialize (IHrt _ _ _ _ H H1 H3 H4 H2).
exact IHrt.
}
inversion H1. inversion H2. subst. clear H1 H2.
Check looser_SearchTree.
pose proof looser_SearchTree_le _ _ _ _ _ H H0 H3. tauto.
Qed.
Theorem preserve: preserves.
Proof.
unfold preserves;intros.
apply lt_le in H.
apply lt_le in H0.
pose proof preserves_le _ _ _ _ _ _ _ H H0 H1 H2 H3 H4.
exact H5.
Qed.
(* ============================================================*)
(* ===================== Proof of correct =====================*)
(* ============================================================*)
Lemma map_eq: forall lm rm: relate_map,
(forall k , lm k = rm k)->
lm=rm.
Proof.
intros.
extensionality k. apply H.
Qed.
Lemma combine_com:
forall m1 m2,
forall k, combine m1 m2 k = combine m2 m1 k.
Proof.
intros.
unfold combine.
destruct (m1 k);destruct (m2 k);reflexivity.
Qed.
Lemma Abs_in:
forall t lo hi m,
Abs t m->
SearchTree lo t hi->
forall k, (m k= None/\ optionZ_lt lo hi) \/ (exists v, m k =Some v /\ optionZ_lt lo (Some k) /\ optionZ_lt (Some k) hi).
Proof.
intros. revert H k. revert m. induction H0;subst.
intros.
inversion H0;subst. left. tauto.
intros. inversion H;subst. specialize (IHSearchTree1 lm H4 k). specialize (IHSearchTree2 rm H5 k). destruct IHSearchTree1; destruct IHSearchTree2; pose proof combine_com (relate_single (key_of_node n) (value_of_node n)) rm;
pose proof map_eq _ _ H2; rewrite H3; unfold combine .
+ destruct H0;rewrite H0;destruct H1; rewrite H1; unfold relate_single. destruct (Z.eq_dec k (key_of_node n)). right. pose proof optionZ_lt_SearchTree _ _ _ H0_. pose proof optionZ_lt_SearchTree _ _ _ H0_0. pose proof lt_le _ _ H8. pose proof lt_le _ _ H9. rewrite e. exists (value_of_node n). split;tauto. left. split. tauto. pose proof optionZ_lt_cong _ _ _ H7 H6. tauto.
+ destruct H0; rewrite H0. destruct H1 as[v[?[? ?]]]. rewrite H1. assert ((key_of_node n)<> k). simpl in H7. lia. unfold relate_single. destruct (Z.eq_dec k (key_of_node n)). rewrite e in H9. tauto. right. exists v. split;[reflexivity| ]. split. pose proof optionZ_lt_SearchTree _ _ _ H0_. pose proof optionZ_lt_cong _ _ _ H7 H10. tauto. tauto.
+ destruct H0 as[v[?[? ?]]]. destruct H1. rewrite H0. rewrite H1. assert ((key_of_node n)<> k). simpl in H7. lia. unfold relate_single. destruct (Z.eq_dec k (key_of_node n)). rewrite e in H9. tauto. right. exists v. split;[reflexivity| ]. split. tauto. pose proof optionZ_lt_SearchTree _ _ _ H0_0. pose proof optionZ_lt_cong _ _ _ H10 H7. tauto.
+ destruct H0 as[vl[?[? ?]]]. destruct H1 as[vr[?[? ?]]]. simpl in H7. simpl in H8. lia.
Qed.
Lemma l_none_le:
forall n r h m k v,
SearchTree_half_in None ((R, n, r) :: h) (Some (key_of_node n)) ->
Abs_half ((R, n, r) :: h) m ->
m k = Some v ->
optionZ_le (Some (key_of_node n)) (Some k).
Proof.
intros.
pose proof l_none_all_R _ _ _ H.
inversion H2;subst. clear H2.
revert n r m v H H0 H1.
induction H4;subst;intros.
+ inversion H0;subst.
inversion H8;subst. clear H8.
inversion H;subst. clear H5.
inversion H8;subst.
{ inversion H7;subst. assert( relate_default k = None ) by (unfold relate_default;reflexivity). assert((relate_single (key_of_node n) (value_of_node n) k) = Some v). { destruct (relate_single (key_of_node n) (value_of_node n) k) eqn:?H. { unfold combine in H1;rewrite H3 in H1;rewrite H4 in H1. exact H1. } unfold combine in H1;rewrite H3 in H1;rewrite H4 in H1. discriminate H1. } unfold relate_single in H4. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H4. } rewrite H5. simpl. simpl. lia. }
pose proof Abs_in _ _ _ _ H7 H8. specialize (H4 k).
destruct H4.
{ destruct H4.
assert( relate_default k = None ) by (unfold relate_default;reflexivity). assert((relate_single (key_of_node n) (value_of_node n) k) = Some v). { destruct (relate_single (key_of_node n) (value_of_node n) k) eqn:?H. { unfold combine in H1;rewrite H4 in H1;rewrite H9 in H1;rewrite H6 in H1. exact H1. } unfold combine in H1;rewrite H4 in H1;rewrite H9 in H1;rewrite H6 in H1. discriminate H1. } unfold relate_single in H9. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H9. } rewrite H10. simpl. simpl. lia. }
destruct H4 as [? [? [? ?]]]. apply lt_le in H5. exact H5.
+ inversion H;subst.
inversion H0;subst.
specialize (IHall_R n r m2).
pose proof Abs_in _ _ _ _ H10 H8.
specialize (H2 k).
destruct H2.
2:{ destruct H2 as [? [? [? ?]]]. apply lt_le in H3. exact H3. }
destruct H2.
destruct ((relate_single (key_of_node n0) (value_of_node n0)) k) eqn:?H.
{ unfold relate_single in H5. assert (k=(key_of_node n0)). { destruct (Z.eq_dec k (key_of_node n0)). tauto. discriminate H5. } rewrite H7. simpl. simpl. lia. }
assert (m2 k = Some v). { unfold combine in H1; rewrite H2 in H1; rewrite H5 in H1. destruct (m2 k);[ exact H1| discriminate H1]. }
clear H1.
inversion H6;subst.
specialize (IHall_R v H6 H11 H7).
pose proof optionZ_let_cong _ _ _ IHall_R H3.
apply lt_le in H1; exact H1.
Qed.
Lemma r_none_le:
forall n l h m k v,
SearchTree_half_in (Some (key_of_node n)) ((L, n, l) :: h) None ->
Abs_half ((L, n, l) :: h) m ->
m k = Some v ->
optionZ_le (Some k) (Some (key_of_node n)).
Proof.
intros.
pose proof r_none_all_L _ _ _ H.
inversion H2;subst. clear H2.
revert n l m v H H0 H1.
induction H4;subst;intros.
+ inversion H0;subst.
inversion H8;subst. clear H8.
inversion H;subst. clear H2. clear H8.
inversion H9;subst.
{ inversion H7;subst. assert( relate_default k = None ) by (unfold relate_default;reflexivity). assert((relate_single (key_of_node n) (value_of_node n) k) = Some v). { destruct (relate_single (key_of_node n) (value_of_node n) k) eqn:?H. { unfold combine in H1;rewrite H3 in H1;rewrite H4 in H1. exact H1. } unfold combine in H1;rewrite H3 in H1;rewrite H4 in H1. discriminate H1. } unfold relate_single in H4. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H4. } rewrite H5. simpl. simpl. lia. }
pose proof Abs_in _ _ _ _ H7 H9. specialize (H4 k).
destruct H4.
{ destruct H4.
assert( relate_default k = None ) by (unfold relate_default;reflexivity). assert((relate_single (key_of_node n) (value_of_node n) k) = Some v). { destruct (relate_single (key_of_node n) (value_of_node n) k) eqn:?H. { unfold combine in H1;rewrite H4 in H1;rewrite H8 in H1;rewrite H6 in H1. exact H1. } unfold combine in H1;rewrite H4 in H1;rewrite H8 in H1;rewrite H6 in H1. discriminate H1. } unfold relate_single in H8. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H8. } rewrite H10. simpl. simpl. lia. }
destruct H4 as [? [? [? ?]]]. apply lt_le in H6. exact H6.
+ inversion H;subst. clear H2.
inversion H0;subst.
specialize (IHall_L n l m2).
pose proof Abs_in _ _ _ _ H10 H9.
specialize (H2 k).
destruct H2.
2:{ destruct H2 as [? [? [? ?]]]. apply lt_le in H5. exact H5. }
destruct H2.
destruct ((relate_single (key_of_node n0) (value_of_node n0)) k) eqn:?H.
{ unfold relate_single in H5. assert (k=(key_of_node n0)). { destruct (Z.eq_dec k (key_of_node n0)). tauto. discriminate H5. } rewrite H6. simpl. simpl. lia. }
assert (m2 k = Some v). { unfold combine in H1; rewrite H2 in H1; rewrite H5 in H1. destruct (m2 k);[ exact H1| discriminate H1]. }
clear H1.
inversion H8;subst.
specialize (IHall_L v H8 H11 H6).
pose proof optionZ_lte_cong _ _ _ H3 IHall_L.
apply lt_le in H1; exact H1.
Qed.
Lemma Abs_in_half:
forall t lo hi m ,
Abs_half t m->
SearchTree_half_in (Some lo) t (Some hi)->
(* SearchTree_half_out LO t HI-> *)
forall k, m k= None \/ (exists v, m k =Some v /\ (optionZ_le (Some k) (Some lo) \/ optionZ_le (Some hi) (Some k))).
Proof.
intros. revert H k. revert m. induction H0;subst;intros.
+ inversion H0;subst. left. tauto.
+ inversion H1;subst.
specialize (IHSearchTree_half_in m2 H8 k).
destruct IHSearchTree_half_in.
{ pose proof Abs_in _ _ _ _ H7 H.
specialize (H3 k).
destruct H3.
{ destruct H3. destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ right. exists v. split;[unfold combine;rewrite H2;rewrite H3;rewrite H5;reflexivity|]. unfold relate_single in H5. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H5. } rewrite H6 in *. left. simpl. lia. }
left. unfold combine;rewrite H2;rewrite H3;rewrite H5;reflexivity. }
destruct H3 as [v [? [? ?]]].
destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ left. unfold combine;rewrite H2;rewrite H3;rewrite H6;reflexivity. }
right. exists v. split;[unfold combine; rewrite H2;rewrite H3;rewrite H6;reflexivity|]. left. apply lt_le in H5;exact H5. }
destruct H2 as [v [? ?]].
destruct (m1 k) eqn: ?H;destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ left. assert(k=(key_of_node n)). {unfold relate_single in H5 . destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H5. } rewrite H6 in *. pose proof Abs_in _ _ _ _ H7 H. specialize (H9 (key_of_node n)). destruct H9. { destruct H9. rewrite H4 in H9. discriminate H9. } destruct H9 as [? [? [? ?]]]. simpl in H11. lia. }
{ left. unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity. }
{ left. unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity. }
{ right. exists v. split;[unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity|].
destruct H3;[|right;exact H3].
destruct lo0 ;[rename k0 into lo0|].
{ left. apply optionZ_lt_SearchTree in H. pose proof optionZ_lte_cong _ _ _ H H3. apply lt_le in H6;exact H6. }
inversion H0;subst.
{ inversion H8. rewrite <- H10 in H2. unfold relate_default in H2. discriminate H2. }
pose proof l_none_le _ _ _ _ _ _ H0 H8 H2.
right. exact H10. }
+ inversion H1;subst.
specialize (IHSearchTree_half_in m2 H8 k).
destruct IHSearchTree_half_in.
{ pose proof Abs_in _ _ _ _ H7 H.
specialize (H3 k).
destruct H3.
{ destruct H3. destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ right. exists v. split;[unfold combine;rewrite H2;rewrite H3;rewrite H5;reflexivity|]. unfold relate_single in H5. assert(k=(key_of_node n)). { destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H5. } rewrite H6 in *. right. simpl. lia. }
left. unfold combine;rewrite H2;rewrite H3;rewrite H5;reflexivity. }
destruct H3 as [v [? [? ?]]].
destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ left. unfold combine;rewrite H2;rewrite H3;rewrite H6;reflexivity. }
right. exists v. split;[unfold combine; rewrite H2;rewrite H3;rewrite H6;reflexivity|]. right. apply lt_le in H4;exact H4. }
destruct H2 as [v [? ?]].
destruct (m1 k) eqn: ?H;destruct ((relate_single (key_of_node n) (value_of_node n)) k) eqn: ?H.
{ left. assert(k=(key_of_node n)). {unfold relate_single in H5 . destruct(Z.eq_dec k (key_of_node n)). tauto. discriminate H5. } rewrite H6 in *. pose proof Abs_in _ _ _ _ H7 H. specialize (H9 (key_of_node n)). destruct H9. { destruct H9. rewrite H4 in H9. discriminate H9. } destruct H9 as [? [? [? ?]]]. simpl in H10. lia. }
{ left. unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity. }
{ left. unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity. }
{ right. exists v. split;[unfold combine; rewrite H2;rewrite H4;rewrite H5;reflexivity|].
destruct H3;[left;exact H3|].
destruct hi0 ;[rename k0 into hi0|].
{ right. apply optionZ_lt_SearchTree in H. pose proof optionZ_let_cong _ _ _ H3 H. apply lt_le in H6;exact H6. }
inversion H0;subst.
{ inversion H8. rewrite <- H10 in H2. unfold relate_default in H2. discriminate H2. }
pose proof r_none_le _ _ _ _ _ _ H0 H8 H2.
left. exact H10. }
Qed.
Lemma step_correct_le:
forall h t h' t' m1 m2 lo hi LO HI,
splay_step (h,t) (h',t') ->
Abs_half h m1 ->
Abs t m2 ->
SearchTree (Some lo) t (Some hi)->
(SearchTree_half_in (Some lo) h (Some hi))->
(SearchTree_half_out (Some LO) h (Some HI))->
optionZ_le (Some LO) (Some lo)->
optionZ_le (Some hi) (Some HI)->
(exists lo' hi' LO' HI' m1' m2',
(SearchTree (Some lo') t' (Some hi') )/\
(SearchTree_half_in (Some lo') h' (Some hi')) /\ (SearchTree_half_out (Some LO') h' (Some HI')) /\ optionZ_le (Some LO') (Some lo') /\
optionZ_le (Some hi') (Some HI') /\ (Abs_half h' m1') /\ (Abs t' m2') /\
(forall k, combine m1' m2' k = combine m1 m2 k)).
Proof.
intros.
pose proof step_preserves _ _ _ _ _ _ _ _ H5 H6 H3 H4 H2 H.
destruct H7 as [lo' [hi' [? [? [? [? ?]]]]]].
exists lo',hi', LO ,HI.
inversion H;subst.
+ inversion H0;subst. inversion H1;subst. inversion H18;subst. exists m2. exists (combine lm (combine (relate_single (key_of_node n1) (value_of_node n1))
(combine rm (combine (relate_single (key_of_node n2) (value_of_node n2))
(combine m0 (combine (relate_single (key_of_node n3) (value_of_node n3))
m1)))))). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split. constructor;try tauto. constructor;try tauto. constructor;try tauto. intros. clear H7 H8 H11 H5 H6 H3 H4 H H0 H18 H1. inversion H9;subst;clear H9. inversion H6;subst;clear H6. inversion H8;subst;clear H8. pose proof Abs_in _ _ _ _ H16 H5 k; pose proof optionZ_lt_SearchTree _ _ _ H5; clear H16 H5. pose proof Abs_in _ _ _ _ H19 H7 k; pose proof optionZ_lt_SearchTree _ _ _ H7; clear H19 H7. pose proof Abs_in _ _ _ _ H17 H6 k; pose proof optionZ_lt_SearchTree _ _ _ H6; clear H17 H6. clear H2. pose proof Abs_in _ _ _ _ H21 H9 k. clear H21 H9.
pose proof Abs_in_half _ _ _ _ H22 H10 k; clear H22 H10. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in H9. simpl in H8.
lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6; destruct H1;destruct H4;destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H6 as [v1[? ?]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. rewrite H1, H4,H2,H6,H. clear H1 H2 H4 H6 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H. clear H7.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H4;destruct H2. unfold combine. rewrite H1, H4,H2,H6,H. clear H1 H2 H4 H6 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H2. unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H2.
2:{destruct H2 as [v[?[? ?] ]]. destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H4. unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H6;destruct H4;destruct H2.
1:{unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. simpl in *. unfold relate_single. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2)). 1:{ destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. } destruct (Z.eq_dec k (key_of_node n3));try tauto. }
destruct H6 as [v[? ?]]. destruct H10;simpl in *;
unfold combine; rewrite H1,H2,H6, H4,H; clear H1 H2 H6 H4 H; simpl in *; unfold relate_single; destruct (Z.eq_dec k (key_of_node n1));try lia; destruct (Z.eq_dec k (key_of_node n2));try lia; destruct (Z.eq_dec k (key_of_node n3));try lia; tauto.
+ inversion H0;subst. inversion H1;subst. inversion H18;subst. exists m2. exists (combine(combine(combine m1(combine (relate_single(key_of_node n1)(value_of_node n1)) m0))(combine(relate_single(key_of_node n2)(value_of_node n2)) lm)) (combine (relate_single(key_of_node n3)(value_of_node n3)) rm)). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split. constructor;try tauto. constructor;try tauto. constructor;try tauto. intros. clear H7 H8 H11 H5 H6 H3 H4 H H0 H18 H1. inversion H9;subst;clear H9. inversion H5;subst;clear H5. inversion H7;subst;clear H7. pose proof Abs_in _ _ _ _ H21 H5 k; pose proof optionZ_lt_SearchTree _ _ _ H5; clear H21 H5. pose proof Abs_in _ _ _ _ H17 H9 k; pose proof optionZ_lt_SearchTree _ _ _ H9; clear H17 H9. pose proof Abs_in _ _ _ _ H16 H8 k; pose proof optionZ_lt_SearchTree _ _ _ H8; clear H16 H8. clear H2. pose proof Abs_in _ _ _ _ H19 H6 k; clear H19 H6.
pose proof Abs_in_half _ _ _ _ H22 H10 k; clear H22 H10. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in H9. simpl in H8.
lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6; destruct H1;destruct H4;destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H6 as [v1[? ?]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. rewrite H1, H4,H2,H6,H. clear H1 H2 H4 H6 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H. clear H7.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H4;destruct H2. unfold combine. rewrite H1, H4,H2,H6,H. clear H1 H2 H4 H6 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H2. unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H2.
2:{destruct H2 as [v[?[? ?] ]]. destruct H6.
2:{ destruct H6 as [v1[? ? ]]. simpl in *. lia. }
destruct H4. unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H6;destruct H4;destruct H2.
1:{unfold combine. rewrite H1,H2,H6, H4,H. clear H1 H2 H6 H4 H. simpl in *. unfold relate_single. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2)). 1:{ destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. } destruct (Z.eq_dec k (key_of_node n3));try tauto. }
destruct H6 as [v[? ?]]. destruct H10;simpl in *;
unfold combine; rewrite H1,H2,H6, H4,H; clear H1 H2 H6 H4 H; simpl in *; unfold relate_single; destruct (Z.eq_dec k (key_of_node n1));try lia; destruct (Z.eq_dec k (key_of_node n2));try lia; destruct (Z.eq_dec k (key_of_node n3));try lia; tauto.
+ inversion H0;subst. inversion H1;subst. inversion H18;subst. exists m2. exists (combine(combine m0 (combine (relate_single(key_of_node n1)(value_of_node n1)) lm))(combine(relate_single(key_of_node n2)(value_of_node n2))(combine rm (combine(relate_single(key_of_node n3)(value_of_node n3)) m1)))). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split. constructor;try tauto. constructor;try tauto. constructor;try tauto. intros. clear H7 H8 H11 H5 H6 H3 H4 H H0 H18 H1. inversion H9;subst;clear H9. inversion H5;subst;clear H5. inversion H6;subst;clear H6. pose proof Abs_in _ _ _ _ H21 H9 k; pose proof optionZ_lt_SearchTree _ _ _ H9; clear H21 H9. pose proof Abs_in _ _ _ _ H17 H7 k; pose proof optionZ_lt_SearchTree _ _ _ H7; clear H17 H7. pose proof Abs_in _ _ _ _ H16 H8 k; pose proof optionZ_lt_SearchTree _ _ _ H8; clear H16 H8. clear H2. pose proof Abs_in _ _ _ _ H19 H5 k; pose proof optionZ_lt_SearchTree _ _ _ H5; clear H19 H5.
pose proof Abs_in_half _ _ _ _ H22 H10 k; clear H22 H10. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5; destruct H1;destruct H4;destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H5 as [v1[? ?]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. rewrite H1, H4,H2,H5,H. clear H1 H2 H4 H5 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H. clear H8.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H4;destruct H2. unfold combine. rewrite H1, H4,H2,H5,H. clear H1 H2 H4 H5 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H2. unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H2.
2:{destruct H2 as [v[?[? ?] ]]. destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H4. unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H5;destruct H4;destruct H2.
1:{unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. simpl in *. unfold relate_single. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2)). 1:{ destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. } destruct (Z.eq_dec k (key_of_node n3));try tauto. }
destruct H5 as [v[? ?]]. destruct H10;simpl in *;
unfold combine; rewrite H1,H2,H5, H4,H; clear H1 H2 H5 H4 H; simpl in *; unfold relate_single; destruct (Z.eq_dec k (key_of_node n1));try lia; destruct (Z.eq_dec k (key_of_node n2));try lia; destruct (Z.eq_dec k (key_of_node n3));try lia; tauto.
+ inversion H0;subst. inversion H1;subst. inversion H18;subst. exists m2. exists (combine (combine m1 (combine(relate_single(key_of_node n1)(value_of_node n1)) lm))(combine (relate_single(key_of_node n2)(value_of_node n2))(combine rm (combine(relate_single(key_of_node n3)(value_of_node n3)) m0)))). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split. constructor;try tauto. constructor;try tauto. constructor;try tauto. intros. clear H7 H8 H11 H5 H6 H3 H4 H H0 H18 H1. inversion H9;subst;clear H9. inversion H5;subst;clear H5. inversion H6;subst;clear H6. pose proof Abs_in _ _ _ _ H17 H9 k; pose proof optionZ_lt_SearchTree _ _ _ H9; clear H17 H9. pose proof Abs_in _ _ _ _ H21 H7 k; pose proof optionZ_lt_SearchTree _ _ _ H7; clear H21 H7. pose proof Abs_in _ _ _ _ H16 H8 k; pose proof optionZ_lt_SearchTree _ _ _ H8; clear H16 H8. clear H2. pose proof Abs_in _ _ _ _ H19 H5 k; pose proof optionZ_lt_SearchTree _ _ _ H5; clear H19 H5.
pose proof Abs_in_half _ _ _ _ H22 H10 k; clear H22 H10. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5; destruct H1;destruct H4;destruct H2.
2:{ destruct H as [v[?[? ? ]]]. destruct H5 as [v1[? ?]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. rewrite H1, H4,H2,H5,H. clear H1 H2 H4 H5 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H. clear H8.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H4;destruct H2. unfold combine. rewrite H1, H4,H2,H5,H. clear H1 H2 H4 H5 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. destruct H2.
2:{destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H2. unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H2.
2:{destruct H2 as [v[?[? ?] ]]. destruct H5.
2:{ destruct H5 as [v1[? ? ]]. simpl in *. lia. }
destruct H4. unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct H5;destruct H4;destruct H2.
1:{unfold combine. rewrite H1,H2,H5, H4,H. clear H1 H2 H5 H4 H. simpl in *. unfold relate_single. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2)). 1:{ destruct (Z.eq_dec k (key_of_node n3));try lia. tauto. } destruct (Z.eq_dec k (key_of_node n3));try tauto. }
destruct H5 as [v[? ?]]. destruct H10;simpl in *;
unfold combine; rewrite H1,H2,H5, H4,H; clear H1 H2 H5 H4 H; simpl in *; unfold relate_single; destruct (Z.eq_dec k (key_of_node n1));try lia; destruct (Z.eq_dec k (key_of_node n2));try lia; destruct (Z.eq_dec k (key_of_node n3));try lia; tauto.
+ inversion H1;subst;clear H1. inversion H0;subst;clear H0. inversion H19;subst;clear H19. exists relate_default. exists (combine lm (combine (relate_single (key_of_node n1)(value_of_node n1)) (combine rm (combine (relate_single(key_of_node n2)(value_of_node n2)) m0)))). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. constructor. split. constructor;try tauto. constructor;try tauto. inversion H9;subst;clear H9. inversion H19;subst;clear H19. intros. clear H H4 H3 H5 H6 H7 H8 H11 H10. pose proof Abs_in _ _ _ _ H16 H15 k; pose proof optionZ_lt_SearchTree _ _ _ H15; clear H16 H15. pose proof Abs_in _ _ _ _ H17 H14 k; pose proof optionZ_lt_SearchTree _ _ _ H14; clear H17 H14. pose proof Abs_in _ _ _ _ H18 H20 k; pose proof optionZ_lt_SearchTree _ _ _ H20; clear H18 H20. clear H2. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. destruct H1;destruct H2;clear H8. rewrite H1, H2,H. clear H1 H2 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H. clear H2.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4. unfold combine. clear H7. rewrite H1, H4,H. clear H1 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. unfold combine,relate_single. rewrite H1,H4,H. clear H1 H4 H. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H4. unfold combine. clear H6. rewrite H1, H4,H. clear H1 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2));try tauto.
+ inversion H1;subst;clear H1. inversion H0;subst;clear H0. inversion H19;subst;clear H19. exists relate_default. exists (combine (combine m0 (combine (relate_single(key_of_node n1)(value_of_node n1)) lm))(combine(relate_single (key_of_node n2)(value_of_node n2)) rm)). split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. split ;try tauto. constructor. split. constructor;try tauto. constructor;try tauto. inversion H9;subst;clear H9. inversion H15;subst;clear H15. intros. clear H H4 H3 H5 H6 H7 H8 H11 H10. pose proof Abs_in _ _ _ _ H16 H20 k; pose proof optionZ_lt_SearchTree _ _ _ H20; clear H16 H20. pose proof Abs_in _ _ _ _ H17 H19 k; pose proof optionZ_lt_SearchTree _ _ _ H19; clear H17 H19. pose proof Abs_in _ _ _ _ H18 H14 k; pose proof optionZ_lt_SearchTree _ _ _ H14; clear H18 H14. clear H2. destruct H.
2:{ destruct H1.
2:{ destruct H as [v[?[? ? ]]]. destruct H1 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4.
2:{ destruct H as [v[?[? ? ]]]. destruct H2 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H as[v[?[? ?]]]. unfold combine. destruct H1;destruct H2;clear H8. rewrite H1, H2,H. clear H1 H2 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H. clear H2.
destruct H1.
2:{ destruct H1 as [v[?[? ? ]]]. destruct H4.
2:{ destruct H4 as [v1[?[? ? ]]]. simpl in *. lia. }
destruct H4. unfold combine. clear H7. rewrite H1, H4,H. clear H1 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H1.
destruct H4.
2:{ destruct H4 as [v[?[? ? ]]]. unfold combine,relate_single. rewrite H1,H4,H. clear H1 H4 H. simpl in *. destruct (Z.eq_dec k (key_of_node n1));try lia. destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct H4. unfold combine. clear H6. rewrite H1, H4,H. clear H1 H4 H. unfold relate_single. simpl in *. destruct (Z.eq_dec k (key_of_node n1)).
1:{ destruct (Z.eq_dec k (key_of_node n2));try lia. tauto. }
destruct (Z.eq_dec k (key_of_node n2));try tauto.
Qed.
Lemma combine_default:
forall m ,
forall k, m k= combine relate_default m k.
Proof.
intros. unfold combine, relate_default. induction m;tauto.
Qed.
Lemma correct_le:
forall h t t' m1 m2 lo hi LO HI,
Abs_half h m1 ->
Abs t m2 ->
splay h t t' ->
SearchTree (Some lo) t (Some hi)->(* new *)
SearchTree_half_in (Some lo) h (Some hi)->(* new *)
SearchTree_half_out (Some LO) h (Some HI)->(* new *)
optionZ_le (Some LO) (Some lo) ->
optionZ_le (Some hi) (Some HI) ->
Abs t' (combine m1 m2).
Proof.
intros.
apply splay_splay' in H1.
revert H H0 H2 H3 H4 H5 H6 .
revert m1 m2 lo hi LO HI.
induction_1n H1;intros.
+ inversion H;subst.
pose proof combine_default m2 . pose proof map_eq _ _ H1. rewrite <-H7. tauto.
+
pose proof step_correct_le _ _ _ _ _ _ _ _ _ _ H H1 H2 H3 H4 H5 H6 H7.
destruct H8 as [lo' [hi'[LO'[HI'[m1'[m2'[?[?[?[?[?[?[? ?]]]]]]]]]]]]].
specialize (IHrt _ _ _ _ _ _ H13 H14 H8 H9 H10 H11 H12). pose proof map_eq _ _ H15. rewrite<- H16 . tauto.
Qed.
Theorem correctness: correct.
Proof.
unfold correct;intros.
pose proof lt_le _ _ H6. pose proof lt_le _ _ H5.
pose proof correct_le _ _ _ _ _ _ _ _ _ H H0 H1 H2 H3 H4 H8 H7. tauto.
Qed.
(* Long may the sun shine! *)
(* 2021-06-08 22:13 *)
|
# Copyright 2022 Yan Yan
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from cumm.gemm.algospec.core import TensorOp
from cumm.gemm.constants import NVRTCConstants, NVRTCMode
from cumm.gemm.dev import GemmHelper
from cumm.gemm.main import gen_gemm_params, gen_gemm_kernels
from cumm.gemm import kernel
from cumm import cudasim
from cumm.nvrtc import CummNVRTCModule, get_cudadevrt_path
from cumm import tensorview as tv
import numpy as np
from cumm.gemm.cutlasstest import cutlass_test_gemm, GemmAlgoParams, CutlassGemm
def cutlass_test_tf32():
params = GemmAlgoParams((128, 128, 16), (64, 64, 16), 2,
"f32,f32,f32,f32,f32", False, True, False,
kernel.GemmAlgo.Ampere, TensorOp((16, 8, 8)))
main_cu = CutlassGemm(params, 128, "Sm80")
cutlass_test_gemm(main_cu)
def dev_tf32():
top = TensorOp((16, 8, 8), "tf32,tf32,f32")
params = gen_gemm_params((128, 128, 16),
(64, 64, 16), 2, "f32,f32,f32,f32,f32",
kernel.GemmAlgo.Turing, top)[0]
# top = TensorOp((16, 8, 8), "f16,f16,f16")
# params = gen_gemm_params((128, 128, 32),
# (64, 64, 32), 2, "f16,f16,f16,f16,f16",
# kernel.GemmAlgo.Turing, top)[0]
# ref: [4, 1], [8, 16]
nvrtc_mode = NVRTCMode.ConstantMemory
with cudasim.enter_debug_context(True, 0):
cutlass_test_tf32()
ker = gen_gemm_kernels(params, nvrtc_mode=nvrtc_mode)
print("start")
ker.namespace = "wtf"
custom_names = []
if nvrtc_mode == NVRTCMode.ConstantMemory:
custom_names = [f"&{ker.namespace}::{NVRTCConstants.CONSTANT_PARAM_KEY}"]
with cudasim.enter_debug_context(True, 0):
with tv.measure_and_print("RTC Compile Time"):
mod = CummNVRTCModule(
[ker],
cudadevrt_path=str(get_cudadevrt_path()),
verbose=False,
custom_names=custom_names)
mod.load()
np.random.seed(12315)
helper = GemmHelper(ker, 128, 128, 8)
params_cpp = helper.get_params()
nvrtc_params = helper.get_nvrtc_params(nvrtc_mode, mod, ker.namespace)
params_cpp.nvrtc_params = nvrtc_params
with tv.measure_and_print("Gemm Time"):
tv.gemm.run_nvrtc_gemm_kernel(params_cpp)
c_cpu = params_cpp.c.cpu().numpy()
print(ker.get_algo_name(), helper.a.mean(), helper.b.mean(), helper.c.mean(),
np.linalg.norm(c_cpu - helper.c))
if __name__ == "__main__":
dev_tf32()
|
function [] = assertEqual(a, b)
if (a != b)
testFailed;
end
function [] = testFailed()
[ST, I] = dbstack(2);
disp(strcat("FAILED: ", ST(1).name));
|
(* This Isabelle theory is produced using the TIP tool offered at the following website:
https://github.com/tip-org/tools
This file was originally provided as part of TIP benchmark at the following website:
https://github.com/tip-org/benchmarks
Yutaka Nagashima at CIIRC, CTU changed the TIP output theory file slightly
to make it compatible with Isabelle2017.
\:w
Some proofs were added by Yutaka Nagashima.*)
theory TIP_sort_nat_HSort2Sorts
imports "../../Test_Base"
begin
datatype 'a list = nil2 | cons2 "'a" "'a list"
datatype Nat = Z | S "Nat"
datatype Heap = Node "Heap" "Nat" "Heap" | Nil
fun le :: "Nat => Nat => bool" where
"le (Z) y = True"
| "le (S z) (Z) = False"
| "le (S z) (S x2) = le z x2"
fun ordered :: "Nat list => bool" where
"ordered (nil2) = True"
| "ordered (cons2 y (nil2)) = True"
| "ordered (cons2 y (cons2 y2 xs)) =
((le y y2) & (ordered (cons2 y2 xs)))"
fun hmerge :: "Heap => Heap => Heap" where
"hmerge (Node z x2 x3) (Node x4 x5 x6) =
(if le x2 x5 then Node (hmerge x3 (Node x4 x5 x6)) x2 z else
Node (hmerge (Node z x2 x3) x6) x5 x4)"
| "hmerge (Node z x2 x3) (Nil) = Node z x2 x3"
| "hmerge (Nil) y = y"
(*fun did not finish the proof*)
function toList :: "Heap => Nat list" where
"toList (Node q y r) = cons2 y (toList (hmerge q r))"
| "toList (Nil) = nil2"
by pat_completeness auto
fun hinsert :: "Nat => Heap => Heap" where
"hinsert x y = hmerge (Node Nil x Nil) y"
fun toHeap2 :: "Nat list => Heap" where
"toHeap2 (nil2) = Nil"
| "toHeap2 (cons2 y xs) = hinsert y (toHeap2 xs)"
fun hsort2 :: "Nat list => Nat list" where
"hsort2 x = toList (toHeap2 x)"
theorem property0 :
"ordered (hsort2 xs)"
oops
end
|
If $M_i$ is a family of measures on a common measurable space $N$, then the supremum of the $M_i$ is the supremum of the suprema of the finite subfamilies of $M_i$. |
Mean=c(2.2, 4.6 ,21.2 ,31.4)
Standard_deviation=c(1.476,2.119,4.733,5.52)
s_min=min(Standard_deviation)
s_max=max(Standard_deviation)
# test statistic
F=s_max^2/s_min^2
print(F)
# The critical value of F > F.alpha
# we reject the hypothesis of homogeneity
#of the population variances.
distance_1km=c(1,5,2,1,2,2,4,3,0,2)
distance_5km=c(4,8,2,3,8,5,6,4,3,3)
distance_10km=c(20,26,24,11,28,20,19,19,21,24)
distance_20km=c(37,30,26,24,41,25,36,31,31,33)
print(Standard_deviation[1]^2/Mean[1])
print(Standard_deviation[2]^2/Mean[2])
print(Standard_deviation[3]^2/Mean[3])
print(Standard_deviation[4]^2/Mean[4])
i=1
while(i<11){
distance_1km[i]=sqrt(distance_1km[i]+0.375)
i=i+1
}
i=1
while(i<11){
distance_5km[i]=sqrt(distance_5km[i]+0.375)
i=i+1
}
i=1
while(i<11){
distance_10km[i]=sqrt(distance_10km[i]+0.375)
i=i+1
}
i=1
while(i<11){
distance_20km[i]=sqrt(distance_20km[i]+0.375)
i=i+1
}
combined_group=data.frame(cbind(distance_1km,distance_5km,distance_10km,distance_20km))
combined_group
|
{-# LANGUAGE ConstraintKinds #-}
module HashedExpression.Internal.Expression
( R,
C,
Covector,
ET (..),
Node (..),
Internal,
NodeID,
ExpressionMap,
Expression (..),
Scalar,
Dimension,
ToShape (..),
DimensionType,
ElementType,
NumType,
VectorSpace,
InnerProductSpace,
PowerOp (..),
PiecewiseOp (..),
VectorSpaceOp (..),
FTOp (..),
ComplexRealOp (..),
RotateOp (..),
Shape,
RotateAmount,
Arg,
Args,
BranchArg,
ConditionArg,
InnerProductSpaceOp (..),
)
where
import Data.Array
import qualified Data.Complex as DC
import Data.IntMap (IntMap)
import qualified Data.IntMap.Strict as IM
import Data.Proxy (Proxy (..))
import Data.Typeable (Typeable, typeRep)
import GHC.TypeLits (KnownNat, Nat, natVal)
import Prelude hiding ((^))
-- | Type representation of elements in the 1D, 2D, 3D, ... grid
data R
deriving (NumType, ElementType, Typeable)
data C
deriving (NumType, ElementType, Typeable)
data Covector
deriving (ElementType, Typeable)
-- | Type representation of vector dimension
data Scalar
deriving (Dimension, Typeable)
-- |
instance (KnownNat n) => Dimension n
instance (KnownNat m, KnownNat n) => Dimension '(m, n)
instance (KnownNat m, KnownNat n, KnownNat p) => Dimension '(m, n, p)
-- | Classes as constraints
class ElementType et
class ElementType et => NumType et
-------------------------------------------------------------------------------
class
(Dimension d) =>
ToShape d where
toShape :: Proxy d -> Shape
instance ToShape Scalar where
toShape _ = []
instance (KnownNat n) => ToShape n where
toShape _ = [nat @n]
instance (KnownNat m, KnownNat n) => ToShape '(m, n) where
toShape _ = [nat @m, nat @n]
instance (KnownNat m, KnownNat n, KnownNat p) => ToShape '(m, n, p) where
toShape _ = [nat @m, nat @n, nat @p]
type DimensionType d = (Dimension d, ToShape d)
-------------------------------------------------------------------------------
-- |
nat :: forall n. (KnownNat n) => Int
nat = fromIntegral $ natVal (Proxy :: Proxy n)
-------------------------------------------------------------------------------
class Dimension d
class VectorSpace d et s
class VectorSpace d s s => InnerProductSpace d s
instance (DimensionType d, ElementType et) => VectorSpace d et R
instance (DimensionType d) => VectorSpace d C C
instance VectorSpace d s s => InnerProductSpace d s
-- | Classes for operations so that both Expression and Pattern (in HashedPattern) can implement
class PowerOp a b | a -> b where
(^) :: a -> b -> a
class VectorSpaceOp a b where
scale :: a -> b -> b
(*.) :: a -> b -> b
(*.) = scale
class ComplexRealOp r c | r -> c, c -> r where
(+:) :: r -> r -> c
xRe :: c -> r
xIm :: c -> r
class InnerProductSpaceOp a b c | a b -> c where
(<.>) :: a -> b -> c
class RotateOp k a | a -> k where
rotate :: k -> a -> a
class PiecewiseOp a b where
piecewise :: [Double] -> a -> [b] -> b
class FTOp a b | a -> b where
ft :: a -> b
infixl 6 +:
infixl 8 *., `scale`, <.>
infixl 8 ^
-- | Shape type:
-- [] --> scalar
-- [n] --> 1D with size n
-- [n, m] --> 2D with size n Γ m
-- [n, m, p] --> 3D with size n Γ m Γ p
type Shape = [Int]
-- | Args - list of indices of arguments in the ExpressionMap
type Args = [NodeID]
type Arg = NodeID
type ConditionArg = NodeID
type BranchArg = NodeID
-- | Rotation in each dimension.
-- | Property: the length of this must match the dimension of the data
type RotateAmount = [Int]
-- | Data representation of element type
data ET
= R
| C
| Covector
deriving (Show, Eq, Ord)
-- | Internal
-- Shape: Shape of the expression
-- we can reconstruct the type of the Expression
type Internal = (Shape, Node)
-- | Hash map of all subexpressions
type ExpressionMap = IntMap Internal
-- | The index/key to look for the node on the hash table
type NodeID = Int
-- | Expression with 2 phantom types (dimension and num type)
data Expression d et
= Expression
{ exRootID :: Int, -- the index this expression
exMap :: ExpressionMap -- all subexpressions
}
deriving (Show, Eq, Ord, Typeable)
type role Expression nominal nominal -- So the users cannot use Data.Coerce.coerce to convert between expression types
-- | Node type
data Node
= Var String
| DVar String -- only contained in **Expression d Covector (1-form)**
| Const Double -- only all elements the same
-- MARK: Basics
| Sum ET Args -- element-wise sum
| Mul ET Args -- multiply --> have different meanings (scale in vector space, multiplication, ...)
| Power Int Arg
| Neg ET Arg
| Scale ET Arg Arg
| -- MARK: only apply to R
Div Arg Arg -- TODO: Delete?
| Sqrt Arg
| Sin Arg
| Cos Arg
| Tan Arg
| Exp Arg
| Log Arg
| Sinh Arg
| Cosh Arg
| Tanh Arg
| Asin Arg
| Acos Arg
| Atan Arg
| Asinh Arg
| Acosh Arg
| Atanh Arg
| -- MARK: Complex related
RealImag Arg Arg -- from real and imagine
| RealPart Arg -- extract real part
| ImagPart Arg -- extract imaginary part
-- MARK: Inner product Space
| InnerProd ET Arg Arg
| -- MARK: Piecewise
Piecewise [Double] ConditionArg [BranchArg]
| Rotate RotateAmount Arg
| -- MARK: Discrete Fourier Transform
ReFT Arg
| ImFT Arg
| -- Need these inside because taking real of FT twice can be very fast
TwiceReFT Arg
| TwiceImFT Arg
deriving (Show, Eq, Ord)
|
# Symbolic regression using cartesian genetic programming
Here we're going to use pycartgp library to find an exact expression of some unknown function, knowing only its data points.
```python
import math
import pycartgp
import matplotlib.pyplot as plt
```
## The dataset
First we need a dataset: a bunch of X and Ys for an "unknown" function.
```python
def unknown_function(x):
return x**3 + 2 * x**2 + 3*x + 5
num_points = 50
all_x = [0.1 * (i - num_points/2) for i in range(num_points)]
all_y = [unknown_function(x) for x in all_x]
```
```python
plt.scatter(all_x, all_y)
```
## Functions and fitness
To evaluate the solutions we need a fitness function. Here we use negative [MSE](https://en.wikipedia.org/wiki/Mean_squared_error).
```python
def fitness(genotype: pycartgp.Genotype, functions) -> float:
error = 0
for x, y in zip(all_x, all_y):
ty = genotype.evaluate(functions, [x])[0]
error += (y - ty)**2
return -error / len(all_x)
```
We also need a list of functions we're going to build the solution from.
```python
available_functions = [
("plus", lambda args: args[0] + args[1]),
("minus", lambda args: args[0] - args[1]),
("mul", lambda args: args[0] * args[1]),
("div", lambda args: (args[0] / args[1]) if args[1] != 0 else 1), # "protected" division
]
```
# Evolving a solution
We create a random genotype, the ancestor of our final solution. Because it's random, it's totally meaningless.
```python
genotype = pycartgp.Genotype(
arity=2,
num_functions=len(available_functions),
num_inputs=1,
num_outputs=1, depth=100)
print(genotype.raw)
print(genotype.explain_outputs(available_functions))
```
[0, 0, 0, 1, 1, 1, 3, 2, 2, 2, 0, 0, 1, 3, 3, 0, 2, 3, 1, 1, 1, 0, 0, 5, 2, 4, 8, 3, 2, 2, 2, 10, 0, 2, 7, 3, 0, 11, 4, 0, 5, 8, 1, 3, 9, 1, 15, 7, 1, 5, 14, 3, 15, 14, 3, 13, 9, 1, 3, 3, 1, 0, 16, 1, 11, 2, 2, 7, 19, 3, 20, 22, 3, 9, 22, 2, 8, 0, 0, 15, 14, 0, 7, 0, 0, 0, 25, 1, 3, 28, 0, 0, 12, 2, 6, 27, 0, 32, 5, 1, 9, 11, 0, 4, 15, 0, 22, 34, 1, 31, 7, 2, 26, 18, 0, 11, 7, 0, 11, 18, 0, 15, 21, 2, 27, 27, 3, 19, 38, 2, 11, 31, 1, 25, 18, 3, 20, 20, 0, 43, 14, 3, 36, 27, 2, 14, 11, 3, 10, 24, 2, 0, 0, 2, 2, 23, 1, 27, 34, 3, 22, 4, 2, 26, 33, 3, 45, 33, 2, 26, 30, 2, 5, 18, 0, 25, 8, 0, 51, 4, 3, 51, 13, 1, 36, 29, 2, 19, 2, 2, 10, 14, 3, 25, 35, 1, 29, 44, 3, 41, 41, 3, 43, 23, 3, 66, 24, 3, 52, 37, 1, 41, 25, 2, 67, 70, 2, 0, 30, 1, 71, 71, 0, 59, 44, 3, 24, 50, 3, 75, 68, 2, 76, 3, 1, 57, 31, 2, 4, 42, 2, 1, 25, 0, 9, 15, 2, 69, 9, 2, 50, 3, 2, 78, 44, 2, 3, 66, 2, 29, 78, 3, 43, 44, 3, 80, 84, 1, 23, 45, 2, 87, 15, 3, 44, 68, 2, 41, 9, 3, 31, 26, 1, 73, 32, 3, 39, 19, 2, 44, 10, 1, 88, 64, 2, 6, 69, 1, 39, 5, 77]
['div(plus(plus(div(mul(mul(In0, In0), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0)))))), minus(mul(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), In0), minus(plus(In0, In0), plus(In0, In0)))), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0)))))), mul(mul(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), In0), plus(In0, mul(minus(plus(In0, In0), plus(In0, In0)), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))))), div(div(div(plus(mul(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), In0), mul(In0, In0)), mul(mul(In0, In0), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))))), mul(mul(plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))), In0), div(minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), mul(mul(In0, In0), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))))), plus(minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0)))), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))))))), mul(minus(plus(In0, In0), plus(In0, In0)), div(plus(mul(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), In0), mul(In0, In0)), mul(mul(In0, In0), plus(In0, minus(div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))), div(minus(plus(In0, In0), plus(In0, In0)), minus(plus(In0, In0), plus(In0, In0))))))))))']
Now we can run the evolution until the solution stabilizes (fitness stops improving).
```python
solution, info = genotype.evolve(
available_functions, num_offsprings=4, stable_margin=1e-6,
steps_to_stabilize=1000, fitness_function=fitness)
print('Steps taken:', info.steps)
print('Final fitness:', info.fitness)
```
Steps taken: 1753
Final fitness: -2.1457016622011523e-30
The evolved expression can be lengthy, because evolution had to "invent" how to construct everything (like constants, etc.) from the only input.
```python
solution.explain_outputs(available_functions)[0]
```
'plus(plus(plus(plus(div(In0, plus(In0, minus(In0, In0))), mul(plus(In0, minus(In0, In0)), plus(In0, minus(In0, In0)))), plus(minus(div(In0, plus(In0, minus(In0, In0))), minus(plus(In0, minus(In0, In0)), minus(In0, In0))), plus(In0, minus(In0, In0)))), mul(plus(div(In0, plus(In0, minus(In0, In0))), minus(plus(In0, minus(In0, In0)), minus(In0, In0))), plus(plus(div(In0, plus(In0, minus(In0, In0))), mul(plus(In0, minus(In0, In0)), plus(In0, minus(In0, In0)))), plus(minus(div(In0, plus(In0, minus(In0, In0))), minus(plus(In0, minus(In0, In0)), minus(In0, In0))), plus(In0, minus(In0, In0)))))), plus(div(plus(In0, minus(In0, In0)), div(minus(In0, In0), minus(In0, In0))), plus(minus(div(In0, plus(In0, minus(In0, In0))), minus(plus(In0, minus(In0, In0)), minus(In0, In0))), plus(In0, minus(In0, In0)))))'
We can plot the evolved solution and see how well it fits the dataset
```python
plt.plot(all_x, [solution.evaluate(available_functions, [x])[0] for x in all_x], 'r')
plt.scatter(all_x, all_y)
```
## Human-readable form
We can evaluate the evolved solution using SymPy symbols instead of numbers and thus get a valid SymPy
expression that is much more readable and can even be further simplified
```python
import sympy
sympy.init_printing()
sympy_expr = solution.evaluate(available_functions, [sympy.symbols('x')])[0]
sympy.simplify(sympy.expand(sympy_expr))
```
```python
```
|
!=====================================================================*
! *
! Software Name : HACApK *
! Version : 1.0.0 *
! *
! License *
! This file is part of HACApK. *
! HACApK is a free software, you can use it under the terms *
! of The MIT License (MIT). See LICENSE file and User's guide *
! for more details. *
! *
! ppOpen-HPC project: *
! Open Source Infrastructure for Development and Execution of *
! Large-Scale Scientific Applications on Post-Peta-Scale *
! Supercomputers with Automatic Tuning (AT). *
! *
! Sponsorship: *
! Japan Science and Technology Agency (JST), Basic Research *
! Programs: CREST, Development of System Software Technologies *
! for post-Peta Scale High Performance Computing. *
! *
! Copyright (c) 2015 <Akihiro Ida and Takeshi Iwashita> *
! *
!=====================================================================*
!C***********************************************************************
!C This file includes routines for utilizing H-matrices, such as solving
!C linear system with an H-matrix as the coefficient matrix and
!C multiplying an H-matrix and a vector,
!C created by Akihiro Ida at Kyoto University on May 2012,
!C last modified by Akihiro Ida on Sep 2014,
!C***********************************************************************
module m_HACApK_solve
use m_HACApK_base
implicit real*8(a-h,o-z)
implicit integer*4(i-n)
contains
!***HACApK_adot_lfmtx_p
subroutine HACApK_adot_lfmtx_p(zau,st_leafmtxp,st_ctl,zu,nd)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(nd),zu(nd)
real*8,dimension(:),allocatable :: wws,wwr
integer*4 :: ISTATUS(MPI_STATUS_SIZE),isct(2),irct(2)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
ndnr_s=lpmd(6); ndnr_e=lpmd(7); ndnr=lpmd(5)
allocate(wws(maxval(lnp(0:nrank-1))),wwr(maxval(lnp(0:nrank-1))))
zau(:)=0.0d0
call HACApK_adot_body_lfmtx(zau,st_leafmtxp,st_ctl,zu,nd)
if(nrank==1) return
wws(1:lnp(mpinr))=zau(lsp(mpinr):lsp(mpinr)+lnp(mpinr)-1)
ncdp=mod(mpinr+1,nrank)
ncsp=mod(mpinr+nrank-1,nrank)
! write(mpilog,1000) 'destination process=',ncdp,'; source process=',ncsp
isct(1)=lnp(mpinr);isct(2)=lsp(mpinr);
! irct=lnp(ncsp)
do ic=1,nrank-1
! idp=mod(mpinr+ic,nrank) ! rank of destination process
! isp=mod(mpinr+nrank+ic-2,nrank) ! rank of source process
call MPI_SENDRECV(isct,2,MPI_INTEGER,ncdp,1, &
irct,2,MPI_INTEGER,ncsp,1,icomm,ISTATUS,ierr)
! write(mpilog,1000) 'ISTATUS=',ISTATUS,'; ierr=',ierr
! write(mpilog,1000) 'ic=',ic,'; isct=',isct(1),'; irct=',irct(1),'; ivsps=',isct(2),'; ivspr=',irct(2)
call MPI_SENDRECV(wws,isct,MPI_DOUBLE_PRECISION,ncdp,1, &
wwr,irct,MPI_DOUBLE_PRECISION,ncsp,1,icomm,ISTATUS,ierr)
! write(mpilog,1000) 'ISTATUS=',ISTATUS,'; ierr=',ierr
zau(irct(2):irct(2)+irct(1)-1)=zau(irct(2):irct(2)+irct(1)-1)+wwr(:irct(1))
wws(:irct(1))=wwr(:irct(1))
isct=irct
! write(mpilog,1000) 'ic=',ic,'; isct=',isct
enddo
deallocate(wws,wwr)
end subroutine HACApK_adot_lfmtx_p
!***HACApK_adot_lfmtx_hyp
subroutine HACApK_adot_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,isct,irct,nd)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(*),zu(*),wws(*),wwr(*)
integer*4 :: isct(*),irct(*)
integer*4 :: ISTATUS(MPI_STATUS_SIZE)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
real*8,pointer :: time(:)
! integer*4,dimension(:),allocatable :: ISTATUS
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
time => st_ctl%time(:)
! allocate(ISTATUS(MPI_STATUS_SIZE))
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
ndnr_s=lpmd(6); ndnr_e=lpmd(7); ndnr=lpmd(5)
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
wws(1:lnp(mpinr))=zau(lsp(mpinr):lsp(mpinr)+lnp(mpinr)-1)
ncdp=mod(mpinr+1,nrank)
ncsp=mod(mpinr+nrank-1,nrank)
isct(1)=lnp(mpinr);isct(2)=lsp(mpinr);
do ic=1,nrank-1
call MPI_SENDRECV(isct,2,MPI_INTEGER,ncdp,1, &
irct,2,MPI_INTEGER,ncsp,1,icomm,ISTATUS,ierr)
call MPI_SENDRECV(wws,isct,MPI_DOUBLE_PRECISION,ncdp,1, &
wwr,irct,MPI_DOUBLE_PRECISION,ncsp,1,icomm,ISTATUS,ierr)
zau(irct(2):irct(2)+irct(1)-1)=zau(irct(2):irct(2)+irct(1)-1)+wwr(:irct(1))
wws(:irct(1))=wwr(:irct(1))
isct(:2)=irct(:2)
enddo
endif
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
!$omp end master
! stop
end subroutine HACApK_adot_lfmtx_hyp
!***HACApK_adot_blr_hyp
subroutine HACApK_adot_blr_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35); nrank_l=lpmd(36); mpinrt=lpmd(33)
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(zw(nd)); zw=0.0d0
call HACApK_adot_blrmtx_hyp4(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
do irl=1,nlfalt; ibll=mod(irl-1,nrank_l)
if(ibll/=mpinrt) cycle
irlstrtl=st_leafmtxp%lbstrtl(irl) ; irlnd=st_leafmtxp%lbndl(irl)
zw(irlstrtl:irlstrtl+irlnd-1)=zau(irlstrtl:irlstrtl+irlnd-1)
enddo
! write(mpilog,*) zw(1:600)
! write(mpilog,*) ' ',zw(1:5)
! write(mpilog,*) ' ',zw(308:312)
! write(mpilog,*) ' ',zw(313:318)
! write(mpilog,*) ' ',zw(596:600)
call MPI_Allreduce(zw, zau, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
! stop
end subroutine HACApK_adot_blr_hyp
!***HACApK_adot_blrmtx_hyp4
subroutine HACApK_adot_blrmtx_hyp4(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
integer ISTATUS(MPI_STATUS_SIZE)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:),lbl2t(:)
integer*4,allocatable :: ireqs(:),ireqr(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:),zw2(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
! print*,'HACApK_adot_blrmtx_hyp4 is called'
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:); lbl2t(0:) => st_leafmtxp%lbl2t(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); nrank_l=lpmd(36); mpinrl=lpmd(37); mpinrt=lpmd(33)
icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(ireqs(0:nrank_l-1),ireqr(0:nrank_l-1))
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
wws(ilp:ilp+irlnd-1)=zau(irlstrtl:irlstrtl+irlnd-1)
ilp=ilp+irlnd
enddo
!!! call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
call MPI_reduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, mpinrl, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd));
is=1
do irl=0,nrank_l-1
itag=1; irlnd=st_leafmtxp%lbndlfs(irl)
if(mpinrl==irl .and. mpinrl==mpinrt) then
zw(is:is+irlnd-1)=wwr(1:irlnd)
else
! print*,'mpinrl=',mpinrl,' ;irl=',irl,' ;is=',is, ' ;irlnd=',irlnd
if(lbl2t(irl)==1) call MPI_IRECV(zw(is), irlnd, MPI_REAL8, irl, itag, icomml, ireqr(irl),ierr)
endif
is=is+irlnd
enddo
if(lbl2t(mpinrl)==1)then
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_ISEND(wwr, ndlfs, MPI_REAL8, irl, itag, icomml, ireqs(irl),ierr)
enddo
endif
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
zau(irlstrtl:irlstrtl+irlnd-1)=wwr(ilp:ilp+irlnd-1)
ilp=ilp+irlnd
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
if(lbl2t(irl)==0) cycle
call MPI_WAIT(ireqr(irl),ISTATUS,ierr)
enddo
is=1
do icl=1,nrank_l; irl=icl-1
do ibl=0,nlfl
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
if(mpinrl/=irl .and. lbl2t(irl)==1) zau(ips:ipe)=zw(is:ie)
is=ie+1
enddo
enddo
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
!$omp end master
! print*,'HACApK_adot_blrmtx_hyp4 end'
! stop
end subroutine
!***HACApK_adot_blrmtx_hyp31
subroutine HACApK_adot_blrmtx_hyp31(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
integer ISTATUS(MPI_STATUS_SIZE)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:),lbl2t(:)
integer*4,allocatable :: ireqs(:),ireqr(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:),zw2(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
! print*,'HACApK_adot_blrmtx_hyp31 is called'
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:); lbl2t(0:) => st_leafmtxp%lbl2t(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); nrank_l=lpmd(36); mpinrl=lpmd(37)
icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(ireqs(0:nrank_l-1),ireqr(0:nrank_l-1))
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
if(st_ctl%param(41)==1)then
call MPI_Allreduce(zau, wwr, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
zau(1:nd)=wwr(1:nd)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
else
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
wws(ilp:ilp+irlnd-1)=zau(irlstrtl:irlstrtl+irlnd-1)
ilp=ilp+irlnd
enddo
call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd));
is=1
do irl=0,nrank_l-1
itag=1; irlnd=st_leafmtxp%lbndlfs(irl)
if(mpinrl==irl) then
zw(is:is+irlnd-1)=wwr(1:irlnd)
else
! print*,'mpinrl=',mpinrl,' ;irl=',irl,' ;is=',is, ' ;irlnd=',irlnd
if(lbl2t(irl)==1) call MPI_IRECV(zw(is), irlnd, MPI_REAL8, irl, itag, icomml, ireqr(irl),ierr)
endif
is=is+irlnd
enddo
if(lbl2t(mpinrl)==1)then
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_ISEND(wwr, ndlfs, MPI_REAL8, irl, itag, icomml, ireqs(irl),ierr)
enddo
endif
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
zau(irlstrtl:irlstrtl+irlnd-1)=wwr(ilp:ilp+irlnd-1)
ilp=ilp+irlnd
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
if(lbl2t(irl)==0) cycle
call MPI_WAIT(ireqr(irl),ISTATUS,ierr)
enddo
is=1
do icl=1,nrank_l; irl=icl-1
do ibl=0,nlfl
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
if(mpinrl/=irl .and. lbl2t(irl)==1) zau(ips:ipe)=zw(is:ie)
is=ie+1
enddo
enddo
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
endif
!$omp end master
! stop
end subroutine
!***HACApK_adot_blrmtx_hyp3
subroutine HACApK_adot_blrmtx_hyp3(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
integer ISTATUS(MPI_STATUS_SIZE)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:),lbl2t(:)
integer*4,allocatable :: ireqs(:),ireqr(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:),zw2(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
print*,'HACApK_adot_blrmtx_hyp3 is called'
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:); lbl2t(0:) => st_leafmtxp%lbl2t(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); nrank_l=lpmd(36); mpinrl=lpmd(37)
icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(ireqs(0:nrank_l-1),ireqr(0:nrank_l-1))
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
if(st_ctl%param(41)==1)then
call MPI_Allreduce(zau, wwr, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
zau(1:nd)=wwr(1:nd)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
else
ilp=1
do ilf=1,nlfl
ip=(ilf-1)*nlft+1
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
wws(ilp:ilp+ndl-1)=zau(nstrtl:nstrtl+ndl-1)
ilp=ilp+ndl
enddo
call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd));
is=1
do irl=0,nrank_l-1
itag=1; irlnd=st_leafmtxp%lbndlfs(irl)
if(mpinrl==irl) then
zw(is:is+irlnd-1)=wwr(1:irlnd)
else
! print*,'mpinrl=',mpinrl,' ;irl=',irl,' ;is=',is, ' ;irlnd=',irlnd
if(lbl2t(irl)==1) call MPI_IRECV(zw(is), irlnd, MPI_REAL8, irl, itag, icomml, ireqr(irl),ierr)
endif
is=is+irlnd
enddo
if(lbl2t(mpinrl)==1)then
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_ISEND(wwr, ndlfs, MPI_REAL8, irl, itag, icomml, ireqs(irl),ierr)
enddo
endif
nlfla=nlfalt/nrank_l
is=1; icl=mpinrl+1
do ibl=0,nlfla
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
zau(ips:ipe)=wwr(is:ie)
is=ie+1
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
if(lbl2t(irl)==0) cycle
call MPI_WAIT(ireqr(irl),ISTATUS,ierr)
enddo
nlfla=nlfalt/nrank_l
is=1
do icl=1,nrank_l; irl=icl-1
do ibl=0,nlfla
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
if(mpinrl/=irl .and. lbl2t(irl)==1) zau(ips:ipe)=zw(is:ie)
is=ie+1
enddo
enddo
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
endif
!$omp end master
! stop
end subroutine
!***HACApK_adot_blrmtx_hyp21
subroutine HACApK_adot_blrmtx_hyp21(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
integer ISTATUS(MPI_STATUS_SIZE)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:)
integer*4,allocatable :: ireqs(:),ireqr(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:),zw2(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); nrank_l=lpmd(36); mpinrl=lpmd(37)
icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(ireqs(0:nrank_l-1),ireqr(0:nrank_l-1))
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
if(st_ctl%param(41)==1)then
call MPI_Allreduce(zau, wwr, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
zau(1:nd)=wwr(1:nd)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
else
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
wws(ilp:ilp+irlnd-1)=zau(irlstrtl:irlstrtl+irlnd-1)
ilp=ilp+irlnd
enddo
call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd));
is=1
do irl=0,nrank_l-1
itag=1; irlnd=st_leafmtxp%lbndlfs(irl)
if(mpinrl==irl) then
zw(is:is+irlnd-1)=wwr(1:irlnd)
else
! print*,'mpinrl=',mpinrl,' ;irl=',irl,' ;is=',is, ' ;irlnd=',irlnd
call MPI_IRECV(zw(is), irlnd, MPI_REAL8, irl, itag, icomml, ireqr(irl),ierr)
endif
is=is+irlnd
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_ISEND(wwr, ndlfs, MPI_REAL8, irl, itag, icomml, ireqs(irl),ierr)
enddo
!! my color
ilp=1
do irl=1,nlfl
ibl=st_leafmtxp%lnlfl2g(1,irl); ibll=(ibl-1)/nlfalt+1
irlstrtl=st_leafmtxp%lbstrtl(ibll) ; irlnd=st_leafmtxp%lbndl(ibll)
zau(irlstrtl:irlstrtl+irlnd-1)=wwr(ilp:ilp+irlnd-1)
ilp=ilp+irlnd
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_WAIT(ireqr(irl),ISTATUS,ierr)
enddo
is=1
do icl=1,nrank_l
do ibl=0,nlfl
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
if(mpinrl/=icl-1) zau(ips:ipe)=zw(is:ie)
is=ie+1
enddo
enddo
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
endif
!$omp end master
! stop
end subroutine
!***HACApK_adot_blrmtx_hyp2
subroutine HACApK_adot_blrmtx_hyp2(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
integer ISTATUS(MPI_STATUS_SIZE)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:)
integer*4,allocatable :: ireqs(:),ireqr(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:),zw2(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); nrank_l=lpmd(36); mpinrl=lpmd(37)
icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft; nlfalt=st_leafmtxp%nlfalt
ndlfs=st_leafmtxp%ndlfs
allocate(ireqs(0:nrank_l-1),ireqr(0:nrank_l-1))
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
if(st_ctl%param(41)==1)then
call MPI_Allreduce(zau, wwr, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
zau(1:nd)=wwr(1:nd)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
else
ilp=1
do ilf=1,nlfl
ip=(ilf-1)*nlft+1
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
wws(ilp:ilp+ndl-1)=zau(nstrtl:nstrtl+ndl-1)
ilp=ilp+ndl
enddo
call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd));
is=1
do irl=0,nrank_l-1
itag=1; irlnd=st_leafmtxp%lbndlfs(irl)
if(mpinrl==irl) then
zw(is:is+irlnd-1)=wwr(1:irlnd)
else
! print*,'mpinrl=',mpinrl,' ;irl=',irl,' ;is=',is, ' ;irlnd=',irlnd
call MPI_IRECV(zw(is), irlnd, MPI_REAL8, irl, itag, icomml, ireqr(irl),ierr)
endif
is=is+irlnd
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
call MPI_ISEND(wwr, ndlfs, MPI_REAL8, irl, itag, icomml, ireqs(irl),ierr)
enddo
nlfla=nlfalt/nrank_l
is=1; icl=mpinrl+1
do ibl=0,nlfla
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
zau(ips:ipe)=wwr(is:ie)
is=ie+1
enddo
do irl=0,nrank_l-1
if(mpinrl==irl) cycle
! call MPI_WAIT(ireqs(irl),ISTATUS,ierr)
call MPI_WAIT(ireqr(irl),ISTATUS,ierr)
enddo
! write(mpilog,*) 'zw='
! write(mpilog,*) zw
! write(mpilog,*) 'wwr='
! write(mpilog,*) wwr(1:ndlfs)
nlfla=nlfalt/nrank_l
is=1
do icl=1,nrank_l
do ibl=0,nlfla
ibpl=ibl*nrank_l+icl; if(ibpl>nlfalt) exit
ips=st_leafmtxp%lbstrtl(ibpl)
ipe=st_leafmtxp%lbstrtl(ibpl+1)-1
ie=is+ipe-ips
if(mpinrl/=icl-1) zau(ips:ipe)=zw(is:ie)
! if(mpinr==0) write(*,1000) 'ibpl=',ibpl,' ;ips=',ips,' ;ipe=',ipe,' ;is=',is,' ;ie=',ie
is=ie+1
enddo
enddo
if(.false.)then
allocate(zw2(nd))
zw(:nd)=0.0d0
ilp=1
do ilf=1,nlfl
ip=(ilf-1)*nlft+1
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
zw(nstrtl:nstrtl+ndl-1)=wwr(ilp:ilp+ndl-1)
ilp=ilp+ndl
enddo
call MPI_Allreduce(zw, zw2, nd, MPI_REAL8, MPI_SUM, icomml, ierr)
zzz=sum(abs(zw2(:nd)-zau(:nd)));
print*,'zzz=',zzz
stop
endif
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
endif
!$omp end master
! stop
end subroutine
!***HACApK_adot_blrmtx_hyp
subroutine HACApK_adot_blrmtx_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:)
real*8,pointer :: time(:)
real*8, allocatable :: zw(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); time => st_ctl%time(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1); icommt=lpmd(31); icomml=lpmd(35);
nlf=st_leafmtxp%nlf; nlfl=st_leafmtxp%nlfl; nlft=st_leafmtxp%nlft
ndlfs=st_leafmtxp%ndlfs
!$omp master
call MPI_Barrier( icomm, ierr )
st_time=MPI_Wtime()
!$omp end master
zau(:nd)=0.0d0
!$omp barrier
call HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
!$omp barrier
!$omp master
call MPI_Barrier( icomm, ierr )
en_time=MPI_Wtime()
time(1)=time(1)+en_time-st_time
if(nrank>1)then
st_time=MPI_Wtime()
if(st_ctl%param(41)==1)then
call MPI_Allreduce(zau, wwr, nd, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
zau(1:nd)=wwr(1:nd)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
else
ilp=1
do ilf=1,nlfl
ip=(ilf-1)*nlft+1
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
wws(ilp:ilp+ndl-1)=zau(nstrtl:nstrtl+ndl-1)
ilp=ilp+ndl
enddo
call MPI_Allreduce(wws, wwr, ndlfs, MPI_REAL8, MPI_SUM, icommt, ierr)
en_time=MPI_Wtime()
time(2)=time(2)+en_time-st_time
allocate(zw(nd)); zw(:nd)=0.0d0
ilp=1
do ilf=1,nlfl
ip=(ilf-1)*nlft+1
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
zw(nstrtl:nstrtl+ndl-1)=wwr(ilp:ilp+ndl-1)
ilp=ilp+ndl
enddo
call MPI_Allreduce(zw, zau, nd, MPI_REAL8, MPI_SUM, icomml, ierr)
en2_time=MPI_Wtime()
time(3)=time(3)+en2_time-en_time
endif
endif
!$omp end master
! stop
end subroutine
!***HACApK_adot_body_lfmtx
RECURSIVE subroutine HACApK_adot_body_lfmtx(zau,st_leafmtxp,st_ctl,zu,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(nd),zu(nd)
real*8,dimension(:),allocatable :: zbu
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
nlf=st_leafmtxp%nlf
do ip=1,nlf
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt ; ns=ndl*ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
if(st_leafmtxp%st_lf(ip)%ltmtx==1)then
kt=st_leafmtxp%st_lf(ip)%kt
allocate(zbu(kt)); zbu(:)=0.0d0
do il=1,kt
do it=1,ndt; itt=it+nstrtt-1
zbu(il)=zbu(il)+st_leafmtxp%st_lf(ip)%a1(it,il)*zu(itt)
enddo
enddo
do il=1,kt
do it=1,ndl; ill=it+nstrtl-1
zau(ill)=zau(ill)+st_leafmtxp%st_lf(ip)%a2(it,il)*zbu(il)
enddo
enddo
deallocate(zbu)
elseif(st_leafmtxp%st_lf(ip)%ltmtx==2)then
do il=1,ndl; ill=il+nstrtl-1
do it=1,ndt; itt=it+nstrtt-1
zau(ill)=zau(ill)+st_leafmtxp%st_lf(ip)%a1(it,il)*zu(itt)
enddo
enddo
endif
enddo
end subroutine HACApK_adot_body_lfmtx
!***HACApK_adot_body_lfmtx_hyp
subroutine HACApK_adot_body_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zau(*),zu(*)
real*8,dimension(:),allocatable :: zbut
real*8,dimension(:),allocatable :: zaut
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),ltmp(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;ltmp(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
nlf=st_leafmtxp%nlf; ktmax=st_leafmtxp%ktmax
ith = omp_get_thread_num()
ith1 = ith+1
nths=ltmp(ith); nthe=ltmp(ith1)-1
allocate(zbut(ktmax),zaut(nd),stat=ierr)
if(ierr.ne.0) then
!$omp critical
write(*,*) 'sub HACApK_adot_body_lfmtx_hyp; zbut,zaut Memory allocation failed !'
!$omp end critical
stop
endif
zaut(:)=0.0d0
ls=nd; le=1
do ip=nths,nthe
ndl =st_leafmtxp%st_lf(ip)%ndl ; ndt =st_leafmtxp%st_lf(ip)%ndt ; ns=ndl*ndt
nstrtl=st_leafmtxp%st_lf(ip)%nstrtl; nstrtt=st_leafmtxp%st_lf(ip)%nstrtt
if(nstrtl<ls) ls=nstrtl; if(nstrtl+ndl-1>le) le=nstrtl+ndl-1
if(st_leafmtxp%st_lf(ip)%ltmtx==1)then
kt=st_leafmtxp%st_lf(ip)%kt
zbut(1:kt)=0.0d0
do il=1,kt
do it=1,ndt; itt=it+nstrtt-1
zbut(il)=zbut(il)+st_leafmtxp%st_lf(ip)%a1(it,il)*zu(itt)
enddo
enddo
do il=1,kt
do it=1,ndl; ill=it+nstrtl-1
zaut(ill)=zaut(ill)+st_leafmtxp%st_lf(ip)%a2(it,il)*zbut(il)
enddo
enddo
elseif(st_leafmtxp%st_lf(ip)%ltmtx==2)then
do il=1,ndl; ill=il+nstrtl-1
do it=1,ndt; itt=it+nstrtt-1
zaut(ill)=zaut(ill)+st_leafmtxp%st_lf(ip)%a1(it,il)*zu(itt)
enddo
enddo
endif
enddo
deallocate(zbut)
do il=ls,le
!$omp atomic
zau(il)=zau(il)+zaut(il)
enddo
end subroutine
!***HACApK_adotsub_lfmtx_p
subroutine HACApK_adotsub_lfmtx_p(zr,st_leafmtxp,st_ctl,zu,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zu(nd),zr(nd)
real*8,dimension(:),allocatable :: zau
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr;
allocate(zau(nd))
call HACApK_adot_lfmtx_p(zau,st_leafmtxp,st_ctl,zu,nd)
zr(1:nd)=zr(1:nd)-zau(1:nd)
deallocate(zau)
end subroutine HACApK_adotsub_lfmtx_p
!***HACApK_adotsub_lfmtx_hyp
subroutine HACApK_adotsub_lfmtx_hyp(zr,zau,st_leafmtxp,st_ctl,zu,wws,wwr,isct,irct,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zr(*),zau(*),zu(*),wws(*),wwr(*)
integer*4 :: isct(*),irct(*)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
call HACApK_adot_lfmtx_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,isct,irct,nd)
!$omp barrier
!$omp workshare
zr(1:nd)=zr(1:nd)-zau(1:nd)
!$omp end workshare
end subroutine HACApK_adotsub_lfmtx_hyp
!***HACApK_adotsub_blrmtx_hyp
subroutine HACApK_adotsub_blrmtx_hyp(zr,zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zr(:),zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
call HACApK_adot_blrmtx_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
!$omp barrier
!$omp workshare
zr(1:nd)=zr(1:nd)-zau(1:nd)
!$omp end workshare
end subroutine HACApK_adotsub_blrmtx_hyp
!***HACApK_adotsub_blr_hyp
subroutine HACApK_adotsub_blr_hyp(zr,zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: zr(:),zau(:),zu(:),wws(:),wwr(:)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
call HACApK_adot_blrmtx_hyp21(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
!!! call HACApK_adot_blr_hyp(zau,st_leafmtxp,st_ctl,zu,wws,wwr,nd)
!$omp barrier
!$omp workshare
zr(1:nd)=zr(1:nd)-zau(1:nd)
!$omp end workshare
end subroutine HACApK_adotsub_blr_hyp
!***HACApK_bicgstab_lfmtx
subroutine HACApK_bicgstab_lfmtx(st_leafmtxp,st_ctl,u,b,param,nd,nstp,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: u(nd),b(nd)
real*8 :: param(*)
real*8,dimension(:),allocatable :: zr,zshdw,zp,zt,zkp,zakp,zkt,zakt
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
call MPI_Barrier( icomm, ierr )
st_measure_time=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'HACApK_bicgstab_lfmtx start'
mstep=param(83)
eps=param(91)
allocate(zr(nd),zshdw(nd),zp(nd),zt(nd),zkp(nd),zakp(nd),zkt(nd),zakt(nd))
zp(1:nd)=0.0d0; zakp(1:nd)=0.0d0
alpha = 0.0; beta = 0.0; zeta = 0.0;
zz=HACApK_dotp_d(nd, b, b); bnorm=dsqrt(zz);
zr(:nd)=b(:nd)
call HACApK_adotsub_lfmtx_p(zr,st_leafmtxp,st_ctl,u,nd)
zshdw(:nd)=zr(:nd)
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'Original relative residual norm =',zrnorm/bnorm
if(zrnorm/bnorm<eps) return
! mstep=1
do in=1,mstep
zp(:nd) =zr(:nd)+beta*(zp(:nd)-zeta*zakp(:nd))
zkp(:nd)=zp(:nd)
call HACApK_adot_lfmtx_p(zakp,st_leafmtxp,st_ctl,zkp,nd)
! exit
znom=HACApK_dotp_d(nd,zshdw,zr); zden=HACApK_dotp_d(nd,zshdw,zakp);
alpha=znom/zden; znomold=znom;
zt(:nd)=zr(:nd)-alpha*zakp(:nd)
zkt(:nd)=zt(:nd)
call HACApK_adot_lfmtx_p(zakt,st_leafmtxp,st_ctl,zkt,nd)
znom=HACApK_dotp_d(nd,zakt,zt); zden=HACApK_dotp_d(nd,zakt,zakt);
zeta=znom/zden;
u(:nd)=u(:nd)+alpha*zkp(:nd)+zeta*zkt(:nd)
zr(:nd)=zt(:nd)-zeta*zakt(:nd)
beta=alpha/zeta*HACApK_dotp_d(nd,zshdw,zr)/znomold;
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,in,time,log10(zrnorm/bnorm)
if(zrnorm/bnorm<eps) exit
enddo
end subroutine HACApK_bicgstab_lfmtx
!***HACApK_bicgstab_lfmtx_hyp
subroutine HACApK_bicgstab_lfmtx_hyp(st_leafmtxp,st_ctl,u,b,param,nd,nstp,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: u(nd),b(nd)
real*8 :: param(*)
real*8,dimension(:),allocatable :: zr,zshdw,zp,zt,zkp,zakp,zkt,zakt
real*8,dimension(:),allocatable :: wws,wwr
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
integer*4 :: isct(2),irct(2)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
call MPI_Barrier( icomm, ierr )
st_measure_time=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'HACApK_bicgstab_lfmtx_hyp start'
mstep=param(83)
eps=param(91)
allocate(wws(maxval(lnp(0:nrank-1))),wwr(maxval(lnp(0:nrank-1))))
allocate(zr(nd),zshdw(nd),zp(nd),zt(nd),zkp(nd),zakp(nd),zkt(nd),zakt(nd))
alpha = 0.0; beta = 0.0; zeta = 0.0;
zz=HACApK_dotp_d(nd, b, b); bnorm=dsqrt(zz);
!$omp parallel
!$omp workshare
zp(1:nd)=0.0d0; zakp(1:nd)=0.0d0
zr(:nd)=b(:nd)
!$omp end workshare
call HACApK_adotsub_lfmtx_hyp(zr,zshdw,st_leafmtxp,st_ctl,u,wws,wwr,isct,irct,nd)
!$omp barrier
!$omp workshare
zshdw(:nd)=zr(:nd)
!$omp end workshare
!$omp single
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
if(mpinr==0) print*,'Original relative residual norm =',zrnorm/bnorm
!$omp end single
do in=1,mstep
if(zrnorm/bnorm<eps) exit
!$omp workshare
zp(:nd) =zr(:nd)+beta*(zp(:nd)-zeta*zakp(:nd))
zkp(:nd)=zp(:nd)
!$omp end workshare
call HACApK_adot_lfmtx_hyp(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,isct,irct,nd)
!!! call HACApK_adot_blrmtx_hyp(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,isct,irct,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zshdw,zr); zden=HACApK_dotp_d(nd,zshdw,zakp);
alpha=znom/zden; znomold=znom;
!$omp end single
!$omp workshare
zt(:nd)=zr(:nd)-alpha*zakp(:nd)
zkt(:nd)=zt(:nd)
!$omp end workshare
call HACApK_adot_lfmtx_hyp(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,isct,irct,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zakt,zt); zden=HACApK_dotp_d(nd,zakt,zakt);
zeta=znom/zden;
!$omp end single
!$omp workshare
u(:nd)=u(:nd)+alpha*zkp(:nd)+zeta*zkt(:nd)
zr(:nd)=zt(:nd)-zeta*zakt(:nd)
!$omp end workshare
!$omp single
beta=alpha/zeta*HACApK_dotp_d(nd,zshdw,zr)/znomold;
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
nstp=in
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,in,time,log10(zrnorm/bnorm)
!$omp end single
enddo
!$omp end parallel
end subroutine HACApK_bicgstab_lfmtx_hyp
!***HACApK_bicgstab_blrmtx_hyp
subroutine HACApK_bicgstab_blrmtx_hyp(st_leafmtxp,st_ctl,u,b,param,nd,nstp,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: u(nd),b(nd)
real*8 :: param(*)
real*8,dimension(:),allocatable :: zr,zshdw,zp,zt,zkp,zakp,zkt,zakt
real*8,dimension(:),allocatable :: wws,wwr
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
integer*4 :: isct(2),irct(2)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
call MPI_Barrier( icomm, ierr )
st_measure_time=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'HACApK_bicgstab_blrmtx_hyp start'
mstep=param(83)
eps=param(91)
ndlfs=st_leafmtxp%ndlfs
allocate(wws(ndlfs),wwr(ndlfs))
allocate(zr(nd),zshdw(nd),zp(nd),zt(nd),zkp(nd),zakp(nd),zkt(nd),zakt(nd))
alpha = 0.0; beta = 0.0; zeta = 0.0;
zz=HACApK_dotp_d(nd, b, b); bnorm=dsqrt(zz);
!$omp parallel
!$omp workshare
zp(1:nd)=0.0d0; zakp(1:nd)=0.0d0
zr(:nd)=b(:nd)
!$omp end workshare
call HACApK_adotsub_blrmtx_hyp(zr,zshdw,st_leafmtxp,st_ctl,u,wws,wwr,nd)
!$omp barrier
!$omp workshare
zshdw(:nd)=zr(:nd)
!$omp end workshare
!$omp single
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
if(mpinr==0) print*,'Original relative residual norm =',zrnorm/bnorm
!$omp end single
do in=1,mstep
if(zrnorm/bnorm<eps) exit
!$omp workshare
zp(:nd) =zr(:nd)+beta*(zp(:nd)-zeta*zakp(:nd))
zkp(:nd)=zp(:nd)
!$omp end workshare
call HACApK_adot_blrmtx_hyp21(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp2(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zshdw,zr); zden=HACApK_dotp_d(nd,zshdw,zakp);
alpha=znom/zden; znomold=znom;
!$omp end single
!$omp workshare
zt(:nd)=zr(:nd)-alpha*zakp(:nd)
zkt(:nd)=zt(:nd)
!$omp end workshare
call HACApK_adot_blrmtx_hyp21(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp2(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zakt,zt); zden=HACApK_dotp_d(nd,zakt,zakt);
zeta=znom/zden;
!$omp end single
!$omp workshare
u(:nd)=u(:nd)+alpha*zkp(:nd)+zeta*zkt(:nd)
zr(:nd)=zt(:nd)-zeta*zakt(:nd)
!$omp end workshare
!$omp single
beta=alpha/zeta*HACApK_dotp_d(nd,zshdw,zr)/znomold;
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
nstp=in
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,in,time,log10(zrnorm/bnorm)
!$omp end single
enddo
!$omp end parallel
end subroutine HACApK_bicgstab_blrmtx_hyp
!***HACApK_bicgstab_blrleaf_hyp
subroutine HACApK_bicgstab_blrleaf_hyp(st_leafmtxp,st_ctl,u,b,param,nd,nstp,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8 :: u(nd),b(nd)
real*8 :: param(*)
real*8,dimension(:),allocatable :: zr,zshdw,zp,zt,zkp,zakp,zkt,zakt
real*8,dimension(:),allocatable :: wws,wwr
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
integer*4 :: isct(2),irct(2)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
call MPI_Barrier( icomm, ierr )
st_measure_time=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'HACApK_bicgstab_blrmtx_hyp start'
mstep=param(83)
eps=param(91)
ndlfs=st_leafmtxp%ndlfs
allocate(wws(ndlfs),wwr(ndlfs))
allocate(zr(nd),zshdw(nd),zp(nd),zt(nd),zkp(nd),zakp(nd),zkt(nd),zakt(nd))
alpha = 0.0; beta = 0.0; zeta = 0.0;
zz=HACApK_dotp_d(nd, b, b); bnorm=dsqrt(zz);
!$omp parallel
!$omp workshare
zp(1:nd)=0.0d0; zakp(1:nd)=0.0d0
zr(:nd)=b(:nd)
!$omp end workshare
call HACApK_adotsub_blr_hyp(zr,zshdw,st_leafmtxp,st_ctl,u,wws,wwr,nd)
!!! call HACApK_adotsub_blrmtx_hyp(zr,zshdw,st_leafmtxp,st_ctl,u,wws,wwr,nd)
!$omp barrier
!$omp workshare
zshdw(:nd)=zr(:nd)
!$omp end workshare
!$omp single
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
if(mpinr==0) print*,'Original relative residual norm =',zrnorm/bnorm
!$omp end single
do in=1,mstep
if(zrnorm/bnorm<eps) exit
!$omp workshare
zp(:nd) =zr(:nd)+beta*(zp(:nd)-zeta*zakp(:nd))
zkp(:nd)=zp(:nd)
!$omp end workshare
call HACApK_adot_blr_hyp(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp21(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp2(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp(zakp,st_leafmtxp,st_ctl,zkp,wws,wwr,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zshdw,zr); zden=HACApK_dotp_d(nd,zshdw,zakp);
alpha=znom/zden; znomold=znom;
!$omp end single
!$omp workshare
zt(:nd)=zr(:nd)-alpha*zakp(:nd)
zkt(:nd)=zt(:nd)
!$omp end workshare
call HACApK_adot_blr_hyp(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp3(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp21(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp2(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!!! call HACApK_adot_blrmtx_hyp(zakt,st_leafmtxp,st_ctl,zkt,wws,wwr,nd)
!$omp barrier
!$omp single
znom=HACApK_dotp_d(nd,zakt,zt); zden=HACApK_dotp_d(nd,zakt,zakt);
zeta=znom/zden;
!$omp end single
!$omp workshare
u(:nd)=u(:nd)+alpha*zkp(:nd)+zeta*zkt(:nd)
zr(:nd)=zt(:nd)-zeta*zakt(:nd)
!$omp end workshare
!$omp single
beta=alpha/zeta*HACApK_dotp_d(nd,zshdw,zr)/znomold;
zrnorm=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm)
nstp=in
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,in,time,log10(zrnorm/bnorm)
!$omp end single
enddo
!$omp end parallel
end subroutine HACApK_bicgstab_blrleaf_hyp
!***HACApK_gcrm_lfmtx
subroutine HACApK_gcrm_lfmtx(st_leafmtxp,st_ctl,st_bemv,u,b,param,nd,nstp,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
type(st_HACApK_calc_entry) :: st_bemv
real*8 :: u(nd),b(nd)
real*8 :: param(*)
real*8,dimension(:),allocatable :: zr,zar,capap
real*8,dimension(:,:),allocatable,target :: zp,zap
real*8,pointer :: zq(:)
real*8,dimension(:),allocatable :: wws,wwr
integer*4 :: isct(2),irct(2)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
call MPI_Barrier( icomm, ierr )
st_measure_time=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) print*,'gcr_lfmtx_hyp start'
mstep=param(83)
mreset=param(87)
eps=param(91)
allocate(wws(maxval(lnp(0:nrank-1))),wwr(maxval(lnp(0:nrank-1))))
allocate(zr(nd),zar(nd),zp(nd,mreset),zap(nd,mreset),capap(mreset))
alpha = 0.0
zz=HACApK_dotp_d(nd, b, b); bnorm=dsqrt(zz);
call HACApK_adot_lfmtx_hyp(zar,st_leafmtxp,st_ctl,u,wws,wwr,isct,irct,nd)
zr(:nd)=b(:nd)-zar(:nd)
zp(:nd,1)=zr(:nd)
zrnorm2=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm2)
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,0,time,log10(zrnorm/bnorm)
if(zrnorm/bnorm<eps) return
call HACApK_adot_lfmtx_hyp(zap(:nd,1),st_leafmtxp,st_ctl,zp(:nd,1),wws,wwr,isct,irct,nd)
do in=1,mstep
ik=mod(in-1,mreset)+1
zq=>zap(:nd,ik)
znom=HACApK_dotp_d(nd,zq,zr); capap(ik)=HACApK_dotp_d(nd,zq,zq)
alpha=znom/capap(ik)
u(:nd)=u(:nd)+alpha*zp(:nd,ik)
zr(:nd)=zr(:nd)-alpha*zq(:nd)
zrnomold=zrnorm2
zrnorm2=HACApK_dotp_d(nd,zr,zr); zrnorm=dsqrt(zrnorm2)
call MPI_Barrier( icomm, ierr )
en_measure_time=MPI_Wtime()
time = en_measure_time - st_measure_time
if(st_ctl%param(1)>0 .and. mpinr==0) print*,in,time,log10(zrnorm/bnorm)
if(zrnorm/bnorm<eps .or. in==mstep) exit
call HACApK_adot_lfmtx_hyp(zar,st_leafmtxp,st_ctl,zr,wws,wwr,isct,irct,nd)
ikn=mod(in,mreset)+1
zp(:nd,ikn)=zr(:nd)
zap(:nd,ikn)=zar(:nd)
do il=1,ik
zq=>zap(:nd,il)
znom=HACApK_dotp_d(nd,zq,zar)
beta=-znom/capap(il)
zp(:nd,ikn) =zp(:nd,ikn)+beta*zp(:nd,il)
zap(:nd,ikn)=zap(:nd,ikn)+beta*zq(:nd)
enddo
enddo
nstp=in
end subroutine
!***HACApK_measurez_time_ax_lfmtx
subroutine HACApK_measurez_time_ax_lfmtx(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
integer*4 :: isct(2),irct(2)
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr; param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
allocate(u(nd),b(nd),wws(maxval(lnp(0:nrank-1))),wwr(maxval(lnp(0:nrank-1))))
st_ctl%time(:)=0.0d0; ztime_ax=1.0d10; ztime_body=1.0d10
nit=param(98)
do it=1,nit
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
!$omp workshare
u(:)=1.0; b(:)=1.0
!$omp end workshare
call HACApK_adot_lfmtx_hyp(u,st_leafmtxp,st_ctl,b,wws,wwr,isct,irct,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
ztime=en_measure_time_ax - st_measure_time_ax
ztime_ax=min(ztime,ztime_ax)
ztime_body=min(st_ctl%time(1),ztime_body)
enddo
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'lfmtx; time_AX_once =',ztime_ax/mstep
write(6,2000) 'lfmtx; body =',ztime_body/mstep
write(6,2000) 'lfmtx; mpi0_t_reduce =',st_ctl%time(2)/(mstep*nit)
endif
deallocate(wws,wwr)
end subroutine HACApK_measurez_time_ax_lfmtx
!***HACApK_measurez_time_ax_blrmtx_
subroutine HACApK_measurez_time_ax_blrmtx_(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
ndlfs=st_leafmtxp%ndlfs
allocate(u(nd),b(nd),wws(ndlfs),wwr(ndlfs))
st_ctl%time(:)=0.0d0
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
u(:)=1.0; b(:)=1.0
call HACApK_adot_blrmtx_hyp(u,st_leafmtxp,st_ctl,b,wws,wwr,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'blrmtx; time_AX_once =',(en_measure_time_ax - st_measure_time_ax)/mstep
write(6,2000) 'blrmtx; body =',st_ctl%time(1)/mstep
write(6,2000) 'blrmtx; t_reduce =',st_ctl%time(2)/mstep
write(6,2000) 'blrmtx; l_reduce =',st_ctl%time(3)/mstep
endif
deallocate(wws,wwr)
end subroutine
!***HACApK_measurez_time_ax_blrmtx
subroutine HACApK_measurez_time_ax_blrmtx(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
ndlfs=st_leafmtxp%ndlfs
allocate(u(nd),b(nd),wws(ndlfs),wwr(ndlfs))
st_ctl%time(:)=0.0d0; ztime_ax=1.0d10; ztime_body=1.0d10
nit=param(98)
do it=1,nit
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
!$omp workshare
u(:)=1.0; b(:)=1.0
!$omp end workshare
call HACApK_adot_blrmtx_hyp(u,st_leafmtxp,st_ctl,b,wws,wwr,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
ztime=en_measure_time_ax - st_measure_time_ax
ztime_ax=min(ztime,ztime_ax)
ztime_body=min(st_ctl%time(1),ztime_body)
enddo
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'blrmtx; time_AX_once =',ztime_ax/mstep
write(6,2000) 'blrmtx; body =',ztime_body/mstep
write(6,2000) 'blrmtx; mpi0_t_reduce =',st_ctl%time(2)/(mstep*nit)
write(6,2000) 'blrmtx; mpi0_l_reduce =',st_ctl%time(3)/(mstep*nit)
endif
deallocate(wws,wwr)
end subroutine
!***HACApK_measurez_time_ax_blrmtx2
subroutine HACApK_measurez_time_ax_blrmtx2(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
ndlfs=st_leafmtxp%ndlfs
allocate(u(nd),b(nd),wws(ndlfs),wwr(ndlfs))
st_ctl%time(:)=0.0d0; ztime_ax=1.0d10; ztime_body=1.0d10
nit=param(98)
do it=1,nit
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
!$omp workshare
u(:)=1.0; b(:)=1.0
!$omp end workshare
call HACApK_adot_blrmtx_hyp21(u,st_leafmtxp,st_ctl,b,wws,wwr,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
ztime=en_measure_time_ax - st_measure_time_ax
ztime_ax=min(ztime,ztime_ax)
ztime_body=min(st_ctl%time(1),ztime_body)
enddo
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'blrmtx2; time_AX_once =',ztime_ax/mstep
write(6,2000) 'blrmtx2; body =',ztime_body/mstep
write(6,2000) 'blrmtx2; mpi0_t_reduce =',st_ctl%time(2)/(mstep*nit)
write(6,2000) 'blrmtx2; mpi0_l_reduce =',st_ctl%time(3)/(mstep*nit)
endif
deallocate(wws,wwr)
end subroutine HACApK_measurez_time_ax_blrmtx2
!***HACApK_measurez_time_ax_blrmtx3
subroutine HACApK_measurez_time_ax_blrmtx3(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
ndlfs=st_leafmtxp%ndlfs
allocate(u(nd),b(nd),wws(ndlfs),wwr(ndlfs))
st_ctl%time(:)=0.0d0; ztime_ax=1.0d10; ztime_body=1.0d10
nit=param(98)
do it=1,nit
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
!$omp workshare
u(:)=1.0; b(:)=1.0
!$omp end workshare
call HACApK_adot_blrmtx_hyp31(u,st_leafmtxp,st_ctl,b,wws,wwr,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
ztime=en_measure_time_ax - st_measure_time_ax
ztime_ax=min(ztime,ztime_ax)
ztime_body=min(st_ctl%time(1),ztime_body)
enddo
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'blrmtx3; time_AX_once =',ztime_ax/mstep
write(6,2000) 'blrmtx3; body =',ztime_body/mstep
write(6,2000) 'blrmtx3; mpi0_t_reduce =',st_ctl%time(2)/(mstep*nit)
write(6,2000) 'blrmtx3; mpi0_l_reduce =',st_ctl%time(3)/(mstep*nit)
endif
deallocate(wws,wwr)
end subroutine HACApK_measurez_time_ax_blrmtx3
!***HACApK_measurez_time_ax_blrmtx4
subroutine HACApK_measurez_time_ax_blrmtx4(st_leafmtxp,st_ctl,nd,lrtrn)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
real*8,dimension(:),allocatable :: wws,wwr,u,b
real*8,pointer :: param(:)
integer*4,pointer :: lpmd(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lpmd => st_ctl%lpmd(:); param=>st_ctl%param(:)
mpinr=lpmd(3); mpilog=lpmd(4); nrank=lpmd(2); icomm=lpmd(1)
mstep=param(99)
ndlfs=st_leafmtxp%ndlfs
allocate(u(nd),b(nd),wws(ndlfs),wwr(ndlfs))
st_ctl%time(:)=0.0d0; ztime_ax=1.0d10; ztime_body=1.0d10
nit=param(98)
do it=1,nit
call MPI_Barrier( icomm, ierr )
st_measure_time_ax=MPI_Wtime()
!$omp parallel private(il)
do il=1,mstep
!$omp workshare
u(:)=1.0; b(:)=1.0
!$omp end workshare
call HACApK_adot_blrmtx_hyp4(u,st_leafmtxp,st_ctl,b,wws,wwr,nd)
enddo
!$omp end parallel
call MPI_Barrier( icomm, ierr )
en_measure_time_ax=MPI_Wtime()
ztime=en_measure_time_ax - st_measure_time_ax
ztime_ax=min(ztime,ztime_ax)
ztime_body=min(st_ctl%time(1),ztime_body)
enddo
if(st_ctl%param(1)>0 .and. mpinr==0) then
write(6,2000) 'blrmtx4; time_AX_once =',ztime_ax/mstep
write(6,2000) 'blrmtx4; body =',ztime_body/mstep
write(6,2000) 'blrmtx4; mpi0_t_reduce =',st_ctl%time(2)/(mstep*nit)
write(6,2000) 'blrmtx4; mpi0_l_reduce =',st_ctl%time(3)/(mstep*nit)
endif
deallocate(wws,wwr)
end subroutine HACApK_measurez_time_ax_blrmtx4
!***HACApK_adot_pmt_lfmtx_p
integer function HACApK_adot_pmt_lfmtx_p(st_leafmtxp,st_bemv,st_ctl,aww,ww)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
type(st_HACApK_calc_entry) :: st_bemv
real*8 :: ww(st_bemv%nd),aww(st_bemv%nd)
real*8,dimension(:),allocatable :: u,au
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:),lod(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lrtrn=0
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr;lod => st_ctl%lod(:)
mpinr=st_ctl%lpmd(3); icomm=st_ctl%lpmd(1); nd=st_bemv%nd
allocate(u(nd),au(nd)); u(:nd)=ww(st_ctl%lod(:nd))
call MPI_Barrier( icomm, ierr )
call HACApK_adot_lfmtx_p(au,st_leafmtxp,st_ctl,u,nd)
aww(st_ctl%lod(:nd))=au(:nd)
HACApK_adot_pmt_lfmtx_p=lrtrn
end function HACApK_adot_pmt_lfmtx_p
!***HACApK_adot_pmt_lfmtx_hyp
integer function HACApK_adot_pmt_lfmtx_hyp(st_leafmtxp,st_bemv,st_ctl,aww,ww)
include 'mpif.h'
type(st_HACApK_leafmtxp) :: st_leafmtxp
type(st_HACApK_lcontrol) :: st_ctl
type(st_HACApK_calc_entry) :: st_bemv
real*8 :: ww(*),aww(*)
real*8,dimension(:),allocatable :: u,au,wws,wwr
integer*4,dimension(:),allocatable :: isct,irct
integer*4,pointer :: lpmd(:),lnp(:),lsp(:),lthr(:),lod(:)
1000 format(5(a,i10)/)
2000 format(5(a,f10.4)/)
lrtrn=0
lpmd => st_ctl%lpmd(:); lnp(0:) => st_ctl%lnp; lsp(0:) => st_ctl%lsp;lthr(0:) => st_ctl%lthr;lod => st_ctl%lod(:)
mpinr=st_ctl%lpmd(3); icomm=st_ctl%lpmd(1); nd=st_bemv%nd; nrank=st_ctl%lpmd(2)
allocate(u(nd),au(nd),isct(2),irct(2)); u(:nd)=ww(st_ctl%lod(:nd))
allocate(wws(maxval(st_ctl%lnp(:nrank))),wwr(maxval(st_ctl%lnp(:nrank))))
call MPI_Barrier( icomm, ierr )
!$omp parallel
!$omp barrier
call HACApK_adot_lfmtx_hyp(au,st_leafmtxp,st_ctl,u,wws,wwr,isct,irct,nd)
!$omp barrier
!$omp end parallel
call MPI_Barrier( icomm, ierr )
aww(st_ctl%lod(:nd))=au(:nd)
HACApK_adot_pmt_lfmtx_hyp=lrtrn
end function HACApK_adot_pmt_lfmtx_hyp
endmodule m_HACApK_solve
|
module Issue1280 where
open import Common.Prelude
open import Common.Reflection
infixr 5 _β·_
data Vec (A : Set) : Nat β Set where
[] : Vec A 0
_β·_ : β {n} β A β Vec A n β Vec A (suc n)
test : Vec _ _
test = 0 β· []
quoteTest : Term
quoteTest = quoteTerm test
unquoteTest = unquote (give quoteTest)
data Foo (A : Set) : Set where
foo : Foo A
ok : Foo Nat
ok = unquote (give (quoteTerm (foo {Nat})))
-- This shouldn't type-check. The term `bad` is type-checked because
-- the implicit argument of `foo` is missing when using quoteTerm.
bad : Foo Bool
bad = unquote (give (quoteTerm (foo {Nat})))
|
open import Reflection hiding (return; _>>=_)
open import Data.List renaming (_++_ to _++l_)
open import Data.Vec as V using (Vec; updateAt)
open import Data.Unit
open import Data.Nat as N
open import Data.Nat.Properties
open import Data.Fin using (Fin; #_; suc; zero)
open import Data.Maybe hiding (_>>=_; map)
open import Function
open import Data.Bool
open import Data.Product hiding (map)
open import Data.String renaming (_++_ to _++s_; concat to sconc; length to slen)
open import Data.Char renaming (_β?_ to _cβ?_)
open import Relation.Binary.PropositionalEquality hiding ([_])
open import Relation.Nullary
open import Relation.Nullary.Decidable hiding (map)
open import Data.Nat.Show renaming (show to showNat)
open import Level renaming (zero to lzero; suc to lsuc)
open import Category.Monad using (RawMonad)
open RawMonad {{...}} public
instance
monadMB : β {f} β RawMonad {f} Maybe
monadMB = record { return = just ; _>>=_ = Data.Maybe._>>=_ }
monadTC : β {f} β RawMonad {f} TC
monadTC = record { return = Reflection.return ; _>>=_ = Reflection._>>=_ }
data Err {a} (A : Set a) : Set a where
error : String β Err A
ok : A β Err A
instance
monadErr : β {f} β RawMonad {f} Err
monadErr = record {
return = ok ;
_>>=_ = Ξ» { (error s) f β error s ; (ok a) f β f a }
}
record RawMonoid {a}(A : Set a) : Set a where
field
_++_ : A β A β A
Ξ΅ : A
++/_ : List A β A
++/ [] = Ξ΅
++/ (x β· a) = x ++ ++/ a
infixr 5 _++_
open RawMonoid {{...}} public
instance
monoidLst : β {a}{A : Set a} β RawMonoid (List A)
monoidLst {A = A} = record {
_++_ = _++l_;
Ξ΅ = []
}
monoidStr : RawMonoid String
monoidStr = record {
_++_ = _++s_;
Ξ΅ = ""
}
monoidErrStr : RawMonoid (Err String)
monoidErrStr = record {
_++_ = Ξ» where
(error s) _ β error s
_ (error s) β error s
(ok sβ) (ok sβ) β ok (sβ ++ sβ)
;
Ξ΅ = ok ""
}
monoidErrLst : β{a}{A : Set a} β RawMonoid (Err $ List A)
monoidErrLst = record {
_++_ = Ξ» where
(error s) _ β error s
_ (error s) β error s
(ok sβ) (ok sβ) β ok (sβ ++ sβ)
;
Ξ΅ = ok []
}
defToTerm : Name β Definition β List (Arg Term) β Term
defToTerm _ (function cs) as = pat-lam cs as
defToTerm _ (constructorβ² d) as = con d as
defToTerm _ _ _ = unknown
derefImmediate : Term β TC Term
derefImmediate (def f args) = getDefinition f >>= Ξ» f' β return (defToTerm f f' args)
derefImmediate x = return x
derefT : Term β TC Term
derefT (def f args) = getType f
derefT (con f args) = getType f
derefT x = return x
defName : Term β Maybe Name
defName (def f args) = just f
defName _ = nothing
Ctx = List $ Arg Type
pi-to-ctx : Term β Ctx
-- we have a Ctx for the entire function, now we want to build
-- a context for the given variables in the clause. To do so
-- we merge function's ctx with patterns of the given clause
-- and we grab the types that correspond to the variables within
-- the patterns.
--pats-ctx : Ctx β (List $ Arg Pattern) β TC $ Maybe Ctx
macro
reflect : Term β Term β TC β€
reflect f a = (derefImmediate f)
>>= quoteTC >>= unify a
reflect-ty : Name β Type β TC β€
reflect-ty f a = getType f >>= quoteTC >>= normalise >>= unify a
rtest : Term β Term β TC β€
rtest f a = do
t β derefT f
v β derefImmediate f
--v β pat-lam-no--rm v (pi-to-ctx t)
--q β quoteTC v
q β quoteTC (pi-to-ctx t)
unify a q
rmkstring : Term β Term β TC β€
rmkstring f a = unify (lit (string "Test")) a
infert : Type β Term β TC β€
infert t a = inferType t >>= quoteTC >>= unify a
-- FIXME we probably want to error out on these two functions.
pi-to-ctx (Ξ [ s βΆ a ] x) = (a β· pi-to-ctx x)
pi-to-ctx _ = []
Prog = Err $ List String
infixl 5 _#p_
_#p_ = _++_
okl : String β Prog
okl s = ok ([ s ])
-- reduce a list of Progs with a delimiter
_/#p_ : List Prog β (delim : String) β Prog
[] /#p d = ok []
(x β· []) /#p d = x
(x β· xs@(_ β· _)) /#p d = x #p okl d #p xs /#p d
-- Normalise the name of functions that we obtain from showName,
-- i.e. remove dots, replace weird symbols by ascii.
nnorm : String β Prog
nnorm s = okl
$ replace '.' '_'
$ replace '-' '_'
$ s
where
repchar : (t f x : Char) β Char
repchar f t x with x cβ? f
... | yes _ = t
... | no _ = x
replace : (from to : Char) β String β String
replace f t s = fromList $ map (repchar f t) $ toList s
data NumClauses : Set where
Many One : NumClauses
record State : Set where
constructor st
field
--arg-ctx : Ctx
--ret-typ : String
var-names : List String
retvar : String
cls : NumClauses
open State
--compile-clause : Clause β State β Prog
-- Pack the information about new variables generated
-- by patterns in the clause, assignments to these,
-- and the list of conditions. E.g.
-- foo : List β β β
-- foo (x β· xs) 2 = ...
--
-- Assume that we named top-level arguments [a, b]
-- Then, new variables for this clause are going to be
-- [x, xs]
-- Assignments are:
-- [x = hd a, xs = tl a]
-- Conditions are:
-- [is-cons a, b = 2]
record ClCond : Set where
constructor clcond
field
vars : List String
assigns : List String
conds : List String
data MbClCond : Set where
ok : ClCond β MbClCond
error : String β MbClCond
_#c_ : MbClCond β MbClCond β MbClCond
error s #c _ = error s
_ #c error s = error s
ok (clcond a b c) #c ok (clcond a' b' c') = ok (clcond (a ++ a') (b ++ b') (c ++ c'))
{-# TERMINATING #-}
clause-ctx-vars : (pats : List $ Arg Pattern) β (vars : List String) β (counter : β) β MbClCond
showLitProg : Literal β Prog
comp-term : Term β (varctx : List String) β Prog
sjoin : List String β String β String
sjoin [] delim = ""
sjoin (x β· []) delim = x
sjoin (x β· xs@(_ β· _)) delim = x ++s delim ++s sjoin xs delim
join' : List String β (delim : String) β (empty : String) β String
join' [] _ e = e
join' x@(_ β· _) d _ = sjoin x d
compile-cls : List Clause β State β Prog
compile-cls [] s = error "comile-cls: expected at least one clause"
compile-cls (clause ps t β· []) s with clause-ctx-vars ps (var-names s) 0
... | error msg = error msg
... | ok (clcond vars assgns conds) = let
as = sconc (map (_++s "\n") assgns)
rv = retvar s ++s " = "
in okl (as ++s rv) #p comp-term t vars #p okl ";"
compile-cls (absurd-clause ps β· []) s with clause-ctx-vars ps (var-names s) 0
... | error msg = error msg
... | ok (clcond vars assgns conds) = okl "unreachable ();"
compile-cls (clause ps t β· xs@(_ β· _)) s with clause-ctx-vars ps (var-names s) 0
... | error msg = error msg
... | ok (clcond vars assgns conds) = let
cs = join' conds " && " "true"
as = sconc (map (_++s "\n") assgns)
rv = retvar s ++s " = "
in okl ("if (" ++s cs ++s ") {" ++s as ++s rv) #p comp-term t vars #p okl "; }"
#p okl "else {"
#p compile-cls xs s
#p okl "}"
compile-cls (absurd-clause ps β· xs@(_ β· _)) s with clause-ctx-vars ps (var-names s) 0
... | error msg = error msg
... | ok (clcond vars assgns conds) = let
-- XXX it would be weird if conds were empty... catch it?
cs = join' conds " && " "true"
in okl ("if (" ++s cs ++s ") { unreachable (); } else {" )
#p compile-cls xs s
#p okl "}"
clause-ctx-vars (arg i (con c ps) β· l) (v β· vars) vcnt with showName c
... | "Agda.Builtin.List.List.[]" =
ok (clcond [] [] [ "emptyvec_p (" ++s v ++s ")" ])
#c clause-ctx-vars l vars vcnt
... | "Agda.Builtin.List.List._β·_" =
ok (clcond [] [] [ "nonemptyvec_p (" ++s v ++s ")" ])
#c clause-ctx-vars (ps ++ l) (("hd (" ++s v ++s ")") β· ("tl (" ++s v ++s ")") β· vars) vcnt
... | "Agda.Builtin.Bool.Bool.true" =
ok (clcond [] [] [ v {- == true -} ])
#c clause-ctx-vars l vars vcnt
... | "Agda.Builtin.Bool.Bool.false" =
ok (clcond [] [] [ "! " ++s v ])
#c clause-ctx-vars l vars vcnt
... | "Agda.Builtin.Nat.Nat.suc" =
ok (clcond [] [] [ v ++s " > 0" ])
#c clause-ctx-vars (ps ++ l) ((v ++s " - 1") β· vars) vcnt
... | "Agda.Builtin.Nat.Nat.zero" =
ok (clcond [] [] [ v ++s " == 0" ])
#c clause-ctx-vars l vars vcnt
... | "Data.Fin.Base.Fin.zero" =
ok (clcond [] [] [ v ++s " == 0" ]) -- XXX can also add v < u
#c clause-ctx-vars l vars vcnt
... | "Data.Fin.Base.Fin.suc" =
ok (clcond [] [] [ v ++s " > 0" ]) -- XXX can also add v < u
#c clause-ctx-vars l vars vcnt
... | "Data.Vec.Base.Vec.[]" =
ok (clcond [] [] [ "emptyvec_p (" ++s v ++s ")" ])
#c clause-ctx-vars l vars vcnt
... | "Data.Vec.Base.Vec._β·_" =
ok (clcond [] [] [ "nonemptyvec_p (" ++s v ++s ")" ])
#c clause-ctx-vars (ps ++ l) (("len (" ++s v ++s ") - 1") β· ("hd (" ++s v ++s ")") β· ("tl (" ++s v ++s ")") β· vars) vcnt
-- Well, let's see how far we can go with this hack
... | "Array.Base.Ar.imap" =
--... | "test-extract.Ar'.imap" =
ok (clcond [ "IMAP_" ++s v ] [ "\n#define IMAP_" ++s v ++s "(__x) " ++s v ++s "[__x]\n" ] [ "true" ])
#c clause-ctx-vars l vars vcnt
... | x = error ("clause-ctx-vars: don't know what to do with `" ++s x ++s "` constructor in patterns")
clause-ctx-vars (arg i dot β· l) (v β· vars) vcnt =
-- Dot patterns are skipped.
clause-ctx-vars l vars vcnt
clause-ctx-vars (arg (arg-info visible r) (var s) β· l) (v β· vars) vcnt =
-- If we have "_" as a variable, we need to insert it
-- into the list, but we don't generate an assignment for it.
let asgn = case s β? "_" of Ξ» where
-- XXX hopefully this is fine, otherwise
-- we can do the same thing as for hidden
-- vars.
(yes p) β []
(no Β¬p) β [ s ++s " = " ++s v ++s ";" ]
in ok (clcond [ s ] asgn [])
#c clause-ctx-vars l vars vcnt
clause-ctx-vars (arg (arg-info hidden r) (var s) β· l) (v β· vars) vcnt =
-- Hidden variables are simply added to the context
-- as regular variables
let s , vcnt = case s β? "_" of Ξ» where
(yes p) β s ++ "_" ++ showNat vcnt , 1 + vcnt
(no Β¬p) β s , vcnt
in ok (clcond [ s ] [ s ++ " = " ++ v ++ ";" ] [])
#c clause-ctx-vars l vars vcnt
clause-ctx-vars (arg (arg-info instanceβ² r) (var s) β· l) (v β· vars) vcnt =
error "FIXME handle instance variables"
clause-ctx-vars (arg i (lit x) β· l) (v β· vars) vcnt =
case showLitProg x of Ξ» where
(error s) β error s
(ok s) β ok (clcond [] [] [ v ++s " == " ++s (sconc s) ])
#c clause-ctx-vars l vars vcnt
clause-ctx-vars (arg i (proj f) β· l) (v β· vars) vcnt =
error "FIXME proj pattern"
clause-ctx-vars (arg i absurd β· l) (v β· vars) vcnt =
-- I assume that absurd can only appear in the
-- absurd clause, therefore, we don't need a condition
-- for this pattern, so we just skip it.
clause-ctx-vars l vars vcnt
clause-ctx-vars [] [] _ =
ok (clcond [] [] [])
clause-ctx-vars _ _ _ =
error "mismatching number of patterns and types"
showLitProg (name x) = error ("Found name `" ++s (showName x) ++s "` as literal")
showLitProg (meta x) = error ("Found meta `" ++s (showMeta x) ++s "` as literal")
showLitProg x = okl (showLiteral x)
var-lkup : List String β β β Prog
var-lkup [] n = error ("Variable not found")
var-lkup (x β· l) zero = okl x
var-lkup (x β· l) (suc n) = var-lkup l n
-- Compile each arg and join them with ", "
comp-arglist : List $ Arg Term β (varctx : List String) β Prog
comp-arglist args varctx = map (Ξ» {(arg i x) β comp-term x varctx}) args /#p ", "
-- Helper for comp-arglist-mask
mk-mask : (n : β) β List (Fin n) β List Bool
mk-mask n xs = V.toList $ go (V.replicate {n = n} false) xs
where go : _ β _ β _
go e [] = e
go e (x β· xs) = go (updateAt x (Ξ» _ β true) e) xs
comp-arglist-mask : List $ Arg Term β (mask : List Bool) β (varctx : List String) β Prog
comp-arglist-mask args mask varctx = go args mask varctx /#p ", "
where
go : List $ Arg Term β (mask : List Bool) β (varctx : List String) β List Prog
go [] [] _ = []
go [] (x β· mask) _ = [ error "Incorrect argument mask" ]
go (x β· args) [] _ = [ error "Incorrect argument mask" ]
go (x β· args) (false β· mask) vars = go args mask vars
go (arg i x β· args) (true β· mask) vars = comp-term x vars β· go args mask vars
comp-term (var x []) vars = var-lkup (reverse vars) x
comp-term (var x args@(_ β· _)) vars = var-lkup (reverse vars) x #p okl "(" #p comp-arglist args vars #p okl ")"
--comp-term (var x (xβ β· args)) vars with var-lkup (reverse vars) x
--comp-term (var x (xβ β· args)) vars | ok l = error ("Variable " ++s (sconc l) ++s " used as a function call")
--comp-term (var x (xβ β· args)) vars | error s = error s
comp-term (lit l) vars = showLitProg l
comp-term exp@(con c args) vars with showName c
... | "Data.Vec.Base.Vec.[]" =
okl "[]"
... | "Data.Vec.Base.Vec._β·_" =
okl "cons (" #p comp-arglist-mask args (mk-mask 5 $ # 3 β· # 4 β· []) vars #p okl ")"
... | "Agda.Builtin.Nat.Nat.suc" =
okl "(1 + " #p comp-arglist-mask args (mk-mask 1 $ # 0 β· []) vars #p okl ")"
... | "Data.Fin.Base.Fin.zero" =
okl "0"
... | "Data.Fin.Base.Fin.suc" =
okl "(1 + " #p comp-arglist-mask args (mk-mask 2 $ # 1 β· []) vars #p okl ")"
... | "Array.Base.Ar.imap" =
--... | "test-extract.Ar'.imap" =
case args of Ξ» where
(_ β· _ β· _ β· arg _ s β· arg _ (vLam x e) β· []) β let
p = comp-term e (vars ++ [ x ])
sh = comp-term s vars --infert exp
in okl ("with { (. <= " ++s x ++s " <= .): ") #p p #p okl "; }: genarray (" #p sh #p okl ")"
_ β
error "comp-term: don't recognize arguments to imap"
... | "Array.Base.Ix.[]" =
okl "[]"
... | "Array.Base.Ix._β·_" =
okl "cons (" #p comp-arglist-mask args (mk-mask 5 $ # 3 β· # 4 β· []) vars #p okl ")"
... | n = error ("comp-term: don't know constructor `" ++s n ++s "`")
comp-term (def f args) vars with showName f
... | "Agda.Builtin.Nat._+_" =
okl "_add_SxS_ (" #p comp-arglist args vars #p okl ")"
... | "Agda.Builtin.Nat._*_" =
okl "_mul_SxS_ (" #p comp-arglist args vars #p okl ")"
... | "Data.Nat.DivMod._/_" =
okl "_div_SxS_ (" #p comp-arglist-mask args (mk-mask 3 $ # 0 β· # 1 β· []) vars #p okl ")"
... | "Data.Fin.#_" =
comp-arglist-mask args (mk-mask 3 $ # 0 β· []) vars
... | "Array.Base.ix-lookup" =
case args of Ξ» where
(_ β· _ β· arg _ iv β· arg _ el β· []) β let
ivβ² = comp-term iv vars
elβ² = comp-term el vars
in ivβ² #p okl "[" #p elβ² #p okl "]"
_ β
error "comp-term: don't recognize arguments to ix-lookup"
... | "Data.Vec.Base.[_]" =
case args of Ξ» where
(_ β· _ β· arg _ x β· []) β okl "[" #p comp-term x vars #p okl "]"
_ β
error "comp-term: don't recognize arguments to Vec.[_]"
... | "Data.Fin.Base.raise" =
-- Note that "raise" is a total junk, as it only makes sure that the
-- Fin value is valid in some context; all we are interested in is the
-- value of that Fin.
case args of Ξ» where
(_ β· _ β· arg _ x β· []) β comp-term x vars
_ β
error "comp-term: don't recognize arguments to Data.Fin.raise"
-- XXX we need to figure out what are we going to do with recursive functions,
-- as clearly its name can't be known in advance. Probably add it to the
-- state? Or maintain a list of functions?
... | n = nnorm n #p okl " (" #p comp-arglist args vars #p okl ")"
--... | n = error ("comp-term: don't know definition `" ++s n ++s "`")
comp-term (lam v t) vars = error "comp-term: lambdas are not supported"
comp-term (pat-lam cs args) vars = error "comp-term: pattern-matching lambdas are not supported"
comp-term (pi a b) vars = error "comp-term: pi types are not supported"
comp-term (sort s) vars = error "comp-term: sorts are not supported"
comp-term (meta x xβ) vars = error "comp-term: metas are not supported"
comp-term unknown vars = error "comp-term: unknowns are not supported"
record Pistate : Set where
constructor pist-vc=_cv=_vctx=_tctx=_rv=_ret=_cons=_
field
var-counter : β
cur-var : String
var-ctx : List String
type-ctx : List String
ret-var : String
ret : Prog
-- XXX come up with a better type for
-- constraints on variables.
cons : List (String Γ Prog)
open Pistate
trav-pi : Type β Pistate β Pistate
trav-pi (Ξ [ s βΆ arg i x ] y) pst
= let
varname = case s of Ξ» where
"_" β "x_" ++s (showNat $ var-counter pst)
n β n
tp = trav-pi x (record pst {cur-var = varname}) -- ; cons = []})
in case ret tp of Ξ» where
(error s) β tp
(ok l) β trav-pi y (record pst {var-counter = 1 + var-counter pst ;
cur-var = ret-var pst ;
var-ctx = var-ctx pst ++ [ varname ] ;
type-ctx = type-ctx pst ++ [ (sjoin l "") ] ;
cons = cons tp}) --cons pst ++ cons tp })
trav-pi (con c args) pst with showName c
... | x = record pst {ret = error ("trav-pi: don't know how to handle `" ++s x ++s "` constructor")}
trav-pi (def f args) pst with showName f
... | "Agda.Builtin.Nat.β" = record pst {ret = okl "int"}
... | "Agda.Builtin.Nat.Nat" = record pst {ret = okl "int"}
... | "Agda.Builtin.Bool.Bool" = record pst {ret = okl "bool"}
... | "Agda.Builtin.List.List" =
-- We encode lists as 1-d arrays of their argument type.
case args of Ξ» where
(_ β· arg i x β· _) β let tp = trav-pi x (record pst {cons = []})
in case ret tp of Ξ» where
(error s) β tp
(ok l) β record tp {ret = okl $ (sjoin l "") ++s "[.]"}
_ β record pst {ret = error "trav-pi: incorrect arguments to List"}
... | "Data.Vec.Base.Vec" =
-- Vectors are also 1-d arrays (such as lists) but we extract
-- a bit of extra infromation from these.
case args of Ξ» where
(_ β· arg _ t β· arg _ x β· []) β let tp = trav-pi t (record pst {cur-var = "" {- XXX well, typeof (cur-var pst) is the thing -}
}) --cons = []})
p = comp-term x (var-ctx pst)
in record tp {ret = ret tp #p okl "[.]" ;
cons = (cons tp)
++ [ cur-var pst ,
okl ("/* assert (shape (" ++s (cur-var pst) ++s ")[[0]] == ") #p p #p okl ") */" ]
}
_ β record pst {ret = error "trav-pi: incorrect arguments to Vec"}
... | "Data.Fin.Base.Fin" =
case args of Ξ» where
(arg _ x β· []) β let
p = comp-term x (var-ctx pst)
in record pst {
ret = okl "int";
cons = (cons pst)
++ [ cur-var pst ,
okl ("/* assert (" ++s (cur-var pst) ++s " < ") #p p #p okl ") */"]
}
_ β
record pst {ret = error "trav-pi: incorrect arguments to Fin"}
... | "Array.Base.Ar" =
--... | "test-extract.Ar'" =
case args of Ξ» where
(_ β· arg _ ty β· arg _ dim β· arg _ sh β· []) β let
tyβ² = trav-pi ty pst
dimβ² = comp-term dim (var-ctx pst)
shβ² = comp-term sh (var-ctx pst)
in record tyβ² {
ret = ret tyβ² #p okl "[*]" ;
cons = cons tyβ²
++ [ cur-var pst ,
okl ("/* assert (dim (" ++s (cur-var pst) ++s ") == ") #p dimβ² #p okl " )*/" ]
++ [ cur-var pst ,
okl ("/* assert (shape (" ++s (cur-var pst) ++s ") == ") #p shβ² #p okl " )*/" ]
}
_ β
record pst {ret = error "trav-pi: incorrect arguments to Ar"}
... | x = record pst {ret = error ("trav-pi: don't know the `" ++s x ++s "` type")}
trav-pi _ pst = record pst {ret = error "trav-pi ERRR"}
find : List String β String β Bool
find [] x = false
find (y β· l) x with x β? y
... | yes _ = true
... | no _ = find l x
collect-var-cons : List (String Γ Prog) β (accum : List String) β List (String Γ Prog)
collect-var-cons [] acc = []
collect-var-cons ((x , v) β· l) acc with find acc x
... | true = collect-var-cons l acc
... | false = (x , v #p collect l x) β· collect-var-cons l (x β· acc)
where
collect : _ β _ β _
collect [] x = ok []
collect ((y , v) β· l) x with y β? x
... | yes _ = v #p collect l x
... | no _ = collect l x
-- Get the value bound to the given variable or return (ok [])
lkup-var-cons : List (String Γ Prog) β String β Prog
lkup-var-cons [] s = ok []
lkup-var-cons ((x , v) β· xs) s with x β? s
... | yes _ = v
... | no _ = lkup-var-cons xs s
fltr : List (String Γ Prog) β (var : String) β List (String Γ Prog)
fltr [] x = []
fltr ((y , v) β· l) x with x β? y
... | yes _ = fltr l x
... | no _ = (y , v) β· fltr l x
mkfun : Name β _ β Pistate β NumClauses β Prog
mkfun n cls ps nc = let
rv = (ret-var ps)
cs = collect-var-cons (cons ps) []
arg-cons = map projβ $ fltr cs rv
ret-cons = lkup-var-cons cs rv
in (okl $ "// Function " ++s (showName n) ++s "\n")
#p ret ps #p okl "\n"
#p (nnorm $ showName n ++s "(")
#p tvl (var-ctx ps) (type-ctx ps)
#p okl ") {\n"
--#p (cons ps) /#p "\n"
#p arg-cons /#p "\n"
#p ret ps #p okl (" " ++s rv ++s ";\n")
#p compile-cls cls (st (var-ctx ps) rv nc)
#p ret-cons
#p okl ("return " ++s rv ++s ";\n}\n\n")
where
tvl : List String β List String β Prog
tvl [] [] = ok []
tvl [] (t β· typs) = error "more types than variables"
tvl (x β· vars) [] = error "more variables than types"
tvl (x β· []) (t β· []) = okl (t ++s " " ++s x)
tvl (x β· []) (_ β· _ β· _) = error "more types than variables"
tvl (_ β· _ β· _) (_ β· []) = error "more variables than types"
tvl (x β· xs@(_ β· _)) (t β· ts@(_ β· _)) = okl (t ++s " " ++s x ++s ", ") #p tvl xs ts
compile' : (lam : Term) β (sig : Type) β (name : Maybe Name) β TC Prog
compile' (pat-lam cs args) t nm with nm
compile' (pat-lam cs args) t nm | nothing =
return $ error "compile' got invalid function name"
compile' (pat-lam [] args) t nm | just x =
return $ error "compile' got zero clauses in the lambda term"
compile' (pat-lam cs@(_ β· []) args) t nm | just x =
-- XXX currently the name `__ret` is hardcoded.
let ps = trav-pi t (pist-vc= 1 cv= "" vctx= [] tctx= [] rv= "__ret" ret= error "" cons= [])
in return (mkfun x cs ps One)
compile' (pat-lam cs@(_ β· _ β· _) args) t nm | just x =
-- XXX currently the name `__ret` is hardcoded.
let ps = trav-pi t (pist-vc= 1 cv= "" vctx= [] tctx= [] rv= "__ret" ret= error "" cons= [])
in return (mkfun x cs ps Many)
compile' x _ _ =
return (error "compile' expected pattern-matching lambda")
macro
compile : Term β Term β TC β€
compile f a = do
t β derefT f
v β derefImmediate f
let ctx = pi-to-ctx t
let n = defName f
--v β pat-lam-norm v ctx
let v = return v
case v of Ξ» where
(ok v) β do
v β compile' v t n
q β quoteTC v
unify a q
e@(error s) β
return e >>= quoteTC >>= unify a
---===== These are just all examples to play around ====---
tst-triv : β β β
tst-triv x = x + 1
-- Test pattern contraction
tst-ss : β β β
tst-ss (suc (suc x)) = x
tst-ss _ = 0
-- Here we have the (+ 0) in the last clause that
-- stays in the generated code.
tst-rew0 : β β Bool β β β β
tst-rew0 x true y = let a = x * x in a + y
tst-rew0 x false y = x + 2 + 0
-- XXX can't do with clauses yet, but that shouldn
tst-with : β β β
tst-with x with x >? 0
tst-with x | yes p = 0
tst-with x | no Β¬p = 1
-- Trivial test with Lists
lst-test : List β β β
lst-test [] = 0
lst-test (_β·_ x y) = x + 1
data Test : Set where
cstctr : {x : β} β x > 0 β Test
test-test : Test β β
test-test (cstctr p) = 1
test-dot : (a : β) β a > 0 β β
test-dot a@(.(suc _)) (sβ€s pf) = a
data Square : β β Set where
sq : (m : β) β Square (m * m)
root : (m : β) (n : β) β Square n β β
root a .(m * m) (sq m) =
-- This is to show that square pattern is skipped
-- from the context. In the above case, the clause is
-- represetned as: a , . , (sq m) ==ctx==> a , m
-- and the expression is (var 0) + (var 1)
m + a
open import Data.Vec hiding (concat)
tst-vec : β {n} β Vec β n β Vec β (n + n * n) β β
tst-vec [] _ = 0
tst-vec (x β· a) b = x
a = (reflect-ty tst-vec)
tst-undsc : _ β β
tst-undsc n = 1 + n
vec-sum : β {n} β Vec β n β Vec β (n) β Vec β n
vec-sum [] _ = []
vec-sum (x β· a) (y β· b) = x + y β· vec-sum a b
vec-and-1 : β {n} β Vec Bool n β Bool
vec-and-1 (x β· xs) = x β§ vec-and-1 xs
vec-and-1 _ = true
vec-and : β {n} β Vec Bool n β Vec Bool n β Vec Bool n
vec-and [] _ = []
vec-and (x β· a) (y β· b) = x β§ y β· vec-and a b
vec-+ : β {n} β Vec β n β β
vec-+ (x β· xs) = x + vec-+ xs
vec-+ _ = 0
f : β β β
f x = x * x
vec-tst : β n β Vec β (n) β β
vec-tst 0 [] = 0
vec-tst (suc n) x = n * 2 -- (x β· xs) = n * 2
def-pst = (pist-vc= 1 cv= "" vctx= [] tctx= [] rv= "__ret" ret= error "" cons= []) --pist 1 [] [] (error "")
q : List String Γ List String Γ Prog
q = let (pist-vc= _ cv= _ vctx= v tctx= t rv= _ ret= r cons= _) = (trav-pi (reflect-ty vec-sum) def-pst) in (v , t , r)
--open import Data.Fin
xxx : Fin 5 β Fin 6
xxx zero = suc zero
xxx (suc _) = zero
fun-in-ty : (f : β β β) β Vec β (f 3) β β
fun-in-ty f x = 1 -- V.replicate 1
open import Array
data Ar' {a} (X : Set a) (d : β) : (Vec β d) β Set a where
imap : β s β (Ix d s β X) β Ar' X d s
add-2v : β {n} β let X = Ar β 1 (n β· []) in X β X β X
add-2v (imap a) (imap b) = imap Ξ» iv β a iv + b iv
postulate
asum : β {n} β Ar β 1 (n β· []) β β
asum' : β {n} β Ar' β 1 (n β· []) β β
--sum (imap a)
mm : β {m n k} β let Mat a b = Ar β 2 (a β· b β· []) in
Mat m n β Mat n k β Mat m k
mm (imap a) (imap b) = imap Ξ» iv β let i = ix-lookup iv (# 0)
j = ix-lookup iv (# 1)
in asum (imap Ξ» kv β let k = ix-lookup kv (# 0)
in a (i β· k β· []) * b (k β· j β· []))
conv : β {n} β let Ar1d n = Ar β 1 V.[ n ] in
Ar1d (3 + n) β Ar1d 3 β Ar1d (1 + n)
conv (imap inp) (imap ker) = imap Ξ» iv β let i = ix-lookup iv (# 0)
in ( inp (raise 2 i β· []) * ker (# 0 β· [])
+ inp (raise 1 (suc i) β· []) * ker (# 1 β· [])
+ inp (raise 0 (suc (suc i)) β· []) * ker (# 2 β· [])
) / 3
where open import Data.Fin using (raise)
open import Data.Nat.DivMod
test-fin : Fin 3 β Fin 4
test-fin x = suc x
w : String
w = case compile mm of Ξ» where
(error s) β s
(ok l) β sjoin l ""
|
\section{Results}
The Content Addressable Parallel Processor was successfully implemented on a TinyFPGA-BX. It was tested both by manual interaction with the serial USB-UART as well as through a python driver which wraps the protocol in a higher-level object-oriented API. The python driver file also contains a simple test program that serves both as a tutorial and a test-suite for the FPGA based CAPP. |
(******************************************)
(* Finite multiset library *)
(* Developped for the PACTOLE project *)
(* L. Rieg *)
(* *)
(* This file is distributed under *)
(* the terms of the CeCILL-C licence *)
(* *)
(******************************************)
Require Import Bool.
Require Import Omega.
Require Import PArith.
Require Import RelationPairs.
Require Import Equalities.
Require Import SetoidList.
Require Import MMultisets.Preliminary.
Require Import MMultisets.MMultisetInterface.
Set Implicit Arguments.
Module Make(E : DecidableType)(M : MMultisetsOn E).
Include M.
(** An experimental tactic that superficially ressembles [fsetdec]. It is by no mean as general. **)
Hint Rewrite empty_spec add_same remove_same diff_spec union_spec inter_spec lub_spec singleton_same : MFsetdec.
Hint Rewrite is_empty_spec nfilter_spec filter_spec npartition_spec_fst npartition_spec_snd : MFsetdec.
Hint Rewrite partition_spec_fst partition_spec_snd for_all_spec exists_spec : MFsetdec.
Hint Unfold In : MFsetdec.
Ltac saturate_Einequalities :=
repeat match goal with
(* remove duplicates *)
| H1 : ~E.eq ?x ?y, H2 : ~E.eq ?x ?y |- _ => clear H2
(* avoid reflexive inequalities *)
| H : ~E.eq ?x ?x |- _ => change (id (~E.eq x x)) in H
(* avoid already saturated inequalities *)
| H1 : ~E.eq ?x ?y, H2 : ~E.eq ?y ?x |- _ => change (id (~E.eq x y)) in H1; change (id (~E.eq y x)) in H2
(* saturate the remaining ones *)
| H : ~E.eq ?x ?y |- _ => let Hneq := fresh "Hneq" in assert (Hneq : ~E.eq y x) by auto
| _ => idtac
end;
(* remove the markers (id) *)
repeat match goal with
| H : id (~E.eq ?x ?y) |- _ => change (~E.eq x y) in H
| _ => idtac
end.
Ltac msetdec_step :=
match goal with
(* Simplifying equalities *)
| H : ?x = ?x |- _ => clear H
| H : E.eq ?x ?x |- _ => clear H
| H : ?x [=] ?x |- _ => clear H
| H : ?x = ?y |- _ => subst x
| Hneq : ~E.eq ?x ?x |- _ => now elim Hneq
| Hneq : ~?x [=] ?x |- _ => now elim Hneq
| Heq : E.eq ?x ?y |- _ => clear x Heq || rewrite Heq in *
| Heq : ?x [=] ?y, Hin : context[?x] |- _ => rewrite Heq in Hin
| Heq : ?x [=] ?y |- context[?x] => rewrite Heq
| Heq : ?x [=] ?y |- _ => clear x Heq
(* Simplifying [singleton], [add] and [remove] *)
| Hneq : ~E.eq ?y ?x |- context[multiplicity ?y (singleton ?x ?n)] => rewrite singleton_other; trivial
| Hneq : ~E.eq ?y ?x |- context[multiplicity ?y (add ?x ?n ?m)] => rewrite add_other; trivial
| Hneq : ~E.eq ?y ?x |- context[multiplicity ?y (remove ?x ?n ?m)] => rewrite remove_other; trivial
| Hneq : ~E.eq ?y ?x, H : context[multiplicity ?y (singleton ?x ?n)] |- _ =>
rewrite singleton_other in H; trivial
| Hneq : ~E.eq ?y ?x, H : context[multiplicity ?y (add ?x ?n ?m)] |- _ => rewrite add_other in H; trivial
| Hneq : ~E.eq ?y ?x, H : context[multiplicity ?y (remove ?x ?n ?m)] |- _ =>
rewrite remove_other in H; trivial
(* Destructing equalities *)
| H : ~E.eq ?x ?y |- context[E.eq ?x ?y] => destruct (E.eq_dec x y) as [| _]; try contradiction
| |- context[E.eq ?x ?y] => destruct (E.eq_dec x y); trivial
| |- context[multiplicity ?y (singleton ?x ?n)] => destruct (E.eq_dec y x)
| |- context[multiplicity ?y (add ?x ?n ?m)] => destruct (E.eq_dec y x)
| |- context[multiplicity ?y (remove ?x ?n ?m)] => destruct (E.eq_dec y x)
| H : context[multiplicity ?y (singleton ?x ?n)] |- _ => destruct (E.eq_dec y x)
| H : context[multiplicity ?y (add ?x ?n ?m)] |- _ => destruct (E.eq_dec y x)
| H : context[multiplicity ?y (remove ?x ?n ?m)] |- _ => destruct (E.eq_dec y x)
| |- context[E.eq_dec ?x ?y] => destruct (E.eq_dec x y)
| Heq : context[E.eq_dec ?x ?y] |- _ => destruct (E.eq_dec x y)
| _ => idtac
end.
Ltac msetdec :=
repeat (autorewrite with MFsetdec in *; unfold In in *; trivial; saturate_Einequalities;
msetdec_step; easy || (try omega)).
Tactic Notation "msetdec_n" integer(n) :=
do n (autorewrite with MFsetdec in *; unfold In in *; trivial; saturate_Einequalities;
msetdec_step; easy || (try omega)).
Lemma subrelation_pair_elt : subrelation eq_pair eq_elt.
Proof. now intros [x n] [y p] [H _]. Qed.
Lemma InA_pair_elt : forall x n p l, InA eq_pair (x, n) l -> InA eq_elt (x, p) l.
Proof.
intros x n p l Hin. induction l as [| [y q] l].
+ rewrite InA_nil in Hin. elim Hin.
+ inversion_clear Hin.
- destruct H as [ H _]. now left.
- right. now apply IHl.
Qed.
Lemma InA_elt_pair : forall x n l, InA eq_elt (x, n) l -> exists n', InA eq_pair (x, n') l.
Proof.
intros x n l Hin. induction l as [| [y p] l].
+ rewrite InA_nil in Hin. elim Hin.
+ inversion_clear Hin.
- compute in H. exists p. left. now rewrite H.
- apply IHl in H. destruct H as [k Hin]. exists k. now right.
Qed.
Lemma pair_dec : forall xn yp, {eq_pair xn yp} + {~eq_pair xn yp}.
Proof.
intros [x n] [y p]. destruct (E.eq_dec x y).
+ destruct (eq_nat_dec n p).
- left. split; assumption.
- right. intros [_ Habs]. contradiction.
+ right. intros [Habs _]. contradiction.
Qed.
Lemma elt_dec : forall xn yp, {eq_elt xn yp} + {~eq_elt xn yp}.
Proof. intros [? ?] [? ?]. apply E.eq_dec. Qed.
(** * Instances for rewriting **)
Existing Instance multiplicity_compat.
Existing Instance fold_compat.
(** ** [Subset] and [eq] are an order relation and the corresponding equivalence relation **)
Instance eq_equiv : Equivalence eq.
Proof. split.
intros m x. reflexivity.
intros m1 m2 H x. now symmetry.
intros m1 m2 m3 H1 H2 x. now transitivity (multiplicity x m2).
Qed.
Instance Subset_PreOrder : PreOrder Subset.
Proof. split; repeat intro. msetdec. etransitivity; eauto. Qed.
Instance Subset_PartialOrder : PartialOrder eq Subset.
Proof.
intros m1 m2. split; intro H.
- now split; intro x; rewrite H.
- destruct H. intro. now apply le_antisym.
Qed.
(** ** Compatibility with respect to [eq] **)
Instance InA_elt_compat : Proper (eq_elt ==> PermutationA eq_pair ==> iff) (InA eq_elt).
Proof.
intros ? ? ? ? ? Hperm. apply (InA_perm_compat _). assumption.
revert Hperm. apply PermutationA_subrelation_compat; trivial. apply subrelation_pair_elt.
Qed.
Instance In_compat : Proper (E.eq ==> eq ==> iff) In.
Proof. repeat intro. msetdec. Qed.
Instance is_empty_compat : Proper (eq ==> Logic.eq) is_empty.
Proof.
intros s1 s2 Heq. destruct (is_empty s2) eqn:Hs2.
+ msetdec.
+ destruct (is_empty s1) eqn:Hs1.
- rewrite <- Hs2. symmetry. msetdec.
- reflexivity.
Qed.
Instance add_compat : Proper (E.eq ==> Logic.eq ==> eq ==> eq) add.
Proof. repeat intro. msetdec. Qed.
Instance singleton_compat : Proper (E.eq ==> Logic.eq ==> eq) singleton.
Proof. repeat intro. msetdec. Qed.
Instance remove_compat : Proper (E.eq ==> Logic.eq ==> eq ==> eq) remove.
Proof. repeat intro. msetdec. Qed.
Instance union_compat : Proper (eq ==> eq ==> eq) union.
Proof. repeat intro. msetdec. Qed.
Instance inter_compat : Proper (eq ==> eq ==> eq) inter.
Proof. repeat intro. msetdec. Qed.
Instance diff_compat : Proper (eq ==> eq ==> eq) diff.
Proof. repeat intro. msetdec. Qed.
Instance lub_compat : Proper (eq ==> eq ==> eq) lub.
Proof. repeat intro. msetdec. Qed.
(* Instance subset_compat : Proper (eq ==> eq ==> iff) Subset.
Proof. intros m1 m1' Heq1 m2 m2' Heq2. now rewrite Heq1, Heq2. Qed.*)
Instance nfilter_compat f : compatb f -> Proper (eq ==> eq) (nfilter f).
Proof. repeat intro. msetdec. Qed.
Instance filter_compat f : Proper (E.eq ==> Logic.eq) f -> Proper (eq ==> eq) (filter f).
Proof. repeat intro. msetdec. Qed.
Instance npartition_compat f : compatb f -> Proper (eq ==> eq * eq) (npartition f).
Proof.
intros Hf s1 s2 Hs. split; intro x.
- msetdec.
- msetdec; repeat intro; now rewrite Hf.
Qed.
Instance partition_compat f : Proper (E.eq ==> Logic.eq) f -> Proper (eq ==> eq * eq) (partition f).
Proof.
intros Hf s1 s2 Hs. split; intro x.
- msetdec.
- msetdec; repeat intro; now rewrite Hf.
Qed.
Instance elements_compat : Proper (eq ==> PermutationA eq_pair) elements.
Proof.
intros s1 s2 Hs. apply NoDupA_equivlistA_PermutationA.
- now apply eq_pair_equiv.
- generalize (elements_NoDupA s1). apply NoDupA_strengthen. now intros [? ?] [? ?] [? _].
- generalize (elements_NoDupA s2). apply NoDupA_strengthen. now intros [? ?] [? ?] [? _].
- intros [x n]. do 2 rewrite elements_spec. now rewrite Hs.
Qed.
Instance support_compat : Proper (eq ==> PermutationA E.eq) support.
Proof.
intros s1 s2 Hs. assert (Equivalence E.eq) by auto with typeclass_instances.
apply NoDupA_equivlistA_PermutationA. assumption.
now apply support_NoDupA. now apply support_NoDupA.
intro x. do 2 rewrite support_spec. unfold In. now rewrite Hs.
Qed.
Instance size_compat : Proper (eq ==> Logic.eq) size.
Proof. intros s1 s2 Hs. do 2 rewrite size_spec. now rewrite Hs. Qed.
Instance for_all_compat : forall f, compatb f -> Proper (eq ==> Logic.eq) (for_all f).
Proof.
intros f Hf s1 s2 Hs. destruct (for_all f s2) eqn:Hs2.
+ rewrite for_all_spec in *; trivial. intros x Hin. rewrite Hs. apply Hs2. now rewrite <- Hs.
+ destruct (for_all f s1) eqn:Hs1.
- rewrite <- Hs2. symmetry. rewrite for_all_spec in *; trivial.
intros x Hin. rewrite <- Hs. apply Hs1. now rewrite Hs.
- reflexivity.
Qed.
Instance exists_compat : forall f, compatb f -> Proper (eq ==> Logic.eq) (exists_ f).
Proof.
intros f Hf s1 s2 Hs. destruct (exists_ f s2) eqn:Hs2.
+ rewrite exists_spec in *; trivial. destruct Hs2 as [x [Hin Hfx]]. exists x. now split; rewrite Hs.
+ destruct (exists_ f s1) eqn:Hs1.
- rewrite <- Hs2. symmetry. rewrite exists_spec in *; trivial.
destruct Hs1 as [x [Hin Hfx]]. exists x. now split; rewrite <- Hs.
- reflexivity.
Qed.
Instance For_all_compat : forall f, Proper (E.eq ==> Logic.eq ==> iff) f -> Proper (eq ==> iff) (For_all f).
Proof.
intros f Hf s1 s2 Hs. split; intros H x Hin.
- rewrite <- Hs. apply H. now rewrite Hs.
- rewrite Hs. apply H. now rewrite <- Hs.
Qed.
Instance Exists_compat : forall f, Proper (E.eq ==> Logic.eq ==> iff) f -> Proper (eq ==> iff) (Exists f).
Proof.
intros f Hf s1 s2 Hs. split; intros [x [Hin Hfx]].
- exists x. now split; rewrite <- Hs.
- exists x. now split; rewrite Hs.
Qed.
Instance cardinal_compat : Proper (eq ==> Logic.eq) cardinal.
Proof.
intros s1 s2 Hs. do 2 rewrite cardinal_spec, fold_spec. rewrite (fold_left_symmetry_PermutationA _ _).
- reflexivity.
- intros x ? ? [? ?] [? ?] [? Heq]. hnf in *. simpl in *. now subst.
- intros [? ?] [? ?] ?. omega.
- now rewrite Hs.
- reflexivity.
Qed.
(** ** Compatibility with respect to [Subset] **)
Instance In_sub_compat : Proper (E.eq ==> Subset ==> impl) In.
Proof. intros ? y ? ? ? Hs H. msetdec. specialize (Hs y). omega. Qed.
Instance add_sub_compat : Proper (E.eq ==> le ==> Subset ==> Subset) add.
Proof. repeat intro. msetdec. now apply plus_le_compat. Qed.
Instance singleton_sub_compat : Proper (E.eq ==> le ==> Subset) singleton.
Proof. repeat intro. msetdec. Qed.
Instance remove_sub_compat : Proper (E.eq ==> le --> Subset ==> Subset) remove.
Proof. intros ? y ? ? ? Hle ? ? Hsub ?. msetdec. specialize (Hsub y). compute in Hle. omega. Qed.
Instance union_sub_compat : Proper (Subset ==> Subset ==> Subset) union.
Proof. intros ? ? Hsub1 ? ? Hsub2 x. specialize (Hsub1 x). specialize (Hsub2 x). msetdec. Qed.
Instance inter_sub_compat : Proper (Subset ==> Subset ==> Subset) inter.
Proof.
intros ? ? Hsub1 ? ? Hsub2 x. specialize (Hsub1 x). specialize (Hsub2 x). msetdec.
apply Min.min_glb. now etransitivity; try apply Min.le_min_l. now etransitivity; try apply Min.le_min_r.
Qed.
Instance diff_sub_compat : Proper (Subset ==> Subset --> Subset) diff.
Proof. intros ? ? Hsub1 ? ? Hsub2 x. specialize (Hsub1 x). specialize (Hsub2 x). msetdec. Qed.
Instance lub_sub_compat : Proper (Subset ==> Subset ==> Subset) lub.
Proof.
intros ? ? Hsub1 ? ? Hsub2 x. specialize (Hsub1 x). specialize (Hsub2 x).
msetdec. now apply Nat.max_le_compat.
Qed.
Instance subset_sub_compat : Proper (Subset --> Subset ==> impl) Subset.
Proof. intros m1 m1' Heq1 m2 m2' Heq2 H. hnf in Heq1. repeat (etransitivity; try eassumption). Qed.
Instance support_sub_compat : Proper (Subset ==> inclA E.eq) support.
Proof. intros ? ? Hsub ? ?. rewrite support_spec in *. now rewrite <- Hsub. Qed.
Instance size_sub_compat : Proper (Subset ==> le) size.
Proof.
intros m1 m2 Hsub. do 2 rewrite size_spec. apply support_sub_compat in Hsub.
apply (inclA_length _); trivial. apply support_NoDupA.
Qed.
(*
(* Wrong if [f] is not monotonous in its second argument *)
Instance filter_sub_compat f : compatb f -> Proper (Subset ==> Subset) (filter f).
Proof. repeat intro. msetdec. Abort.
Instance partition_compat f : compatb f -> Proper (eq ==> eq * eq) (partition f).
Proof.
intros Hf s1 s2 Hs. split; intro x.
msetdec.
repeat rewrite partition_spec_snd; trivial. rewrite filter_compat; trivial. repeat intro. now rewrite Hf.
Qed.
Instance elements_compat : Proper (Subset ==> inclA eq_pair) elements.
Proof.
intros s1 s2 Hs. apply NoDupA_equivlistA_PermutationA.
now apply eq_pair_equiv.
generalize (elements_NoDupA s1). apply NoDupA_strengthen. now intros [? ?] [? ?] [? _].
generalize (elements_NoDupA s2). apply NoDupA_strengthen. now intros [? ?] [? ?] [? _].
intro x. do 2 rewrite elements_spec. now rewrite Hs.
Qed.
Instance for_all_sub_compat : forall f, compatb f -> Proper (Subset --> Bool.le) (for_all f).
Proof.
intros f Hf s1 s2 Hs. destruct (for_all f s2) eqn:Hs2.
rewrite for_all_spec in *; trivial. intros x Hin. rewrite Hs. apply Hs2. now rewrite <- Hs.
destruct (for_all f s1) eqn:Hs1.
rewrite <- Hs2. symmetry. rewrite for_all_spec in *; trivial.
intros x Hin. rewrite <- Hs. apply Hs1. now rewrite Hs.
reflexivity.
Qed.
Instance exists_compat : forall f, compatb f -> Proper (eq ==> Logic.eq) (exists_ f).
Proof.
intros f Hf s1 s2 Hs. destruct (exists_ f s2) eqn:Hs2.
rewrite exists_spec in *; trivial. destruct Hs2 as [x [Hin Hfx]]. exists x. now split; rewrite Hs.
destruct (exists_ f s1) eqn:Hs1.
rewrite <- Hs2. symmetry. rewrite exists_spec in *; trivial.
destruct Hs1 as [x [Hin Hfx]]. exists x. now split; rewrite <- Hs.
reflexivity.
Qed.
Instance For_all_sub_compat : forall f, compatb f -> Proper (Subset --> impl) (For_all f).
Proof.
intros f Hf s1 s2 Hs H x Hin.
rewrite <- Hs. apply H. now rewrite Hs.
rewrite Hs. apply H. now rewrite <- Hs.
Qed.
Instance Exists_compat : forall f, compatb f -> Proper (Subset ==> impl) (Exists f).
Proof.
intros f Hf s1 s2 Hs. split; intros [x [Hin Hfx]].
exists x. now split; rewrite <- Hs.
exists x. now split; rewrite Hs.
Qed.
*)
(** * Complementary results **)
Lemma eq_dec : forall m1 m2, {m1 [=] m2} + {~m1 [=] m2}.
Proof.
intros m1 m2. destruct (equal m1 m2) eqn:Heq.
- left. now rewrite <- equal_spec.
- right. intro Habs. rewrite <- equal_spec, Heq in Habs. discriminate.
Qed.
(** ** Results about [In] **)
Lemma not_In : forall x m, ~In x m <-> multiplicity x m = 0.
Proof. intros. msetdec. Qed.
Lemma In_dec : forall x m, {In x m} + {~In x m}.
Proof.
intros x m. destruct (multiplicity x m) eqn:Hn.
- right. msetdec.
- left. msetdec.
Qed.
(** ** Results about [empty] **)
Lemma empty_is_empty : is_empty empty = true.
Proof. rewrite is_empty_spec. reflexivity. Qed.
Lemma In_empty : forall x, ~In x empty.
Proof. intro. msetdec. Qed.
Lemma subset_empty_l : forall m, empty [<=] m.
Proof. repeat intro. msetdec. Qed.
Lemma subset_empty_r : forall m, m [<=] empty <-> m [=] empty.
Proof.
repeat intro. split; intro H.
- apply antisymmetry. assumption. apply subset_empty_l.
- now rewrite H.
Qed.
Lemma add_empty : forall x n, add x n empty [=] singleton x n.
Proof. repeat intro. msetdec. Qed.
Lemma remove_empty : forall x n, remove x n empty [=] empty.
Proof. repeat intro. msetdec. Qed.
Lemma elements_empty : elements empty = nil.
Proof.
destruct (elements empty) as [| [x n] l] eqn:Habs. reflexivity.
assert (Hin : InA eq_pair (x, n) ((x, n) :: l)) by now left.
rewrite <- Habs, elements_spec, empty_spec in Hin. omega.
Qed.
Corollary fold_empty : forall A f (i : A), fold f empty i = i.
Proof. intros A f i. now rewrite fold_spec, elements_empty. Qed.
Corollary cardinal_empty : cardinal empty = 0.
Proof. now rewrite cardinal_spec, fold_empty. Qed.
Corollary support_empty : support empty = nil.
Proof.
destruct (support empty) as [| e l] eqn:Habs. reflexivity.
cut (InA E.eq e (e :: l)). rewrite <- Habs, support_spec. unfold In. rewrite empty_spec. omega. now left.
Qed.
Corollary size_empty : size empty = 0.
Proof. now rewrite size_spec, support_empty. Qed.
Lemma nfilter_empty : forall f, compatb f -> nfilter f empty [=] empty.
Proof. repeat intro. msetdec. now destruct (f x 0). Qed.
Lemma filter_empty : forall f, Proper (E.eq ==> Logic.eq) f -> filter f empty [=] empty.
Proof. repeat intro. msetdec. now destruct (f x). Qed.
Lemma for_all_empty : forall f, compatb f -> for_all f empty = true.
Proof. repeat intro. msetdec. intro. msetdec. Qed.
Lemma exists_empty : forall f, compatb f -> exists_ f empty = false.
Proof.
intros. destruct (exists_ f empty) eqn:Habs; trivial.
rewrite exists_spec in Habs; trivial. destruct Habs. msetdec.
Qed.
Lemma npartition_empty_fst : forall f, compatb f -> fst (npartition f empty) [=] empty.
Proof. intros. msetdec. now apply nfilter_empty. Qed.
Lemma npartition_empty_snd : forall f, compatb f -> snd (npartition f empty) [=] empty.
Proof. intros f Hf. msetdec. apply nfilter_empty. repeat intro. now rewrite Hf. Qed.
Lemma partition_empty_fst : forall f, Proper (E.eq ==> Logic.eq) f -> fst (partition f empty) [=] empty.
Proof. intros. msetdec. now apply filter_empty. Qed.
Lemma partition_empty_snd : forall f, Proper (E.eq ==> Logic.eq) f -> snd (partition f empty) [=] empty.
Proof. intros f Hf. msetdec. apply filter_empty. repeat intro. now rewrite Hf. Qed.
Lemma choose_empty : choose empty = None.
Proof. rewrite choose_None. reflexivity. Qed.
(** ** Results about [singleton] **)
Lemma singleton_spec : forall x y n, multiplicity y (singleton x n) = if E.eq_dec y x then n else 0.
Proof. repeat intro. msetdec. Qed.
Lemma singleton_0 : forall x, singleton x 0 [=] empty.
Proof. repeat intro. msetdec. Qed.
Lemma subset_singleton_l : forall x n m, n <= multiplicity x m -> singleton x n [<=] m.
Proof. repeat intro. msetdec. Qed.
Lemma subset_singleton_r : forall x n m,
m [<=] singleton x n <-> multiplicity x m <= n /\ m [=] singleton x (multiplicity x m).
Proof.
repeat intro. split; intro H.
+ split.
- specialize (H x). msetdec.
- intro y. specialize (H y). msetdec.
+ intro y. destruct H as [H1 H2]. rewrite H2. clear H2. msetdec.
Qed.
Lemma singleton_empty : forall x n, singleton x n [=] empty <-> n = 0.
Proof.
intros x n. split; intro H.
+ destruct n. reflexivity. symmetry. rewrite <- (empty_spec x), <- H. msetdec.
+ subst. apply singleton_0.
Qed.
Lemma In_singleton : forall x n y, In y (singleton x n) <-> E.eq y x /\ n > 0.
Proof. intros. unfold In. rewrite singleton_spec. destruct (E.eq_dec y x); intuition. omega. Qed.
Lemma add_singleton_same : forall x n p, add x p (singleton x n) [=] singleton x (n + p).
Proof. repeat intro. msetdec. Qed.
Lemma add_singleton_other_comm : forall x y n p, p > 0 -> add y p (singleton x n) [=] add x n (singleton y p).
Proof. repeat intro. msetdec. Qed.
Lemma remove_singleton_same : forall x n p, remove x n (singleton x p) [=] singleton x (p - n).
Proof. repeat intro. msetdec. Qed.
Lemma remove_singleton_other : forall x y n p, ~E.eq y x -> remove y n (singleton x p) [=] singleton x p.
Proof. repeat intro. msetdec. Qed.
Theorem elements_singleton : forall x n, n > 0 -> eqlistA eq_pair (elements (singleton x n)) ((x, n) :: nil).
Proof.
intros x n Hn. apply (PermutationA_length1 _). apply (NoDupA_equivlistA_PermutationA _).
+ apply (NoDupA_strengthen (eqA' := eq_elt) _). apply elements_NoDupA.
+ apply NoDupA_singleton.
+ intros [y p]. rewrite elements_spec. simpl. split; intro H.
- destruct H as [H1 H2]. msetdec. now left.
- inversion_clear H.
compute in H0. destruct H0. msetdec.
now rewrite InA_nil in H0.
Qed.
Lemma singleton_injective : forall x y n p, n > 0 -> singleton x n [=] singleton y p -> E.eq x y /\ n = p.
Proof.
intros x y n p Hn Heq.
assert (p > 0) by (destruct p; try rewrite singleton_0, singleton_empty in Heq; omega).
assert (Hel := elements_singleton x Hn). apply eqlistA_PermutationA_subrelation in Hel.
rewrite Heq in Hel. apply (PermutationA_length1 _) in Hel. rewrite elements_singleton in Hel; trivial.
inversion_clear Hel. now destruct H0.
Qed.
Lemma fold_singleton : forall A eqA, Reflexive eqA -> forall f, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f ->
forall (acc : A) x n, n > 0 -> eqA (fold f (singleton x n) acc) (f x n acc).
Proof.
intros A eqA HeqA f Hf acc x n Hn. rewrite fold_spec.
change (f x n acc) with (fold_left (fun acc xn => f (fst xn) (snd xn) acc) ((x, n) :: nil) acc).
assert (Hf2 : Proper (eqA ==> eq_pair ==> eqA) (fun acc xn => f (fst xn) (snd xn) acc)).
{ intros ? ? Heq [y p] [z q] [Hxy Hnp]. simpl. apply Hf; assumption. }
apply (fold_left_compat Hf2); trivial. now apply elements_singleton.
Qed.
Lemma cardinal_singleton : forall x n, cardinal (singleton x n) = n.
Proof.
intros. destruct n.
- rewrite singleton_0. apply cardinal_empty.
- rewrite cardinal_spec, fold_singleton; omega || now repeat intro; subst.
Qed.
Lemma support_singleton : forall x n, n > 0 -> PermutationA E.eq (support (singleton x n)) (x :: nil).
Proof.
intros x n Hn. apply (NoDupA_equivlistA_PermutationA _).
+ apply support_NoDupA.
+ apply NoDupA_singleton.
+ intro. rewrite support_spec. unfold In. split; intro Hin.
- left. msetdec.
- inversion_clear Hin. msetdec. inversion H.
Qed.
Corollary size_singleton : forall x n, n > 0 -> size (singleton x n) = 1.
Proof. intros. now rewrite size_spec, support_singleton. Qed.
Lemma nfilter_singleton_true : forall f x n, compatb f -> n > 0 ->
(nfilter f (singleton x n) [=] singleton x n <-> f x n = true).
Proof.
intros f x n Hf Hn. split; intro H.
- specialize (H x). msetdec. destruct (f x n); reflexivity || omega.
- intro y. msetdec. now rewrite H. now destruct (f y 0).
Qed.
Lemma nfilter_singleton_false : forall f x n, compatb f -> n > 0 ->
(nfilter f (singleton x n) [=] empty <-> f x n = false).
Proof.
intros f x n Hf Hn. split; intro H.
- specialize (H x). msetdec. destruct (f x n); reflexivity || omega.
- intro y. msetdec. now rewrite H. now destruct (f y 0).
Qed.
Lemma filter_singleton_true : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(filter f (singleton x n) [=] singleton x n <-> f x = true).
Proof.
intros f x n Hf Hn. split; intro H.
- specialize (H x). msetdec. destruct (f x); reflexivity || omega.
- intro y. msetdec. now rewrite H. now destruct (f y).
Qed.
Lemma filter_singleton_false : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(filter f (singleton x n) [=] empty <-> f x = false).
Proof.
intros f x n Hf Hn. split; intro H.
- specialize (H x). msetdec. destruct (f x); reflexivity || omega.
- intro y. msetdec. now rewrite H. now destruct (f y).
Qed.
Lemma for_all_singleton_true : forall f x n, compatb f -> n > 0 ->
(for_all f (singleton x n) = true <-> f x n = true).
Proof.
intros f x n Hf Hn. rewrite for_all_spec; trivial. split; intro H.
- unfold For_all in H. setoid_rewrite In_singleton in H. specialize (H x). msetdec. now apply H.
- intro. msetdec.
Qed.
Lemma for_all_singleton_false : forall f x n, compatb f -> n > 0 ->
(for_all f (singleton x n) = false <-> f x n = false).
Proof.
intros. destruct (f x n) eqn:Hfxn.
- rewrite <- for_all_singleton_true in Hfxn; trivial. now rewrite Hfxn.
- destruct (for_all f (singleton x n)) eqn:Habs; try reflexivity.
rewrite for_all_singleton_true, Hfxn in Habs; trivial. discriminate Habs.
Qed.
Lemma for_all_exists_singleton : forall f x n, compatb f -> n > 0 ->
exists_ f (singleton x n) = for_all f (singleton x n).
Proof.
intros f x n Hf Hn. destruct (for_all f (singleton x n)) eqn:Hall.
+ rewrite for_all_singleton_true in Hall; trivial. rewrite exists_spec; trivial. exists x. msetdec.
+ rewrite for_all_singleton_false in Hall; trivial. destruct (exists_ f (singleton x n)) eqn:Hex; trivial.
rewrite exists_spec in Hex; trivial. destruct Hex as [y [Hin Hy]]. rewrite In_singleton in Hin.
destruct Hin. msetdec. rewrite Hy in Hall. discriminate Hall.
Qed.
Corollary exists_singleton_true : forall f x n, compatb f -> n > 0 ->
(exists_ f (singleton x n) = true <-> f x n = true).
Proof. intros. now rewrite for_all_exists_singleton, for_all_singleton_true. Qed.
Corollary exists_singleton_false : forall f x n, compatb f -> n > 0 ->
(exists_ f (singleton x n) = false <-> f x n = false).
Proof. intros. now rewrite for_all_exists_singleton, for_all_singleton_false. Qed.
Lemma npartition_singleton_true_fst : forall f x n, compatb f -> n > 0 ->
(fst (npartition f (singleton x n)) [=] singleton x n <-> f x n = true).
Proof. intros. msetdec. now rewrite nfilter_singleton_true. Qed.
Lemma npartition_singleton_true_snd : forall f x n, compatb f -> n > 0 ->
(snd (npartition f (singleton x n)) [=] empty <-> f x n = true).
Proof.
intros. msetdec. rewrite nfilter_singleton_false; trivial.
- apply negb_false_iff.
- intros ? ? Heq1 ? ? Heq2. now rewrite Heq1, Heq2.
Qed.
Lemma npartition_singleton_false_fst : forall f x n, compatb f -> n > 0 ->
(fst (npartition f (singleton x n)) [=] empty <-> f x n = false).
Proof. intros. msetdec. now rewrite nfilter_singleton_false. Qed.
Lemma npartition_singleton_false_snd : forall f x n, compatb f -> n > 0 ->
(snd (npartition f (singleton x n)) [=] singleton x n <-> f x n = false).
Proof.
intros. msetdec. rewrite nfilter_singleton_true; trivial.
- apply negb_true_iff.
- intros ? ? Heq1 ? ? Heq2. now rewrite Heq1, Heq2.
Qed.
Lemma partition_singleton_true_fst : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(fst (partition f (singleton x n)) [=] singleton x n <-> f x = true).
Proof. intros. msetdec. now rewrite filter_singleton_true. Qed.
Lemma partition_singleton_true_snd : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(snd (partition f (singleton x n)) [=] empty <-> f x = true).
Proof.
intros. msetdec. rewrite filter_singleton_false; trivial.
- apply negb_false_iff.
- intros ? ? Heq. now rewrite Heq.
Qed.
Lemma partition_singleton_false_fst : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(fst (partition f (singleton x n)) [=] empty <-> f x = false).
Proof. intros. msetdec. now rewrite filter_singleton_false. Qed.
Lemma partition_singleton_false_snd : forall f x n, Proper (E.eq ==> Logic.eq) f -> n > 0 ->
(snd (partition f (singleton x n)) [=] singleton x n <-> f x = false).
Proof.
intros. msetdec. rewrite filter_singleton_true; trivial.
- apply negb_true_iff.
- intros ? ? Heq. now rewrite Heq.
Qed.
Lemma choose_singleton : forall x n, n > 0 -> exists y, E.eq x y /\ choose (singleton x n) = Some y.
Proof.
intros x n Hn. destruct (choose (singleton x n)) eqn:Hx.
+ exists e. repeat split. apply choose_Some in Hx. rewrite In_singleton in Hx. now destruct Hx as [Hx _].
+ rewrite choose_None, singleton_empty in Hx. omega.
Qed.
(** ** Results about [add] **)
Lemma add_spec : forall x y n m,
multiplicity y (add x n m) = if E.eq_dec y x then multiplicity y m + n else multiplicity y m.
Proof. repeat intro. msetdec. Qed.
Lemma add_0 : forall x m, add x 0 m [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma add_comm : forall x1 x2 n1 n2 m, add x1 n1 (add x2 n2 m) [=] add x2 n2 (add x1 n1 m).
Proof. repeat intro. msetdec. Qed.
Lemma add_multiplicity_inf_bound : forall x n m, n <= multiplicity x (add x n m).
Proof. intros. msetdec. Qed.
Lemma add_disjoint : forall x n m, add x n m [=] add x (n + multiplicity x m) (remove x (multiplicity x m) m).
Proof. repeat intro. msetdec. Qed.
Lemma add_merge : forall x n p m, add x n (add x p m) [=] add x (n + p) m.
Proof. repeat intro. msetdec. Qed.
Lemma add_is_empty : forall x n m, add x n m [=] empty <-> n = 0 /\ m [=] empty.
Proof.
intros x n m. split; intro H.
+ split.
- specialize (H x). msetdec.
- intro y. specialize (H y). msetdec.
+ intro y. destruct H as [H1 H2]. specialize (H2 y). msetdec.
Qed.
Lemma add_is_singleton : forall x y n p m, add x n m [=] singleton y p -> m [=] singleton y (p - n).
Proof.
intros x y n p m Hadd z. destruct n.
+ rewrite add_0 in Hadd. now rewrite Hadd, <- minus_n_O.
+ assert (Heq := Hadd x). msetdec. rewrite <- (add_other y z (S n)), Hadd; trivial. msetdec.
Qed.
Lemma add_subset : forall x n m, m [<=] add x n m.
Proof. repeat intro. msetdec. Qed.
Lemma add_subset_remove : forall x n m1 m2, add x n m1 [<=] m2 -> m1 [<=] remove x n m2.
Proof. intros x n m1 m2 Hsub y. specialize (Hsub y). msetdec. Qed.
Lemma add_In : forall x y n m, In x (add y n m) <-> In x m \/ n <> 0 /\ E.eq x y.
Proof.
intros x y n m. unfold In. destruct (E.eq_dec x y) as [Heq | Heq].
- repeat rewrite (multiplicity_compat _ _ Heq _ _ (reflexivity _)). rewrite add_same. destruct n; intuition.
- rewrite add_other; trivial. intuition.
Qed.
Lemma add_injective : forall x n, injective eq eq (add x n).
Proof. intros ? ? ? ? Heq y. specialize (Heq y). msetdec. Qed.
(** ** Results about [remove] **)
Lemma remove_spec : forall x y n m,
multiplicity y (remove x n m) = if E.eq_dec y x then multiplicity y m - n else multiplicity y m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_0 : forall x m, remove x 0 m [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_out : forall x n m, ~In x m -> remove x n m [=] m.
Proof. unfold In. repeat intro. msetdec. Qed.
Lemma remove_comm : forall x1 x2 n1 n2 m, remove x1 n1 (remove x2 n2 m) [=] remove x2 n2 (remove x1 n1 m).
Proof. repeat intro. msetdec. Qed.
Lemma remove_merge : forall x n p m, remove x n (remove x p m) [=] remove x (n + p) m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_cap : forall x n m, multiplicity x m <= n -> remove x n m [=] remove x (multiplicity x m) m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_add_comm : forall x1 x2 n1 n2 m, ~E.eq x1 x2 ->
remove x1 n1 (add x2 n2 m) [=] add x2 n2 (remove x1 n1 m).
Proof. repeat intro. msetdec. Qed.
Lemma remove_add1 : forall x n p m, n <= p -> remove x n (add x p m) [=] add x (p - n) m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_add2 : forall x n p m, p <= n -> remove x n (add x p m) [=] remove x (n - p) m.
Proof. repeat intro. msetdec. Qed.
Corollary remove_add_cancel : forall x n m, remove x n (add x n m) [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma add_remove1 : forall x n p m, p <= multiplicity x m -> p <= n -> add x n (remove x p m) [=] add x (n - p) m.
Proof. repeat intro. msetdec. Qed.
Lemma add_remove2 : forall x n p m, multiplicity x m <= p -> multiplicity x m <= n ->
add x n (remove x p m) [=] add x (n - multiplicity x m) m.
Proof. repeat intro. msetdec. Qed.
Lemma add_remove3 : forall x n p m, n <= multiplicity x m <= p ->
add x n (remove x p m) [=] remove x (multiplicity x m - n) m.
Proof. repeat intro. msetdec. Qed.
Lemma add_remove4 : forall x n p m, n <= p <= multiplicity x m ->
add x n (remove x p m) [=] remove x (p - n) m.
Proof. repeat intro. msetdec. Qed.
Corollary add_remove_cancel : forall x n m, n <= multiplicity x m -> add x n (remove x n m) [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma add_remove_id : forall x n m, multiplicity x m <= n -> add x (multiplicity x m) (remove x n m) [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_is_empty : forall x n m,
remove x n m [=] empty <-> m [=] singleton x (multiplicity x m) /\ multiplicity x m <= n.
Proof.
intros x n m. split; intro H.
+ split.
- intro y. specialize (H y). msetdec.
- specialize (H x). msetdec.
+ destruct H as [H1 H2]. rewrite H1. intro y. destruct (E.eq_dec y x) as [Heq | Hneq].
- rewrite Heq, remove_same, singleton_spec, empty_spec. destruct (E.eq_dec x x); omega.
- rewrite remove_other, singleton_spec, empty_spec; trivial. now destruct (E.eq_dec y x).
Qed.
Lemma remove_is_singleton : forall n x m,
(exists p, remove x n m [=] singleton x p) <-> m [=] singleton x (multiplicity x m).
Proof.
intros n x m. split; intro H.
+ destruct H as [p H]. intro y. msetdec. erewrite <- remove_other. rewrite H. msetdec. assumption.
+ exists (multiplicity x m - n). intro y. rewrite H at 1. clear H. msetdec.
Qed.
Lemma remove_subset : forall x n m, remove x n m [<=] m.
Proof. repeat intro. msetdec. Qed.
Lemma remove_subset_add : forall x n m1 m2, remove x n m1 [<=] m2 -> m1 [<=] add x n m2.
Proof. intros x n m1 m2 Hsub y. specialize (Hsub y). msetdec. Qed.
Lemma remove_In : forall x y n m,
In x (remove y n m) <-> E.eq x y /\ n < multiplicity x m \/ ~E.eq x y /\ In x m.
Proof.
intros x y n m. unfold In. destruct (E.eq_dec x y) as [Heq | Heq].
+ repeat rewrite (multiplicity_compat _ _ Heq _ _ (reflexivity _)). rewrite remove_same. intuition.
+ rewrite remove_other; trivial. split; intro; try right; try intuition omega.
Qed.
Lemma remove_injective : forall x n m1 m2, n <= multiplicity x m1 -> n <= multiplicity x m2 ->
remove x n m1 [=] remove x n m2 -> m1 [=] m2.
Proof. intros x n m1 m2 Hm1 Hm2 Heq y. specialize (Heq y). msetdec. Qed.
(** ** Results about [union]. **)
Lemma union_empty_l : forall m, union empty m [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma union_empty_r : forall m, union m empty [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma union_comm : forall m1 m2, union m1 m2 [=] union m2 m1.
Proof. repeat intro. msetdec. Qed.
Lemma union_assoc : forall m1 m2 m3, union m1 (union m2 m3) [=] union (union m1 m2) m3.
Proof. repeat intro. msetdec. Qed.
Lemma add_union_singleton_l : forall x n m, union (singleton x n) m [=] add x n m.
Proof. repeat intro. msetdec. Qed.
Lemma add_union_singleton_r : forall x n m, union m (singleton x n) [=] add x n m.
Proof. repeat intro. msetdec. Qed.
Lemma union_add_comm_l : forall x n m1 m2, union (add x n m1) m2 [=] add x n (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma union_add_comm_r : forall x n m1 m2, union m1 (add x n m2) [=] add x n (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma union_remove_comm_l1 : forall x n m1 m2, n <= multiplicity x m1 ->
union (remove x n m1) m2 [=] remove x n (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma union_remove_comm_l2 : forall x n m1 m2, multiplicity x m1 <= n ->
union (remove x n m1) m2 [=] remove x (multiplicity x m1) (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma union_remove_comm_r1 : forall x n m1 m2, n <= multiplicity x m2 ->
union m1 (remove x n m2) [=] remove x n (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma union_remove_comm_r2 : forall x n m1 m2, multiplicity x m2 <= n ->
union m1 (remove x n m2) [=] remove x (multiplicity x m2) (union m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma empty_union : forall m1 m2, union m1 m2 [=] empty <-> m1 [=] empty /\ m2 [=] empty.
Proof.
intros m1 m2. split; intro H.
+ split; intro x; specialize (H x); msetdec.
+ intro. destruct H. msetdec.
Qed.
Lemma union_singleton : forall x n m1 m2, union m1 m2 [=] singleton x n <->
m1 [=] singleton x (multiplicity x m1) /\ m2 [=] singleton x (multiplicity x m2)
/\ n = multiplicity x m1 + multiplicity x m2.
Proof.
intros x n m1 m2. split; intro H.
+ repeat split.
- intro y. specialize (H y). msetdec.
- intro y. specialize (H y). msetdec.
- specialize (H x). msetdec.
+ destruct H as [H1 [H2 H3]]. intro y. subst n.
rewrite union_spec, H1, H2 at 1. repeat rewrite singleton_spec.
now destruct (E.eq_dec y x).
Qed.
Lemma union_In : forall x m1 m2, In x (union m1 m2) <-> In x m1 \/ In x m2.
Proof. intros. unfold In. rewrite union_spec. omega. Qed.
Lemma union_out : forall x m1 m2, ~In x (union m1 m2) <-> ~In x m1 /\ ~In x m2.
Proof. intros x m1 m2. rewrite union_In. tauto. Qed.
Lemma union_subset_l : forall m1 m2, m1 [<=] union m1 m2.
Proof. repeat intro. msetdec. Qed.
Lemma union_subset_r : forall m1 m2, m2 [<=] union m1 m2.
Proof. repeat intro. msetdec. Qed.
Lemma union_injective_l : forall m, injective eq eq (union m).
Proof. intros m1 m2 m3 Heq x. specialize (Heq x). msetdec. Qed.
Lemma union_injective_r : forall m, injective eq eq (fun m' => union m' m).
Proof. intros m1 m2 m3 Heq x. specialize (Heq x). msetdec. Qed.
(** ** Results about [inter] **)
Lemma inter_empty_l : forall m, inter empty m [=] empty.
Proof. repeat intro. msetdec. Qed.
Lemma inter_empty_r : forall m, inter m empty [=] empty.
Proof. repeat intro. msetdec. auto with arith. Qed.
Lemma inter_comm : forall m1 m2, inter m1 m2 [=] inter m2 m1.
Proof. repeat intro. msetdec. apply Min.min_comm. Qed.
Lemma inter_assoc : forall m1 m2 m3, inter m1 (inter m2 m3) [=] inter (inter m1 m2) m3.
Proof. repeat intro. msetdec. apply Min.min_assoc. Qed.
Lemma add_inter_distr : forall x n m1 m2, add x n (inter m1 m2) [=] inter (add x n m1) (add x n m2).
Proof. repeat intro. msetdec. symmetry. apply Min.plus_min_distr_r. Qed.
Lemma remove_inter_distr : forall x n m1 m2, remove x n (inter m1 m2) [=] inter (remove x n m1) (remove x n m2).
Proof. repeat intro. msetdec. symmetry. apply Nat.sub_min_distr_r. Qed.
Lemma inter_singleton_l : forall x n m, inter (singleton x n) m [=] singleton x (min n (multiplicity x m)).
Proof. intros x n m y. msetdec. Qed.
(*Qed.*)
Lemma inter_singleton_r : forall x n m, inter m (singleton x n) [=] singleton x (min n (multiplicity x m)).
Proof. intros. rewrite inter_comm. apply inter_singleton_l. Qed.
Lemma inter_is_singleton : forall x m1 m2,
(exists n, inter m1 m2 [=] singleton x n) <-> forall y, ~E.eq y x -> ~In y m1 \/ ~In y m2.
Proof.
intros x m1 m2. split; intro Hin.
* intros y Hy. destruct Hin as [n Hin]. destruct (multiplicity y m1) eqn:Hm1.
+ msetdec.
+ right. specialize (Hin y). msetdec. rewrite Hm1 in Hin. destruct (multiplicity y m2). omega. discriminate Hin.
* exists (min (multiplicity x m1) (multiplicity x m2)).
intro y. msetdec. apply Hin in n. destruct (multiplicity y m1).
+ msetdec.
+ destruct (multiplicity y m2); trivial. destruct n; omega.
Qed.
Lemma inter_In : forall x m1 m2, In x (inter m1 m2) <-> In x m1 /\ In x m2.
Proof. intros. unfold In. rewrite inter_spec. unfold gt. rewrite Nat.min_glb_lt_iff. omega. Qed.
Lemma inter_out : forall x m1 m2, ~In x (inter m1 m2) <-> ~In x m1 \/ ~In x m2.
Proof. intros x m1 m2. rewrite inter_In. destruct (In_dec x m1); intuition. Qed.
Lemma empty_inter : forall m1 m2, inter m1 m2 [=] empty <->
forall x, ~In x m1 /\ ~In x m2 \/ In x m1 /\ ~In x m2 \/ ~In x m1 /\ In x m2.
Proof.
intros m1 m2. split; intro Hin.
+ intro x. destruct (In_dec x m1) as [Hin1 | Hin1], (In_dec x m2) as [Hin2 | Hin2]; auto.
assert (Habs : In x (inter m1 m2)). { rewrite inter_In. auto. }
rewrite Hin in Habs. apply In_empty in Habs. elim Habs.
+ intro x. rewrite empty_spec, inter_spec. destruct (Hin x) as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]];
rewrite not_In in *; try rewrite Hin1; try rewrite Hin2; auto with arith.
Qed.
Lemma inter_add_l1 : forall x n m1 m2, n <= multiplicity x m2 ->
inter (add x n m1) m2 [=] add x n (inter m1 (remove x n m2)).
Proof. intros x n m1 m2 Hn. rewrite <- (add_remove_cancel Hn) at 1. symmetry. apply add_inter_distr. Qed.
Lemma inter_add_l2 : forall x n m1 m2, multiplicity x m2 <= n ->
inter (add x n m1) m2 [=] add x (multiplicity x m2) (inter m1 (remove x n m2)).
Proof.
intros x n m1 m2 Hn. assert (Heq : n = multiplicity x m2 + (n - multiplicity x m2)) by omega.
rewrite <- (add_remove_cancel (reflexivity (multiplicity x m2))) at 1.
rewrite Heq, <- add_merge, <- add_inter_distr. f_equiv. intro. msetdec. Psatz.lia. Qed.
Corollary inter_add_r1 : forall x n m1 m2, n <= multiplicity x m1 ->
inter m1 (add x n m2) [=] add x n (inter (remove x n m1) m2).
Proof. intros. setoid_rewrite inter_comm. now apply inter_add_l1. Qed.
Corollary inter_add_r2 : forall x n m1 m2, multiplicity x m1 <= n ->
inter m1 (add x n m2) [=] add x (multiplicity x m1) (inter (remove x n m1) m2).
Proof. intros. setoid_rewrite inter_comm. now apply inter_add_l2. Qed.
Lemma remove_inter_add_l : forall x n m1 m2, inter (remove x n m1) m2 [=] remove x n (inter m1 (add x n m2)).
Proof.
repeat intro. msetdec. rewrite <- Nat.sub_min_distr_r.
assert (Heq : multiplicity x m2 + n - n = multiplicity x m2) by omega. now rewrite Heq.
Qed.
Lemma remove_inter_add_r : forall x n m1 m2, inter m1 (remove x n m2) [=] remove x n (inter (add x n m1) m2).
Proof.
repeat intro. msetdec. rewrite <- Nat.sub_min_distr_r.
assert (Heq : multiplicity x m1 + n - n = multiplicity x m1) by omega. now rewrite Heq.
Qed.
Lemma inter_subset_l : forall m1 m2, inter m1 m2 [<=] m1.
Proof. repeat intro. msetdec. apply Min.le_min_l. Qed.
Lemma inter_subset_r : forall m1 m2, inter m1 m2 [<=] m2.
Proof. repeat intro. msetdec. apply Min.le_min_r. Qed.
Lemma inter_eq_subset_l : forall m1 m2, inter m1 m2 [=] m1 <-> m1 [<=] m2.
Proof.
intros. split; intros Hm y; specialize (Hm y); msetdec.
+ rewrite <- Hm. apply Min.le_min_r.
+ apply Min.min_l. assumption.
Qed.
Lemma inter_eq_subset_r : forall m1 m2, inter m1 m2 [=] m2 <-> m2 [<=] m1.
Proof. intros. rewrite inter_comm. apply inter_eq_subset_l. Qed.
(** ** Results about [diff] **)
Lemma diff_In : forall x m1 m2, In x (diff m1 m2) <-> multiplicity x m1 > multiplicity x m2.
Proof. intros. unfold In. rewrite diff_spec. omega. Qed.
Lemma diff_empty_l : forall m, diff empty m [=] empty.
Proof. repeat intro. msetdec. Qed.
Lemma diff_empty_r : forall m, diff m empty [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma diff_empty_subset : forall m1 m2, diff m1 m2 [=] empty <-> m1 [<=] m2.
Proof. intros. split; intros Hm y; specialize (Hm y); msetdec. Qed.
Corollary diff_refl : forall m, diff m m [=] empty.
Proof. repeat intro. msetdec. Qed.
Lemma diff_subset : forall m1 m2, diff m1 m2 [<=] m1.
Proof. repeat intro. msetdec. Qed.
Lemma diff_singleton_l : forall x n m, diff (singleton x n) m [=] singleton x (n - multiplicity x m).
Proof. repeat intro. msetdec. Qed.
Lemma diff_singleton_r : forall x n m, diff m (singleton x n) [=] remove x n m.
Proof. repeat intro. msetdec. Qed.
Lemma diff_singleton_subset : forall x n m1 m2, diff m1 m2 [=] singleton x n -> m1 [<=] add x n m2.
Proof. intros x n m1 m2 H y. specialize (H y). msetdec. Qed.
Lemma diff_add_r : forall x n m1 m2, n <= multiplicity x m2 -> multiplicity x m1 <= multiplicity x m2 ->
diff m1 (add x n m2) [=] remove x n (diff m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma diff_add_l1 : forall x n m1 m2, n <= multiplicity x m2 -> diff (add x n m1) m2 [=] diff m1 (remove x n m2).
Proof. repeat intro. msetdec. Qed.
Lemma diff_add_l2 : forall x n m1 m2, multiplicity x m2 <= n ->
diff (add x n m1) m2 [=] add x (n - multiplicity x m2) (diff m1 (remove x (multiplicity x m2) m2)).
Proof. repeat intro. msetdec. Qed.
Lemma diff_remove_l : forall x n m1 m2, diff (remove x n m1) m2 [=] remove x n (diff m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma diff_remove_r1 : forall x n m1 m2, multiplicity x m1 <= n -> multiplicity x m2 <= n ->
diff m1 (remove x n m2) [=] add x (min (multiplicity x m1) (multiplicity x m2)) (diff m1 m2).
Proof.
repeat intro. msetdec. destruct (le_lt_dec (multiplicity x m1) (multiplicity x m2)).
- rewrite min_l; omega.
- rewrite min_r; omega.
Qed.
Lemma diff_remove_r2 : forall x n m1 m2, n <= multiplicity x m1 -> multiplicity x m2 <= multiplicity x m1 ->
diff m1 (remove x n m2) [=] add x (min n (multiplicity x m2)) (diff m1 m2).
Proof.
repeat intro. msetdec. destruct (le_lt_dec n (multiplicity x m2)).
- rewrite min_l; omega.
- rewrite min_r; omega.
Qed.
Lemma diff_remove_r3 : forall x n m1 m2, n <= multiplicity x m2 -> multiplicity x m1 <= multiplicity x m2 ->
diff m1 (remove x n m2) [=] add x (n + multiplicity x m1 - multiplicity x m2) (diff m1 m2).
Proof. repeat intro. msetdec. Qed.
Lemma diff_union_comm_l : forall m1 m2 m3, m3 [<=] m1 -> union (diff m1 m3) m2 [=] diff (union m1 m2) m3.
Proof. intros ? ? ? Hle x. specialize (Hle x). msetdec. Qed.
Lemma diff_union_comm_r : forall m1 m2 m3, m3 [<=] m2 -> union m1 (diff m2 m3) [=] diff (union m1 m2) m3.
Proof. intros ? ? ? Hle x. specialize (Hle x). msetdec. Qed.
Lemma inter_diff_l : forall m1 m2 m3, inter (diff m1 m2) m3 [=] diff (inter m1 (union m2 m3)) m2.
Proof. repeat intro. msetdec. setoid_rewrite <- minus_plus at 5. rewrite Nat.sub_min_distr_r. reflexivity. Qed.
Lemma inter_diff_r : forall m1 m2 m3, inter m1 (diff m2 m3) [=] diff (inter m2 (union m1 m3)) m3.
Proof. intros. rewrite inter_comm, union_comm. apply inter_diff_l. Qed.
Lemma diff_inter_distr_l : forall m1 m2 m3, diff (inter m1 m2) m3 [=] inter (diff m1 m3) (diff m2 m3).
Proof. repeat intro. msetdec. rewrite Nat.sub_min_distr_r. reflexivity. Qed.
Lemma diff_inter_r : forall m1 m2 m3, diff m1 (inter m2 m3) [=] lub (diff m1 m2) (diff m1 m3).
Proof. repeat intro. msetdec. symmetry. apply Nat.sub_max_distr_l. Qed.
Lemma diff_inter : forall m1 m2, diff m1 m2 [=] diff m1 (inter m1 m2).
Proof. repeat intro. msetdec. apply Nat.min_case_strong; omega. Qed.
Corollary diff_disjoint : forall m1 m2, inter m1 m2 [=] empty -> diff m1 m2 [=] m1.
Proof. intros m1 m2 Hm. rewrite diff_inter, Hm. apply diff_empty_r. Qed.
Lemma diff_injective : forall m1 m2 m3, m3 [<=] m1 -> m3 [<=] m2 -> diff m1 m3 [=] diff m2 m3 -> m1 [=] m2.
Proof. intros ? ? ? Hle1 Hle2 Heq x. specialize (Heq x). specialize (Hle1 x). specialize (Hle2 x). msetdec. Qed.
Lemma inter_as_diff : forall m1 m2, inter m1 m2 [=] diff m1 (diff m1 m2).
Proof. intros ? ? ?. msetdec. apply Nat.min_case_strong; omega. Qed.
(** ** Results about [lub] **)
Lemma lub_In : forall x m1 m2, In x (lub m1 m2) <-> In x m1 \/ In x m2.
Proof. intros. unfold In, gt. rewrite lub_spec. apply Nat.max_lt_iff. Qed.
Lemma lub_comm : forall m1 m2, lub m1 m2 [=] lub m2 m1.
Proof. repeat intro. msetdec. apply Nat.max_comm. Qed.
Lemma lub_assoc : forall m1 m2 m3, lub m1 (lub m2 m3) [=] lub (lub m1 m2) m3.
Proof. repeat intro. msetdec. apply Nat.max_assoc. Qed.
Lemma lub_empty_l : forall m, lub empty m [=] m.
Proof. repeat intro. msetdec. Qed.
Lemma lub_empty_r : forall m, lub m empty [=] m.
Proof. intros. rewrite lub_comm. apply lub_empty_l. Qed.
Lemma lub_subset_l : forall m1 m2, m1 [<=] lub m1 m2.
Proof. repeat intro. msetdec. auto with arith. Qed.
Lemma lub_subset_r : forall m1 m2, m2 [<=] lub m1 m2.
Proof. repeat intro. msetdec. auto with arith. Qed.
Lemma lub_is_empty : forall m1 m2, lub m1 m2 [=] empty <-> m1 [=] empty /\ m2 [=] empty.
Proof.
intros m1 m2. split; intro H.
+ split; intro x; specialize (H x); msetdec;
destruct (multiplicity x m1), (multiplicity x m2); trivial; discriminate.
+ intro. destruct H. msetdec.
Qed.
Lemma lub_eq_l : forall m1 m2, lub m1 m2 [=] m2 <-> m1 [<=] m2.
Proof.
intros m1 m2. split; intro H.
- rewrite <- H. apply lub_subset_l.
- intro x. msetdec. apply Nat.max_r, H.
Qed.
Lemma subset_lub_r : forall m1 m2, m2 [<=] m1 <-> lub m1 m2 [=] m1.
Proof. intros. now rewrite lub_comm, lub_eq_l. Qed.
Lemma add_lub_distr : forall x n m1 m2, add x n (lub m1 m2) [=] lub (add x n m1) (add x n m2).
Proof. repeat intro. msetdec. symmetry. apply Nat.add_max_distr_r. Qed.
Lemma lub_add_l : forall x n m1 m2, lub (add x n m1) m2 [=] add x n (lub m1 (remove x n m2)).
Proof.
intros x n m1 m2 y. msetdec. destruct (le_lt_dec (multiplicity x m2) n).
+ replace (multiplicity x m2 - n) with 0 by omega. rewrite Nat.max_0_r. apply max_l. omega.
+ rewrite <- Nat.add_max_distr_r. now replace (multiplicity x m2 - n + n) with (multiplicity x m2) by omega.
Qed.
Lemma lub_add_r : forall x n m1 m2, lub m1 (add x n m2) [=] add x n (lub (remove x n m1) m2).
Proof. intros. setoid_rewrite lub_comm. apply lub_add_l. Qed.
Lemma lub_singleton_l : forall x n m, lub (singleton x n) m [=] add x (n - multiplicity x m) m.
Proof. repeat intro. msetdec. apply Max.max_case_strong; omega. Qed.
Lemma lub_singleton_r : forall x n m, lub m (singleton x n) [=] add x (n - multiplicity x m) m.
Proof. intros. rewrite lub_comm. apply lub_singleton_l. Qed.
Lemma lub_is_singleton : forall x n m1 m2, lub m1 m2 [=] singleton x n
<-> m1 [=] singleton x (multiplicity x m1) /\ m2 [=] singleton x (multiplicity x m2)
/\ n = max (multiplicity x m1) (multiplicity x m2).
Proof.
intros x n m1 m2. split; intro H.
+ repeat split; try intro y.
- specialize (H y). msetdec. destruct (multiplicity y m1), (multiplicity y m2); trivial; discriminate.
- specialize (H y). msetdec. destruct (multiplicity y m1), (multiplicity y m2); trivial; discriminate.
- specialize (H x). msetdec.
+ destruct H as [Hm1 [Hm2 Hn]]. rewrite Hm1, Hm2, lub_singleton_l, add_singleton_same.
intro. clear -Hn. msetdec. apply Max.max_case_strong; omega.
Qed.
Lemma remove_lub : forall x n m1 m2, remove x n (lub m1 m2) [=] lub (remove x n m1) (remove x n m2).
Proof. repeat intro. msetdec. symmetry. apply Nat.sub_max_distr_r. Qed.
Lemma lub_remove_l : forall x n m1 m2, lub (remove x n m1) m2 [=] remove x n (lub m1 (add x n m2)).
Proof. repeat intro. msetdec. rewrite <- Nat.sub_max_distr_r. f_equal. omega. Qed.
Lemma lub_remove_r : forall x n m1 m2, lub m1 (remove x n m2) [=] remove x n (lub (add x n m1) m2).
Proof. intros. setoid_rewrite lub_comm. apply lub_remove_l. Qed.
Lemma union_lub_distr_l : forall m1 m2 m3, union (lub m1 m2) m3 [=] lub (union m1 m3) (union m2 m3).
Proof. repeat intro. msetdec. symmetry. apply Nat.add_max_distr_r. Qed.
Lemma union_lub_distr_r : forall m1 m2 m3, union m1 (lub m2 m3) [=] lub (union m1 m2) (union m1 m3).
Proof. repeat intro. msetdec. symmetry. apply Nat.add_max_distr_l. Qed.
Lemma lub_union_l : forall m1 m2 m3, lub (union m1 m2) m3 [=] union m1 (lub m2 (diff m3 m1)).
Proof.
repeat intro. msetdec. rewrite <- Nat.add_max_distr_l.
destruct (le_lt_dec (multiplicity x m1) (multiplicity x m3)).
+ now replace (multiplicity x m1 + (multiplicity x m3 - multiplicity x m1)) with (multiplicity x m3) by omega.
+ replace (multiplicity x m1 + (multiplicity x m3 - multiplicity x m1)) with (multiplicity x m1) by omega.
repeat rewrite max_l; omega.
Qed.
Lemma lub_union_r : forall m1 m2 m3, lub m1 (union m2 m3) [=] union (lub m3 (diff m1 m2)) m2.
Proof. intros. rewrite lub_comm. setoid_rewrite union_comm at 2. apply lub_union_l. Qed.
Lemma lub_inter_distr_l : forall m1 m2 m3, lub m1 (inter m2 m3) [=] inter (lub m1 m2) (lub m1 m3).
Proof. repeat intro. msetdec. apply Nat.max_min_distr. Qed.
Lemma lub_inter_distr_r : forall m1 m2 m3, lub (inter m1 m2) m3 [=] inter (lub m1 m3) (lub m2 m3).
Proof. intros. setoid_rewrite lub_comm. apply lub_inter_distr_l. Qed.
Lemma inter_lub_distr_l : forall m1 m2 m3, inter m1 (lub m2 m3) [=] lub (inter m1 m2) (inter m1 m3).
Proof. repeat intro. msetdec. apply Nat.min_max_distr. Qed.
Lemma inter_lub_distr_r : forall m1 m2 m3, inter (lub m1 m2) m3 [=] lub (inter m1 m3) (inter m2 m3).
Proof. intros. setoid_rewrite inter_comm. apply inter_lub_distr_l. Qed.
Lemma lub_diff_l : forall m1 m2 m3, lub (diff m1 m2) m3 [=] diff (lub m1 (union m2 m3)) m2.
Proof. repeat intro. msetdec. rewrite <- Nat.sub_max_distr_r, minus_plus. reflexivity. Qed.
Lemma lub_diff_r : forall m1 m2 m3, lub m1 (diff m2 m3) [=] diff (lub (union m1 m3) m2) m3.
Proof. intros. setoid_rewrite lub_comm. rewrite union_comm. apply lub_diff_l. Qed.
Lemma diff_lub_distr_r : forall m1 m2 m3, diff (lub m1 m2) m3 [=] lub (diff m1 m3) (diff m2 m3).
Proof. repeat intro. msetdec. symmetry. apply Nat.sub_max_distr_r. Qed.
Lemma diff_lub_l : forall m1 m2 m3, diff m1 (lub m2 m3) [=] inter (diff m1 m2) (diff m1 m3).
Proof. repeat intro. msetdec. symmetry. apply Nat.sub_min_distr_l. Qed.
Lemma lub_subset_union : forall m1 m2, lub m1 m2 [<=] union m1 m2.
Proof. intros m1 m2 ?. msetdec. Psatz.lia. Qed.
(** ** Results about [elements] **)
Lemma elements_pos : forall x n m, InA eq_pair (x, n) (elements m) -> n > 0.
Proof. intros x n m Hin. now rewrite elements_spec in Hin. Qed.
Theorem elements_In : forall x n m, InA eq_elt (x, n) (elements m) <-> In x m.
Proof.
intros x n m. split; intro H.
+ apply InA_elt_pair in H. destruct H as [p Hp]. simpl in Hp. rewrite elements_spec in Hp.
destruct Hp as [Heq Hpos]. unfold In. simpl in *. now subst.
+ apply InA_pair_elt with (multiplicity x m). rewrite elements_spec. split; trivial.
Qed.
Lemma elements_elt_strengthen : forall x n m,
InA eq_elt (x, n) (elements m) -> InA eq_pair (x, multiplicity x m) (elements m).
Proof. intros ? ? ? Hin. rewrite elements_spec. simpl. rewrite elements_In in Hin. intuition. Qed.
Theorem elements_eq_equiv : forall mβ mβ, equivlistA eq_pair (elements mβ) (elements mβ) <-> mβ [=] mβ.
Proof.
intros mβ mβ. split; intro H.
+ assert (Hdupβ := NoDupA_strengthen subrelation_pair_elt (elements_NoDupA mβ)).
assert (Hdupβ := NoDupA_strengthen subrelation_pair_elt (elements_NoDupA mβ)).
apply (NoDupA_equivlistA_PermutationA _) in H; trivial. clear Hdupβ Hdupβ.
intro x. destruct (multiplicity x mβ) eqn:Hmβ.
- assert (Hin : forall n, ~InA eq_pair (x, n) (elements mβ)).
{ intros n Habs. rewrite elements_spec in Habs. destruct Habs as [Heq Habs]. simpl in *. omega. }
destruct (multiplicity x mβ) eqn:Hmβ. reflexivity.
specialize (Hin (S n)). rewrite <- H in Hin. rewrite elements_spec in Hin.
elim Hin. split; simpl. assumption. omega.
- assert (Hin : InA eq_pair (x, S n) (elements mβ)). { rewrite elements_spec. split; simpl. assumption. omega. }
rewrite <- H in Hin. rewrite elements_spec in Hin. now destruct Hin.
+ intros [x n]. now rewrite H.
Qed.
Corollary elements_eq : forall mβ mβ, PermutationA eq_pair (elements mβ) (elements mβ) <-> mβ [=] mβ.
Proof.
intros mβ mβ. rewrite <- elements_eq_equiv. split; intro H.
- now apply (PermutationA_equivlistA _).
- apply (NoDupA_equivlistA_PermutationA _); trivial; apply (NoDupA_strengthen _ (elements_NoDupA _)).
Qed.
Lemma elements_pair_subset : forall x n mβ mβ,
mβ [<=] mβ -> InA eq_pair (x, n) (elements mβ) -> exists n', n <= n' /\ InA eq_pair (x, n') (elements mβ).
Proof.
intros x n mβ mβ Hm. setoid_rewrite elements_spec. simpl. intros [Heq Hpos].
exists (multiplicity x mβ); repeat split.
- rewrite <- Heq. apply Hm.
- specialize (Hm x). omega.
Qed.
Lemma elements_elt_subset : forall xn mβ mβ,
mβ [<=] mβ -> InA eq_elt xn (elements mβ) -> InA eq_elt xn (elements mβ).
Proof. intros [? ?] * ?. do 2 rewrite elements_In. now apply In_sub_compat. Qed.
Lemma elements_nil : forall m, elements m = nil <-> m [=] empty.
Proof.
intro m. split; intro H.
- unfold elements in H. intro x.
assert (~multiplicity x m > 0).
{ intro Habs. apply (conj (eq_refl (multiplicity x m))) in Habs.
change x with (fst (x, multiplicity x m)) in Habs at 1.
change (multiplicity x m) with (snd (x, multiplicity x m)) in Habs at 2 3.
rewrite <- M.elements_spec in Habs. rewrite H in Habs. now rewrite InA_nil in Habs. }
rewrite empty_spec. omega.
- apply (@PermutationA_nil _ eq_pair _). now rewrite H, elements_empty.
Qed.
Lemma elements_add : forall x y n p m, InA eq_pair (x, n) (elements (add y p m))
<-> E.eq x y /\ n = p + multiplicity y m /\ n > 0
\/ ~E.eq x y /\ InA eq_pair (x, n) (elements m).
Proof.
intros x y n p m. rewrite elements_spec. simpl. split; intro H.
+ destruct H as [H1 H2]. destruct (E.eq_dec x y) as [Heq | Hneq].
- left. repeat split; try assumption. subst n. rewrite <- Heq. rewrite add_same. apply plus_comm.
- right. split. assumption. rewrite elements_spec. rewrite add_other in H1. simpl. now split. auto.
+ destruct H as [[H1 [H2 H3]] | [H1 H2]].
- rewrite H1, add_same. split; omega.
- rewrite elements_spec in H2. destruct H2. simpl in *. rewrite add_other. now split. auto.
Qed.
Lemma elements_remove : forall x y n p m, InA eq_pair (x, n) (elements (remove y p m))
<-> E.eq x y /\ n = multiplicity y m - p /\ n > 0
\/ ~E.eq x y /\ InA eq_pair (x, n) (elements m).
Proof.
intros x y n p m. rewrite elements_spec. simpl. split; intro H.
+ destruct H as [H1 H2]. destruct (E.eq_dec x y) as [Heq | Hneq].
- left. repeat split; try assumption. now rewrite Heq, remove_same in H1.
- right. split. assumption. rewrite elements_spec. rewrite remove_other in H1; auto.
+ destruct H as [[H1 [H2 H3]] | [H1 H2]].
- rewrite H1, remove_same. now split.
- rewrite elements_spec in H2. destruct H2. simpl in *. rewrite remove_other; trivial. now split.
Qed.
Lemma elements_union : forall x n mβ mβ, InA eq_pair (x, n) (elements (union mβ mβ))
<-> (In x mβ \/ In x mβ) /\ n = multiplicity x mβ + multiplicity x mβ.
Proof.
intros x n mβ mβ. rewrite elements_spec, union_spec. simpl. unfold In.
split; intros [Heq Hpos]; split; now symmetry || omega.
Qed.
Lemma elements_inter : forall x n mβ mβ, InA eq_pair (x, n) (elements (inter mβ mβ))
<-> (In x mβ /\ In x mβ) /\ n = min (multiplicity x mβ) (multiplicity x mβ).
Proof.
intros x n mβ mβ. rewrite elements_spec, inter_spec. unfold In. simpl.
split; intros [Heq Hpos]; split; try (now symmetry).
- rewrite <- Heq in *. unfold gt in *. now rewrite Nat.min_glb_lt_iff in *.
- rewrite Hpos. unfold gt in *. now rewrite Nat.min_glb_lt_iff in *.
Qed.
Lemma elements_diff : forall x n mβ mβ, InA eq_pair (x, n) (elements (diff mβ mβ))
<-> n = multiplicity x mβ - multiplicity x mβ /\ n > 0.
Proof. intros. rewrite elements_spec, diff_spec. intuition. Qed.
Lemma elements_lub : forall x n mβ mβ, InA eq_pair (x, n) (elements (lub mβ mβ))
<-> (In x mβ \/ In x mβ) /\ n = max (multiplicity x mβ) (multiplicity x mβ).
Proof.
intros x n mβ mβ. rewrite elements_spec, lub_spec. unfold In. simpl.
split; intros [Heq Hpos]; split; try (now symmetry).
- rewrite <- Heq in *. unfold gt in *. now rewrite Nat.max_lt_iff in *.
- rewrite Hpos. unfold gt in *. now rewrite Nat.max_lt_iff in *.
Qed.
Lemma support_elements : forall x m, InA E.eq x (support m) <-> InA eq_pair (x, multiplicity x m) (elements m).
Proof. intros. rewrite support_spec, elements_spec. simpl. msetdec. Qed.
Lemma elements_add_out : forall x n m, n > 0 -> ~In x m ->
PermutationA eq_pair (elements (add x n m)) ((x, n) :: elements m).
Proof.
intros x n m Hn Hin. apply (NoDupA_equivlistA_PermutationA _).
* apply (NoDupA_strengthen _ (elements_NoDupA _)).
* constructor.
+ rewrite elements_spec. simpl. intros [H1 H2]. apply Hin. unfold In. omega.
+ apply (NoDupA_strengthen _ (elements_NoDupA _)).
* intros [y p]. rewrite elements_add. split; intro H.
+ destruct H as [[H1 [H2 Hpos]] | [H1 H2]]; simpl in *.
- unfold In in Hin. left. split. assumption. compute. omega.
- now right.
+ simpl. inversion_clear H.
- destruct H0 as [H1 H2]. compute in H1, H2. left. subst. unfold In in Hin. repeat split; trivial. omega.
- right. split; trivial. intro Habs. apply Hin. rewrite <- Habs. rewrite <- support_spec, support_elements.
assert (H1 := H0). rewrite elements_spec in H1. destruct H1 as [H1 _]. simpl in H1. now subst.
Qed.
Lemma elements_remove1 : forall x n m, multiplicity x m <= n ->
PermutationA eq_pair (elements (remove x n m)) (removeA pair_dec (x, multiplicity x m) (elements m)).
Proof.
intros x n m Hn. apply (NoDupA_equivlistA_PermutationA _).
+ apply (NoDupA_strengthen _ (elements_NoDupA _)).
+ apply removeA_NoDupA; refine _. apply (NoDupA_strengthen _ (elements_NoDupA _)).
+ intros [y p]. rewrite removeA_InA_iff; refine _. rewrite elements_remove, elements_spec. simpl. intuition.
- destruct H1. contradiction.
- destruct (E.eq_dec y x) as [Heq | Heq]; auto. elim H1. split; msetdec.
Qed.
Lemma elements_remove2 : forall x n m, n < multiplicity x m ->
PermutationA eq_pair (elements (remove x n m))
((x, multiplicity x m - n) :: removeA elt_dec (x, multiplicity x m) (elements m)).
Proof.
intros x n m Hn. apply (NoDupA_equivlistA_PermutationA _).
+ apply (NoDupA_strengthen _ (elements_NoDupA _)).
+ constructor.
- intro Habs. eapply InA_pair_elt in Habs. rewrite removeA_InA_iff in Habs; refine _.
destruct Habs as [_ Habs]. now elim Habs.
- eapply (NoDupA_strengthen subrelation_pair_elt). apply removeA_NoDupA, elements_NoDupA; refine _.
+ intros [y p]. rewrite elements_remove, elements_spec. simpl. intuition.
- rewrite H. left. split. compute. reflexivity. assumption.
- right. rewrite removeA_InA_iff_strong; refine _. split; trivial.
rewrite elements_spec. auto.
- { destruct (E.eq_dec y x) as [Heq | Heq].
+ inversion_clear H.
- left. destruct H0. repeat split; auto. hnf in *. simpl in *. omega.
- apply (InA_pair_elt (multiplicity x m)) in H0. rewrite Heq, removeA_InA in H0; refine _.
destruct H0 as [_ Habs]. elim Habs. reflexivity.
+ right. split; trivial. inversion_clear H.
- elim Heq. destruct H0. auto.
- apply removeA_InA_weak in H0. rewrite elements_spec in H0. assumption. }
Qed.
(** [is_elements] tests wether a given list can be the elements of a multiset **)
Definition is_elements l := NoDupA eq_elt l /\ Forall (fun xn => snd xn > 0) l.
Lemma is_elements_nil : is_elements nil.
Proof. split; constructor. Qed.
Lemma is_elements_cons : forall xn l, is_elements l -> ~InA eq_elt xn l -> snd xn > 0 -> is_elements (xn :: l).
Proof.
unfold is_elements. setoid_rewrite Forall_forall. intros xn l [Hdup Hpos] Hx Hn. split.
- now constructor.
- intros [y p] Hin. inversion_clear Hin.
inversion H. now subst.
now apply Hpos.
Qed.
Lemma is_elements_cons_inv : forall xn l, is_elements (xn :: l) -> is_elements l.
Proof. intros xn l [Hdup Hpos]. inversion_clear Hpos. inversion_clear Hdup. now split. Qed.
Lemma elements_is_elements : forall m, is_elements (elements m).
Proof.
intro m. split.
- now apply elements_NoDupA.
- rewrite Forall_forall. intros [x n] Hx. apply (@elements_pos x n m). now apply (In_InA _).
Qed.
Instance is_elements_compat : Proper (PermutationA eq_pair ==> iff) is_elements.
Proof.
intros l1 l2 perm. induction perm as [| [x n] [y p] ? ? [Heq1 Heq2] | x y l | l1 l2 l3].
+ reflexivity.
+ compute in Heq1, Heq2. subst p. split; intros [Hdup Hpos]; inversion_clear Hdup; inversion_clear Hpos.
- apply is_elements_cons.
apply IHperm. now split.
now rewrite perm, Heq1 in H.
assumption.
- apply is_elements_cons.
apply IHperm. now split.
now rewrite perm, Heq1.
assumption.
+ split; intros [Hdup Hpos]; inversion_clear Hdup; inversion_clear Hpos;
inversion_clear H0; inversion_clear H2; repeat apply is_elements_cons; trivial.
- now split.
- intro Habs. apply H. now right.
- intros Habs. inversion_clear Habs.
apply H. now left.
contradiction.
- now split.
- intro Habs. apply H. now right.
- intros Habs. inversion_clear Habs.
apply H. now left.
contradiction.
+ now rewrite IHperm1.
Qed.
Theorem is_elements_spec : forall l, is_elements l <-> exists m, PermutationA eq_pair l (elements m).
Proof.
induction l as [| [x n] l].
+ split; intro H.
- exists empty. now rewrite elements_empty.
- apply is_elements_nil.
+ split; intro H.
- destruct H as [Hdup Hpos].
assert (Hel : is_elements l). { split. now inversion_clear Hdup. now inversion_clear Hpos. }
rewrite IHl in Hel. destruct Hel as [m Hm]. exists (add x n m). symmetry. rewrite Hm. apply elements_add_out.
now inversion_clear Hpos.
inversion_clear Hdup. rewrite <- support_spec, support_elements. intro Habs. apply H.
rewrite <- Hm in Habs. eapply InA_pair_elt. apply Habs.
- destruct H as [m Hperm]. rewrite Hperm. apply elements_is_elements.
Qed.
(** A variant of the previous theorem, but with conclusion in [Type] rather than [Prop]. **)
Theorem is_elements_build : forall l, is_elements l -> {m | PermutationA eq_pair l (elements m)}.
Proof.
induction l as [| [x n] l]; intro H.
+ exists empty. now rewrite elements_empty.
+ destruct H as [Hdup Hpos].
assert (Hel : is_elements l). { split. now inversion_clear Hdup. now inversion_clear Hpos. }
apply IHl in Hel. destruct Hel as [m Hm]. exists (add x n m). symmetry. rewrite Hm. apply elements_add_out.
- now inversion_clear Hpos.
- inversion_clear Hdup. rewrite <- support_spec, support_elements. intro Habs. apply H.
rewrite <- Hm in Habs. eapply InA_pair_elt. apply Habs.
Defined.
(** [from_elements] builds back a multiset from its elements **)
Fixpoint from_elements l := (* List.fold_left (fun acc xn => add (fst xn) (snd xn) acc) l empty. *)
match l with
| nil => empty
| (x, n) :: l => add x n (from_elements l)
end.
Instance from_elements_compat : Proper (PermutationA eq_pair ==> eq) from_elements.
Proof.
intros l1 l2 perm. induction perm as [| [x n] [y p] ? ? [Hxy Hnp] | [x n] [y p] |]; simpl.
+ reflexivity.
+ intro z. compute in Hxy, Hnp. now rewrite Hxy, Hnp, IHperm.
+ apply add_comm.
+ now transitivity (from_elements lβ).
Qed.
Lemma from_elements_nil : from_elements nil = empty.
Proof. reflexivity. Qed.
Lemma from_elements_cons : forall x n l, from_elements ((x, n) :: l) = add x n (from_elements l).
Proof. reflexivity. Qed.
Lemma from_elements_valid_empty : forall l, is_elements l -> from_elements l [=] empty <-> l = nil.
Proof.
intros [| [x n] l] Hl; simpl.
- intuition.
- destruct Hl as [_ Hl]. inversion_clear Hl. simpl in *. rewrite add_is_empty. intuition (omega || discriminate).
Qed.
Lemma from_elements_empty : forall l, from_elements l [=] empty <-> Forall (fun xn => snd xn = 0) l.
Proof.
induction l as [| [x n] l]; simpl.
+ intuition.
+ split; intro Hl; rewrite add_is_empty, IHl in *; inversion_clear Hl; intuition.
Qed.
Lemma from_elements_singleton : forall x n l, is_elements l -> n > 0 ->
from_elements l [=] singleton x n <-> eqlistA eq_pair l ((x, n) :: nil).
Proof.
intros x n l Hl Hn. destruct l as [| [y p] [| [z q] l]]; simpl.
+ split; intro Hin.
- symmetry in Hin. rewrite singleton_empty in Hin. omega.
- inversion_clear Hin.
+ rewrite add_empty. split; intro Heq.
- symmetry in Heq. apply singleton_injective in Heq; trivial. destruct Heq. now repeat constructor.
- inversion_clear Heq. compute in H. destruct H as [Heq1 Heq2]. now rewrite Heq1, Heq2.
+ split; intro Hin.
- assert (Heq : E.eq y x /\ E.eq z x).
{ split.
+ specialize (Hin y). msetdec. destruct Hl as [_ Hl]. inversion_clear Hl. simpl in *. omega.
+ apply add_is_singleton in Hin. specialize (Hin z). msetdec. destruct Hl as [_ Hl].
inversion_clear Hl. inversion_clear H0. simpl in *. omega. }
destruct Heq as [Heq1 Heq2]. destruct Hl as [Hl _]. inversion_clear Hl.
elim H. left. compute. now transitivity x.
- inversion_clear Hin. inversion_clear H0.
Qed.
Lemma from_elements_out : forall x n l, ~InA eq_elt (x, n) l -> multiplicity x (from_elements l) = 0.
Proof.
intros x n l Hin. induction l as [| [y p] l]; simpl.
+ apply empty_spec.
+ rewrite add_other.
apply IHl. intro Habs. apply Hin. now right.
intro Habs. apply Hin. now left.
Qed.
Lemma from_elements_in : forall x n l,
NoDupA eq_elt l -> InA eq_pair (x, n) l -> multiplicity x (from_elements l) = n.
Proof.
intros x n l Hl Hin. induction l as [| [y p] l].
+ rewrite InA_nil in Hin. elim Hin.
+ simpl. inversion_clear Hin.
- destruct H as [Hx Hn]. compute in Hx, Hn. inversion Hl. now rewrite Hx, add_same, (@from_elements_out y p).
- inversion_clear Hl. rewrite add_other. now apply IHl.
intro Habs. apply H0. apply InA_pair_elt with n. now rewrite <- Habs.
Qed.
Lemma from_elements_elements : forall m, from_elements (elements m) [=] m.
Proof.
intros m x. destruct (multiplicity x m) eqn:Hn.
- apply from_elements_out with 0. intro Habs. apply InA_elt_pair in Habs.
destruct Habs as [n Habs]. rewrite elements_spec in Habs. simpl in Habs. omega.
- apply from_elements_in. apply elements_NoDupA. rewrite elements_spec. simpl. omega.
Qed.
Lemma elements_from_elements : forall l, is_elements l -> PermutationA eq_pair (elements (from_elements l)) l.
Proof.
intros l Hl. rewrite is_elements_spec in Hl. destruct Hl as [m Hm]. now rewrite Hm, from_elements_elements.
Qed.
Lemma elements_injective : forall m1 m2, PermutationA eq_pair (elements m1) (elements m2) -> m1 [=] m2.
Proof. intros. setoid_rewrite <- from_elements_elements. now f_equiv. Qed.
Lemma from_elements_injective : forall l1 l2, is_elements l1 -> is_elements l2 ->
from_elements l1 [=] from_elements l2 -> PermutationA eq_pair l1 l2.
Proof. intros. setoid_rewrite <- elements_from_elements; trivial. now f_equiv. Qed.
(* If [l] contains duplicates of [x], we need to sum all their contribution. *)
Theorem from_elements_spec : forall x n l, multiplicity x (from_elements l) = n <->
List.fold_left (fun acc yp => if E.eq_dec (fst yp) x then (snd yp) + acc else acc) l 0 = n.
Proof.
intros x n l. rewrite <- Nat.add_0_r at 1. generalize 0. revert n. induction l as [| [y p] l]; intros n q; simpl.
+ msetdec.
+ destruct (E.eq_dec y x) as [Heq | Heq].
- rewrite Heq, add_same, <- Nat.add_assoc, IHl. reflexivity.
- rewrite add_other; msetdec.
Qed.
Lemma from_elements_In : forall l x, In x (from_elements l) <-> exists n, InA eq_pair (x, n) l /\ n > 0.
Proof.
intros l x. induction l as [| [y p] l].
* simpl. split; intro Hin.
+ elim (In_empty Hin).
+ destruct Hin as [? [Hin _]]. rewrite InA_nil in Hin. elim Hin.
* simpl. rewrite add_In, IHl; trivial. split; intros Hin.
+ destruct Hin as [[n [Hin Hn]] | [? Heq]].
- exists n. split; trivial. now right.
- exists p. split; try (left; split); auto; omega.
+ destruct Hin as [n [Hin Hn]]. inversion_clear Hin.
- destruct H. right. compute in *. split; trivial. omega.
- left. exists n. now split.
Qed.
Corollary from_elements_In_valid : forall x l, is_elements l ->
In x (from_elements l) <-> forall n, InA eq_elt (x, n) l.
Proof.
intros x l Hl. rewrite from_elements_In. split; intro Hin.
+ destruct Hin as [n [Hin Hn]]. intro m. revert Hin. apply InA_pair_elt.
+ specialize (Hin 0). apply InA_elt_pair in Hin. destruct Hin as [n Hin].
exists n. split; trivial. destruct Hl as [_ Hl]. rewrite Forall_forall in Hl.
rewrite InA_alt in Hin. destruct Hin as [[y p] [[Heq Hnp] Hin]].
compute in Hnp. subst. change p with (snd (y, p)). now apply Hl.
Qed.
Theorem from_elements_nodup_spec : forall l x n, n > 0 -> NoDupA eq_elt l ->
multiplicity x (from_elements l) = n <-> InA eq_pair (x, n) l.
Proof.
induction l as [| [y p] l]; intros x n Hn Hnodup.
* simpl. rewrite InA_nil, empty_spec. omega.
* simpl. destruct (E.eq_dec x y) as [Heq | Heq].
+ assert (Hin : multiplicity y (from_elements l) = 0).
{ setoid_replace (multiplicity y (from_elements l) = 0) with (~In y (from_elements l)) by (unfold In; omega).
destruct l as [| z l]; try apply In_empty. inversion_clear Hnodup.
rewrite from_elements_In; trivial. intros [q [Hin Hq]]. apply H. revert Hin. apply InA_pair_elt. }
rewrite Heq, add_same, Hin. simpl. split; intro H.
subst. now repeat left.
inversion_clear H.
destruct H0; auto.
inversion_clear Hnodup. elim H. revert H0. apply InA_pair_elt.
+ rewrite add_other; trivial. destruct l as [| z l].
- simpl. rewrite empty_spec. intuition; try omega.
inversion_clear H. destruct H0; try contradiction. rewrite InA_nil in H0. elim H0.
- inversion_clear Hnodup. rewrite IHl; discriminate || trivial. intuition. inversion_clear H0; trivial.
inversion_clear H1; trivial. destruct H0. contradiction.
Qed.
Corollary from_elements_valid_spec : forall l x n, n > 0 -> is_elements l ->
multiplicity x (from_elements l) = n <-> InA eq_pair (x, n) l.
Proof. intros ? ? ? ? [? _]. now apply from_elements_nodup_spec. Qed.
Lemma from_elements_append : forall l1 l2,
from_elements (l1 ++ l2) [=] union (from_elements l1) (from_elements l2).
Proof.
induction l1 as [| [x n] l]; intro l2; simpl.
- now rewrite union_empty_l.
- rewrite IHl. symmetry. apply union_add_comm_l.
Qed.
Lemma elements_add_in : forall x n m, In x m ->
PermutationA eq_pair (elements (add x n m))
((x, n + multiplicity x m) :: removeA pair_dec (x, multiplicity x m) (elements m)).
Proof.
intros x n m Hin.
rewrite <- (elements_In x 0) in Hin. apply elements_elt_strengthen, PermutationA_split in Hin; refine _.
destruct Hin as [l' Hin]. rewrite <- (from_elements_elements m), Hin at 1.
assert (Hl' : is_elements ((x, multiplicity x m) :: l')). { rewrite <- Hin. apply elements_is_elements. }
assert (Hout : ~InA eq_elt (x, (multiplicity x m)) l'). { apply proj1 in Hl'. now inversion_clear Hl'. }
rewrite from_elements_cons, add_merge. rewrite elements_add_out.
+ constructor; try reflexivity. apply is_elements_cons_inv in Hl'.
rewrite Hin, elements_from_elements; trivial. simpl.
destruct pair_dec as [? | Habs]; try now elim Habs.
rewrite removeA_out; try reflexivity. intro Habs. apply Hout. revert Habs. apply InA_pair_elt.
+ apply proj2 in Hl'. inversion_clear Hl'. simpl in *. omega.
+ apply is_elements_cons_inv in Hl'. rewrite <- elements_In, elements_from_elements; eauto.
Qed.
(*
Lemma from_elements_remove : forall x n l, countA_occ eq_pair (x, n) l = 1 ->
from_elements (removeA pair_dec (x, n) l) [=] remove x n (from_elements l).
Proof.
intros x n l Hl y. induction l as [| [z p] l]; simpl.
+ discriminate Hl.
+ destruct (pair_dec (x, n) (z, p)) as [Heq | Heq].
- compute in Heq. destruct Heq as [Heq ?]. subst p. rewrite <- Heq. rewrite remove_add_cancel.
-
Qed.
Lemma from_elements_remove_all : forall x n l,
from_elements (removeA elt_dec (x, n) l) [=] remove_all x (from_elements l)
*)
(** ** Results about [fold] **)
Section Fold_results.
Variables (A : Type) (eqA : relation A).
Context (HeqA : Equivalence eqA).
Variable f : elt -> nat -> A -> A.
Hypotheses (Hf : Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f) (Hf2 : transpose2 eqA f).
Instance fold_f_compat : Proper (eq ==> eqA ==> eqA) (fold f) := fold_compat _ _ _ Hf Hf2.
Definition fold_rect : forall (P : t -> A -> Type) (i : A) (m : t),
(forall m1 m2 acc, m1 [=] m2 -> P m1 acc -> P m2 acc) -> P empty i ->
(forall x n m' acc, In x m -> n > 0 -> ~In x m' -> P m' acc -> P (add x n m') (f x n acc)) -> P m (fold f m i).
Proof.
intros P i m HP H0 Hrec. rewrite fold_spec. rewrite <- fold_left_rev_right.
assert (Hrec' : forall x n acc m', InA eq_pair (x, n) (rev (elements m)) -> ~In x m' ->
P m' acc -> P (add x n m') (f x n acc)).
{ intros ? ? ? ? ? Hin. rewrite (InA_rev _), elements_spec in H. simpl in H.
destruct H. apply Hrec; trivial. unfold In. omega. }
assert (Helt : is_elements (rev (elements m))).
{ rewrite <- (PermutationA_rev _). apply (elements_is_elements _). }
clear Hrec. pose (l := rev (elements m)). fold l in Hrec', Helt. change (rev (elements m)) with l.
eapply HP. rewrite <- from_elements_elements. rewrite (PermutationA_rev _). reflexivity.
fold l. clearbody l. induction l as [| [x n] l]; simpl.
+ (* elements m = nil *)
assumption.
+ (* elements m = (x, n) :: l *)
assert (Hdup := Helt). destruct Hdup as [Hdup _]. apply is_elements_cons_inv in Helt.
apply Hrec'.
- now left.
- intro. inversion_clear Hdup. apply H1. rewrite <- elements_from_elements; trivial. now rewrite elements_In.
- apply IHl.
intros. apply Hrec'; trivial. now right.
assumption.
Qed.
Lemma fold_rect_weak : forall (P : t -> A -> Type) (i : A) (m : t),
(forall m1 m2 acc, m1 [=] m2 -> P m1 acc -> P m2 acc) -> P empty i ->
(forall x n m' acc, n > 0 -> ~In x m' -> P m' acc -> P (add x n m') (f x n acc)) -> P m (fold f m i).
Proof. intros * ? ? Hrec. apply fold_rect; trivial. intros. now apply Hrec. Qed.
Lemma fold_rect_nodep : forall (P : A -> Type) (f : elt -> nat -> A -> A) (i : A) (m : t),
P i -> (forall x n acc, In x m -> P acc -> P (f x n acc)) -> P (fold f m i).
Proof.
intros P ff i m H0 Hrec. rewrite fold_spec.
assert (Hrec' : forall x n k acc, InA eq_elt (x, k) (rev (elements m)) -> P acc -> P (ff x n acc)).
{ intros ? ? ? ? Hin. apply Hrec. change x with (fst (x, k)).
rewrite <- elements_In, <- (InA_rev _). eassumption. }
rewrite <- fold_left_rev_right. induction (rev (elements m)) as [| [x n] l]; simpl.
+ assumption.
+ eapply Hrec'. now left. apply IHl. intros. apply Hrec' with k; trivial. now right.
Qed.
Theorem fold_add : forall x n m (i : A), n > 0 -> ~In x m -> eqA (fold f (add x n m) i) (f x n (fold f m i)).
Proof.
intros x n m i Hn Hin. do 2 rewrite fold_spec.
assert (Hperm : PermutationA eq_pair (elements (add x n m)) ((elements m) ++ (x, n) :: nil)).
{ rewrite elements_add_out; trivial. apply (PermutationA_cons_append _). }
erewrite (fold_left_symmetry_PermutationA _ _); try apply Hperm || reflexivity.
- do 2 rewrite <- fold_left_rev_right. now rewrite rev_unit.
- intros ? ? ? [? ?] [? ?] [Heq ?]. now apply Hf.
- intros [? ?] [? ?] ?. simpl. apply Hf2.
Qed.
Theorem fold_add_additive : additive2 eqA f ->
forall x n m (i : A), n > 0 -> eqA (fold f (add x n m) i) (f x n (fold f m i)).
Proof.
intros Hfadd x n m i Hn.
destruct (multiplicity x m) eqn:Hm.
+ (* If [In x m], then we can simply use [fold_add] *)
apply fold_add. assumption. unfold In. omega.
+ (* Otherwise, the real proof starts *)
assert (Hperm : PermutationA eq_pair (elements (add x n m))
(elements (remove x (multiplicity x m) m) ++ (x, n + multiplicity x m) :: nil)).
{ etransitivity; try apply (PermutationA_cons_append _).
rewrite <- elements_add_out; try omega.
rewrite add_remove1; try omega. do 2 f_equiv. omega.
unfold In. rewrite remove_same. omega. }
rewrite fold_spec. erewrite (fold_left_symmetry_PermutationA _ _); try apply Hperm || reflexivity.
- rewrite <- fold_left_rev_right. rewrite rev_unit. simpl. rewrite <- Hfadd. f_equiv.
rewrite fold_left_rev_right, <- fold_spec. etransitivity.
symmetry. apply fold_add. omega. unfold In. rewrite remove_same. omega.
rewrite add_remove1; trivial. now rewrite minus_diag, add_0.
- intros ? ? ? [? ?] [? ?] [Heq ?]. now apply Hf.
- intros [? ?] [? ?] ?. simpl. apply Hf2.
Qed.
(* Wrong in general when m1 and m2 are not disjoint. *)
Lemma fold_union : forall m1 m2 (i : A), (forall x, In x m1 -> In x m2 -> False) ->
eqA (fold f (union m1 m2) i) (fold f m1 (fold f m2 i)).
Proof.
intros m1 m2 i Hm. apply fold_rect with (m := m1); trivial.
+ intros * Heq. now rewrite Heq.
+ now rewrite union_empty_l.
+ intros. rewrite union_add_comm_l, <- H2. apply fold_add. assumption.
unfold In in *. rewrite union_spec. intro Habs. apply (Hm x). assumption. omega.
Qed.
Lemma fold_union_additive : additive2 eqA f ->
forall m1 m2 (i : A), eqA (fold f (union m1 m2) i) (fold f m1 (fold f m2 i)).
Proof.
intros Hfadd m1 m2 i. apply fold_rect with (m := m1).
+ intros * Heq. now rewrite Heq.
+ now rewrite union_empty_l.
+ intros. rewrite union_add_comm_l, <- H2. now apply fold_add_additive.
Qed.
End Fold_results.
Lemma fold_extensionality_compat (A : Type) (eqA : relation A) `(Equivalence A eqA) :
forall f : elt -> nat -> A -> A, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f -> transpose2 eqA f ->
forall g, (forall x v acc, eqA (f x v acc) (g x v acc)) ->
forall m i, eqA (fold f m i) (fold g m i).
Proof.
intros f Hf Hf2 g Hext m i.
assert (Hg : Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) g).
{ repeat intro. repeat rewrite <- Hext. apply Hf; assumption. }
assert (Hg2 : transpose2 eqA g). { repeat intro. repeat rewrite <- Hext. apply Hf2. }
apply fold_rect.
+ intros m1 m2 acc Hm Heq. rewrite Heq. apply (fold_compat _ _ g Hg Hg2); assumption || reflexivity.
+ rewrite fold_empty. reflexivity.
+ intros x n m' acc Hin Hn Hout Heq. rewrite Hext, Heq. rewrite fold_add; reflexivity || assumption.
Qed.
Lemma union_fold_add : forall m1 m2, fold (fun x n acc => add x n acc) m1 m2 [=] union m1 m2.
Proof.
intros m1 m2 x. apply fold_rect with (m := m1).
+ intros * Heq1 Heq2. now rewrite <- Heq1, Heq2.
+ now rewrite union_empty_l.
+ intros. rewrite union_add_comm_l. destruct (E.eq_dec x x0) as [Heq | Hneq].
- rewrite <- Heq. do 2 rewrite add_same. now rewrite H2.
- now repeat rewrite add_other.
Qed.
Corollary fold_add_id : forall m, fold (fun x n acc => add x n acc) m empty [=] m.
Proof. intro. now rewrite union_fold_add, union_empty_r. Qed.
(** ** Generic induction principles on multisets **)
Definition rect : forall P, (forall m1 m2, m1 [=] m2 -> P m1 -> P m2) ->
(forall m x n, ~In x m -> n > 0 -> P m -> P (add x n m)) ->
P empty -> forall m, P m.
Proof. intros P HP ? ? ?. apply (@fold_rect _ (fun _ _ _ => tt) (fun m _ => P m) tt); eauto. Defined.
Definition ind : forall P, Proper (eq ==> iff) P ->
(forall m x n, ~In x m -> n > 0 -> P m -> P (add x n m)) ->
P empty -> forall m, P m.
Proof. intros. apply rect; trivial. intros ? ? Heq. now rewrite Heq. Qed.
(** Direct definition by induction on [elements m], which does not use [fold]. **)
Definition rect2 : forall P, (forall m1 m2, m1 [=] m2 -> P m1 -> P m2) ->
(forall m x n, ~In x m -> P m -> P (add x n m)) ->
P empty -> forall m, P m.
Proof.
intros P Heq Hadd H0.
(* The proof goes by induction on [elements m]
so we first replace all occurences of [m] by [elements m] and prove the premises. *)
pose (P' := fun l => P (fold_left (fun acc xn => add (fst xn) (snd xn) acc) l empty)).
assert (Pequiv1 : forall m, P m -> P' (elements m)).
{ intro. unfold P'. apply Heq. rewrite <- fold_spec. symmetry. apply fold_add_id. }
assert (Pequiv2 : forall m, P' (elements m) -> P m).
{ intro. unfold P'. apply Heq. rewrite <- fold_spec. apply fold_add_id. }
assert (HP' : forall l1 l2, PermutationA eq_pair l1 l2 -> P' l1 -> P' l2).
{ intros l1 l2 Hl. unfold P'.
assert (Hf : Proper (eq ==> eq_pair ==> eq) (fun acc xn => add (fst xn) (snd xn) acc)).
{ repeat intro. now rewrite H1, H. }
apply Heq. apply (@fold_left_symmetry_PermutationA _ _ eq_pair eq _ _ _ Hf); reflexivity || trivial.
intros [x n] [y p] acc. simpl. apply add_comm. }
assert (Hadd' : forall l x n, is_elements l -> n > 0 -> ~InA eq_elt (x, n) l -> P' l -> P' ((x, n) :: l)).
{ intros l x n Hl Hn Hin. apply is_elements_build in Hl. destruct Hl as [m Hm]. rewrite Hm in Hin.
assert (Hx : ~In x m).
{ rewrite <- support_spec, support_elements. intro Habs. apply Hin. eapply InA_pair_elt. eassumption. }
intro Hl. apply (HP' _ _ Hm), Pequiv2, Hadd with m x n, Pequiv1 in Hl; trivial. revert Hl.
apply HP'. etransitivity. now apply elements_add_out. now apply PermutationA_cons. }
(* The real proof starts. *)
intro m. apply Pequiv2. generalize (elements_is_elements m).
induction (elements m) as [| [x n] l]; intro Hl.
+ unfold P'. simpl. apply H0.
+ apply Hadd'.
- eapply is_elements_cons_inv. eassumption.
- destruct Hl as [_ Hpos]. now inversion_clear Hpos.
- destruct Hl as [Hdup _]. now inversion_clear Hdup.
- apply IHl. eapply is_elements_cons_inv. eassumption.
Qed.
Corollary not_empty_In : forall m, ~m [=] empty <-> exists x, In x m.
Proof.
intro m. split.
+ pattern m. apply ind; clear m.
- intros m1 m2 Hm. setoid_rewrite Hm. reflexivity.
- intros m x n Hm Hn Hrec _. exists x. apply add_In. right. split; omega || reflexivity.
- intro Habs. now elim Habs.
+ intros [x Hin]. intro Habs. revert Hin. rewrite Habs. apply In_empty.
Qed.
Corollary empty_or_In_dec : forall m, {m [=] empty} + {exists x, In x m}.
Proof.
intro m. destruct (equal m empty) eqn:Heq.
+ rewrite equal_spec in Heq. now left.
+ right. rewrite <- not_empty_In. rewrite <- equal_spec, Heq. discriminate.
Qed.
(** ** Results about [support] **)
Lemma support_nil : forall m, support m = nil <-> m [=] empty.
Proof.
intro m. split; intro H.
+ intro x. rewrite empty_spec. destruct (multiplicity x m) eqn:Hin. reflexivity.
assert (Hm : In x m). { unfold In. rewrite Hin. omega. }
rewrite <- support_spec in Hm. rewrite H in Hm. inversion Hm.
+ apply (@PermutationA_nil _ E.eq _). rewrite H. now rewrite support_empty.
Qed.
Lemma support_1 : forall x m, PermutationA E.eq (support m) (x :: nil)
<-> m [=] singleton x (multiplicity x m) /\ (multiplicity x m) > 0.
Proof.
intros x m. split; intro Hm.
+ assert (Hin : In x m). { rewrite <- support_spec, Hm. now left. }
unfold In in Hin. split; try omega. intro y. rewrite singleton_spec.
destruct (E.eq_dec y x) as [Heq | Hneq]. now rewrite Heq.
destruct (multiplicity y m) eqn:Hy. reflexivity.
assert (Hiny : In y m). { unfold In. rewrite Hy. omega. }
rewrite <- support_spec, Hm in Hiny. inversion_clear Hiny. contradiction. inversion H.
+ destruct Hm as [Hm Hmult]. rewrite Hm. apply support_singleton. omega.
Qed.
Lemma support_In : forall x m, InA E.eq x (support m) <-> In x m.
Proof. intros. rewrite support_elements, elements_spec. unfold In. intuition. Qed.
Lemma support_add : forall x n m, n > 0 ->
PermutationA E.eq (support (add x n m)) (if In_dec x m then support m else x :: support m).
Proof.
intros x n m Hn. apply (NoDupA_equivlistA_PermutationA _).
* apply support_NoDupA.
* destruct (In_dec x m) as [Hin | Hin].
+ apply support_NoDupA.
+ constructor. now rewrite support_spec. apply support_NoDupA.
* intro z. destruct (In_dec x m) as [Hin | Hin].
+ do 2 rewrite support_spec. unfold In in *. msetdec.
+ rewrite support_spec. unfold In in *. msetdec.
- split; intro H. now left. omega.
- split; intro H.
right. now rewrite support_spec.
inversion H; subst. contradiction. now rewrite support_spec in H1.
Qed.
Lemma support_remove : forall x n m,
PermutationA E.eq (support (remove x n m))
(if le_dec (multiplicity x m) n then removeA E.eq_dec x (support m) else support m).
Proof.
intros x n m. apply (NoDupA_equivlistA_PermutationA _).
+ apply support_NoDupA.
+ destruct (le_dec (multiplicity x m) n) as [Hin | Hin].
- apply (removeA_NoDupA _). apply support_NoDupA.
- apply support_NoDupA.
+ intro z. destruct (le_dec (multiplicity x m) n) as [Hle | Hlt].
- rewrite (removeA_InA _). do 2 rewrite support_spec. unfold In in *. split; intro H.
destruct (E.eq_dec z x).
exfalso. revert H. msetdec.
split; msetdec.
destruct H. msetdec.
- do 2 rewrite support_spec. unfold In in *. msetdec.
Qed.
Lemma support_union : forall x m1 m2,
InA E.eq x (support (union m1 m2)) <-> InA E.eq x (support m1) \/ InA E.eq x (support m2).
Proof. intros. repeat rewrite support_spec. apply union_In. Qed.
(* The more global versions (PermutationA with union/inter/...)
would require ListSet operations on a setoid equality. Absent from the std lib...
Lemma support_union2 : forall m1 m2,
PermutationA E.eq (support (union m1 m2)) (ListSet.set_union (support m1) (support m2)).
Proof.
Qed. *)
Lemma support_inter : forall x m1 m2,
InA E.eq x (support (inter m1 m2)) <-> InA E.eq x (support m1) /\ InA E.eq x (support m2).
Proof. intros. repeat rewrite support_spec. apply inter_In. Qed.
Lemma support_diff : forall x m1 m2, InA E.eq x (support (diff m1 m2)) <-> multiplicity x m2 < multiplicity x m1.
Proof. intros. rewrite support_In, diff_In. intuition. Qed.
Lemma support_lub : forall k m1 m2,
InA E.eq k (support (lub m1 m2)) <-> InA E.eq k (support m1) \/ InA E.eq k (support m2).
Proof. intros. repeat rewrite support_spec. apply lub_In. Qed.
Lemma support_map_elements : forall m, PermutationA E.eq (support m) (map (@fst E.t nat) (elements m)).
Proof.
intro m. apply (NoDupA_equivlistA_PermutationA _).
+ apply support_NoDupA.
+ assert (Hm := elements_NoDupA m).
induction Hm as [| [x n] l].
- constructor.
- simpl. constructor; trivial.
intro Habs. apply H. clear -Habs. induction l as [| [y p] l].
now inversion Habs.
inversion_clear Habs. now left. right. now apply IHl.
+ intro x. rewrite support_elements. rewrite (InA_map_iff _ _). split; intro Hin.
- exists (x, multiplicity x m). now split.
- destruct Hin as [[y p] [Heq Hin]]. rewrite elements_spec in *. simpl in *.
split. reflexivity. destruct Hin. subst. now rewrite <- Heq.
- clear. intros [x n] [y p] [? ?]. apply H.
Qed.
(** ** Results about [cardinal] **)
Lemma cardinal_lower_aux : forall (l : list (E.t * nat)) acc, acc <= fold_left (fun acc xn => snd xn + acc) l acc.
Proof.
induction l; intro acc; simpl.
- omega.
- transitivity (snd a + acc). omega. apply IHl.
Qed.
Lemma fold_left_cardinal : Proper (PermutationA eq_pair ==> Logic.eq ==> Logic.eq)
(fold_left (fun (acc : nat) (xn : elt * nat) => snd xn + acc)).
Proof.
apply (fold_left_symmetry_PermutationA _ _).
- intros ? ? ? [? ?] [? ?] [? Heq]. hnf in *. simpl in *. now subst.
- intros [? ?] [? ?] ?. omega.
Qed.
Corollary cardinal_lower : forall x m, multiplicity x m <= cardinal m.
Proof.
intros x m. destruct (multiplicity x m) eqn:Hm. omega.
assert (Hin : InA eq_pair (x, S n) (elements m)). { rewrite elements_spec. split; simpl. assumption. omega. }
rewrite cardinal_spec, fold_spec.
apply (PermutationA_split _) in Hin. destruct Hin as [l Hperm]. assert (H0 := eq_refl 0).
rewrite fold_left_cardinal; try eassumption. simpl. rewrite plus_0_r. now apply cardinal_lower_aux.
Qed.
Corollary cardinal_In : forall x m, In x m -> 0 < cardinal m.
Proof. intros. apply lt_le_trans with (multiplicity x m). assumption. apply cardinal_lower. Qed.
Lemma cardinal_0 : forall m, cardinal m = 0 <-> m [=] empty.
Proof.
intro m. split; intro Hm.
+ intro y. rewrite empty_spec, <- empty_spec with y. revert y. change (m [=] empty). rewrite <- elements_nil.
destruct (elements m) as [| [x n] l] eqn:Helt. reflexivity.
simpl in Hm. elim (lt_irrefl 0). apply lt_le_trans with n.
- apply elements_pos with x m. rewrite Helt. now left.
- assert (Hn : multiplicity x m = n). { eapply proj1. rewrite <- (elements_spec x n), Helt. now left. }
rewrite <- Hn, <- Hm. apply cardinal_lower.
+ rewrite Hm. apply cardinal_empty.
Qed.
Instance fold_cardinal_compat : Proper (eq ==> Logic.eq ==> Logic.eq) (fold (fun x n acc => n + acc)).
Proof.
intros mβ mβ Hm ? ? ?. apply (fold_compat _ _); trivial.
- now repeat intro; subst.
- repeat intro. omega.
Qed.
Theorem cardinal_union : forall mβ mβ, cardinal (union mβ mβ) = cardinal mβ + cardinal mβ.
Proof.
intros mβ mβ. do 2 rewrite cardinal_spec. rewrite (fold_union_additive _).
+ rewrite <- cardinal_spec. revert mβ. apply ind.
- intros ? ? Heq. now rewrite Heq.
- intros. destruct n.
now rewrite add_0.
repeat rewrite (fold_add _); trivial; repeat intro; omega || now repeat intro; subst.
- now do 2 rewrite fold_empty.
+ now repeat intro; subst.
+ repeat intro. omega.
+ repeat intro. omega.
Qed.
Corollary cardinal_add : forall x n m, cardinal (add x n m) = n + cardinal m.
Proof. intros. now rewrite <- add_union_singleton_l, cardinal_union, cardinal_singleton. Qed.
Theorem cardinal_remove : forall x n m, cardinal (remove x n m) = cardinal m - min n (multiplicity x m).
Proof.
intros x n m. destruct (le_dec n (multiplicity x m)) as [Hle | Hlt].
+ setoid_rewrite <- (add_0 x) at 3. erewrite <- (minus_diag n).
rewrite <- (@add_remove1 x n n m), cardinal_add, min_l; trivial. omega.
+ assert (Hle : multiplicity x m <= n) by omega.
setoid_rewrite <- (add_0 x) at 3. erewrite <- minus_diag.
rewrite <- (@add_remove2 x _ n m Hle (le_refl _)), cardinal_add, min_r; trivial. omega.
Qed.
Instance cardinal_sub_compat : Proper (Subset ==> le) cardinal.
Proof.
intro s. pattern s. apply ind; clear s.
+ intros ? ? Hm. now setoid_rewrite Hm.
+ intros m x n Hin Hn Hrec m' Hsub. rewrite (cardinal_add _).
assert (n <= multiplicity x m'). { transitivity (n + multiplicity x m). omega. specialize (Hsub x). msetdec. }
assert (n <= cardinal m'). { etransitivity; try eassumption. apply cardinal_lower. }
apply add_subset_remove in Hsub. apply Hrec in Hsub. rewrite cardinal_remove in Hsub.
etransitivity. apply plus_le_compat. reflexivity. apply Hsub. rewrite min_l; trivial. omega.
+ intros. rewrite cardinal_empty. omega.
Qed.
Lemma cardinal_inter_le_min : forall m1 m2, cardinal (inter m1 m2) <= min (cardinal m1) (cardinal m2).
Proof.
intro m1; pattern m1. apply ind; clear m1.
* intros m1 m1' Heq. split; intros Hle m2; rewrite Heq || rewrite <- Heq; apply Hle.
* intros m x n Hout Hn Hind m2. destruct (le_lt_dec n (multiplicity x m2)) as [Hle | Hlt].
+ rewrite inter_add_l1; trivial. rewrite <- (add_remove_cancel Hle) at 2.
do 3 rewrite cardinal_add. rewrite Min.plus_min_distr_l. apply plus_le_compat_l, Hind.
+ rewrite inter_add_l2; try omega.
transitivity (Init.Nat.min (cardinal (add x (multiplicity x m2) m)) (cardinal m2)).
- rewrite <- (add_remove_cancel (reflexivity (multiplicity x m2))) at 4.
do 3 rewrite cardinal_add. rewrite Min.plus_min_distr_l. apply plus_le_compat_l.
rewrite remove_cap; try omega. apply Hind.
- do 2 rewrite cardinal_add. apply Nat.min_le_compat_r. omega.
* intro. rewrite inter_empty_l, cardinal_empty. omega.
Qed.
Lemma cardinal_diff_upper : forall m1 m2, cardinal (diff m1 m2) <= cardinal m1.
Proof. intros. apply cardinal_sub_compat, diff_subset. Qed.
Lemma cardinal_diff_lower : forall m1 m2, cardinal m1 - cardinal m2 <= cardinal (diff m1 m2).
Proof.
intro m1. pattern m1. apply ind; clear m1.
+ intros m1 m1' Heq. now setoid_rewrite Heq.
+ intros m x n Hout Hn Hind m2. destruct (le_lt_dec n (multiplicity x m2)) as [Hle | Hlt].
- rewrite diff_add_l1; trivial. rewrite <- (add_remove_cancel Hle) at 1. do 2 rewrite cardinal_add.
replace (n + cardinal m - (n + cardinal(remove x n m2))) with (cardinal m - cardinal(remove x n m2)) by omega.
apply Hind.
- rewrite diff_add_l2; try omega. rewrite <- (add_remove_cancel (reflexivity (multiplicity x m2))) at 1.
do 3 rewrite cardinal_add. rewrite <- (@remove_cap x n); try omega.
transitivity ((n - multiplicity x m2) + (cardinal m - cardinal(remove x n m2))); try omega.
apply plus_le_compat_l, Hind.
+ intro. now rewrite diff_empty_l, cardinal_empty.
Qed.
Lemma cardinal_lub_upper : forall m1 m2, cardinal (lub m1 m2) <= cardinal m1 + cardinal m2.
Proof. intros. rewrite <- cardinal_union. apply cardinal_sub_compat, lub_subset_union. Qed.
Lemma cardinal_lub_lower : forall m1 m2, max (cardinal m1) (cardinal m2) <= cardinal (lub m1 m2).
Proof.
intro m1. pattern m1. apply ind; clear m1.
+ intros m1 m1' Heq. now setoid_rewrite Heq.
+ intros m x n Hout Hn Hind m2. rewrite lub_add_l. do 2 rewrite cardinal_add.
transitivity (n + Init.Nat.max (cardinal m) (cardinal (remove x n m2))).
- rewrite <- Max.plus_max_distr_l. apply Nat.max_le_compat_l. rewrite <- (cardinal_add x).
apply cardinal_sub_compat. intro. msetdec.
- apply plus_le_compat_l, Hind.
+ intro. now rewrite lub_empty_l, cardinal_empty.
Qed.
Lemma cardinal_fold_elements : forall m, cardinal m = List.fold_left (fun acc xn => snd xn + acc) (elements m) 0.
Proof. intro. now rewrite cardinal_spec, fold_spec. Qed.
Lemma cardinal_from_elements : forall l,
cardinal (from_elements l) = List.fold_left (fun acc xn => snd xn + acc) l 0.
Proof.
intro l. rewrite <- plus_0_l at 1. generalize 0. induction l as [| [x n] l]; intro p; simpl.
- now rewrite cardinal_empty.
- rewrite cardinal_add, plus_assoc. rewrite (plus_comm p n). apply IHl.
Qed.
Lemma cardinal_total_sub_eq : forall m1 m2, m1 [<=] m2 -> cardinal m1 = cardinal m2 -> m1 [=] m2.
Proof.
intro m. pattern m. apply ind; clear m.
+ intros m1 m1' Heq. now setoid_rewrite Heq.
+ intros m1 x n Hout Hn Hrec m2 Hsub Heq.
assert (n <= multiplicity x m2).
{ transitivity (n + multiplicity x m1); try omega.
specialize (Hsub x). rewrite add_same in Hsub. omega. }
rewrite <- (@add_remove_cancel x n m2); trivial. f_equiv. apply Hrec.
- now apply add_subset_remove.
- rewrite cardinal_add in Heq. rewrite cardinal_remove, <- Heq, Nat.min_l; omega.
+ intros m _ Heq. symmetry. rewrite <- cardinal_0, <- Heq. apply cardinal_empty.
Qed.
(** ** Results about [size] **)
Lemma size_0 : forall m, size m = 0 <-> m [=] empty.
Proof.
intro m. split; intro Heq.
- now rewrite size_spec, length_zero_iff_nil, support_nil in Heq.
- rewrite Heq. apply size_empty.
Qed.
Lemma size_1 : forall m, size m = 1 <-> exists x, m [=] singleton x (multiplicity x m) /\ multiplicity x m > 0.
Proof.
intro m. split; intro Heq.
- rewrite size_spec, length_1 in Heq. destruct Heq as [x Heq]. exists x. rewrite <- support_1. now rewrite Heq.
- destruct Heq as [x [Heq Hmul]]. rewrite Heq. now apply size_singleton.
Qed.
Lemma size_In : forall m, size m > 0 <-> exists x, In x m.
Proof.
intro m. split; intro Hin.
+ rewrite size_spec in Hin. destruct (support m) as [| x l] eqn:Heq.
- inversion Hin.
- exists x. rewrite <- support_In, Heq. now left.
+ destruct Hin as [x Hin]. destruct (size m) eqn:Hsize.
- rewrite size_0 in Hsize. rewrite Hsize in Hin. elim (In_empty Hin).
- auto with arith.
Qed.
Lemma size_add : forall x n m, n > 0 -> size (add x n m) = if In_dec x m then size m else S (size m).
Proof.
intros x n m Hn. do 2 rewrite size_spec. rewrite support_add; trivial. destruct (In_dec x m); reflexivity.
Qed.
Lemma size_remove : forall x n m,
In x m -> size (remove x n m) = if le_dec (multiplicity x m) n then pred (size m) else size m.
Proof.
intros x n m Hin. do 2 rewrite size_spec. rewrite support_remove.
destruct (le_dec (multiplicity x m) n) as [Hle | ?]; trivial.
rewrite <- support_In in Hin. apply PermutationA_split in Hin; refine _. destruct Hin as [l Hin].
assert (Hnodup : NoDupA E.eq (x :: l)). { rewrite <- Hin. apply support_NoDupA. }
rewrite Hin. simpl. destruct (E.eq_dec x x) as [_ | Hneq]; try now elim Hneq.
inversion_clear Hnodup. now rewrite removeA_out.
Qed.
Corollary size_remove_eq : forall x n m, n < multiplicity x m -> size (remove x n m) = size m.
Proof.
intros x n m ?. rewrite size_remove; try (unfold In; omega). destruct (le_dec (multiplicity x m) n); omega.
Qed.
Lemma size_union_lower : forall m1 m2, max (size m1) (size m2) <= size (union m1 m2).
Proof. intros. apply Max.max_case; apply size_sub_compat; apply union_subset_l || apply union_subset_r. Qed.
Lemma size_union_upper : forall m1 m2, size (union m1 m2) <= size m1 + size m2.
Proof.
intros m1 m2. pattern m1. apply ind; clear m1.
+ intros m1 m1' Heq. rewrite Heq. reflexivity.
+ intros m1 x n Hin Hn Hrec. rewrite union_add_comm_l. repeat rewrite size_add; trivial.
destruct (In_dec x m1); try contradiction. destruct (In_dec x (union m1 m2)); omega.
+ rewrite size_empty, union_empty_l. reflexivity.
Qed. (* the most straigthforward way to express this would be by using set_union, hence requiring ListSetA. *)
Lemma size_inter_upper : forall m1 m2, size (inter m1 m2) <= min (size m1) (size m2).
Proof. intros. apply Min.min_case; apply size_sub_compat; apply inter_subset_l || apply inter_subset_r. Qed.
Lemma size_diff_upper : forall m1 m2, size (diff m1 m2) <= size m1.
Proof. intros. apply size_sub_compat, diff_subset. Qed.
Lemma size_lub_lower : forall m1 m2, max (size m1) (size m2) <= size (lub m1 m2).
Proof. intros. apply Max.max_case; apply size_sub_compat; apply lub_subset_l || apply lub_subset_r. Qed.
Lemma size_lub_upper : forall m1 m2, size (lub m1 m2) <= size m1 + size m2.
Proof.
intros. transitivity (size (union m1 m2)).
- apply size_sub_compat. apply lub_subset_union.
- apply size_union_upper.
Qed.
Lemma size_elements : forall m, size m = length (elements m).
Proof. intro. now rewrite size_spec, support_map_elements, map_length. Qed.
Lemma size_from_elements : forall l, size (from_elements l) <= length l.
Proof.
induction l as [| [x n] l].
+ rewrite from_elements_nil, size_empty. reflexivity.
+ simpl. destruct n.
- rewrite add_0. now apply le_S.
- rewrite size_add; try omega. destruct (In_dec x (from_elements l)); (now apply le_S) || now apply le_n_S.
Qed.
Lemma size_from_elements_valid : forall l, is_elements l -> size (from_elements l) = length l.
Proof. intros. now rewrite size_elements, elements_from_elements. Qed.
Lemma size_cardinal : forall m, size m <= cardinal m.
Proof.
apply ind.
+ intros ? ? Heq. now rewrite Heq.
+ intros m x n Hin Hn Hrec. rewrite size_add, cardinal_add; trivial. destruct (In_dec x m); omega.
+ rewrite size_empty, cardinal_empty. reflexivity.
Qed.
(** ** Results about [nfilter] **)
Section nFilter_results.
Variable f : E.t -> nat -> bool.
Hypothesis Hf : compatb f.
Lemma nfilter_In : forall x m, In x (nfilter f m) <-> In x m /\ f x (multiplicity x m) = true.
Proof.
intros x m. unfold In. rewrite nfilter_spec; trivial.
destruct (f x (multiplicity x m)); intuition; discriminate.
Qed.
Corollary In_nfilter : forall x m, In x (nfilter f m) -> In x m.
Proof. intros x m Hin. rewrite nfilter_In in Hin; intuition. Qed.
Lemma nfilter_subset : forall m, nfilter f m [<=] m.
Proof. intros m x. rewrite nfilter_spec; trivial. destruct (f x (multiplicity x m)); omega. Qed.
Lemma nfilter_add_true : forall x n m, ~In x m -> n > 0 ->
(nfilter f (add x n m) [=] add x n (nfilter f m) <-> f x n = true).
Proof.
intros x n m Hin Hn. assert (Hm : multiplicity x m = 0) by (unfold In in Hin; omega). split; intro H.
+ specialize (H x). rewrite nfilter_spec, add_same, add_same, nfilter_spec in H; trivial.
rewrite Hm in H. simpl in H. destruct (f x n). reflexivity. omega.
+ intro y. msetdec. rewrite Hm. simpl. rewrite H. now destruct (f x 0).
Qed.
Lemma nfilter_add_false : forall x n m, ~In x m -> n > 0 ->
(nfilter f (add x n m) [=] nfilter f m <-> f x n = false).
Proof.
intros x n m Hin Hn. assert (Hm : multiplicity x m = 0) by (unfold In in Hin; omega). split; intro H.
+ specialize (H x). rewrite nfilter_spec, add_same, nfilter_spec in H; trivial.
rewrite Hm in H. simpl in H. destruct (f x n). destruct (f x 0); omega. reflexivity.
+ intro y. msetdec. rewrite Hm. simpl. rewrite H. now destruct (f x 0).
Qed.
Theorem nfilter_add : forall x n m, ~In x m -> n > 0 ->
nfilter f (add x n m) [=] if f x n then add x n (nfilter f m) else nfilter f m.
Proof.
intros x n m Hin Hn. destruct (f x n) eqn:Hfxn.
- now rewrite nfilter_add_true.
- now rewrite nfilter_add_false.
Qed.
Instance nfilter_sub_compat : Proper (E.eq ==> le ==> Bool.leb) f -> Proper (Subset ==> Subset) (nfilter f).
Proof.
intros Hf2 m1 m2. revert m1. pattern m2. apply ind; clear m2.
+ intros ? ? Hm. now setoid_rewrite Hm.
+ intros m x n Hm Hn Hrec m' Hsub. rewrite nfilter_add; trivial. intro y. specialize (Hsub y).
assert (multiplicity x m = 0) by msetdec. assert (Hbool := Hf2 y y (reflexivity _) _ _ Hsub).
destruct (f x n) eqn:Hfxn.
- msetdec; try rewrite H in *.
destruct (f x (multiplicity x m')), (f x 0); omega.
destruct (f y (multiplicity y m')); omega || now rewrite Hbool.
- msetdec; try rewrite H in *.
simpl in Hbool. rewrite Hfxn in Hbool. now destruct (f x (multiplicity x m')), (f x 0).
destruct (f y (multiplicity y m)), (f y (multiplicity y m')); omega || inversion Hbool.
+ intros m Hm. rewrite subset_empty_r in Hm. now rewrite Hm.
Qed.
Lemma nfilter_extensionality_compat : forall g, (forall x n, g x n = f x n) ->
forall m, nfilter f m [=] nfilter g m.
Proof.
intros g Hext m x.
assert (Hg : Proper (E.eq ==> Logic.eq ==> Logic.eq) g). { repeat intro. repeat rewrite Hext. now apply Hf. }
repeat rewrite nfilter_spec; trivial. rewrite Hext. reflexivity.
Qed.
Lemma nfilter_dependent_extensionality_compat : forall g, Proper (E.eq ==> Logic.eq ==> Logic.eq) g ->
forall m, (forall x n, In x m -> g x n = f x n) -> nfilter f m [=] nfilter g m.
Proof.
intros g Hg m Hext x. repeat rewrite nfilter_spec; trivial.
destruct (eq_nat_dec m[x] 0) as [Heq | Hneq].
- rewrite Heq. destruct (f x 0), (g x 0); reflexivity.
- rewrite Hext. reflexivity. unfold In. omega.
Qed.
Lemma elements_nfilter : forall m,
PermutationA eq_pair (elements (nfilter f m)) (List.filter (fun xn => f (fst xn) (snd xn)) (elements m)).
Proof.
intro m. apply NoDupA_equivlistA_PermutationA; refine _.
* eapply NoDupA_strengthen, elements_NoDupA. apply subrelation_pair_elt.
* apply NoDupA_filter_compat.
+ intros [x n] [y p] [? ?]; compute in *. auto.
+ eapply NoDupA_strengthen, elements_NoDupA. apply subrelation_pair_elt.
* intros [x n]. split; intro Hin.
+ rewrite elements_spec in Hin. destruct Hin as [Hin Hpos]. simpl in *. subst.
rewrite filter_InA; simpl in *.
- rewrite nfilter_spec in *; trivial. destruct (f x (multiplicity x m)) eqn:Hfx; trivial; try omega.
split; trivial. rewrite elements_spec; intuition.
- intros [? ?] [? ?] [? ?]. compute in *. auto.
+ rewrite filter_InA in Hin.
- rewrite elements_spec in *. destruct Hin as [[Hin Hpos] Hfx]. simpl in *. split; trivial.
rewrite nfilter_spec; trivial. subst. now rewrite Hfx.
- intros [? ?] [? ?] [? ?]. compute in *. auto.
Qed.
Lemma nfilter_from_elements : forall l, is_elements l ->
nfilter f (from_elements l) [=] from_elements (List.filter (fun xn => f (fst xn) (snd xn)) l).
Proof.
intros l Hl. rewrite <- elements_eq. rewrite elements_nfilter; trivial.
setoid_rewrite elements_from_elements at 2.
* apply filter_PermutationA_compat; refine _.
+ intros [] [] []. compute in *. auto.
+ now apply elements_from_elements.
* destruct Hl as [Hnodup Hpos]. induction l as [| [x n] l]; try (split; assumption).
inversion_clear Hnodup. inversion_clear Hpos.
destruct IHl as [Hnodup Hpos]. assumption. assumption.
split; simpl.
+ destruct (f x n); trivial. constructor; trivial. intro Hin. apply H.
apply InA_elt_pair in Hin. destruct Hin as [n' Hin]. simpl in *. rewrite filter_InA in Hin.
- destruct Hin. eapply InA_pair_elt; eassumption.
- intros [] [] []. compute in *. auto.
+ destruct (f x n); trivial. now constructor.
Qed.
Lemma support_nfilter : forall m, inclA E.eq (support (nfilter f m)) (support m).
Proof. intro. apply support_sub_compat, nfilter_subset. Qed.
Lemma cardinal_nfilter : forall m, cardinal (nfilter f m) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, nfilter_subset. Qed.
Lemma size_nfilter : forall m, size (nfilter f m) <= size m.
Proof. intro. apply size_sub_compat, nfilter_subset. Qed.
End nFilter_results.
Lemma nfilter_merge : forall f g, compatb f -> compatb g ->
forall m, nfilter f (nfilter g m) [=] nfilter (fun x n => f x n && g x n) m.
Proof.
intros f g Hf Hg m x. repeat rewrite nfilter_spec; trivial.
+ destruct (g x (multiplicity x m)), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
Qed.
Lemma nfilter_comm : forall f g, compatb f -> compatb g ->
forall m, nfilter f (nfilter g m) [=] nfilter g (nfilter f m).
Proof.
intros. repeat rewrite nfilter_merge; trivial. apply nfilter_extensionality_compat.
+ intros x y Hxy ? n ?. subst. now rewrite Hxy.
+ intros. apply andb_comm.
Qed.
Lemma fold_nfilter_fold_left A eqA `{Equivalence A eqA} :
forall f g, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f -> transpose2 eqA f -> compatb g ->
forall m i, eqA (fold f (nfilter g m) i)
(fold_left (fun acc xn => f (fst xn) (snd xn) acc)
(List.filter (fun xn => g (fst xn) (snd xn)) (elements m))
i).
Proof.
intros. rewrite fold_spec, fold_left_symmetry_PermutationA; refine _; try reflexivity.
+ intros ? ? ? [] [] []. compute in *. auto.
+ auto.
+ now apply elements_nfilter.
Qed.
Lemma fold_nfilter A eqA `{Equivalence A eqA} :
forall f g, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f -> transpose2 eqA f -> compatb g ->
forall m i, eqA (fold f (nfilter g m) i) (fold (fun x n acc => if g x n then f x n acc else acc) m i).
Proof.
intros f g Hf Hf2 Hg m.
assert (Hf' : Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) (fun x n acc => if g x n then f x n acc else acc)).
{ clear -Hf Hg. intros x1 x2 Hx n1 n2 Hn m1 m2 Hm. subst.
destruct (g x1 n2) eqn:Hgx; rewrite Hx in Hgx; rewrite Hgx; trivial. apply Hf; trivial. }
assert (Hf2' : transpose2 eqA (fun x n acc => if g x n then f x n acc else acc)).
{ intros x1 x2 y1 y2 z. now destruct (g x1 y1), (g x2 y2); trivial. }
pattern m. apply ind.
* intros m1 m2 Hm. split; intros Heq i.
+ rewrite <- (fold_compat _ _ _ _ Hf2 _ _ (nfilter_compat Hg Hm) _ _ (reflexivity i)), Heq.
apply fold_compat; trivial. reflexivity.
+ rewrite (fold_compat _ _ _ _ Hf2 _ _ (nfilter_compat Hg Hm) _ _ (reflexivity i)), Heq.
apply fold_compat; trivial.
- now symmetry.
- reflexivity.
* clear m. intros m x n Hin Hn Hrec i.
assert (Hadd := nfilter_add Hg Hin Hn). rewrite fold_compat; try eassumption || reflexivity.
rewrite fold_add; trivial. destruct (g x n); trivial. rewrite fold_add; trivial.
+ now apply Hf.
+ intro Hx. apply Hin. now apply In_nfilter in Hx.
* intro i. rewrite fold_empty. rewrite fold_compat; trivial.
+ rewrite fold_empty. reflexivity.
+ now apply nfilter_empty.
+ reflexivity.
Qed.
Lemma cardinal_nfilter_is_multiplicity : forall x m,
cardinal (nfilter (fun y _ => if E.eq_dec y x then true else false) m) = multiplicity x m.
Proof.
intros x m.
assert (Hf : Proper (E.eq ==> Logic.eq ==> Logic.eq) (fun y (_ : nat) => if E.eq_dec y x then true else false)).
{ intros y1 y2 Hy ? ? ?. subst. destruct (E.eq_dec y1 x), (E.eq_dec y2 x); auto; rewrite Hy in *; contradiction. }
pattern m. apply ind; clear m.
+ intros m1 m2 Hm. now setoid_rewrite Hm.
+ intros m y n Hm Hn Hrec. rewrite nfilter_add; trivial. destruct (E.eq_dec y x) as [Heq | Heq].
- rewrite cardinal_add, Hrec, Heq, add_same. apply plus_comm.
- rewrite add_other; msetdec.
+ now rewrite nfilter_empty, cardinal_empty, empty_spec.
Qed.
Lemma nfilter_mono_compat : forall f g, compatb f -> compatb g -> (forall x n, Bool.leb (f x n) (g x n)) ->
forall m, nfilter f m [<=] nfilter g m.
Proof.
intros f g Hf Hg Hfg. apply ind.
+ intros m1 m2 Hm. now rewrite Hm.
+ intros m x n Hm Hn Hrec. repeat rewrite nfilter_add; trivial. destruct (f x n) eqn:Hfx.
- specialize (Hfg x n). rewrite Hfx in Hfg. simpl in Hfg. rewrite Hfg. now f_equiv.
- destruct (g x n); trivial. etransitivity; try eassumption. apply add_subset.
+ repeat rewrite nfilter_empty; trivial. reflexivity.
Qed.
(** ** Results about [nfilter] **)
Section Filter_results.
Variable f : E.t -> bool.
Hypothesis Hf : Proper (E.eq ==> Logic.eq) f.
Theorem filter_nfilter : forall m, filter f m [=] nfilter (fun x _ => f x) m.
Proof. repeat intro. rewrite nfilter_spec, filter_spec; trivial. repeat intro. now apply Hf. Qed.
Lemma filter_In : forall x m, In x (filter f m) <-> In x m /\ f x = true.
Proof. intros x m. unfold In. rewrite filter_spec; trivial. destruct (f x); intuition; discriminate. Qed.
Corollary In_filter : forall x m, In x (filter f m) -> In x m.
Proof. intros x m Hin. rewrite filter_In in Hin; intuition. Qed.
Lemma filter_subset : forall m, filter f m [<=] m.
Proof. intros m x. rewrite filter_spec; trivial. destruct (f x); omega. Qed.
Lemma filter_add_true : forall x n m, ~In x m -> n > 0 ->
(filter f (add x n m) [=] add x n (filter f m) <-> f x = true).
Proof.
repeat intro. do 2 rewrite filter_nfilter. apply nfilter_add_true; trivial. repeat intro. now apply Hf.
Qed.
Lemma filter_add_false : forall x n m, ~In x m -> n > 0 ->
(filter f (add x n m) [=] filter f m <-> f x = false).
Proof.
repeat intro. do 2 rewrite filter_nfilter. apply nfilter_add_false; trivial. repeat intro. now apply Hf.
Qed.
Theorem filter_add : forall x n m, ~In x m -> n > 0 ->
filter f (add x n m) [=] if f x then add x n (filter f m) else filter f m.
Proof.
intros x n m Hin Hn. destruct (f x) eqn:Hfxn.
- now rewrite filter_add_true.
- now rewrite filter_add_false.
Qed.
Instance filter_sub_compat : Proper (Subset ==> Subset) (filter f).
Proof.
repeat intro. do 2 rewrite filter_nfilter. apply nfilter_sub_compat.
- repeat intro. now apply Hf.
- repeat intro. rewrite Hf; try eassumption. apply Bleb_refl.
- assumption.
Qed.
Lemma filter_extensionality_compat : forall g, (forall x, g x = f x) -> forall m, filter f m [=] filter g m.
Proof.
intros g Hext m x.
assert (Hg : Proper (E.eq ==> Logic.eq) g). { repeat intro. repeat rewrite Hext. now apply Hf. }
repeat rewrite filter_spec; trivial. rewrite Hext. reflexivity.
Qed.
Lemma elements_filter : forall m,
PermutationA eq_pair (elements (filter f m)) (List.filter (fun xn => f (fst xn)) (elements m)).
Proof.
intro m. rewrite filter_nfilter, elements_nfilter.
- reflexivity.
- repeat intro. now apply Hf.
Qed.
Lemma filter_from_elements : forall l, is_elements l ->
filter f (from_elements l) [=] from_elements (List.filter (fun xn => f (fst xn)) l).
Proof.
intros l Hl. rewrite filter_nfilter, nfilter_from_elements.
- reflexivity.
- repeat intro. now apply Hf.
- assumption.
Qed.
Lemma support_filter : forall m, inclA E.eq (support (filter f m)) (support m).
Proof. intro. apply support_sub_compat, filter_subset. Qed.
Lemma cardinal_filter : forall m, cardinal (filter f m) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, filter_subset. Qed.
Lemma size_filter : forall m, size (filter f m) <= size m.
Proof. intro. apply size_sub_compat, filter_subset. Qed.
End Filter_results.
Lemma filter_merge : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
forall m, filter f (filter g m) [=] filter (fun x => f x && g x) m.
Proof.
intros f g Hf Hg m x. repeat rewrite filter_spec; trivial.
+ now destruct (f x).
+ clear x m. intros x y Hxy. now rewrite Hxy.
Qed.
Lemma filter_filtern_merge : forall f g, Proper (E.eq ==> Logic.eq) f -> compatb g ->
forall m, filter f (nfilter g m) [=] nfilter (fun x n => f x && g x n) m.
Proof.
intros f g Hf Hg m x. rewrite filter_spec, nfilter_spec, nfilter_spec; trivial.
+ now destruct (f x).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
Qed.
Lemma nfilter_filter_merge : forall f g, compatb f -> Proper (E.eq ==> Logic.eq) g ->
forall m, nfilter f (filter g m) [=] nfilter (fun x n => f x n && g x) m.
Proof.
intros f g Hf Hg m x. rewrite nfilter_spec, nfilter_spec, filter_spec; trivial.
+ destruct (g x), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
Qed.
Lemma filter_comm : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
forall m, filter f (filter g m) [=] filter g (filter f m).
Proof.
intros. repeat rewrite filter_merge; trivial. apply filter_extensionality_compat.
+ intros x y Hxy. subst. now rewrite Hxy.
+ intros. apply andb_comm.
Qed.
Lemma nfilter_filter_comm : forall f g, compatb f -> Proper (E.eq ==> Logic.eq) g ->
forall m, nfilter f (filter g m) [=] filter g (nfilter f m).
Proof.
intros ** x. repeat rewrite filter_spec, nfilter_spec; trivial.
destruct (g x), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
Qed.
Lemma fold_filter_fold_left A eqA `{Equivalence A eqA} :
forall f g, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f -> transpose2 eqA f -> Proper (E.eq ==> Logic.eq) g ->
forall m i, eqA (fold f (filter g m) i)
(fold_left (fun acc xn => f (fst xn) (snd xn) acc)
(List.filter (fun xn => g (fst xn)) (elements m))
i).
Proof.
intros. rewrite fold_spec, fold_left_symmetry_PermutationA; refine _; try reflexivity.
+ intros ? ? ? [] [] []. compute in *. auto.
+ auto.
+ now apply elements_filter.
Qed.
Lemma fold_filter A eqA `{Equivalence A eqA} :
forall f g, Proper (E.eq ==> Logic.eq ==> eqA ==> eqA) f -> transpose2 eqA f -> Proper (E.eq ==> Logic.eq) g ->
forall m i, eqA (fold f (filter g m) i) (fold (fun x n acc => if g x then f x n acc else acc) m i).
Proof.
intros f g Hf Hf2 Hg m i. rewrite (fold_compat _ _ f Hf Hf2 _ _ (filter_nfilter Hg m) i i (reflexivity i)).
apply fold_nfilter; trivial. repeat intro. now apply Hg.
Qed.
Lemma cardinal_filter_is_multiplicity : forall x m,
cardinal (filter (fun y => if E.eq_dec y x then true else false) m) = multiplicity x m.
Proof.
intros x m. rewrite filter_nfilter.
- apply cardinal_nfilter_is_multiplicity.
- intros x' y' Heq. destruct (E.eq_dec x' x), (E.eq_dec y' x); trivial; rewrite Heq in *; contradiction.
Qed.
Lemma filter_mono_compat : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
(forall x, Bool.leb (f x) (g x)) -> forall m, filter f m [<=] filter g m.
Proof.
intros f g Hf Hg Hfg m. repeat rewrite filter_nfilter; trivial. apply nfilter_mono_compat.
- repeat intro. now apply Hf.
- repeat intro. now apply Hg.
- repeat intro. apply Hfg.
Qed.
(** ** Results about [npartition] **)
Section nPartition_results.
Variable f : E.t -> nat -> bool.
Hypothesis Hf : compatb f.
Lemma negf_compatb : Proper (E.eq ==> Logic.eq ==> Logic.eq) (fun x n => negb (f x n)).
Proof. repeat intro. now rewrite Hf. Qed.
Lemma npartition_In_fst : forall x m, In x (fst (npartition f m)) <-> In x m /\ f x (multiplicity x m) = true.
Proof. intros. rewrite npartition_spec_fst; trivial. now apply nfilter_In. Qed.
Lemma npartition_In_snd : forall x m, In x (snd (npartition f m)) <-> In x m /\ f x (multiplicity x m) = false.
Proof.
intros. rewrite npartition_spec_snd, <- negb_true_iff; trivial. apply nfilter_In.
repeat intro. now rewrite Hf.
Qed.
Corollary In_npartition_fst : forall x m, In x (fst (npartition f m)) -> In x m.
Proof. intros x m Hin. rewrite npartition_In_fst in Hin; intuition. Qed.
Corollary In_npartition_snd : forall x m, In x (snd (npartition f m)) -> In x m.
Proof. intros x m Hin. rewrite npartition_In_snd in Hin; intuition. Qed.
Lemma npartition_subset_fst : forall m, fst (npartition f m) [<=] m.
Proof. intro. rewrite npartition_spec_fst; trivial. now apply nfilter_subset. Qed.
Lemma npartition_subset_snd : forall m, snd (npartition f m) [<=] m.
Proof. intro. rewrite npartition_spec_snd; trivial. apply nfilter_subset, negf_compatb. Qed.
Lemma npartition_add_true_fst : forall x n m, ~In x m -> n > 0 ->
(fst (npartition f (add x n m)) [=] add x n (fst (npartition f m)) <-> f x n = true).
Proof. intros. repeat rewrite npartition_spec_fst; trivial. now apply nfilter_add_true. Qed.
Lemma npartition_add_true_snd : forall x n m, ~In x m -> n > 0 ->
(snd (npartition f (add x n m)) [=] snd (npartition f m) <-> f x n = true).
Proof.
intros. repeat rewrite npartition_spec_snd; trivial. rewrite nfilter_add_false; trivial.
apply negb_false_iff. repeat intro. f_equal. now apply Hf.
Qed.
Lemma npartition_add_false_fst : forall x n m, ~In x m -> n > 0 ->
(fst (npartition f (add x n m)) [=] fst (npartition f m) <-> f x n = false).
Proof. intros. repeat rewrite npartition_spec_fst; trivial. now apply nfilter_add_false. Qed.
Lemma npartition_add_false_snd : forall x n m, ~In x m -> n > 0 ->
(snd (npartition f (add x n m)) [=] add x n (snd (npartition f m)) <-> f x n = false).
Proof.
intros. repeat rewrite npartition_spec_snd; trivial. rewrite nfilter_add_true; trivial.
apply negb_true_iff. repeat intro. f_equal. now apply Hf.
Qed.
Theorem npartition_add_fst : forall x n m, ~In x m -> n > 0 ->
fst (npartition f (add x n m)) [=] if f x n then add x n (fst (npartition f m)) else fst (npartition f m).
Proof.
intros x n m Hin Hn. destruct (f x n) eqn:Hfn.
- now rewrite npartition_add_true_fst.
- now rewrite npartition_add_false_fst.
Qed.
Theorem npartition_add_snd : forall x n m, ~In x m -> n > 0 ->
snd (npartition f (add x n m)) [=] if f x n then snd (npartition f m) else add x n (snd (npartition f m)).
Proof.
intros x n m Hin Hn. destruct (f x n) eqn:Hfn.
- now rewrite npartition_add_true_snd.
- now rewrite npartition_add_false_snd.
Qed.
Lemma npartition_swap_fst : forall m, fst (npartition (fun x n => negb (f x n)) m) [=] snd (npartition f m).
Proof.
intros m x. rewrite npartition_spec_fst, npartition_spec_snd; trivial.
repeat intro. rewrite Hf; try eassumption. reflexivity.
Qed.
Lemma npartition_swap_snd : forall m, snd (npartition (fun x n => negb (f x n)) m) [=] fst (npartition f m).
Proof.
intros m x. rewrite npartition_spec_fst, npartition_spec_snd; trivial.
- symmetry. rewrite nfilter_extensionality_compat; trivial. setoid_rewrite negb_involutive. reflexivity.
- repeat intro. rewrite Hf; try eassumption. reflexivity.
Qed.
Lemma npartition_sub_compat_fst :
Proper (E.eq ==> le ==> Bool.leb) f -> Proper (Subset ==> Subset@@1) (npartition f).
Proof. repeat intro. repeat rewrite npartition_spec_fst; trivial. now apply nfilter_sub_compat. Qed.
Lemma npartition_sub_compat_snd :
Proper (E.eq ==> le --> Bool.leb) f -> Proper (Subset ==> Subset@@2) (npartition f).
Proof.
repeat intro. repeat rewrite npartition_spec_snd; trivial. apply nfilter_sub_compat.
- repeat intro. f_equal. now apply Hf.
- clear -H Hf. intros x y Hxy n p Hnp. destruct (f x n) eqn:Hfxn, (f y p) eqn:Hfyp; simpl; auto.
assert (Himpl := H _ _ (symmetry Hxy) _ _ Hnp). rewrite Hfyp, Hfxn in Himpl. discriminate.
- assumption.
Qed.
Lemma npartition_extensionality_compat_fst : forall g, (forall x n, g x n = f x n) ->
forall m, fst (npartition g m) [=] fst (npartition f m).
Proof.
intros ? Hext ? ?. setoid_rewrite npartition_spec_fst at 2; trivial.
rewrite nfilter_extensionality_compat; trivial. apply npartition_spec_fst.
repeat intro. repeat rewrite Hext. apply Hf; assumption.
Qed.
Lemma npartition_extensionality_compat_snd : forall g, (forall x n, g x n = f x n) ->
forall m, snd (npartition g m) [=] snd (npartition f m).
Proof.
intros g Hext m. intro. repeat rewrite npartition_spec_snd; trivial.
+ apply nfilter_extensionality_compat; trivial.
- repeat intro. f_equal. repeat rewrite Hext. apply Hf; assumption.
- repeat intro. f_equal. symmetry. apply Hext.
+ repeat intro. repeat rewrite Hext. apply Hf; assumption.
Qed.
Lemma elements_npartition_fst : forall m,
PermutationA eq_pair (elements (fst (npartition f m)))
(List.filter (fun xn => f (fst xn) (snd xn)) (elements m)).
Proof. intro. rewrite npartition_spec_fst; trivial. now apply elements_nfilter. Qed.
Lemma elements_npartition_snd : forall m,
PermutationA eq_pair (elements (snd (npartition f m)))
(List.filter (fun xn => negb (f (fst xn) (snd xn))) (elements m)).
Proof. intro. rewrite npartition_spec_snd; trivial. apply elements_nfilter, negf_compatb. Qed.
Lemma npartition_from_elements_fst : forall l, is_elements l ->
fst (npartition f (from_elements l)) [=] from_elements (List.filter (fun xn => f (fst xn) (snd xn)) l).
Proof. intros. rewrite npartition_spec_fst; trivial. now apply nfilter_from_elements. Qed.
Lemma npartition_from_elements_snd : forall l, is_elements l ->
snd (npartition f (from_elements l)) [=] from_elements (List.filter (fun xn => negb (f (fst xn) (snd xn))) l).
Proof. intros. rewrite npartition_spec_snd; auto. now apply nfilter_from_elements; try apply negf_compatb. Qed.
Lemma support_npartition_fst : forall m, inclA E.eq (support (fst (npartition f m))) (support m).
Proof. intro. apply support_sub_compat, npartition_subset_fst. Qed.
Lemma support_npartition_snd : forall m, inclA E.eq (support (snd (npartition f m))) (support m).
Proof. intro. apply support_sub_compat, npartition_subset_snd. Qed.
Lemma cardinal_npartition_fst : forall m, cardinal (fst (npartition f m)) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, npartition_subset_fst. Qed.
Lemma cardinal_npartition_snd : forall m, cardinal (snd (npartition f m)) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, npartition_subset_snd. Qed.
Lemma npartition_nfilter_fst : forall m, size (fst (npartition f m)) <= size m.
Proof. intro. apply size_sub_compat, npartition_subset_fst. Qed.
Lemma npartition_nfilter_snd : forall m, size (snd (npartition f m)) <= size m.
Proof. intro. apply size_sub_compat, npartition_subset_snd. Qed.
Lemma npartition_injective : injective eq (eq * eq)%signature (npartition f).
Proof.
intros m1 m2 [Heq1 Heq2] x. specialize (Heq1 x). specialize (Heq2 x).
do 2 rewrite npartition_spec_fst, nfilter_spec in *; trivial.
do 2 rewrite npartition_spec_snd, nfilter_spec in *; trivial; try now apply negf_compatb.
destruct (f x (multiplicity x m1)), (f x (multiplicity x m2)); simpl in *; omega.
Qed.
End nPartition_results.
Section nPartition2_results.
Variable f g : E.t -> nat -> bool.
Hypothesis (Hf : compatb f) (Hg : compatb g).
Lemma npartition_nfilter_merge_fst :
forall m, fst (npartition f (nfilter g m)) [=] nfilter (fun x n => f x n && g x n) m.
Proof.
intros m x. rewrite npartition_spec_fst; trivial. repeat rewrite nfilter_spec; trivial.
+ destruct (g x (multiplicity x m)), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
Qed.
Lemma npartition_nfilter_merge_snd :
forall m, snd (npartition f (nfilter g m)) [=] nfilter (fun x n => negb (f x n) && g x n) m.
Proof.
intros m x. rewrite npartition_spec_snd; trivial. repeat rewrite nfilter_spec; trivial.
+ destruct (g x (multiplicity x m)), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
+ now apply negf_compatb.
Qed.
Lemma nfilter_npartition_merge_fst :
forall m, nfilter f (fst (npartition g m)) [=] nfilter (fun x n => f x n && g x n) m.
Proof.
intros m x. rewrite npartition_spec_fst; trivial. repeat rewrite nfilter_spec; trivial.
+ destruct (g x (multiplicity x m)), (f x (multiplicity x m)); simpl; trivial; now destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
Qed.
Lemma nfilter_npartition_merge_snd :
forall m, nfilter f (snd (npartition g m)) [=] nfilter (fun x n => f x n && negb (g x n)) m.
Proof.
intros m x. rewrite npartition_spec_snd; trivial. repeat rewrite nfilter_spec; trivial.
+ destruct (f x (multiplicity x m)) eqn:Hfx, (g x (multiplicity x m));
simpl; trivial; now rewrite Hfx || destruct (f x 0).
+ clear x m. intros x y Hxy n m Hnm. subst. now rewrite Hxy.
+ now apply negf_compatb.
Qed.
Lemma npartition_merge_fst_fst :
forall m, fst (npartition f (fst (npartition g m))) [=] nfilter (fun x n => f x n && g x n) m.
Proof. intro. repeat rewrite npartition_spec_fst; trivial. now apply nfilter_merge. Qed.
Lemma npartition_merge_fst_snd :
forall m, snd (npartition f (fst (npartition g m))) [=] nfilter (fun x n => negb (f x n) && g x n) m.
Proof.
intro. repeat rewrite npartition_spec_fst, npartition_spec_snd; trivial.
apply negf_compatb in Hf. now rewrite nfilter_merge.
Qed.
Lemma npartition_merge_snd_fst :
forall m, fst (npartition f (snd (npartition g m))) [=] nfilter (fun x n => f x n && negb (g x n)) m.
Proof.
intro. repeat rewrite npartition_spec_fst, npartition_spec_snd; trivial.
apply negf_compatb in Hg. now rewrite nfilter_merge.
Qed.
End nPartition2_results.
Lemma npartition_merge_snd_snd : forall f g, compatb f -> compatb g ->
forall m, snd (npartition f (snd (npartition g m))) [=] nfilter (fun x n => negb (f x n) && negb (g x n)) m.
Proof.
intros f g Hf Hg m. repeat rewrite npartition_spec_snd; trivial. rewrite nfilter_npartition_merge_snd; trivial.
- reflexivity.
- now apply negf_compatb.
Qed.
Lemma npartition_comm_fst : forall f g, compatb f -> compatb g ->
forall m, fst (npartition f (fst (npartition g m))) [=] fst (npartition g (fst (npartition f m))).
Proof.
intros. repeat rewrite npartition_merge_fst_fst; trivial. apply nfilter_extensionality_compat.
- intros x y Hxy ? n ?. subst. now rewrite Hxy.
- intros. apply andb_comm.
Qed.
Lemma npartition_comm_snd : forall f g, compatb f -> compatb g ->
forall m, snd (npartition f (snd (npartition g m))) [=] snd (npartition g (snd (npartition f m))).
Proof.
intros. repeat rewrite npartition_merge_snd_snd; trivial. apply nfilter_extensionality_compat.
- intros x y Hxy ? n ?. subst. now rewrite Hxy.
- intros. apply andb_comm.
Qed.
(** ** Results about [partition] **)
Section Partition_results.
Variable f : E.t -> bool.
Hypothesis Hf : Proper (E.eq ==> Logic.eq) f.
Lemma negf_proper : Proper (E.eq ==> Logic.eq) (fun x => negb (f x)).
Proof. repeat intro. now rewrite Hf. Qed.
Lemma partition_In_fst : forall x m, In x (fst (partition f m)) <-> In x m /\ f x = true.
Proof. intros. rewrite partition_spec_fst; trivial. now apply filter_In. Qed.
Lemma partition_In_snd : forall x m, In x (snd (partition f m)) <-> In x m /\ f x = false.
Proof.
intros. rewrite partition_spec_snd, <- negb_true_iff; trivial. apply filter_In. repeat intro. now rewrite Hf.
Qed.
Corollary In_partition_fst : forall x m, In x (fst (partition f m)) -> In x m.
Proof. intros x m Hin. rewrite partition_In_fst in Hin; intuition. Qed.
Corollary In_partition_snd : forall x m, In x (snd (partition f m)) -> In x m.
Proof. intros x m Hin. rewrite partition_In_snd in Hin; intuition. Qed.
Lemma partition_subset_fst : forall m, fst (partition f m) [<=] m.
Proof. intro. rewrite partition_spec_fst; trivial. now apply filter_subset. Qed.
Lemma partition_subset_snd : forall m, snd (partition f m) [<=] m.
Proof. intro. rewrite partition_spec_snd; trivial. apply filter_subset, negf_proper. Qed.
Lemma partition_add_true_fst : forall x n m, ~In x m -> n > 0 ->
(fst (partition f (add x n m)) [=] add x n (fst (partition f m)) <-> f x = true).
Proof. intros. repeat rewrite partition_spec_fst; trivial. now apply filter_add_true. Qed.
Lemma partition_add_true_snd : forall x n m, ~In x m -> n > 0 ->
(snd (partition f (add x n m)) [=] snd (partition f m) <-> f x = true).
Proof.
intros. repeat rewrite partition_spec_snd; trivial. rewrite filter_add_false; trivial.
apply negb_false_iff. repeat intro. f_equal. now apply Hf.
Qed.
Lemma partition_add_false_fst : forall x n m, ~In x m -> n > 0 ->
(fst (partition f (add x n m)) [=] fst (partition f m) <-> f x = false).
Proof. intros. repeat rewrite partition_spec_fst; trivial. now apply filter_add_false. Qed.
Lemma partition_add_false_snd : forall x n m, ~In x m -> n > 0 ->
(snd (partition f (add x n m)) [=] add x n (snd (partition f m)) <-> f x = false).
Proof.
intros. repeat rewrite partition_spec_snd; trivial. rewrite filter_add_true; trivial.
apply negb_true_iff. repeat intro. f_equal. now apply Hf.
Qed.
Theorem partition_add_fst : forall x n m, ~In x m -> n > 0 ->
fst (partition f (add x n m)) [=] if f x then add x n (fst (partition f m)) else fst (partition f m).
Proof.
intros x n m Hin Hn. destruct (f x) eqn:Hfn.
- now rewrite partition_add_true_fst.
- now rewrite partition_add_false_fst.
Qed.
Theorem partition_add_snd : forall x n m, ~In x m -> n > 0 ->
snd (partition f (add x n m)) [=] if f x then snd (partition f m) else add x n (snd (partition f m)).
Proof.
intros x n m Hin Hn. destruct (f x) eqn:Hfn.
- now rewrite partition_add_true_snd.
- now rewrite partition_add_false_snd.
Qed.
Lemma partition_swap_fst : forall m, fst (partition (fun x => negb (f x)) m) [=] snd (partition f m).
Proof.
intros m x. rewrite partition_spec_fst, partition_spec_snd; trivial.
repeat intro. rewrite Hf; try eassumption. reflexivity.
Qed.
Lemma partition_swap_snd : forall m, snd (partition (fun x => negb (f x)) m) [=] fst (partition f m).
Proof.
intros m x. rewrite partition_spec_fst, partition_spec_snd; trivial.
- symmetry. rewrite filter_extensionality_compat; trivial. setoid_rewrite negb_involutive. reflexivity.
- repeat intro. rewrite Hf; try eassumption. reflexivity.
Qed.
Lemma partition_sub_compat_fst :
Proper (E.eq ==> Bool.leb) f -> Proper (Subset ==> Subset@@1) (partition f).
Proof. repeat intro. repeat rewrite partition_spec_fst; trivial. now apply filter_sub_compat. Qed.
Lemma partition_sub_compat_snd :
Proper (E.eq --> Bool.leb) f -> Proper (Subset ==> Subset@@2) (partition f).
Proof.
repeat intro. repeat rewrite partition_spec_snd; trivial. apply filter_sub_compat.
- repeat intro. f_equal. now apply Hf.
- assumption.
Qed.
Lemma partition_extensionality_compat_fst : forall g, (forall x, g x = f x) ->
forall m, fst (partition g m) [=] fst (partition f m).
Proof.
intros ? Hext ? ?. setoid_rewrite partition_spec_fst at 2; trivial.
rewrite filter_extensionality_compat; trivial. apply partition_spec_fst.
repeat intro. repeat rewrite Hext. apply Hf; assumption.
Qed.
Lemma partition_extensionality_compat_snd : forall g, (forall x, g x = f x) ->
forall m, snd (partition g m) [=] snd (partition f m).
Proof.
intros g Hext m. intro. repeat rewrite partition_spec_snd; trivial.
+ apply filter_extensionality_compat; trivial.
- repeat intro. f_equal. repeat rewrite Hext. apply Hf; assumption.
- repeat intro. f_equal. symmetry. apply Hext.
+ repeat intro. repeat rewrite Hext. apply Hf; assumption.
Qed.
Lemma elements_partition_fst : forall m,
PermutationA eq_pair (elements (fst (partition f m)))
(List.filter (fun xn => f (fst xn)) (elements m)).
Proof. intro. rewrite partition_spec_fst; trivial. now apply elements_filter. Qed.
Lemma elements_partition_snd : forall m,
PermutationA eq_pair (elements (snd (partition f m)))
(List.filter (fun xn => negb (f (fst xn))) (elements m)).
Proof. intro. rewrite partition_spec_snd; trivial. apply elements_filter, negf_proper. Qed.
Lemma partition_from_elements_fst : forall l, is_elements l ->
fst (partition f (from_elements l)) [=] from_elements (List.filter (fun xn => f (fst xn)) l).
Proof. intros. rewrite partition_spec_fst; trivial. now apply filter_from_elements. Qed.
Lemma partition_from_elements_snd : forall l, is_elements l ->
snd (partition f (from_elements l)) [=] from_elements (List.filter (fun xn => negb (f (fst xn))) l).
Proof. intros. rewrite partition_spec_snd; auto. now apply filter_from_elements; try apply negf_proper. Qed.
Lemma support_partition_fst : forall m, inclA E.eq (support (fst (partition f m))) (support m).
Proof. intro. apply support_sub_compat, partition_subset_fst. Qed.
Lemma support_partition_snd : forall m, inclA E.eq (support (snd (partition f m))) (support m).
Proof. intro. apply support_sub_compat, partition_subset_snd. Qed.
Lemma cardinal_partition_fst : forall m, cardinal (fst (partition f m)) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, partition_subset_fst. Qed.
Lemma cardinal_partition_snd : forall m, cardinal (snd (partition f m)) <= cardinal m.
Proof. intro. apply cardinal_sub_compat, partition_subset_snd. Qed.
Lemma partition_nfilter_fst : forall m, size (fst (partition f m)) <= size m.
Proof. intro. apply size_sub_compat, partition_subset_fst. Qed.
Lemma partition_nfilter_snd : forall m, size (snd (partition f m)) <= size m.
Proof. intro. apply size_sub_compat, partition_subset_snd. Qed.
Lemma partition_injective : injective eq (eq * eq)%signature (partition f).
Proof.
intros m1 m2 [Heq1 Heq2] x. specialize (Heq1 x). specialize (Heq2 x).
do 2 rewrite partition_spec_fst, filter_spec in *; trivial.
do 2 rewrite partition_spec_snd, filter_spec in *; trivial; try now apply negf_proper.
destruct (f x); simpl in *; omega.
Qed.
End Partition_results.
Section Partition2_results.
Variable f g : E.t -> bool.
Hypothesis (Hf : Proper (E.eq ==> Logic.eq) f) (Hg : Proper (E.eq ==> Logic.eq) g).
Lemma partition_filter_merge_fst :
forall m, fst (partition f (filter g m)) [=] filter (fun x => f x && g x) m.
Proof.
intros m x. rewrite partition_spec_fst; trivial. repeat rewrite filter_spec; trivial.
- now destruct (g x), (f x).
- clear x m. intros x y Hxy. now rewrite Hxy.
Qed.
Lemma partition_filter_merge_snd :
forall m, snd (partition f (filter g m)) [=] filter (fun x => negb (f x) && g x) m.
Proof.
intros m x. rewrite partition_spec_snd; trivial. repeat rewrite filter_spec; trivial.
- now destruct (g x), (f x).
- clear x m. intros x y Hxy. now rewrite Hxy.
- now apply negf_proper.
Qed.
Lemma filter_partition_merge_fst :
forall m, filter f (fst (partition g m)) [=] filter (fun x => f x && g x) m.
Proof.
intros m x. rewrite partition_spec_fst; trivial. repeat rewrite filter_spec; trivial.
- now destruct (g x), (f x).
- clear x m. intros x y Hxy. now rewrite Hxy.
Qed.
Lemma filter_partition_merge_snd :
forall m, filter f (snd (partition g m)) [=] filter (fun x => f x && negb (g x)) m.
Proof.
intros m x. rewrite partition_spec_snd; trivial. repeat rewrite filter_spec; trivial.
- now destruct (f x), (g x).
- clear x m. intros x y Hxy. now rewrite Hxy.
- now apply negf_proper.
Qed.
Lemma partition_merge_fst_fst :
forall m, fst (partition f (fst (partition g m))) [=] filter (fun x => f x && g x) m.
Proof. intro. repeat rewrite partition_spec_fst; trivial. now apply filter_merge. Qed.
Lemma partition_merge_fst_snd :
forall m, snd (partition f (fst (partition g m))) [=] filter (fun x => negb (f x) && g x) m.
Proof.
intro. repeat rewrite partition_spec_fst, partition_spec_snd; trivial.
apply negf_proper in Hf. now rewrite filter_merge.
Qed.
Lemma partition_merge_snd_fst :
forall m, fst (partition f (snd (partition g m))) [=] filter (fun x => f x && negb (g x)) m.
Proof.
intro. repeat rewrite partition_spec_fst, partition_spec_snd; trivial.
apply negf_proper in Hg. now rewrite filter_merge.
Qed.
End Partition2_results.
Lemma partition_merge_snd_snd : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
forall m, snd (partition f (snd (partition g m))) [=] filter (fun x => negb (f x) && negb (g x)) m.
Proof.
intros f g Hf Hg m. rewrite partition_spec_snd, filter_partition_merge_snd; trivial.
- reflexivity.
- now apply negf_proper.
Qed.
Lemma partition_comm_fst : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
forall m, fst (partition f (fst (partition g m))) [=] fst (partition g (fst (partition f m))).
Proof.
intros. repeat rewrite partition_merge_fst_fst; trivial. apply filter_extensionality_compat.
- intros x y Hxy. now rewrite Hxy.
- intros. apply andb_comm.
Qed.
Lemma partition_comm_snd : forall f g, Proper (E.eq ==> Logic.eq) f -> Proper (E.eq ==> Logic.eq) g ->
forall m, snd (partition f (snd (partition g m))) [=] snd (partition g (snd (partition f m))).
Proof.
intros. repeat rewrite partition_merge_snd_snd; trivial. apply filter_extensionality_compat.
- intros x y Hxy. now rewrite Hxy.
- intros. apply andb_comm.
Qed.
(** ** Results about [choose] **)
Lemma choose_In : forall m, (exists x, In x m) <-> exists x, choose m = Some x.
Proof.
intro m. split; intros [x Hin].
- destruct (choose m) eqn:Hm; eauto. exfalso. rewrite choose_None in Hm. rewrite Hm in Hin. apply (In_empty Hin).
- exists x. now apply choose_Some.
Qed.
Lemma choose_not_None : forall m, choose m <> None <-> ~m [=] empty.
Proof. intro. now rewrite choose_None. Qed.
Lemma choose_sub_Some : forall m1 m2, m1 [<=] m2 -> choose m1 <> None -> choose m2 <> None.
Proof. intros ? ? Hle Hm1 Habs. apply Hm1. rewrite choose_None in *. now rewrite <- subset_empty_r, <- Habs. Qed.
Lemma choose_add_None : forall x n m, n > 0 -> choose (add x n m) <> None.
Proof. intros. rewrite choose_None, add_is_empty. omega. Qed.
(*
Lemma choose_union : forall m1 m2, choose (union m1 m2) = None <-> m1 [=] empty /\ m2 [=] empty.
Proof. intros. rewrite choose_None. apply empty_union. Qed.
Lemma choose_inter : forall m1 m2, choose (inter m1 m2) = None <->
forall x, ~In x m1 /\ ~In x m2 \/ In x m1 /\ ~In x m2 \/ ~In x m1 /\ In x m2.
Proof. intros. rewrite choose_None. apply empty_inter. Qed.
Lemma choose_diff : forall m1 m2, choose (diff m1 m2) = None <-> m1 [<=] m2.
Proof. intros. rewrite choose_None. apply diff_empty_subset. Qed.
Lemma choose_lub : forall m1 m2, choose (lub m1 m2) = None <-> m1 [=] empty /\ m2 [=] empty.
Proof. intros. rewrite choose_None. apply lub_is_empty. Qed.
*)
(** ** Results about [for_all] and [For_all] **)
Section for_all_results.
Variable f : E.t -> nat -> bool.
Hypothesis Hf : compatb f.
Lemma for_all_false : forall m, for_all f m = false <-> ~For_all (fun x n => f x n = true) m.
Proof.
intro m. destruct (for_all f m) eqn:Hfm.
- rewrite for_all_spec in Hfm; trivial. intuition.
- rewrite <- for_all_spec; trivial. intuition. rewrite Hfm in *. discriminate.
Qed.
Lemma for_all_add : forall x n m, n > 0 -> ~In x m -> for_all f (add x n m) = f x n && for_all f m.
Proof.
intros x n m Hn Hin. destruct (for_all f (add x n m)) eqn:Hm.
+ rewrite for_all_spec in Hm; trivial. symmetry. rewrite andb_true_iff. split.
- specialize (Hm x). msetdec. assert (Hx : multiplicity x m = 0) by omega. rewrite Hx in *. now apply Hm.
- rewrite for_all_spec; trivial. intros y Hy. rewrite <- (add_other x y n).
apply Hm. msetdec.
intro Heq. apply Hin. now rewrite <- Heq.
+ symmetry. rewrite andb_false_iff. destruct (f x n) eqn:Hfn; intuition. right.
rewrite for_all_false in *; trivial. intro Habs. apply Hm. intros y Hy. msetdec.
- assert (multiplicity x m = 0) by omega. now rewrite H.
- now apply Habs.
Qed.
(** Compatibility with [\[<=\]] does not hold because new bindings can appear. *)
Lemma for_all_sub_compat : Proper (E.eq ==> le ==> Bool.leb) f -> Proper (Subset ==> Bool.leb) (for_all f).
Proof. Abort.
Lemma for_all_disjoint_union : forall m1 m2,
inter m1 m2 [=] empty -> for_all f (union m1 m2) = for_all f m1 && for_all f m2.
Proof.
intros m1 m2 Hm. rewrite empty_inter in Hm.
destruct (for_all f m1) eqn:Hfm1; [destruct (for_all f m2) eqn:Hfm2 |];
simpl; try rewrite for_all_spec in *; try rewrite for_all_false in *; trivial.
+ intros x Hin. rewrite union_In in Hin. specialize (Hm x). destruct Hin as [Hin | Hin].
- destruct Hm as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction. apply Hfm1 in Hin.
rewrite not_In in Hin2. now rewrite union_spec, Hin2, plus_0_r.
- destruct Hm as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction. apply Hfm2 in Hin.
rewrite not_In in Hin1. now rewrite union_spec, Hin1.
+ intro Habs. apply Hfm2. intros x Hin.
destruct (Hm x) as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction.
rewrite not_In in Hin1. setoid_rewrite <- plus_0_l. rewrite <- Hin1, <- union_spec.
apply Habs. rewrite union_In. auto.
+ intro Habs. apply Hfm1. intros x Hin.
destruct (Hm x) as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction.
rewrite not_In in Hin2. setoid_rewrite <- plus_0_r. rewrite <- Hin2, <- union_spec.
apply Habs. rewrite union_In. auto.
Qed.
Lemma for_all_inter : forall m1 m2,
for_all f m1 = true -> for_all f m2 = true -> for_all f (inter m1 m2) = true.
Proof.
intros m1 m2 Hm1 Hm2. rewrite for_all_spec in *; trivial. intros x Hin. rewrite inter_In in Hin.
destruct Hin. rewrite inter_spec. now apply Nat.min_case; apply Hm1 || apply Hm2.
Qed.
Lemma for_all_lub : forall m1 m2, for_all f m1 = true -> for_all f m2 = true -> for_all f (lub m1 m2) = true.
Proof.
intros m1 m2 Hm1 Hm2. rewrite for_all_spec in *; trivial. intros x Hin. rewrite lub_In in Hin. rewrite lub_spec.
apply Nat.max_case_strong; intro; apply Hm1 || apply Hm2; destruct Hin; unfold In in *; omega.
Qed.
Lemma for_all_choose : forall m x, for_all f m = true -> choose m = Some x -> f x (multiplicity x m) = true.
Proof. intros m x Hm Hx. rewrite for_all_spec in Hm; trivial. now apply Hm, choose_Some. Qed.
End for_all_results.
Lemma For_all_elements : forall f, Proper (E.eq ==> Logic.eq ==> iff) f ->
forall m, For_all f m <-> List.Forall (fun xn => f (fst xn) (snd xn)) (elements m).
Proof.
intros f Hf m. rewrite List.Forall_forall. split; intro Hall.
+ intros [x n] Hin. simpl. apply (@In_InA _ eq_pair _) in Hin.
assert (In x m). { rewrite <- (elements_In x 0). eapply InA_pair_elt; eassumption. }
rewrite elements_spec in Hin. destruct Hin as [? _]. simpl in *. subst. now apply Hall.
+ intros x Hin. rewrite <- (elements_In x 0) in Hin. apply InA_elt_pair in Hin. destruct Hin as [n Hin].
assert (Hin' : exists y, List.In (y, n) (elements m) /\ E.eq y x).
{ rewrite InA_alt in Hin. destruct Hin as [[y p] [[Heqx Heqn] Hin]]. compute in Heqx, Heqn. subst. now exists y. }
rewrite elements_spec in Hin. destruct Hin as [Heq Hpos]. simpl in *. subst.
destruct Hin' as [y [Hin' Heq]]. rewrite <- Heq at 1. now apply (Hall (y, multiplicity x m)).
Qed.
Corollary For_all_from_elements_valid : forall f, Proper (E.eq ==> Logic.eq ==> iff) f ->
forall l, is_elements l -> For_all f (from_elements l) <-> List.Forall (fun xn => f (fst xn) (snd xn)) l.
Proof.
intros f Hf l Hl.
assert (Hf' : Proper (eq_pair ==> iff) (fun xn => f (fst xn) (snd xn))).
{ intros [? ?] [? ?] [Heq Hn]. compute in Heq, Hn. subst. simpl. now rewrite Heq. }
rewrite <- (elements_from_elements Hl) at 2. now apply For_all_elements.
Qed.
Section for_all2_results.
Variable f g : E.t -> nat -> bool.
Hypothesis (Hf : compatb f) (Hg : compatb g).
Lemma for_all_andb : forall m, for_all (fun x n => f x n && g x n) m = for_all f m && for_all g m.
Proof.
intro m.
assert (Hfg : compatb (fun x n => f x n && g x n)). { intros ? ? Heq ? ? ?. subst. now rewrite Heq. }
destruct (for_all f m) eqn:Hfm; [destruct (for_all g m) eqn:Hgm |];
simpl; try rewrite for_all_spec in *; try rewrite for_all_false in *; trivial.
- intros x Hin. now rewrite Hfm, Hgm.
- intro Habs. apply Hgm. intros x Hin. apply Habs in Hin. now rewrite andb_true_iff in Hin.
- intro Habs. apply Hfm. intros x Hin. apply Habs in Hin. now rewrite andb_true_iff in Hin.
Qed.
Lemma for_all_nfilter : forall m, for_all f m = true -> for_all f (nfilter g m) = true.
Proof.
intros m Hm. rewrite for_all_spec in *; trivial. intros x Hin. unfold In in Hin.
rewrite nfilter_spec in *; trivial. now destruct (g x (multiplicity x m)); apply Hm || omega.
Qed.
Lemma for_all_nfilter_merge : forall m,
for_all f (nfilter g m) = for_all (fun x n => if g x n then f x n else true) m.
Proof.
assert (Hfg : compatb (fun x n => if g x n then f x n else true)).
{ intros x y Hxy n p Hnp. subst. rewrite Hxy. destruct (g y p); trivial. now rewrite Hxy. }
intro m. destruct (for_all f (nfilter g m)) eqn:Hfgm; symmetry.
+ rewrite for_all_spec in *; trivial. intros x Hin. destruct (g x (multiplicity x m)) eqn:Hgm; trivial.
specialize (Hfgm x). rewrite nfilter_spec, Hgm in Hfgm; trivial. apply Hfgm. rewrite nfilter_In; auto.
+ rewrite for_all_false in *; trivial. intros Habs. apply Hfgm. intros x Hin. rewrite nfilter_In in Hin; auto.
destruct Hin as [Hin Hgm]. apply Habs in Hin. rewrite nfilter_spec; trivial. now rewrite Hgm in *.
Qed.
End for_all2_results.
(*
Lemma for_all_partition_fst : forall m, for_all f m = true -> for_all f (fst (partition g m)) = true.
Proof. intros. setoid_rewrite partition_spec_fst; trivial. now apply for_all_nfilter. Qed.
Lemma for_all_partition_snd : forall f g, compatb f -> compatb g ->
forall m, for_all f m = true -> for_all f (snd (partition g m)) = true.
Proof. intros. rewrite partition_spec_snd; trivial. apply for_all_nfilter; trivial. now apply negf_compatb. Qed.
*)
(** ** Results about [exists_] and [Exists] **)
Section exists_results.
Variable f : E.t -> nat -> bool.
Hypothesis Hf : compatb f.
Lemma exists_not_empty : forall m, exists_ f m = true -> ~m [=] empty.
Proof.
intros m Hm. rewrite exists_spec in Hm; trivial. rewrite not_empty_In. destruct Hm as [x [? ?]]. now exists x.
Qed.
Lemma exists_false : forall m, exists_ f m = false <-> ~Exists (fun x n => f x n = true) m.
Proof.
intro m. destruct (exists_ f m) eqn:Hfm.
- rewrite exists_spec in Hfm; trivial. intuition.
- rewrite <- exists_spec; trivial. intuition. rewrite Hfm in *. discriminate.
Qed.
Lemma exists_add : forall x n m, n > 0 -> ~In x m -> exists_ f (add x n m) = f x n || exists_ f m.
Proof.
intros x n m Hn Hin. destruct (exists_ f (add x n m)) eqn:Hm.
+ rewrite exists_spec in Hm; trivial. symmetry. rewrite orb_true_iff. destruct Hm as [y [Hy Hfy]]. msetdec.
- left. assert (Hm : multiplicity x m = 0) by omega. now rewrite Hm in Hfy.
- right. exists y. now split.
+ symmetry. rewrite orb_false_iff. rewrite exists_false in *; trivial.
assert (Hxm : multiplicity x m = 0) by (unfold In in Hin; omega). split.
- destruct (f x n) eqn:Hfxn; trivial. elim Hm. exists x. split; msetdec. now rewrite Hxm.
- intros [y [Hy Hfy]]. apply Hm. exists y. unfold In in *. split; msetdec.
Qed.
Lemma exists_sub_compat : Proper (E.eq ==> le ==> Bool.leb) f -> Proper (Subset ==> Bool.leb) (exists_ f).
Proof.
intros Hf2 m1. pattern m1. apply ind; clear m1.
* intros m1 m2 Hm. setoid_rewrite Hm. reflexivity.
* intros m x n Hm Hn Hrec m2 Hle. destruct (exists_ f (add x n m)) eqn:Hall; try now intuition.
simpl. rewrite exists_add in Hall; trivial. rewrite orb_true_iff in Hall. destruct Hall as [Hall | Hall].
+ specialize (Hle x). rewrite not_In in Hm. rewrite add_same, Hm in Hle.
rewrite <- (@add_remove_cancel x), exists_add; trivial.
- apply (Hf2 _ _ (reflexivity x)) in Hle. simpl in Hle. rewrite Hall in Hle. simpl in Hle. now rewrite Hle.
- omega.
- rewrite remove_In. intros [[_ Habs] | [Habs _]]; omega || now elim Habs.
+ setoid_rewrite Hall in Hrec. simpl in Hrec. apply Hrec. etransitivity; try eassumption. apply add_subset.
* intros. rewrite exists_empty; trivial. intuition.
Qed.
Lemma exists_disjoint_union : forall m1 m2,
inter m1 m2 [=] empty -> exists_ f (union m1 m2) = exists_ f m1 || exists_ f m2.
Proof.
intros m1 m2 Hm. rewrite empty_inter in Hm.
destruct (exists_ f m1) eqn:Hfm1; [| destruct (exists_ f m2) eqn:Hfm2];
simpl; try rewrite exists_spec in *; try rewrite exists_false in *; trivial;
try destruct Hfm1 as [x [Hin Hfm1]] || destruct Hfm2 as [x [Hin Hfm2]].
+ exists x. specialize (Hm x). destruct Hm as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction.
rewrite union_In. split; auto. rewrite union_spec. rewrite not_In in Hin2. now rewrite Hin2, plus_0_r.
+ exists x. specialize (Hm x). destruct Hm as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction.
rewrite union_In. split; auto. rewrite union_spec. rewrite not_In in Hin1. now rewrite Hin1.
+ intro Habs. destruct Habs as [x [Hin Habs]]. rewrite union_In in Hin. specialize (Hm x).
destruct Hin; destruct Hm as [[Hin1 Hin2] | [[Hin1 Hin2] | [Hin1 Hin2]]]; try contradiction.
- apply Hfm1. exists x. rewrite not_In in Hin2. rewrite union_spec, Hin2, plus_0_r in Habs. now split.
- apply Hfm2. exists x. rewrite not_In in Hin1. rewrite union_spec, Hin1 in Habs. now split.
Qed.
Lemma exists_inter : forall m1 m2, exists_ f (inter m1 m2) = true -> exists_ f m1 = true \/ exists_ f m2 = true.
Proof.
intros m1 m2. repeat rewrite exists_spec; trivial. intros [x [Hin Hfx]].
rewrite inter_spec in Hfx. rewrite inter_In in Hin. destruct Hin.
destruct (Min.min_dec (multiplicity x m1) (multiplicity x m2)) as [Hmin | Hmin];
rewrite Hmin in Hfx; left + right; now exists x.
Qed.
Lemma exists_lub : forall m1 m2, exists_ f (lub m1 m2) = true -> exists_ f m1 = true \/ exists_ f m2 = true.
Proof.
intros m1 m2. repeat rewrite exists_spec; trivial. intros [x [Hin Hfx]]. unfold In in *. rewrite lub_spec in *.
destruct (Max.max_dec (multiplicity x m1) (multiplicity x m2)) as [Hmax | Hmax];
rewrite Hmax in *; left + right; now exists x.
Qed.
Lemma exists_false_for_all : forall m, exists_ f m = false <-> for_all (fun x n => negb (f x n)) m = true.
Proof.
intro m. rewrite exists_false, for_all_spec; try now apply negf_compatb.
split; intro Hm.
- intros x Hin. destruct (f x (multiplicity x m)) eqn:Habs; trivial. exfalso. apply Hm. now exists x.
- intros [x [Hin Hx]]. apply Hm in Hin. rewrite Hx in Hin. discriminate.
Qed.
Lemma for_all_false_exists : forall m, for_all f m = false <-> exists_ (fun x n => negb (f x n)) m = true.
Proof.
assert (Hnegf := negf_compatb Hf).
assert (Hf' : Proper (E.eq ==> Logic.eq ==> iff) (fun x n => f x n = true)).
{ intros ? ? Heq ? ? ?. subst. now rewrite Heq. }
assert (Hnegf' : Proper (E.eq ==> Logic.eq ==> iff) (fun x n => negb (f x n) = true)).
{ intros ? ? Heq ? ? ?. subst. now rewrite Heq. }
intro m. rewrite for_all_false, exists_spec; trivial. split; intro Hm.
* revert Hm. pattern m. apply ind; clear m.
+ intros m1 m2 Hm. now rewrite Hm.
+ intros m x n Hm Hn Hrec Hall. destruct (f x n) eqn:Hfxn.
- { destruct Hrec as [y [Hin Hy]].
+ intro Habs. apply Hall. intros y Hin. rewrite add_In in Hin. destruct (E.eq_dec y x) as [Heq | Heq].
- rewrite not_In in Hm. now rewrite Heq, add_same, Hm.
- destruct Hin as [Hin | [_ ?]]; try contradiction. apply Habs in Hin. now rewrite add_other.
+ exists y. split.
- rewrite add_In. now left.
- rewrite add_other; trivial. intro Heq. apply Hm. now rewrite <- Heq. }
- { exists x. split.
+ rewrite add_In. right. split. omega. reflexivity.
+ rewrite not_In in Hm. rewrite add_same, Hm. simpl. now rewrite Hfxn. }
+ intro Habs. elim Habs. intros x Hin. elim (In_empty Hin).
* intro Habs. destruct Hm as [x [Hin Hx]]. apply Habs in Hin. rewrite Hin in Hx. discriminate.
Qed.
Lemma exists_choose : forall m x, choose m = Some x -> f x (multiplicity x m) = true -> exists_ f m = true.
Proof. intros m x Hm Hx. apply choose_Some in Hm. rewrite exists_spec; trivial. now exists x. Qed.
End exists_results.
Lemma Exists_elements : forall f, Proper (E.eq ==> Logic.eq ==> iff) f ->
forall m, Exists f m <-> List.Exists (fun xn => f (fst xn) (snd xn)) (elements m).
Proof.
intros f Hf m. rewrite List.Exists_exists. split; intro Hm.
+ destruct Hm as [x [Hin Hfx]]. rewrite <- (elements_In x 0) in Hin.
apply InA_elt_pair in Hin. destruct Hin as [n Hin].
assert (n = multiplicity x m). { rewrite elements_spec in Hin. intuition. }
rewrite InA_alt in Hin. destruct Hin as [[y p] [[Heqx Heqn] Hin]].
compute in Heqx, Heqn. subst. rewrite Heqx in *. clear Heqx x. subst.
exists (y, multiplicity y m). auto.
+ destruct Hm as [[x n] [Hin Hfx]]. apply (@In_InA _ eq_pair _) in Hin. rewrite elements_spec in Hin.
destruct Hin as [Heq Hpos]. simpl in *. subst. now exists x.
Qed.
Corollary Exists_from_elements_valid : forall f, Proper (E.eq ==> Logic.eq ==> iff) f ->
forall l, is_elements l -> Exists f (from_elements l) <-> List.Exists (fun xn => f (fst xn) (snd xn)) l.
Proof.
intros f Hf l Hl.
assert (Hf' : Proper (eq_pair ==> iff) (fun xn => f (fst xn) (snd xn))).
{ intros [? ?] [? ?] [Heq Hn]. compute in Heq, Hn. subst. simpl. now rewrite Heq. }
rewrite <- (elements_from_elements Hl) at 2. now apply Exists_elements.
Qed.
Lemma nfilter_none : forall f, compatb f ->
forall m, nfilter f m [=] empty <-> for_all (fun x n => negb (f x n)) m = true.
Proof.
intros f Hf m.
assert (Hf2 : Proper (E.eq ==> Logic.eq ==> Logic.eq) (fun x n => negb (f x n))).
{ intros x y Hxy ? n ?. subst. now rewrite Hxy. }
assert (Hf3 : Proper (E.eq ==> Logic.eq ==> Logic.eq) (fun x n => negb (negb (f x n)))).
{ intros x y Hxy ? n ?. subst. now rewrite Hxy. }
split; intros Hall.
+ destruct (for_all (fun (x : elt) (n : nat) => negb (f x n)) m) eqn:Hforall; trivial.
rewrite for_all_false_exists, exists_spec in Hforall; trivial.
destruct Hforall as [x [Hin Hfx]]. rewrite negb_involutive in Hfx.
elim (@In_empty x). rewrite <- Hall, nfilter_In; auto.
+ rewrite for_all_spec in Hall; trivial.
destruct (empty_or_In_dec (nfilter f m)) as [? | [x Hin]]; trivial.
rewrite nfilter_In in Hin; trivial. destruct Hin as [Hin Hfx]. apply Hall in Hin.
rewrite Hfx in Hin. discriminate.
Qed.
Section exists2_results.
Variable f g : E.t -> nat -> bool.
Hypothesis (Hf : compatb f) (Hg : compatb g).
Lemma exists_orb : forall m, exists_ (fun x n => f x n || g x n) m = exists_ f m || exists_ g m.
Proof.
intro m.
assert (Hfg : compatb (fun x n => f x n || g x n)). { intros ? ? Heq ? ? ?. subst. now rewrite Heq. }
destruct (exists_ f m) eqn:Hfm; [| destruct (exists_ g m) eqn:Hgm];
simpl; try rewrite exists_spec in *; try rewrite exists_false in *; trivial.
- destruct Hfm as [x [Hin Hfm]]. exists x. now rewrite Hfm.
- destruct Hgm as [x [Hin Hgm]]. exists x. now rewrite Hgm, orb_b_true.
- intros [x [Hin Habs]]. rewrite orb_true_iff in Habs. destruct Habs; apply Hfm + apply Hgm; now exists x.
Qed.
Lemma exists_nfilter : forall m, exists_ f (nfilter g m) = true -> exists_ f m = true.
Proof.
intros m Hm. rewrite exists_spec in *; trivial. destruct Hm as [x [Hin Hfm]]. rewrite nfilter_In in *; trivial.
destruct Hin as [HIn Hgm]. rewrite nfilter_spec, Hgm in Hfm; trivial. now exists x.
Qed.
Lemma exists_nfilter_merge : forall m, exists_ f (nfilter g m) = exists_ (fun x n => f x n && g x n) m.
Proof.
assert (Hfg : compatb (fun x n => f x n && g x n)). { intros ? ? Heq ? ? ?. subst. now rewrite Heq. }
intro m. destruct (exists_ f (nfilter g m)) eqn:Hfgm; symmetry.
+ rewrite exists_spec in *; trivial. destruct Hfgm as [x [Hin Hfm]]. rewrite nfilter_spec in Hfm; trivial.
rewrite nfilter_In in *; trivial. destruct Hin as [Hin Hgm]. exists x. rewrite Hgm, Hfm in *. now split.
+ rewrite exists_false in *; trivial. intros [x [Hin Hm]]. rewrite andb_true_iff in Hm.
destruct Hm as [? Hm]. apply Hfgm. exists x. rewrite nfilter_In, nfilter_spec, Hm; auto.
Qed.
End exists2_results.
(*
Lemma exists_partition_fst : forall m, for_all f m = true -> for_all f (fst (partition g m)) = true.
Proof. intros. setoid_rewrite partition_spec_fst; trivial. now apply for_all_nfilter. Qed.
Lemma for_all_partition_snd : forall f g, compatb f -> compatb g ->
forall m, for_all f m = true -> for_all f (snd (partition g m)) = true.
Proof. intros. rewrite partition_spec_snd; trivial. apply for_all_nfilter; trivial. now apply negf_compatb. Qed.
*)
End Make.
|
State Before: R : Type u_1
instβ : AddMonoidWithOne R
n : β
β’ β(bit1 n) = bit1 βn State After: R : Type u_1
instβ : AddMonoidWithOne R
n : β
β’ bit0 βn + 1 = bit1 βn Tactic: rw [bit1, cast_add_one, cast_bit0] State Before: R : Type u_1
instβ : AddMonoidWithOne R
n : β
β’ bit0 βn + 1 = bit1 βn State After: no goals Tactic: rfl |
% Digital Video Stabilization and Rolling Shutter Correction using Gyroscopes
% Copyright (C) 2011 Alexandre Karpenko
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
function y = lininterp(t1, x, t2)
if (t2(1) < t1(1))
t1 = [t2(1); t1];
x = [x(1,:); x];
end
if (t2(end) > t1(end))
t1 = [t1; t2(end)];
x = [x; x(end,:)];
end
y = interp1(t1, x, t2); |
(* Title: HOL/Auth/n_mutualEx_lemma_on_inv__3.thy
Author: Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
Copyright 2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)
header{*The n_mutualEx Protocol Case Study*}
theory n_mutualEx_lemma_on_inv__3 imports n_mutualEx_base
begin
section{*All lemmas on causal relation between inv__3 and some rule r*}
lemma n_TryVsinv__3:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_Try i)" and
a2: "(\<exists> p__Inv0 p__Inv1. p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_Try i" apply fastforce done
from a2 obtain p__Inv0 p__Inv1 where a2:"p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1" apply fastforce done
have "(i=p__Inv0)\<or>(i=p__Inv1)\<or>(i~=p__Inv0\<and>i~=p__Inv1)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv0)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i=p__Inv1)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv0\<and>i~=p__Inv1)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_CritVsinv__3:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_Crit i)" and
a2: "(\<exists> p__Inv0 p__Inv1. p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_Crit i" apply fastforce done
from a2 obtain p__Inv0 p__Inv1 where a2:"p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1" apply fastforce done
have "(i=p__Inv0)\<or>(i=p__Inv1)\<or>(i~=p__Inv0\<and>i~=p__Inv1)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv0)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Ident ''n'') p__Inv1)) (Const E)) (eqn (IVar (Ident ''x'')) (Const true))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i=p__Inv1)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv0\<and>i~=p__Inv1)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_ExitVsinv__3:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_Exit i)" and
a2: "(\<exists> p__Inv0 p__Inv1. p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_Exit i" apply fastforce done
from a2 obtain p__Inv0 p__Inv1 where a2:"p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1" apply fastforce done
have "(i=p__Inv0)\<or>(i=p__Inv1)\<or>(i~=p__Inv0\<and>i~=p__Inv1)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv0)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i=p__Inv1)"
have "?P3 s"
apply (cut_tac a1 a2 b1, simp, rule_tac x="(neg (andForm (eqn (IVar (Para (Ident ''n'') p__Inv0)) (Const C)) (eqn (IVar (Para (Ident ''n'') p__Inv1)) (Const C))))" in exI, auto) done
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv0\<and>i~=p__Inv1)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
lemma n_IdleVsinv__3:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_Idle i)" and
a2: "(\<exists> p__Inv0 p__Inv1. p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_Idle i" apply fastforce done
from a2 obtain p__Inv0 p__Inv1 where a2:"p__Inv0\<le>N\<and>p__Inv1\<le>N\<and>p__Inv0~=p__Inv1\<and>f=inv__3 p__Inv0 p__Inv1" apply fastforce done
have "(i=p__Inv0)\<or>(i=p__Inv1)\<or>(i~=p__Inv0\<and>i~=p__Inv1)" apply (cut_tac a1 a2, auto) done
moreover {
assume b1: "(i=p__Inv0)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i=p__Inv1)"
have "?P1 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
assume b1: "(i~=p__Inv0\<and>i~=p__Inv1)"
have "?P2 s"
proof(cut_tac a1 a2 b1, auto) qed
then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed
end
|
import Effects
import Effect.Logging.Category
func : Nat -> Eff () [LOG String]
func x = do
warn Nil $ unwords ["I do nothing with", show x]
pure ()
doubleFunc : Nat -> Eff Nat [LOG String]
doubleFunc x = do
logN 40 ["NumOPS"] $ unwords ["Doing the double with", show x ]
func x
pure (x+x)
eMain : Eff Nat [LOG String]
eMain = do
initLogger ALL ["NumOPS"]
doubleFunc 3
main : IO ()
main = do
x <- run eMain
printLn x
|
proposition nullhomotopic_from_sphere_extension: fixes f :: "'M::euclidean_space \<Rightarrow> 'a::real_normed_vector" shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) \<longleftrightarrow> (\<exists>g. continuous_on (cball a r) g \<and> g ` (cball a r) \<subseteq> S \<and> (\<forall>x \<in> sphere a r. g x = f x))" (is "?lhs = ?rhs") |
If $u$ is locally connected, $S$ is closed in $u$, and $c$ is a component of $u - S$, then $S \cup c$ is closed in $u$. |
function download_algorithms(simulate_no_download)
% Downloads and compiles/preps different external codes needed for RIGOR
%
% @authors: Ahmad Humayun
% @contact: [email protected]
% @affiliation: Georgia Institute of Technology
% @date: Fall 2013 - Summer 2014
clc
% if you don't want to download the files (for testing this script)
if ~exist('simulate_no_download', 'var')
simulate_no_download = false;
end
no_download_dir = 'pre_downloaded';
fprintf(2, '---------------------------------------------------------------------------\n Download Notice \n---------------------------------------------------------------------------\nBy running this script, you accept all licensing agreements accompanied\nwith the 3rd party softwares that will now be downloaded and used later in\nour scripts.\n\nPress ''y'' to accept (any other key to stop the script): ');
user_inp = input('','s');
if isempty(user_inp) || ~strncmpi(user_inp(1), 'y', 1)
return;
end
% destination algorithms directory
curr_dir = pwd;
code_root_dir = fullfile(fileparts(which(mfilename)), '..');
extern_src_rel = 'extern_src';
algos_dir = fullfile(code_root_dir, extern_src_rel);
dont_delete = {'DataHash', 'extra_gb_code', 'para_pseudoflow', 'fuxin_lib_src', 'stein_boundaryprocessing'};
if exist(algos_dir,'dir')
% only delete the dir/files not in dont_delete
d = dir(algos_dir);
for idx = 1:length(d)
if strcmp(d(idx).name,'.') || strcmp(d(idx).name,'..')
continue;
end
if ~any(strcmp(d(idx).name, dont_delete))
if d(idx).isdir
rmdir(fullfile(algos_dir,d(idx).name), 's');
else
delete(fullfile(algos_dir,d(idx).name));
end
end
end
else
mkdir(algos_dir);
end
% create temp download directory
if ~simulate_no_download
temp_dir = fullfile(code_root_dir, sprintf('temp%d', randi(1e8)));
mkdir(temp_dir);
else
temp_dir = no_download_dir;
end
try
% Point Fuxin library to the right directory
fuxin_lib_rel = fullfile(extern_src_rel, 'fuxin_lib_src');
replaceInTextFile(fullfile(code_root_dir, 'internal_params.m'), '''/home/ahumayun/videovolumes/fuxin_lib_src''', ['fullfile(fp.code_root_dir, ''', fuxin_lib_rel, ''')'], true);
classregtree_dir = fullfile(fuxin_lib_rel, '@classregtree_fuxin', 'private');
fprintf(1, 'mex''ing fuxin library code\n');
eval(sprintf('mex -O %s/regtreeEval.cpp -output %s/regtreeEval', classregtree_dir, classregtree_dir));
eval(sprintf('mex -O %s/regtree_findbestsplit.cpp -output %s/regtree_findbestsplit', classregtree_dir, classregtree_dir));
fprintf(1, 'Done mex''ing\n');
% move stein boundary processing code to the segmentation folder
mkdir(fullfile(algos_dir, 'segmentation'));
copyfile(fullfile(algos_dir, 'stein_boundaryprocessing'), fullfile(algos_dir, 'segmentation', 'stein_boundaryprocessing'));
% download Leordeanu''s GB code
success = download_code('http://109.101.234.42/documente/code/doc_8.zip', ...
fullfile(temp_dir, 'doc_8.zip'), 'Leordeanu''s GB flow code', simulate_no_download);
if success
mkdir(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb'));
unzip(fullfile(temp_dir, 'doc_8.zip'), fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb'));
movefile(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb', 'Gb_Code_Oct2012', '*'), fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb'));
rmdir(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb', 'Gb_Code_Oct2012'), 's');
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb', 'Gb_data_lambda.m'), 'gb\(f\) = T/2 \+ sqrt\(\(T \.\^ 2 \) /4 \- D\);', 'gb(f) = T(f)/2 + sqrt((T(f) .^ 2 ) /4 - D(f));');
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb', 'Gb_data_lambda.m'), 'or_C\(f, 1\) = -Ms\(:, 2\);', 'or_C(f, 1) = -Ms(f, 2);');
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb', 'Gb_data_lambda.m'), 'or_C\(f, 2\) = gb\(:\) - Ms\(:, 1\);', 'or_C(f, 2) = gb(f) - Ms(f, 1);');
adjustAttributes(fullfile(algos_dir, 'segmentation', 'boundaries--leordeanu_ECCV_2012_gb'));
end
% download Joseph Lim's SketchTokens code
success = download_code('https://github.com/joelimlimit/SketchTokens/archive/master.zip', ...
fullfile(temp_dir, 'master.zip'), 'Joseph Lim''s SketchTokens code', simulate_no_download);
success = success & download_code('http://people.csail.mit.edu/lim/lzd_cvpr2013/st_data.tgz', ...
fullfile(temp_dir, 'st_data.tgz'), 'Joseph Lim''s SketchTokens data', simulate_no_download);
if success
mkdir(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens'));
unzip(fullfile(temp_dir, 'master.zip'), fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens'));
untar(fullfile(temp_dir, 'st_data.tgz'), fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens'));
rmdir(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens', 'SketchTokens-master', 'models'), 's');
movefile(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens', 'models'), fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens', 'SketchTokens-master'));
movefile(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens', 'SketchTokens-master', '*'), fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens'));
rmdir(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens', 'SketchTokens-master'), 's');
cd(fullfile(algos_dir, 'segmentation', 'boundaries--lim_CVPR_2013_sketchtokens'));
mex stDetectMex.cpp 'OPTIMFLAGS="$OPTIMFLAGS' '/openmp"'
fprintf(1, 'Done mex''ing\n');
cd(curr_dir);
end
% download Piotr's toolbox code
success = download_code('http://vision.ucsd.edu/~pdollar/toolbox/piotr_toolbox.zip', ...
fullfile(temp_dir, 'piotr_toolbox.zip'), 'Piotr''s toolbox code', simulate_no_download);
if success
mkdir(fullfile(algos_dir, 'toolboxes', 'piotr_toolbox'));
unzip(fullfile(temp_dir, 'piotr_toolbox.zip'), fullfile(algos_dir, 'toolboxes', 'piotr_toolbox'));
movefile(fullfile(algos_dir, 'toolboxes', 'piotr_toolbox', 'toolbox', '*'), fullfile(algos_dir, 'toolboxes', 'piotr_toolbox'));
rmdir(fullfile(algos_dir, 'toolboxes', 'piotr_toolbox', 'toolbox'), 's');
% do not compile for linux - there is some problem when running
% struct edges with compiled code (rather than with the compiled
% code that comes with the library)
if ~isunix
cd(fullfile(algos_dir, 'toolboxes', 'piotr_toolbox', 'external'));
toolboxCompile;
fprintf(1, 'Done mex''ing\n');
cd(curr_dir);
end
end
% download Piotr's Struct Edges code
success = download_code('http://ftp.research.microsoft.com/downloads/389109f6-b4e8-404c-84bf-239f7cbf4e3d/releaseV3.zip', ...
fullfile(temp_dir, 'release.zip'), 'Piotr''s Struct Edges code', simulate_no_download);
if success
mkdir(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges'));
unzip(fullfile(temp_dir, 'release.zip'), fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges'));
movefile(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges', 'releaseV3', '*'), fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges'));
rmdir(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges', 'releaseV3'), 's');
cd(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges'));
if isunix
mex private/edgesDetectMex.cpp -outdir private '-DUSEOMP' CFLAGS="\$CFLAGS -fopenmp" LDFLAGS="\$LDFLAGS -fopenmp"
mex private/edgesNmsMex.cpp -outdir private '-DUSEOMP' CFLAGS="\$CFLAGS -fopenmp" LDFLAGS="\$LDFLAGS -fopenmp"
else
% if windows
mex private/edgesDetectMex.cpp -outdir private '-DUSEOMP' 'OPTIMFLAGS="$OPTIMFLAGS' '/openmp"'
mex private/edgesNmsMex.cpp -outdir private '-DUSEOMP' 'OPTIMFLAGS="$OPTIMFLAGS' '/openmp"'
end
fprintf(1, 'Done mex''ing\n');
cd(curr_dir);
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges', 'edgesDetect.m'), '\[E,O,inds,segs\]', '[E,T,O,inds,segs]');
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges', 'edgesDetect.m'), '% perform nms', 'T = [];\n% perform nms');
replaceInTextFile(fullfile(algos_dir, 'segmentation', 'boundaries--dollar_ICCV_2013_structedges', 'edgesDetect.m'), 'E\s*=\s*edgesNmsMex', 'T=edgesNmsMex');
end
% download Woodford's Image-Based Rendering and Stereo code
success = download_code('http://www.robots.ox.ac.uk/~ojw/files/imrender_v2.4.zip', ...
fullfile(temp_dir, 'imrender_v2.4.zip'), 'Woodford''s Image-Based Rendering and Stereo code', simulate_no_download);
if success
unzip(fullfile(temp_dir, 'imrender_v2.4.zip'), fullfile(algos_dir, 'segmentation'));
cd(fullfile(algos_dir, 'segmentation', 'imrender', 'vgg'));
% segment as a test - which also compiles the file
temp = vgg_segment_gb(imread('peppers.png'), 0.5, 10, 10);
fprintf(1, 'Done mex''ing\n');
cd(curr_dir);
end
% download export_fig code
success = download_code('https://codeload.github.com/ojwoodford/export_fig/legacy.zip/master', ...
fullfile(temp_dir, 'ojwoodford-export_fig.zip'), 'Woodford''s matlab export fig code', simulate_no_download);
if success
mkdir(fullfile(algos_dir, 'utils'));
unzip(fullfile(temp_dir, 'ojwoodford-export_fig.zip'), fullfile(algos_dir, 'utils'));
d = dir(fullfile(algos_dir, 'utils', '*export_fig*'));
movefile(fullfile(algos_dir, 'utils', d(1).name), fullfile(algos_dir, 'utils', 'export_fig'));
replaceInTextFile(fullfile(code_root_dir, 'utils', 'drawFigFrames.m'), '~/videovolumes/extern_src', algos_dir, true);
end
% download Vedaldi's vlfeat code
success = download_code('https://github.com/vlfeat/vlfeat/archive/master.zip', ...
fullfile(temp_dir, 'vlfeat.zip'), 'Vedaldi''s vlfeat code', simulate_no_download);
if success
unzip(fullfile(temp_dir, 'vlfeat.zip'), fullfile(algos_dir, 'toolboxes'));
d = dir(fullfile(algos_dir, 'toolboxes', '*vlfeat*'));
movefile(fullfile(algos_dir, 'toolboxes', d(1).name), fullfile(algos_dir, 'toolboxes', 'vlfeat'));
end
% change the extern code path in the internal parameters script
replaceInTextFile(fullfile(code_root_dir, 'internal_params.m'), '''/home/ahumayun/videovolumes/extern_src''', ['fullfile(fp.code_root_dir, ''', extern_src_rel, ''')'], true);
% delete temp directory
if ~simulate_no_download
rmdir(temp_dir, 's');
end
catch exception
% remove temp dir
rmdir(temp_dir, 's');
rethrow(exception)
end
end
function success = download_code(url, filepath, desc, simulate_no_download)
try
fprintf(1, 'Downloading %s ...\n', desc);
if ~simulate_no_download
[f, status] = urlwrite(url, filepath);
success = status == 1;
else
success = true;
end
catch exception
success = false;
end
if success
if ~simulate_no_download
fprintf(1, 'Done downloading\n');
else
fprintf(1, 'Simulated download call\n');
end
else
fprintf(2, 'Downloading failure\n');
end
end
function transferMatlabFromGNU(makefilepath, dest_dir)
fd = fopen(makefilepath, 'r');
str = fread(fd);
str = char(str');
fclose(fd);
files = {};
re_matches = regexp(str, '(?:matlab)\s+:=\s+(\S+.(?:(mat)|m)(?:\s*\\?\s*((\$\(wildcard)|(\)))?\s+))*', 'tokens');
for idx = 1:length(re_matches)
f = regexp(re_matches{idx}{1}, '(\S+.(?:(mat)|m))', 'tokens');
files = [files cellfun(@(x)x, f)];
end
for idx = 1:length(files)
movefile(fullfile(fileparts(makefilepath), files{idx}), dest_dir);
end
end
function adjustAttributes(folder_path)
% dont need to change file permissions on windows
if ispc == 1
return;
end
d = dir(folder_path);
for idx = 1:length(d)
if strcmp(d(idx).name,'.') || strcmp(d(idx).name,'..')
continue;
end
curr_path = fullfile(folder_path,d(idx).name);
if d(idx).isdir == 1
adjustAttributes(curr_path);
else
unix(['chmod 0644 "' curr_path '"']);
end
end
end |
lemma filterlim_at_split: "filterlim f F (at x) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)" for x :: "'a::linorder_topology" |
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
qβ qβ : Basis A cβ cβ
hi : qβ.i = qβ.i
hj : qβ.j = qβ.j
β’ qβ = qβ
[PROOFSTEP]
cases qβ
[GOAL]
case mk
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
qβ : Basis A cβ cβ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
i_mul_jβ : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ }.i =
qβ.i
hj :
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ }.j =
qβ.j
β’ { i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ } = qβ
[PROOFSTEP]
rename_i qβ_i_mul_j _
[GOAL]
case mk
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
qβ : Basis A cβ cβ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
qβ_i_mul_j : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.i =
qβ.i
hj :
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.j =
qβ.j
β’ { i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j, j_mul_i := j_mul_iβ } =
qβ
[PROOFSTEP]
cases qβ
[GOAL]
case mk.mk
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
iβΒΉ jβΒΉ kβΒΉ : A
i_mul_iβΒΉ : iβΒΉ * iβΒΉ = cβ β’ 1
j_mul_jβΒΉ : jβΒΉ * jβΒΉ = cβ β’ 1
qβ_i_mul_j : iβΒΉ * jβΒΉ = kβΒΉ
j_mul_iβΒΉ : jβΒΉ * iβΒΉ = -kβΒΉ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
i_mul_jβ : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.i =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ }.i
hj :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.j =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ }.j
β’ { i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ } =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := i_mul_jβ, j_mul_i := j_mul_iβ }
[PROOFSTEP]
rename_i qβ_i_mul_j _
[GOAL]
case mk.mk
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
iβΒΉ jβΒΉ kβΒΉ : A
i_mul_iβΒΉ : iβΒΉ * iβΒΉ = cβ β’ 1
j_mul_jβΒΉ : jβΒΉ * jβΒΉ = cβ β’ 1
qβ_i_mul_j : iβΒΉ * jβΒΉ = kβΒΉ
j_mul_iβΒΉ : jβΒΉ * iβΒΉ = -kβΒΉ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
qβ_i_mul_j : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.i =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.i
hj :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.j =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.j
β’ { i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ } =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j, j_mul_i := j_mul_iβ }
[PROOFSTEP]
congr
[GOAL]
case mk.mk.e_k
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
iβΒΉ jβΒΉ kβΒΉ : A
i_mul_iβΒΉ : iβΒΉ * iβΒΉ = cβ β’ 1
j_mul_jβΒΉ : jβΒΉ * jβΒΉ = cβ β’ 1
qβ_i_mul_j : iβΒΉ * jβΒΉ = kβΒΉ
j_mul_iβΒΉ : jβΒΉ * iβΒΉ = -kβΒΉ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
qβ_i_mul_j : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.i =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.i
hj :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.j =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.j
β’ kβΒΉ = kβ
[PROOFSTEP]
rw [β qβ_i_mul_j, β qβ_i_mul_j]
[GOAL]
case mk.mk.e_k
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
iβΒΉ jβΒΉ kβΒΉ : A
i_mul_iβΒΉ : iβΒΉ * iβΒΉ = cβ β’ 1
j_mul_jβΒΉ : jβΒΉ * jβΒΉ = cβ β’ 1
qβ_i_mul_j : iβΒΉ * jβΒΉ = kβΒΉ
j_mul_iβΒΉ : jβΒΉ * iβΒΉ = -kβΒΉ
iβ jβ kβ : A
i_mul_iβ : iβ * iβ = cβ β’ 1
j_mul_jβ : jβ * jβ = cβ β’ 1
qβ_i_mul_j : iβ * jβ = kβ
j_mul_iβ : jβ * iβ = -kβ
hi :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.i =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.i
hj :
{ i := iβΒΉ, j := jβΒΉ, k := kβΒΉ, i_mul_i := i_mul_iβΒΉ, j_mul_j := j_mul_jβΒΉ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβΒΉ }.j =
{ i := iβ, j := jβ, k := kβ, i_mul_i := i_mul_iβ, j_mul_j := j_mul_jβ, i_mul_j := qβ_i_mul_j,
j_mul_i := j_mul_iβ }.j
β’ iβΒΉ * jβΒΉ = iβ * jβ
[PROOFSTEP]
congr
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } = cβ β’ 1
[PROOFSTEP]
ext
[GOAL]
case re
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).re = (cβ β’ 1).re
[PROOFSTEP]
simp
[GOAL]
case imI
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imI = (cβ β’ 1).imI
[PROOFSTEP]
simp
[GOAL]
case imJ
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imJ = (cβ β’ 1).imJ
[PROOFSTEP]
simp
[GOAL]
case imK
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imK = (cβ β’ 1).imK
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } = cβ β’ 1
[PROOFSTEP]
ext
[GOAL]
case re
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).re = (cβ β’ 1).re
[PROOFSTEP]
simp
[GOAL]
case imI
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imI = (cβ β’ 1).imI
[PROOFSTEP]
simp
[GOAL]
case imJ
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imJ = (cβ β’ 1).imJ
[PROOFSTEP]
simp
[GOAL]
case imK
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imK = (cβ β’ 1).imK
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } =
{ re := 0, imI := 0, imJ := 0, imK := 1 }
[PROOFSTEP]
ext
[GOAL]
case re
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).re =
{ re := 0, imI := 0, imJ := 0, imK := 1 }.re
[PROOFSTEP]
simp
[GOAL]
case imI
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imI =
{ re := 0, imI := 0, imJ := 0, imK := 1 }.imI
[PROOFSTEP]
simp
[GOAL]
case imJ
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imJ =
{ re := 0, imI := 0, imJ := 0, imK := 1 }.imJ
[PROOFSTEP]
simp
[GOAL]
case imK
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imK =
{ re := 0, imI := 0, imJ := 0, imK := 1 }.imK
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } =
-{ re := 0, imI := 0, imJ := 0, imK := 1 }
[PROOFSTEP]
ext
[GOAL]
case re
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).re =
(-{ re := 0, imI := 0, imJ := 0, imK := 1 }).re
[PROOFSTEP]
simp
[GOAL]
case imI
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imI =
(-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imI
[PROOFSTEP]
simp
[GOAL]
case imJ
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imJ =
(-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imJ
[PROOFSTEP]
simp
[GOAL]
case imK
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
β’ ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imK =
(-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imK
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ q.i * q.k = cβ β’ q.j
[PROOFSTEP]
rw [β i_mul_j, β mul_assoc, i_mul_i, smul_mul_assoc, one_mul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ q.k * q.i = -cβ β’ q.j
[PROOFSTEP]
rw [β i_mul_j, mul_assoc, j_mul_i, mul_neg, i_mul_k, neg_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ q.k * q.j = cβ β’ q.i
[PROOFSTEP]
rw [β i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ q.j * q.k = -cβ β’ q.i
[PROOFSTEP]
rw [β i_mul_j, β mul_assoc, j_mul_i, neg_mul, k_mul_j, neg_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ q.k * q.k = -((cβ * cβ) β’ 1)
[PROOFSTEP]
rw [β i_mul_j, mul_assoc, β mul_assoc q.j _ _, j_mul_i, β i_mul_j, β mul_assoc, mul_neg, β mul_assoc, i_mul_i,
smul_mul_assoc, one_mul, neg_mul, smul_mul_assoc, j_mul_j, smul_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ lift q 0 = 0
[PROOFSTEP]
simp [lift]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ lift q 1 = 1
[PROOFSTEP]
simp [lift]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ lift q (x + y) = lift q x + lift q y
[PROOFSTEP]
simp [lift, add_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ β(algebraMap R A) x.re + β(algebraMap R A) y.re + (x.imI β’ q.i + y.imI β’ q.i) + (x.imJ β’ q.j + y.imJ β’ q.j) +
(x.imK β’ q.k + y.imK β’ q.k) =
β(algebraMap R A) x.re + x.imI β’ q.i + x.imJ β’ q.j + x.imK β’ q.k +
(β(algebraMap R A) y.re + y.imI β’ q.i + y.imJ β’ q.j + y.imK β’ q.k)
[PROOFSTEP]
abel
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ β(algebraMap R A) x.re + β(algebraMap R A) y.re + (x.imI β’ q.i + y.imI β’ q.i) + (x.imJ β’ q.j + y.imJ β’ q.j) +
(x.imK β’ q.k + y.imK β’ q.k) =
β(algebraMap R A) x.re + x.imI β’ q.i + x.imJ β’ q.j + x.imK β’ q.k +
(β(algebraMap R A) y.re + y.imI β’ q.i + y.imJ β’ q.j + y.imK β’ q.k)
[PROOFSTEP]
abel
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ lift q (x * y) = lift q x * lift q y
[PROOFSTEP]
simp only [lift, Algebra.algebraMap_eq_smul_one]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re β’ 1 + x.imI β’ q.i + x.imJ β’ q.j + x.imK β’ q.k) * (y.re β’ 1 + y.imI β’ q.i + y.imJ β’ q.j + y.imK β’ q.k)
[PROOFSTEP]
simp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k +
((x.imI * y.re) β’ q.i + (x.imI * y.imI) β’ (q.i * q.i) + (x.imI * y.imJ) β’ (q.i * q.j) +
(x.imI * y.imK) β’ (q.i * q.k)) +
((x.imJ * y.re) β’ q.j + (x.imJ * y.imI) β’ (q.j * q.i) + (x.imJ * y.imJ) β’ (q.j * q.j) +
(x.imJ * y.imK) β’ (q.j * q.k)) +
((x.imK * y.re) β’ q.k + (x.imK * y.imI) β’ (q.k * q.i) + (x.imK * y.imJ) β’ (q.k * q.j) +
(x.imK * y.imK) β’ (q.k * q.k))
[PROOFSTEP]
simp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k +
((x.imI * y.re) β’ q.i + (x.imI * y.imI) β’ cβ β’ 1 + (x.imI * y.imJ) β’ q.k + (x.imI * y.imK) β’ cβ β’ q.j) +
((x.imJ * y.re) β’ q.j + (x.imJ * y.imI) β’ -q.k + (x.imJ * y.imJ) β’ cβ β’ 1 + (x.imJ * y.imK) β’ -cβ β’ q.i) +
((x.imK * y.re) β’ q.k + (x.imK * y.imI) β’ -cβ β’ q.j + (x.imK * y.imJ) β’ cβ β’ q.i +
(x.imK * y.imK) β’ -((cβ * cβ) β’ 1))
[PROOFSTEP]
simp only [smul_smul, smul_neg, sub_eq_add_neg, add_smul, β add_assoc, mul_neg, neg_smul]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(x.imI * y.imI * cβ) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(x.imI * y.imK * cβ) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(x.imJ * y.imJ * cβ) β’ 1 +
-((x.imJ * y.imK * cβ) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((x.imK * y.imI * cβ) β’ q.j) +
(x.imK * y.imJ * cβ) β’ q.i +
-((x.imK * y.imK * (cβ * cβ)) β’ 1)
[PROOFSTEP]
simp only [mul_right_comm _ _ (cβ * cβ), mul_comm _ (cβ * cβ)]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(x.imI * y.imI * cβ) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(x.imI * y.imK * cβ) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(x.imJ * y.imJ * cβ) β’ 1 +
-((x.imJ * y.imK * cβ) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((x.imK * y.imI * cβ) β’ q.j) +
(x.imK * y.imJ * cβ) β’ q.i +
-((cβ * cβ * (x.imK * y.imK)) β’ 1)
[PROOFSTEP]
simp only [mul_comm _ cβ, mul_right_comm _ _ cβ]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(cβ * (x.imI * y.imI)) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(cβ * (x.imI * y.imK)) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(x.imJ * y.imJ * cβ) β’ 1 +
-((x.imJ * y.imK * cβ) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((cβ * (x.imK * y.imI)) β’ q.j) +
(x.imK * y.imJ * cβ) β’ q.i +
-((cβ * cβ * (x.imK * y.imK)) β’ 1)
[PROOFSTEP]
simp only [mul_comm _ cβ, mul_right_comm _ _ cβ]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(cβ * (x.imI * y.imI)) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(cβ * (x.imI * y.imK)) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(cβ * (x.imJ * y.imJ)) β’ 1 +
-((cβ * (x.imJ * y.imK)) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((cβ * (x.imK * y.imI)) β’ q.j) +
(cβ * (x.imK * y.imJ)) β’ q.i +
-((cβ * cβ * (x.imK * y.imK)) β’ 1)
[PROOFSTEP]
simp only [β mul_comm cβ cβ, β mul_assoc]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x * y).re β’ 1 + (x * y).imI β’ q.i + (x * y).imJ β’ q.j + (x * y).imK β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(cβ * x.imI * y.imI) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(cβ * x.imI * y.imK) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(cβ * x.imJ * y.imJ) β’ 1 +
-((cβ * x.imJ * y.imK) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((cβ * x.imK * y.imI) β’ q.j) +
(cβ * x.imK * y.imJ) β’ q.i +
-((cβ * cβ * x.imK * y.imK) β’ 1)
[PROOFSTEP]
simp [sub_eq_add_neg, add_smul, β add_assoc]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x.re * y.re) β’ 1 + (cβ * x.imI * y.imI) β’ 1 + (cβ * x.imJ * y.imJ) β’ 1 + -((cβ * cβ * x.imK * y.imK) β’ 1) +
(x.re * y.imI) β’ q.i +
(x.imI * y.re) β’ q.i +
-((cβ * x.imJ * y.imK) β’ q.i) +
(cβ * x.imK * y.imJ) β’ q.i +
(x.re * y.imJ) β’ q.j +
(cβ * x.imI * y.imK) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((cβ * x.imK * y.imI) β’ q.j) +
(x.re * y.imK) β’ q.k +
(x.imI * y.imJ) β’ q.k +
-((x.imJ * y.imI) β’ q.k) +
(x.imK * y.re) β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(cβ * x.imI * y.imI) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(cβ * x.imI * y.imK) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(cβ * x.imJ * y.imJ) β’ 1 +
-((cβ * x.imJ * y.imK) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((cβ * x.imK * y.imI) β’ q.j) +
(cβ * x.imK * y.imJ) β’ q.i +
-((cβ * cβ * x.imK * y.imK) β’ 1)
[PROOFSTEP]
abel
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
x y : β[R,cβ,cβ]
β’ (x.re * y.re) β’ 1 + (cβ * x.imI * y.imI) β’ 1 + (cβ * x.imJ * y.imJ) β’ 1 + -((cβ * cβ * x.imK * y.imK) β’ 1) +
(x.re * y.imI) β’ q.i +
(x.imI * y.re) β’ q.i +
-((cβ * x.imJ * y.imK) β’ q.i) +
(cβ * x.imK * y.imJ) β’ q.i +
(x.re * y.imJ) β’ q.j +
(cβ * x.imI * y.imK) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((cβ * x.imK * y.imI) β’ q.j) +
(x.re * y.imK) β’ q.k +
(x.imI * y.imJ) β’ q.k +
-((x.imJ * y.imI) β’ q.k) +
(x.imK * y.re) β’ q.k =
(x.re * y.re) β’ 1 + (x.re * y.imI) β’ q.i + (x.re * y.imJ) β’ q.j + (x.re * y.imK) β’ q.k + (x.imI * y.re) β’ q.i +
(cβ * x.imI * y.imI) β’ 1 +
(x.imI * y.imJ) β’ q.k +
(cβ * x.imI * y.imK) β’ q.j +
(x.imJ * y.re) β’ q.j +
-((x.imJ * y.imI) β’ q.k) +
(cβ * x.imJ * y.imJ) β’ 1 +
-((cβ * x.imJ * y.imK) β’ q.i) +
(x.imK * y.re) β’ q.k +
-((cβ * x.imK * y.imI) β’ q.j) +
(cβ * x.imK * y.imJ) β’ q.i +
-((cβ * cβ * x.imK * y.imK) β’ 1)
[PROOFSTEP]
abel
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
r : R
x : β[R,cβ,cβ]
β’ lift q (r β’ x) = r β’ lift q x
[PROOFSTEP]
simp [lift, mul_smul, β Algebra.smul_def]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
F : A ββ[R] B
β’ βF q.i * βF q.i = cβ β’ 1
[PROOFSTEP]
rw [β F.map_mul, q.i_mul_i, F.map_smul, F.map_one]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
F : A ββ[R] B
β’ βF q.j * βF q.j = cβ β’ 1
[PROOFSTEP]
rw [β F.map_mul, q.j_mul_j, F.map_smul, F.map_one]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
F : A ββ[R] B
β’ βF q.i * βF q.j = βF q.k
[PROOFSTEP]
rw [β F.map_mul, q.i_mul_j]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
F : A ββ[R] B
β’ βF q.j * βF q.i = -βF q.k
[PROOFSTEP]
rw [β F.map_mul, q.j_mul_i, F.map_neg]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ Basis.compHom (Basis.self R) (Basis.liftHom q) = q
[PROOFSTEP]
ext
[GOAL]
case hi
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ (Basis.compHom (Basis.self R) (Basis.liftHom q)).i = q.i
[PROOFSTEP]
simp [Basis.lift]
[GOAL]
case hj
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
q : Basis A cβ cβ
β’ (Basis.compHom (Basis.self R) (Basis.liftHom q)).j = q.j
[PROOFSTEP]
simp [Basis.lift]
[GOAL]
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
β’ Basis.liftHom (Basis.compHom (Basis.self R) F) = F
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ β(Basis.liftHom (Basis.compHom (Basis.self R) F)) xβ = βF xβ
[PROOFSTEP]
dsimp [Basis.lift]
[GOAL]
case H
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ β(algebraMap R A) xβ.re + xβ.imI β’ βF { re := 0, imI := 1, imJ := 0, imK := 0 } +
xβ.imJ β’ βF { re := 0, imI := 0, imJ := 1, imK := 0 } +
xβ.imK β’ βF { re := 0, imI := 0, imJ := 0, imK := 1 } =
βF xβ
[PROOFSTEP]
rw [β F.commutes]
[GOAL]
case H
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ βF (β(algebraMap R β[R,cβ,cβ]) xβ.re) + xβ.imI β’ βF { re := 0, imI := 1, imJ := 0, imK := 0 } +
xβ.imJ β’ βF { re := 0, imI := 0, imJ := 1, imK := 0 } +
xβ.imK β’ βF { re := 0, imI := 0, imJ := 0, imK := 1 } =
βF xβ
[PROOFSTEP]
simp only [β F.commutes, β F.map_smul, β F.map_add, mk_add_mk, smul_mk, smul_zero, algebraMap_eq]
[GOAL]
case H
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ βF
{ re := xβ.re + 0 + 0 + 0, imI := 0 + xβ.imI β’ 1 + 0 + 0, imJ := 0 + 0 + xβ.imJ β’ 1 + 0,
imK := 0 + 0 + 0 + xβ.imK β’ 1 } =
βF xβ
[PROOFSTEP]
congr
[GOAL]
case H.h.e_6.h.e_re
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ xβ.re + 0 + 0 + 0 = xβ.re
[PROOFSTEP]
simp
[GOAL]
case H.h.e_6.h.e_imI
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ 0 + xβ.imI β’ 1 + 0 + 0 = xβ.imI
[PROOFSTEP]
simp
[GOAL]
case H.h.e_6.h.e_imJ
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ 0 + 0 + xβ.imJ β’ 1 + 0 = xβ.imJ
[PROOFSTEP]
simp
[GOAL]
case H.h.e_6.h.e_imK
R : Type u_1
A : Type u_2
B : Type u_3
instββ΄ : CommRing R
instβΒ³ : Ring A
instβΒ² : Ring B
instβΒΉ : Algebra R A
instβ : Algebra R B
cβ cβ : R
F : β[R,cβ,cβ] ββ[R] A
xβ : β[R,cβ,cβ]
β’ 0 + 0 + 0 + xβ.imK β’ 1 = xβ.imK
[PROOFSTEP]
simp
|
\chapter*{Abstract}
Maximising the economic effectiveness of a wind farm is essential in making wind a more economic source of energy. This effectiveness can be increased through the reduction of operation and maintenance costs, which can be achieved through continuously monitoring the condition of wind turbines. An alternative to expensive condition monitoring systems, which can be uneconomical especially for older wind turbines, is to implement classification algorithms on supervisory control and data acquisition (SCADA) signals, which are collected in most wind turbines. Several publications were reviewed, which were all found to use separate algorithms to predict specific faults in advance. In reality, wind turbines tend to have multiple faults which may happen simultaneously and have correlations with one another. This project focusses on developing a methodology to predict multiple wind turbine faults in advance simultaneously by implementing classification algorithms on SCADA signals for a wind farm with 25 turbines rated at 2,500 kW, spanning a period of 30 months. The data, which included measurements of wind speed, active power and pitch angle, was labelled using corresponding downtime data to detect normal behaviour, faults and varying timescales before a fault occurs. Three different classification algorithms, namely decision trees, random forests and k nearest neighbours were tested using imbalanced and balanced training data, initially to optimise a number of hyperparameters. The random forest classifier produced the best results. Upon conducting a more detailed analysis on the performance of specific faults, it was found that the classifier was unable to detect the varying timescales before a fault with accuracy comparable to that of normal or faulty behaviour. This could have been due to the SCADA data, which are used as features, being unsuitable for detecting the faults, and there is potential to improve this by balancing only these classes.
\\[.5cm]
\noindent\textbf{\textit{Keywords:}} \keywords
|
(** Studies the actegory stemming from the self action of the endofunctors on [C] by precomposition
author: Ralph Matthes, 2023
*)
Require Import UniMath.Foundations.PartD.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.FunctorCategory.
Require Import UniMath.CategoryTheory.BicatOfCatsElementary.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors.
Require Import UniMath.CategoryTheory.Monoidal.Categories.
Require Import UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary.
Require Import UniMath.CategoryTheory.Actegories.Examples.ActionOfEndomorphismsInCATElementary.
Require Import UniMath.CategoryTheory.Actegories.Actegories.
Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories.
Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories.
Require Import UniMath.CategoryTheory.limits.bincoproducts.
Require Import UniMath.CategoryTheory.limits.coproducts.
Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories.
Local Open Scope cat.
Section FixACategory.
Context (C : category).
Local Definition Mon_endo : monoidal [C, C] := monendocat_monoidal C.
Definition SelfActCAT : actegory Mon_endo [C, C] := actegory_with_canonical_self_action Mon_endo.
End FixACategory.
Section LineatorForPostcomposition.
Context (C : category) (G : functor C C).
Definition lax_lineator_postcomp_SelfActCAT_data :
lineator_data (Mon_endo C) (SelfActCAT C) (SelfActCAT C) (post_comp_functor G).
Proof.
intros F K. cbn.
apply rassociator_CAT.
Defined.
Lemma lax_lineator_postcomp_SelfActCAT_laws :
lineator_laxlaws (Mon_endo C) (SelfActCAT C) (SelfActCAT C) (post_comp_functor G)
lax_lineator_postcomp_SelfActCAT_data.
Proof.
split4.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left; apply id_right.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left; apply id_right.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
do 3 rewrite id_left.
apply functor_id.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left.
apply functor_id.
Qed.
(** the following definition may not be usable because of its tight typing *)
Definition lax_lineator_postcomp_SelfActCAT :
lineator_lax (Mon_endo C) (SelfActCAT C) (SelfActCAT C) (post_comp_functor G) :=
_,,lax_lineator_postcomp_SelfActCAT_laws.
End LineatorForPostcomposition.
Section LineatorForPostcomposition_alt.
Context (C D : category) (G : functor D C).
Definition lax_lineator_postcomp_SelfActCAT_alt_data :
lineator_data (Mon_endo C) (actegory_from_precomp_CAT C D) (SelfActCAT C) (post_comp_functor G).
Proof.
intros F K. cbn.
apply rassociator_CAT.
Defined.
Lemma lax_lineator_postcomp_SelfActCAT_alt_laws :
lineator_laxlaws (Mon_endo C) (actegory_from_precomp_CAT C D) (SelfActCAT C) (post_comp_functor G)
lax_lineator_postcomp_SelfActCAT_alt_data.
Proof.
split4.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left; apply id_right.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left; apply id_right.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
do 3 rewrite id_left.
apply functor_id.
- intro; intros.
apply (nat_trans_eq C).
intro c.
cbn.
rewrite id_left.
apply functor_id.
Qed.
(** the following definition is peculiar since it relates different constructions of actegories *)
Definition lax_lineator_postcomp_SelfActCAT_alt :
lineator_lax (Mon_endo C) (actegory_from_precomp_CAT C D) (SelfActCAT C) (post_comp_functor G) :=
_,,lax_lineator_postcomp_SelfActCAT_alt_laws.
End LineatorForPostcomposition_alt.
Section DistributionOfCoproducts.
Context (C : category).
Section BinaryCoproduct.
Context (BCP : BinCoproducts C).
Let BCPCD : BinCoproducts [C, C] := BinCoproducts_functor_precat C C BCP.
Definition SelfActCAT_bincoprod_distributor_data :
actegory_bincoprod_distributor_data (Mon_endo C) BCPCD (SelfActCAT C).
Proof.
intro F.
apply precomp_bincoprod_distributor_data.
Defined.
Goal β F G1 G2 c, pr1 (SelfActCAT_bincoprod_distributor_data F G1 G2) c = identity _.
Proof.
intros.
apply idpath.
Qed.
Lemma SelfActCAT_bincoprod_distributor_law :
actegory_bincoprod_distributor_iso_law _ _ _ SelfActCAT_bincoprod_distributor_data.
Proof.
intro F.
apply precomp_bincoprod_distributor_law.
Qed.
Definition SelfActCAT_bincoprod_distributor :
actegory_bincoprod_distributor (Mon_endo C) BCPCD (SelfActCAT C) :=
_,,SelfActCAT_bincoprod_distributor_law.
End BinaryCoproduct.
Section Coproduct.
Context {I : UU} (CP : Coproducts I C).
Let CPCD : Coproducts I [C, C] := Coproducts_functor_precat I C C CP.
Definition SelfActCAT_coprod_distributor_data :
actegory_coprod_distributor_data (Mon_endo C) CPCD (SelfActCAT C).
Proof.
intros F Gs.
apply precomp_coprod_distributor_data.
Defined.
Lemma SelfActCAT_coprod_distributor_law :
actegory_coprod_distributor_iso_law _ _ _ SelfActCAT_coprod_distributor_data.
Proof.
intros F Gs.
apply precomp_coprod_distributor_law.
Qed.
Definition SelfActCAT_CAT_coprod_distributor :
actegory_coprod_distributor (Mon_endo C) CPCD (SelfActCAT C) :=
_,,SelfActCAT_coprod_distributor_law.
End Coproduct.
End DistributionOfCoproducts.
|
State Before: C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
β’ functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) = π (C β₯€ Karoubi D) State After: case refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
β’ β (X : C β₯€ Karoubi D),
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =
(π (C β₯€ Karoubi D)).obj X
case refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
β’ β (X Y : C β₯€ Karoubi D) (f : X βΆ Y),
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map f =
eqToHom (_ : ?m.30774.obj X = ?m.30775.obj X) β«
(π (C β₯€ Karoubi D)).map f β«
eqToHom
(_ :
(π (C β₯€ Karoubi D)).obj Y =
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj Y) Tactic: refine' Functor.ext _ _ State Before: case refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
β’ β (X : C β₯€ Karoubi D),
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =
(π (C β₯€ Karoubi D)).obj X State After: case refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ (functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =
(π (C β₯€ Karoubi D)).obj F Tactic: intro F State Before: case refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ (functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =
(π (C β₯€ Karoubi D)).obj F State After: case refine'_1.refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ β (X : C),
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =
((π (C β₯€ Karoubi D)).obj F).obj X
case refine'_1.refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ β (X Y : C) (f : X βΆ Y),
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).map f =
eqToHom (_ : ?m.31081.obj X = ?m.31082.obj X) β«
((π (C β₯€ Karoubi D)).obj F).map f β«
eqToHom
(_ :
((π (C β₯€ Karoubi D)).obj F).obj Y =
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj Y) Tactic: refine' Functor.ext _ _ State Before: case refine'_1.refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ β (X : C),
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =
((π (C β₯€ Karoubi D)).obj F).obj X State After: case refine'_1.refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ ((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =
((π (C β₯€ Karoubi D)).obj F).obj X Tactic: intro X State Before: case refine'_1.refine'_1
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ ((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =
((π (C β₯€ Karoubi D)).obj F).obj X State After: case refine'_1.refine'_1.h_p
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p β«
eqToHom ?refine'_1.refine'_1.h_X =
eqToHom ?refine'_1.refine'_1.h_X β« (((π (C β₯€ Karoubi D)).obj F).obj X).p
case refine'_1.refine'_1.h_X
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =
(((π (C β₯€ Karoubi D)).obj F).obj X).X Tactic: ext State Before: case refine'_1.refine'_1.h_p
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p β«
eqToHom ?refine'_1.refine'_1.h_X =
eqToHom ?refine'_1.refine'_1.h_X β« (((π (C β₯€ Karoubi D)).obj F).obj X).p
case refine'_1.refine'_1.h_X
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =
(((π (C β₯€ Karoubi D)).obj F).obj X).X State After: case refine'_1.refine'_1.h_X
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =
(((π (C β₯€ Karoubi D)).obj F).obj X).X Tactic: . simp State Before: case refine'_1.refine'_1.h_X
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =
(((π (C β₯€ Karoubi D)).obj F).obj X).X State After: no goals Tactic: . simp State Before: case refine'_1.refine'_1.h_p
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p β«
eqToHom ?refine'_1.refine'_1.h_X =
eqToHom ?refine'_1.refine'_1.h_X β« (((π (C β₯€ Karoubi D)).obj F).obj X).p State After: no goals Tactic: simp State Before: case refine'_1.refine'_1.h_X
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
X : C
β’ (((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =
(((π (C β₯€ Karoubi D)).obj F).obj X).X State After: no goals Tactic: simp State Before: case refine'_1.refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F : C β₯€ Karoubi D
β’ β (X Y : C) (f : X βΆ Y),
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).map f =
eqToHom
(_ :
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =
((π (C β₯€ Karoubi D)).obj F).obj X) β«
((π (C β₯€ Karoubi D)).obj F).map f β«
eqToHom
(_ :
((π (C β₯€ Karoubi D)).obj F).obj Y =
((functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj Y) State After: no goals Tactic: aesop_cat State Before: case refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
β’ β (X Y : C β₯€ Karoubi D) (f : X βΆ Y),
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map f =
eqToHom
(_ :
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =
(π (C β₯€ Karoubi D)).obj X) β«
(π (C β₯€ Karoubi D)).map f β«
eqToHom
(_ :
(π (C β₯€ Karoubi D)).obj Y =
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj Y) State After: case refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F G : C β₯€ Karoubi D
Ο : F βΆ G
β’ (functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map Ο =
eqToHom
(_ :
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =
(π (C β₯€ Karoubi D)).obj F) β«
(π (C β₯€ Karoubi D)).map Ο β«
eqToHom
(_ :
(π (C β₯€ Karoubi D)).obj G =
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj G) Tactic: intro F G Ο State Before: case refine'_2
C : Type u_1
D : Type u_2
E : Type ?u.30040
instβΒ² : Category C
instβΒΉ : Category D
instβ : Category E
F G : C β₯€ Karoubi D
Ο : F βΆ G
β’ (functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map Ο =
eqToHom
(_ :
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =
(π (C β₯€ Karoubi D)).obj F) β«
(π (C β₯€ Karoubi D)).map Ο β«
eqToHom
(_ :
(π (C β₯€ Karoubi D)).obj G =
(functorExtensionβ C D β (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj G) State After: no goals Tactic: aesop_cat |
Require Import Coq.Lists.List.
Import ListNotations.
Require Import Tactics.
Require Import Sequence.
Require Import Syntax.
Require Import Subst.
Require Import SimpSub.
Require Import Hygiene.
Require Import ContextHygiene.
Require Import System.
Require Import Judgement.
Lemma upper_bound_all_list :
forall (l : list nat),
exists n,
list_rect (fun _ => Prop)
True
(fun i _ P => i <= n /\ P)
l.
Proof.
intros l.
cut (exists n,
forall m,
n <= m
-> list_rect (fun _ => Prop)
True
(fun i _ P => i <= m /\ P)
l).
{
intros (n & Hn).
exists n.
apply Hn; auto.
}
induct l.
(* nil *)
{
exists 0.
cbn.
auto.
}
(* cons *)
{
intros i l IH.
destruct IH as (j & Hj).
exists (max i j).
intros n Hn.
cbn.
split.
{
so (Nat.le_max_l i j).
omega.
}
{
apply Hj.
so (Nat.le_max_r i j).
omega.
}
}
Qed.
Lemma upper_bound_all :
forall (n : nat),
nat_rect (fun _ => list nat -> Prop)
(fun l =>
exists n,
list_rect (fun _ => Prop)
True
(fun i _ P => i <= n /\ P)
(List.rev l))
(fun _ P l =>
forall (i : nat), P (cons i l))
n
nil.
Proof.
intro n.
generalize (@nil nat).
induct n.
(* 0 *)
{
intros l.
cbn.
apply upper_bound_all_list.
}
(* S *)
{
intros n IH l.
cbn.
intro i.
apply IH.
}
Qed.
Section object.
Variable object : Type.
Lemma can_hygiene :
forall (m : @term object),
exists i, hygiene (fun j => j < i) m.
Proof.
intros m.
pattern m.
apply (term_mut_ind _ _
(fun a r =>
exists i, hygiene_row (fun j => j < i) r)); clear m.
(* var *)
{
intros i.
exists (S i).
apply hygiene_var.
omega.
}
(* oper *)
{
intros a th r IH.
destruct IH as (i & Hi).
exists i.
apply hygiene_oper; auto.
}
(* nil *)
{
exists 0.
apply hygiene_nil.
}
(* cons *)
{
intros j a m IH1 r IH2.
destruct IH1 as (i1 & Hi1).
destruct IH2 as (i2 & Hi2).
exists (max i1 i2). (* higher than necessary, but that's fine *)
apply hygiene_cons.
{
eapply hygiene_weaken; eauto.
intros k Hk.
so (le_gt_dec j k) as [Hjk | Hkj]; auto.
right.
split; auto.
so (Nat.le_max_l i1 i2).
omega.
}
{
eapply hygiene_row_weaken; eauto.
intros k Hk.
so (Nat.le_max_r i1 i2).
omega.
}
}
Qed.
Lemma can_hygiene' :
forall (m : @term object),
exists i, forall j, i <= j -> hygiene (fun k => k < j) m.
Proof.
intros m.
so (can_hygiene m) as (i & Hi).
exists i.
intros j Hj.
eapply hygiene_weaken; eauto.
intros k.
omega.
Qed.
Lemma can_hygieneh :
forall (h : @hyp object),
exists i, hygieneh (fun j => j < i) h.
Proof.
intros h.
induct h.
{
exists 0.
apply hygieneh_tpl.
}
{
exists 0.
apply hygieneh_tp.
}
{
intros a.
so (can_hygiene a) as (i & Hi).
exists i.
apply hygieneh_tml; auto.
}
{
intros a.
so (can_hygiene a) as (i & Hi).
exists i.
apply hygieneh_tm; auto.
}
{
exists 0.
apply hygieneh_emp.
}
Qed.
Lemma can_hygieneh' :
forall (h : @hyp object),
exists i, forall j, i <= j -> hygieneh (fun k => k < j) h.
Proof.
intros h.
so (can_hygieneh h) as (i & Hi).
exists i.
intros j Hj.
eapply hygieneh_weaken; eauto.
intros k.
omega.
Qed.
Lemma can_hygienej :
forall (J : @judgement object),
exists i, hygienej (fun j => j < i) J.
Proof.
intros J.
induct J.
intros m n a.
so (can_hygiene m) as (i1 & Hm).
so (can_hygiene n) as (i2 & Hn).
so (can_hygiene a) as (i3 & Ha).
so (upper_bound_all 3 i1 i2 i3) as (i & Hi1 & Hi2 & Hi3 & _).
exists i.
apply hygienej_deq; eapply hygiene_weaken; eauto; intros j Hj; omega.
Qed.
Lemma can_hygienej' :
forall (J : @judgement object),
exists i, forall j, i <= j -> hygienej (fun k => k < j) J.
Proof.
intros J.
so (can_hygienej J) as (i & Hi).
exists i.
intros j Hj.
eapply hygienej_weaken; eauto.
intros k.
omega.
Qed.
Fixpoint shut n : @context object :=
match n with
| 0 => nil
| S n' => hyp_tm unittp :: shut n'
end.
Lemma length_shut :
forall n, length (shut n) = n.
Proof.
intro n.
induct n; auto; intros; cbn; f_equal; auto.
Qed.
Lemma promote_shut :
forall n, @promote object (shut n) = shut n.
Proof.
intros n.
induct n; intros; cbn; f_equal; auto.
Qed.
Lemma shut_term :
forall (G : @context object) (m : @term object),
exists i,
forall j,
i <= j
-> hygiene (ctxpred (G ++ shut j)) m.
Proof.
intros G m.
so (can_hygiene m) as (i & Hi).
exists i.
intros j Hj.
eapply hygiene_weaken; eauto.
intros k Hk.
rewrite ctxpred_length, app_length, length_shut.
omega.
Qed.
Lemma shut_hyp :
forall (G : @context object) (h : @hyp object),
exists i,
forall j,
i <= j
-> hygieneh (ctxpred (G ++ shut j)) h.
Proof.
intros G m.
so (can_hygieneh m) as (i & Hi).
exists i.
intros j Hj.
eapply hygieneh_weaken; eauto.
intros k Hk.
rewrite ctxpred_length, app_length, length_shut.
omega.
Qed.
Lemma shut_judgement :
forall (G : @context object) (J : @judgement object),
exists i,
forall j,
i <= j
-> hygienej (ctxpred (G ++ shut j)) J.
Proof.
intros G J.
so (can_hygienej J) as (i & Hi).
exists i.
intros j Hj.
eapply hygienej_weaken; eauto.
intros k Hk.
rewrite ctxpred_length, app_length, length_shut.
omega.
Qed.
End object.
Arguments shut {object}.
Definition pseq (G : scontext) (J : judgement) : Prop :=
exists i,
forall j,
i <= j
-> seq (G ++ shut j) J.
Definition obj := Candidate.obj Page.stop.
(* Given a goal of the form:
forall G,
pseq (G1 ++ G) J1
-> ...
-> pseq (Gn ++ G) Jn
-> pseq G J
refine (seq_pseq m H1 p1 ... Hm pm n G1 J1 ... Gn Jn J _)
will produce a subgoal of the form:
forall G,
hygiene (ctxpred (Hi ++ G)) pi)
-> ...
-> hygiene (ctxpred (Hm ++ G)) pm)
-> seq (G1 ++ G) J1
-> ...
-> seq (Gn ++ G) Jn
-> seq G J
Note that one can fill in all the Jis with _, and all the
Gis with [_, ..., _].
*)
Lemma seq_pseq :
forall m,
nat_rect (fun _ => list (scontext * term _) -> Prop)
(fun T =>
forall n,
nat_rect (fun _ => list (scontext * judgement) -> Prop)
(fun L =>
forall J,
(forall G,
list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(seq G J)
(fun X _ P =>
match X with
| pair G' J' => seq (G' ++ G) J' -> P
end)
(rev L))
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G)) p -> P
end)
(rev T))
->
forall G,
list_rect (fun _ => Prop)
(pseq G J)
(fun X _ P =>
match X with
| pair G' J' => pseq (G' ++ G) J' -> P
end)
(rev L))
(fun _ P L =>
forall (G : scontext) (J : judgement), P ((G, J) :: L))
n
nil)
(fun _ P T =>
forall (G : scontext) (p : term obj), P ((G, p) :: T))
m
nil.
Proof.
intro m.
set (T := @nil (scontext * term obj)).
clearbody T.
revert T.
induct m.
2:{
intros m IH T.
cbn.
intros G p.
apply IH.
}
intros T.
cbn.
set (T' := rev T).
clearbody T'.
renameover T' into T.
intro n.
set (L := nil).
clearbody L.
revert L.
induct n.
2:{
intros n IH L.
cbn.
intros G J.
apply IH.
}
intros L.
cbn.
set (L' := rev L).
clearbody L'.
renameover L' into L.
intros J Hseqs G.
set (i := 0).
assert (forall j,
i <= j
-> list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(seq (G ++ shut j) J)
(fun X _ P =>
match X with
| pair G' J' => seq (G' ++ G ++ shut j) J' -> P
end)
L)
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G ++ shut j)) p -> P
end)
T).
{
intros j Hj.
cbn in Hseqs.
exact (Hseqs (G ++ shut j)).
}
clear Hseqs.
clearbody i.
revert i H.
induct T.
(* nil *)
{
intros i1 Hseq.
cbn in Hseq.
revert i1 Hseq.
induct L.
(* nil *)
{
cbn.
intros i Hseq.
exists i.
intros j Hj.
apply Hseq; auto.
}
(* cons *)
{
intros (G', J') L IH i Hseqs.
cbn.
intros (j & Hseq).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
rewrite -> app_assoc.
apply Hseq.
so (Nat.le_max_r i j).
omega.
}
}
(* cons *)
{
intros (G', p) T IH i Hseqs.
so (shut_term _ (G' ++ G) p) as (j & Hclp).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
rewrite -> app_assoc.
apply Hclp.
so (Nat.le_max_r i j).
omega.
}
Qed.
(* Given a goal of the form:
forall G,
pseq (G1a ++ G1b ++ G) J1
-> ...
-> pseq (Gna ++ Gnb ++ G) Jn
-> pseq (Ga ++ Gb ++ G) J
refine (seq_pseq_hyp m H1 p1 ... Hm pm n G1a G1b J1 ... Gna Gnb Jn Ga Gb J _)
will produce a subgoal of the form:
forall G,
hygiene (ctxpred (Hi ++ G) pi)
-> ...
-> hygiene (ctxpred (Hm ++ G) pm)
-> seq (G1a ++ G1b ++ G) J1
-> ...
-> seq (Gna ++ Gnb ++ G) Jn
-> hygienej (ctxpred (Ga ++ Gb ++ G)) J
-> seq (Ga ++ Gb ++ G) J
Note that one can fill in all the Jis with _, and all the
Gias and Gibs with [_; ...; _].
*)
Lemma seq_pseq_hyp :
forall m,
nat_rect (fun _ => list (scontext * term _) -> Prop)
(fun T =>
forall n,
nat_rect (fun _ => list (scontext * scontext * judgement) -> Prop)
(fun L =>
forall G1 G2 J,
(forall G,
list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(hygienej (ctxpred (G1 ++ G2 ++ G)) J -> seq (G1 ++ G2 ++ G) J)
(fun X _ P =>
match X with
| pair (pair G1' G2') J' => seq (G1' ++ G2' ++ G) J' -> P
end)
(rev L))
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G)) p -> P
end)
(rev T))
->
forall G,
list_rect (fun _ => Prop)
(pseq (G1 ++ G2 ++ G) J)
(fun X _ P =>
match X with
| pair (pair G1' G2') J' => pseq (G1' ++ G2' ++ G) J' -> P
end)
(rev L))
(fun _ P L =>
forall (G1 G2 : scontext) (J : judgement), P (((G1, G2), J) :: L))
n
nil)
(fun _ P T =>
forall (G : scontext) (p : term obj), P ((G, p) :: T))
m
nil.
Proof.
intro m.
set (T := @nil (scontext * term obj)).
clearbody T.
revert T.
induct m.
2:{
intros m IH T.
cbn.
intros G p.
apply IH.
}
intros T.
cbn.
set (T' := rev T).
clearbody T'.
renameover T' into T.
intro n.
set (L := nil).
clearbody L.
revert L.
induct n.
2:{
intros n IH L.
cbn.
intros G1 G2 J.
apply IH.
}
intros L.
cbn.
set (L' := rev L).
clearbody L'.
renameover L' into L.
intros G1 G2 J Hseqs G.
set (i := 0).
assert (forall j,
i <= j
-> list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(hygienej (ctxpred (G1 ++ G2 ++ G ++ shut j)) J -> seq (G1 ++ G2 ++ G ++ shut j) J)
(fun X _ P =>
match X with
| pair (pair G1' G2') J' => seq (G1' ++ G2' ++ G ++ shut j) J' -> P
end)
L)
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G ++ shut j)) p -> P
end)
T) as H.
{
intros j Hj.
exact (Hseqs (G ++ shut j)).
}
clear Hseqs.
clearbody i.
revert i H.
induct T.
(* nil *)
{
intros i1 Hseq.
cbn in Hseq.
revert i1 Hseq.
induct L.
(* nil *)
{
cbn.
intros i1 Hseq.
so (shut_judgement _ (G1 ++ G2 ++ G) J) as (i2 & HclJ).
so (upper_bound_all 2 i1 i2) as (i & Hi1 & Hi2 & _).
exists i.
intros j Hj.
rewrite <- !app_assoc.
apply Hseq; eauto using le_trans.
lapply (HclJ j); [| omega].
intro H.
autorewrite with canonlist in H.
exact H.
}
(* cons *)
{
intros ((G1', G2'), J') L IH i Hseqs.
cbn.
intros (j & Hseq).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
setoid_rewrite -> app_assoc.
setoid_rewrite -> app_assoc.
setoid_rewrite <- app_assoc at 2.
apply Hseq.
so (Nat.le_max_r i j).
omega.
}
}
(* cons *)
{
intros (G', p) T IH i Hseqs.
so (shut_term _ (G' ++ G) p) as (j & Hclp).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
rewrite -> app_assoc.
apply Hclp.
so (Nat.le_max_r i j).
omega.
}
Qed.
(* Given a goal of the form:
forall G,
pseq (G1 ++ if b1 then promote G else G) J1
-> ...
-> pseq (Gn ++ if bn then promote G else G) Jn
-> pseq G J
refine (seq_pseq_promote m H1 p1 ... Hm pm n b1 G1 J1 ... bn Gn Jn J _)
will produce a subgoal of the form:
forall G,
hygiene (ctxpred (Hi ++ G) pi)
-> ...
-> hygiene (ctxpred (Hm ++ G) pm)
-> seq (G1 ++ if b1 then promote G else G) J1
-> ...
-> seq (Gn ++ if bn then promote G else G) Jn
-> seq G J
Note that one can fill in all the Jis with _, and all the
Gis with [_, ..., _].
*)
Lemma seq_pseq_promote :
forall m,
nat_rect (fun _ => list (scontext * term _) -> Prop)
(fun T =>
forall n,
nat_rect (fun _ => list (bool * scontext * judgement) -> Prop)
(fun L =>
forall J,
(forall G,
list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(seq G J)
(fun X _ P =>
match X with
| pair (pair b G') J' => seq (G' ++ match b with true => promote G | false => G end) J' -> P
end)
(rev L))
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G)) p -> P
end)
(rev T))
->
forall G,
list_rect (fun _ => Prop)
(pseq G J)
(fun X _ P =>
match X with
| pair (pair b G') J' => pseq (G' ++ match b with true => promote G | false => G end) J' -> P
end)
(rev L))
(fun _ P L =>
forall (b : bool) (G : scontext) (J : judgement), P ((b, G, J) :: L))
n
nil)
(fun _ P T =>
forall (G : scontext) (p : term obj), P ((G, p) :: T))
m
nil.
Proof.
intro m.
set (T := @nil (scontext * term obj)).
clearbody T.
revert T.
induct m.
2:{
intros m IH T.
cbn.
intros G p.
apply IH.
}
intros T.
cbn.
set (T' := rev T).
clearbody T'.
renameover T' into T.
intro n.
set (L := nil).
clearbody L.
revert L.
induct n.
2:{
intros n IH L.
cbn.
intros b G J.
apply IH.
}
intros L.
cbn.
set (L' := rev L).
clearbody L'.
renameover L' into L.
cbn.
intros J Hseqs G.
set (i := 0).
assert (forall j,
i <= j
-> list_rect (fun _ => Prop)
(list_rect (fun _ => Prop)
(seq (G ++ shut j) J)
(fun X _ P =>
match X with
| pair (pair b G') J' => seq (G' ++ match b with true => promote (G ++ shut j) | false => G ++ shut j end) J' -> P
end)
L)
(fun X _ P =>
match X with
| pair G' p => hygiene (ctxpred (G' ++ G ++ shut j)) p -> P
end)
T) as H.
{
intros j Hj.
cbn in Hseqs.
exact (Hseqs (G ++ shut j)).
}
clear Hseqs.
clearbody i.
revert i H.
induct T.
(* nil *)
{
intros i1 Hseq.
cbn in Hseq.
revert i1 Hseq.
induct L.
(* nil *)
{
cbn.
intros i Hseq.
exists i.
intros j Hj.
apply Hseq; auto.
}
(* cons *)
{
intros ((b, G'), J') L IH i Hseqs.
cbn.
intros (j & Hseq).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
destruct b.
{
rewrite -> promote_append.
rewrite -> promote_shut.
rewrite -> app_assoc.
apply Hseq.
so (Nat.le_max_r i j).
omega.
}
{
rewrite -> app_assoc.
apply Hseq.
so (Nat.le_max_r i j).
omega.
}
}
}
(* cons *)
{
intros (G', p) T IH i Hseqs.
so (shut_term _ (G' ++ G) p) as (j & Hclp).
apply (IH (max i j)).
intros k Hk.
exploit (Hseqs k) as H.
{
so (Nat.le_max_l i j).
omega.
}
cbn in H.
apply H.
rewrite -> app_assoc.
apply Hclp.
so (Nat.le_max_r i j).
omega.
}
Qed.
Ltac finish_pseq j :=
eauto using le_trans;
match goal with
| H : forall (x : nat), _ <= x -> _ |- _ =>
let H' := fresh
in
lapply (H j); [| eauto using le_trans];
intro H';
autorewrite with canonlist in H';
exact H'
end.
|
Prepared by: Manjula Mishra
# Learning Goals
By the end of this class students will:
1. know what is logsitic regression.
2. know why logistic regression is still so poplular.
3. understand the usecases/applications of logistic regression.
4. be able to implement logistic regression in Python from scratch using SKLearn and Statsmodels.
5. know its limitations.
## Keywords
Supervised learning
Classification
Sigmoid function
Gradient Descent
Loss function/Cost function
# Sections:
This notebook is divided into 5 different sections which covers the main idea behind the logistic regression.
1. What is logistic regression?
2. Why is logistic regression still so poplular?
3. Applications of logistic regression
4. Implement logistic regression in Python from scratch
5. Limitations of logistic regression
## 1. What is logistic regression?
Logistic regression is a supervised learning method to model the relationship between a categorical outcome (dependent variable) and one or more independent variable. The independent variables or explanatory variables can be discrete and/or continuous.
The logistic function was invented in the 19TH century for the description of growth of human populations and the course of autocatalytic chemical reactions.
Logistic regression is not a typical linear regression model. It belongs to the family of GLMs. Some of the things to notice about the logistic regression:
1. Logistic regression does not assume a linear relationship between the dependent and independent variables. It's linear in parameters.
2. The dependent variable must be categorical.
3. the independent variables need not be normally distributed, nor linearly related, nor of equal variance within each group, and lastly, the categories (groups) must be mutually exclusive and exhaustive.
4. The logistic regression has the power to accommodate both categorical and continuous independent variables.
5. Although the power of the analysis is increased if the independent variables are normally distributed and do have a linear relationship with the dependent variable.
## 2. Why is logistic regression still so poplular?
* Logistic regression is easier to implement, interpret, and very efficient to train.
* It makes no assumptions about distributions of classes in feature space.
* It can easily extend to multiple classes(multinomial regression) and a natural probabilistic view of class predictions.
* It not only provides a measure of how appropriate a predictor(coefficient size)is, but also its direction of association (positive or negative).
* It is very fast at classifying unknown records.
* Good accuracy for many simple data sets and it performs well when the dataset is linearly separable.
* It can interpret model coefficients as indicators of feature importance.
* Logistic regression is less inclined to over-fitting but it can overfit in high dimensional datasets.One may consider Regularization (L1 and L2) techniques to avoid over-fitting in these scenarios.
## 3. Applications of logistic regression
The logistic regression has many applications in wide variety of fields. The types of problems it deals with are of the following nature:
Binary (Pass/Fail)
Multi (Cats, Dogs, Sheep)
Ordinal (Low, Medium, High)
Some of the usecases in different business settings are:
**Fraud detection**: Detection of credit card frauds or banking fraud is the objective of this use case.
**Email spam or ham**: Classifying the email as spam or ham and putting it in either Inbox or Spam folder is the objective of this use case.
**Sentiment Analysis**: Analyzing the sentiment using the review or tweets is the objective of this use case. Most of the brands and companies use this to increase customer experience.
**Image segmentation, recognition and classification**: The objective of all these use cases is to identify the object in the image and classify it.
**Object detection**: This use case is to detect objects and classify them not in the image but the video.
Handwriting recognition: Recognizing the letters written is the objective of this use case.
**Disease (diabetes, cancer etc.) prediction**: Predicting whether the patient has disease or not is the objective of this use case.
## Quiz
**1. True-False: Is Logistic regression a supervised machine learning algorithm?**
A. TRUE
B. FALSE
**2. True-False: Is Logistic regression mainly used for Regression?**
A. TRUE
B. FALSE
**3.Logistic regression assumes a:**
A. Linear relationship between continuous predictor variables and the outcome variable.
B. Linear relationship between continuous predictor variables and the logit of the outcome variable.
C. Linear relationship between continuous predictor variables.
D. Linear relationship between observations.
**4. True-False: Is it possible to apply a logistic regression algorithm on a 3-class Classification problem?**
A. TRUE
B. FALSE
**5. Logistic regression is used when you want to:**
A. Predict a dichotomous variable from continuous or dichotomous variables.
B. Predict a continuous variable from dichotomous variables.
C. Predict any categorical variable from several other categorical variables.
D. Predict a continuous variable from dichotomous or continuous variables.
**6. In binary logistic regression:**
A. The dependent variable is continuous.
B. The dependent variable is divided into two equal subcategories.
C. The dependent variable consists of two categories.
D. There is no dependent variable.
# 4. Implement logistic regression in Python from scratch
Here, we will be implementing a basic logsitic regression model to improve our understanding of how it works.
To acomplish that, we will be using a famous dataset called 'iris'. The description and the data can be found here: https://archive.ics.uci.edu/ml/datasets/iris. Basically, the dataset contains information (features) about three different types of flowers. We will be focusing on accuratly classifying the correct flowers.
We wil be taking the first two features into account and two non-linearly flowers are classified as one, so leaving them as binary class.
## Loading the data and importing the libraries
```python
# let's import the important libraries we will need
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import sklearn
from sklearn import datasets
```
Although we will directly be loading a preprocessed iris data from sklearn, I wanted to you to know that the data can also be loaded like this to have a first look at the data
```python
url = "https://gist.githubusercontent.com/curran/a08a1080b88344b0c8a7/raw/0e7a9b0a5d22642a06d3d5b9bcbad9890c8ee534/iris.csv"
```
```python
df = pd.read_csv(url)
df.head()
```
<div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>sepal_length</th>
<th>sepal_width</th>
<th>petal_length</th>
<th>petal_width</th>
<th>species</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>5.1</td>
<td>3.5</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>1</th>
<td>4.9</td>
<td>3.0</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>2</th>
<td>4.7</td>
<td>3.2</td>
<td>1.3</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>3</th>
<td>4.6</td>
<td>3.1</td>
<td>1.5</td>
<td>0.2</td>
<td>setosa</td>
</tr>
<tr>
<th>4</th>
<td>5.0</td>
<td>3.6</td>
<td>1.4</td>
<td>0.2</td>
<td>setosa</td>
</tr>
</tbody>
</table>
</div>
Let's load the data from sklearn and encode the target variable 0 and 1
```python
from sklearn.datasets import load_iris
data = load_iris()
# Load digits dataset
iris = sklearn.datasets.load_iris()
# Create feature matrix
X = iris.data[:, :2]
# Create target vector
y = (iris.target != 0) * 1 # so that 2 converts into 1
y
# y1 = iris.target #this will have category 2 as well that's why we use the above method
# y1
```
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
```python
print(iris.feature_names) #column names
```
['sepal length (cm)', 'sepal width (cm)', 'petal length (cm)', 'petal width (cm)']
```python
print(iris.target_names) # target varieties
```
['setosa' 'versicolor' 'virginica']
We will plot the 0 and 1 classes to see how it looks
```python
plt.figure(figsize=(10, 6))
plt.scatter(X[y == 0][:, 0], X[y == 0][:, 1], color='b', label='0')
plt.scatter(X[y == 1][:, 0], X[y == 1][:, 1], color='r', label='1')
plt.legend();
```
## Algorithm
To iterate it once again, given a set of inputs X, our goal is to correctly identify the class (0 or 1).As we know, we will use a logistic regression model to predict the probability for each flower to know which catergory it belongs to.
### Hypothesis
Any function takes inputs and returns the coresponding outputs. Logistic regression uses a function that generates probabilities an gives outputs between 0 and 1 for all values of X. The function that we will to achieve that is called Sigmoid function.
```python
# Import matplotlib, numpy and math
import matplotlib.pyplot as plt
import numpy as np
import math
plt.figure(figsize=(10, 6))
x = np.linspace(-10, 10, 100)
z = 1/(1 + np.exp(-x))
plt.plot(x, z)
plt.xlabel("x")
plt.ylabel("Sigmoid(X)")
plt.show()
```
### The Sigmoid function:
\begin{equation}
\begin{array}{l}
h_{\theta}(x)=g\left(\theta^{T} x\right) \\
z=\theta^{T} x \\
g(z)=\frac{1}{1+e^{-z}}
\end{array}
\end{equation}
### Explanation
The binary dependent variable has the values of 0 and 1 and the predicted value (probability) must be bounded to fall within the same range. To define a relationship bounded by 0 and 1, the logistic regression uses the logistic curve to represent the relationship between the independent and dependent variable. At very low levels of the independent variable, the probability approaches 0, but never reaches 0. Likewise, if the independent variable increases, the predicted values increase up the curve and approach 1 but never equal to 1.
The logistic transformation ensures that estimated values do not fall outside the range of 0 and 1. This is achieved in two steps, firstly the probability is re-stated as odds which is defined as the ratio of the probability of the event occurring to the probability of it not occurring. For example, if a horse has a probability of 0.8 of winning a race, the odds of it winning are 0.8/(1 β 0.8) = 4:1. To constrain the predicted values to within 0 and 1, the odds value can be converted back into a probability; thus,
$$
\text { Probability (event) }=\frac{\text { odds(event) }}{1+\text { odds(event) }}
$$
It can therefore be shown that the corresponding probability is 4/(1 + 4) = 0.8. Also, to keep the odds values form going below 0, which is the lower limit (there is no upper limit), the logit value which is calculated by taking the logarithm of the odds, must be computed. Odds less than 1 have a negative logit value, odds ratio greater than 1.0 have positive logit values and the odds ratio of 1.0 (corresponding to a probability of 0.5) have a logit value of 0.
Let's create a function in Python:
```python
# sigmoin function
def sigmoid(z):
return 1 / (1 + np.exp(-z))
theta = np.zeros(X.shape[1])
z = np.dot(X, theta)
h = sigmoid(z)
```
### Loss Function
Functions (rememeber linear regression) have weights also called parameters what we are represnting by theta in our case. The goal is to find the best value for them that can make most accurate predcitions.
We start by picking random values. Now we need a way to measure how well the algorithm performs using the random weights that we chose. We compute this by the following loss funtion:
$$
\begin{array}{l}
h=g(X \theta) \\
J(\theta)=\frac{1}{m} \cdot\left(-y^{T} \log (h)-(1-y)^{T} \log (1-h)\right)
\end{array}
$$
```python
# Let' code our loss fucntion
def loss(h, y):
return (-y * np.log(h) - (1 - y) * np.log(1 - h)).mean()
```
### Gradient Descent
Now that we have our loss function, our goal is to minimize it by increasing/decreasing teh weights i.e. fititng them. One obvious questions is how do we know which weights should be increased or decreased or how big what parameters should be bigger or smaller?
To achieve this, we calculate the partial derivatives of the loss function with respect to each weight. It will tell us how loss changes if we modify the parameters.
$$
\frac{\delta J(\theta)}{\delta \theta_{j}}=\frac{1}{m} X^{T}(g(X \theta)-y)
$$
```python
gradient = np.dot(X.T, (h - y)) / y.shape[0]
```
To update the weights, we subtract to them the derivative times the learning rate. And we do taht several times until we reach the optimal solution.
```python
lr = 0.01
theta -= lr * gradient
```
### predcitions
To make the predcitions, let's set our threshold (which can change depending on the business prbolem) to 0.5. If the predicted probability is >= 0.5 then it's considered class 1 vs < 0.5 belong to 0.
```python
def predict_probs(X, theta):
return sigmoid(np.dot(X, theta))
def predict(X, theta, threshold=0.5):
return predict_probs(X, theta) >= threshold
```
# Putting the code together
Now that we have already learned Logistic regression step by step, it makes sense to put it all together in one single code block. It is important for reproducibility reasons.
Here, we are creating a class because all the other functions belong to the same class. It's a neater way to consolidate your code.
```python
# create a class
class LogisticRegression:
def __init__(self, lr=0.01, num_iter=100000, fit_intercept=True, verbose=False):
self.lr = lr
self.num_iter = num_iter
self.fit_intercept = fit_intercept
self.verbose = verbose
# add intercept or the bias term
def __add_intercept(self, X):
intercept = np.ones((X.shape[0], 1))
return np.concatenate((intercept, X), axis=1)
# the sigmoid function that predcits output in terms of probabilites
def __sigmoid(self, z):
return 1 / (1 + np.exp(-z))
#the loss function
def __loss(self, h, y):
return (-y * np.log(h) - (1 - y) * np.log(1 - h)).mean()
# fitting the model
def fit(self, X, y):
if self.fit_intercept:
X = self.__add_intercept(X)
# weights initialization
self.theta = np.zeros(X.shape[1])
for i in range(self.num_iter):
z = np.dot(X, self.theta)
h = self.__sigmoid(z)
gradient = np.dot(X.T, (h - y)) / y.size
self.theta -= self.lr * gradient
z = np.dot(X, self.theta)
h = self.__sigmoid(z)
loss = self.__loss(h, y)
if(self.verbose ==True and i % 10000 == 0):
print(f'loss: {loss} \t')
# predciting the probablities for instances
def predict_prob(self, X):
if self.fit_intercept:
X = self.__add_intercept(X)
return self.__sigmoid(np.dot(X, self.theta))
# output the predcited probablities
def predict(self, X):
return self.predict_prob(X).round()
```
### Evaluation
```python
model = LogisticRegression(lr=0.1, num_iter=300000)
%time model.fit(X, y)
```
CPU times: user 10.6 s, sys: 31.2 ms, total: 10.6 s
Wall time: 10.6 s
```python
preds = model.predict(X)
# accuracy
(preds == y).mean()
1.0
```
1.0
### Resulting Weights
```python
model.theta
```
array([-25.89066442, 12.523156 , -13.40150447])
```python
plt.figure(figsize=(10, 6))
plt.scatter(X[y == 0][:, 0], X[y == 0][:, 1], color='b', label='0')
plt.scatter(X[y == 1][:, 0], X[y == 1][:, 1], color='r', label='1')
plt.legend()
x1_min, x1_max = X[:,0].min(), X[:,0].max(),
x2_min, x2_max = X[:,1].min(), X[:,1].max(),
xx1, xx2 = np.meshgrid(np.linspace(x1_min, x1_max), np.linspace(x2_min, x2_max))
grid = np.c_[xx1.ravel(), xx2.ravel()]
probs = model.predict_prob(grid).reshape(xx1.shape)
plt.contour(xx1, xx2, probs, [0.5], linewidths=1, colors='black');
```
## Logistic Regression Using SKLearn
## Import the relevant library
```python
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import accuracy_score
```
```python
model = LogisticRegression(C=1e20)
```
```python
from sklearn.datasets import load_iris
data = load_iris()
iris = sklearn.datasets.load_iris()
X = iris.data[:, :2]
y = (iris.target != 0) * 1
```
```python
model.fit(X, y)
```
LogisticRegression(C=1e+20)
```python
model.score(X,y)
```
1.0
```python
preds = model.predict(X)
# (preds == y).mean()
```
1.0
```python
print(f'Train Accuracy: {accuracy_score(y, preds)}')
```
Train Accuracy: 1.0
```python
model.intercept_, model.coef_
```
(array([-276.67727715]), array([[ 134.80324426, -147.37951668]]))
# Logistic Regression using Statsmodel
```python
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor
```
```python
# This is like using np.ones to add a vector of ones
X = sm.add_constant(X)
#obust standard errors essentially correct heteroskedasticity in our data
#cov_type = "HC3" is to compute robust standard error
model = sm.Logit(y, X).fit()
predictions = model.predict(X)
print_model = model.summary()
print(print_model)
```
Warning: Maximum number of iterations has been exceeded.
Current function value: 0.000000
Iterations: 35
Logit Regression Results
==============================================================================
Dep. Variable: y No. Observations: 150
Model: Logit Df Residuals: 147
Method: MLE Df Model: 2
Date: Fri, 23 Oct 2020 Pseudo R-squ.: 1.000
Time: 13:27:10 Log-Likelihood: -7.5605e-08
converged: False LL-Null: -95.477
Covariance Type: nonrobust LLR p-value: 3.426e-42
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
const -436.7700 5.43e+05 -0.001 0.999 -1.06e+06 1.06e+06
x1 166.7834 2.13e+05 0.001 0.999 -4.17e+05 4.17e+05
x2 -143.5493 1.89e+05 -0.001 0.999 -3.71e+05 3.71e+05
==============================================================================
Complete Separation: The results show that there iscomplete separation.
In this case the Maximum Likelihood Estimator does not exist and the parameters
are not identified.
/Users/manjulamishra/Library/Python/3.7/lib/python/site-packages/statsmodels/base/model.py:568: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
"Check mle_retvals", ConvergenceWarning)
# Your Turn
1. Load the data from this url
2. Take three numerical features to predict whether a passanger survived or not by implementing Logistic regression in sklearn and Statsmodel.
3. Choose X and y.
4. Run the model and make predictions.
5. Explain your results
6. Give at least two examples of real world problems where you can use logistic regression.
### part 1: Load the data
```python
#Train and test datasets
url_train = "https://raw.githubusercontent.com/agconti/kaggle-titanic/master/data/train.csv"
url_test = "https://raw.githubusercontent.com/agconti/kaggle-titanic/master/data/test.csv"
```
```python
# Be mindful of missing values, you might want to drop them
```
```python
# check the first few rows of the dataframes
```
### part 2: Take three numerical feature
```python
# choose three numerical variables to train the model
```
### part 3. Choose X and y
```python
# X matrix and y
```
### part 4. Run the model and make predictions
```python
# run the model
```
```python
# make the predictions
```
### part 5. Interpret your results
```python
# How would you interpret the coefficents?
```
```python
# which feature are negatively related to Y?
```
```python
# which feature are positively related to Y?
```
### 6. Give at least two examples of real world problems where you can use logistic regression.
```python
# write down the examples where you would use logistic regression to solve the problem.
```
## 5. Limitations of logistic regression
* Overfitting: If the number of observations is lesser than the number of features, Logistic Regression should not be used, otherwise, it may lead to overfitting.
* The major limitation of Logistic Regression is the assumption of linearity between the dependent variable and the independent variables.
* It can only be used to predict discrete functions. Hence, the dependent variable of Logistic Regression is bound to the discrete number set.
* Non-linear problems canβt be solved with logistic regression because it has a linear decision surface. Linearly separable data is rarely found in real-world scenarios.
* Logistic Regression requires average or no multicollinearity between independent variables.
* It is tough to obtain complex relationships using logistic regression. More powerful and compact algorithms such as Neural Networks can easily outperform this algorithm.
* In Linear Regression independent and dependent variables are related linearly. But Logistic Regression needs that independent variables are linearly related to the log odds (log(p/(1-p)).
# Some questions to think about!
* What is overfiting? How can we solve this problem?
* What does intercept/constant term represent?
* How would you interpret coefficients?
* What if the classes are imbalanced ( the bianry classes are not equally reprenseted)?
* What acurancy metric we we use in classification problems?
* How can we use logstic regression for multiclass problems?
## References
https://en.wikipedia.org/wiki/Sigmoid_function
https://www.scirp.org/journal/paperinformation.aspx?paperid=95655
https://ml-cheatsheet.readthedocs.io/en/latest/logistic_regression.htm
https://www.quora.com/What-are-applications-of-linear-and-logistic-regression
https://papers.tinbergen.nl/02119.pdf
https://online.stat.psu.edu/stat504/node/149/.
http://www.stat.cmu.edu/~ryantibs/advmethods/notes/glm.pdf
https://www.geeksforgeeks.org/advantages-and-disadvantages-of-logistic-regression/
############################################ The End ########################################################
|
-- Record modules for no-eta records should not be irrelevant in
-- record even if all fields are irrelevant (cc #392).
open import Agda.Builtin.Equality
postulate A : Set
record Unit : Set where
no-eta-equality
postulate a : A
record Irr : Set where
no-eta-equality
field .unit : Unit
postulate a : A
record Coind : Set where
coinductive
postulate a : A
typeOf : {A : Set} β A β Set
typeOf {A} _ = A
checkUnit : typeOf Unit.a β‘ (Unit β A)
checkUnit = refl
checkIrr : typeOf Irr.a β‘ (Irr β A)
checkIrr = refl
checkCoind : typeOf Coind.a β‘ (Coind β A)
checkCoind = refl
|
[STATEMENT]
lemma regular_implies_setmonotone: "regular f \<Longrightarrow> setmonotone f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. regular f \<Longrightarrow> setmonotone f
[PROOF STEP]
by (simp add: regular_def) |
= = Awards and nominations = =
|
Formal statement is: lemma big_small_asymmetric: "f \<in> L F (g) \<Longrightarrow> g \<in> l F (f) \<Longrightarrow> eventually (\<lambda>x. f x = 0) F" Informal statement is: If $f$ is big-O of $g$ and $g$ is little-o of $f$, then $f$ is eventually zero. |
If $s$ is a $k$-simplex and $a \in s$, then there is a unique $k$-simplex $s'$ such that $s' - \{b\} = s - \{a\}$ for some $b \in s'$. |
module Issue728 where
open import Common.MAlonzo using () renaming (main to mainDefault)
main = mainDefault
|
```python
import sympy as sp
import warnings
warnings.filterwarnings('ignore')
sp.init_printing()
```
```python
N = sp.Symbol("N")
x = sp.IndexedBase("x")
y = sp.IndexedBase("y")
z = sp.IndexedBase("z")
i = sp.Symbol("i")
j = sp.Symbol("j")
k = sp.Symbol("k")
dx = x[i] - x[j]
dy = y[i] - y[j]
dz = z[i] - z[j]
r = sp.sqrt(dx**2 + dy**2 + dz**2)
eps = sp.Symbol("Ξ΅")
sig = sp.Symbol("Ο")
```
```python
#energy = (1/2)*sp.Sum(sp.Sum(4 * eps * ((sig/r)**12 - (sig/r)**6), (i, 1, N)), (j, 1, N))
# It is easier to work with just the summand
energy = 4 * eps * ((sig/r)**12 - (sig/r)**6)
```
```python
energy
```
$\displaystyle 4 Ξ΅ \left(\frac{Ο^{12}}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{6}} - \frac{Ο^{6}}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{3}}\right)$
```python
r.diff(x[k])
```
$\displaystyle \frac{\left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({x}_{i} - {x}_{j}\right)}{2 \sqrt{\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}}}$
```python
energy.diff(x[k])
```
$\displaystyle 4 Ξ΅ \left(- \frac{6 Ο^{12} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({x}_{i} - {x}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{7}} + \frac{3 Ο^{6} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({x}_{i} - {x}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{4}}\right)$
```python
energy.diff(y[k])
```
$\displaystyle 4 Ξ΅ \left(- \frac{6 Ο^{12} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({y}_{i} - {y}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{7}} + \frac{3 Ο^{6} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({y}_{i} - {y}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{4}}\right)$
```python
energy.diff(z[k])
```
$\displaystyle 4 Ξ΅ \left(- \frac{6 Ο^{12} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({z}_{i} - {z}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{7}} + \frac{3 Ο^{6} \left(2 \delta_{i k} - 2 \delta_{j k}\right) \left({z}_{i} - {z}_{j}\right)}{\left(\left({x}_{i} - {x}_{j}\right)^{2} + \left({y}_{i} - {y}_{j}\right)^{2} + \left({z}_{i} - {z}_{j}\right)^{2}\right)^{4}}\right)$
```python
import sympy as sp
import warnings
warnings.filterwarnings('ignore')
sp.init_printing()
eps = sp.Symbol("Ξ΅")
sig = sp.Symbol("Ο")
rad = sp.Symbol("r")
energyRad = 4 * eps * ((sig/rad)**12 - (sig/rad)**6)
energyRad.diff(rad)
```
$\displaystyle 4 Ξ΅ \left(\frac{6 Ο^{6}}{r^{7}} - \frac{12 Ο^{12}}{r^{13}}\right)$
```python
```
```python
```
|
||| Spec: https://webassembly.github.io/spec/core/syntax/instructions.html#numeric-instructions
module WebAssembly.Structure.Instructions.Definition
import WebAssembly.Structure.Values
import WebAssembly.Structure.Types
import WebAssembly.Structure.Modules.Indices
import WebAssembly.Structure.Instructions.Control
import WebAssembly.Structure.Instructions.Memory
import Decidable.Equality
-- Definition
public export
data Instr : Type where
Unreachable : Instr
Nop : Instr
Block : BlockType -> (List Instr) -> Instr
Loop : BlockType -> (List Instr) -> Instr
If : BlockType -> (List Instr) -> (List Instr) -> Instr
Else : Instr
End : Instr
Br : LabelIdx -> Instr
BrIf : LabelIdx -> Instr
BrTable : List LabelIdx -> LabelIdx -> Instr
Return : Instr
Call : FuncIdx -> Instr
CallIndirect : TypeIdx -> Instr
Drop : Instr
Select : Instr
LocalGet : LocalIdx -> Instr
LocalSet : LocalIdx -> Instr
LocalTee : LocalIdx -> Instr
GlobalGet : GlobalIdx -> Instr
GlobalSet : GlobalIdx -> Instr
I32Load : MemArg -> Instr
I64Load : MemArg -> Instr
F32Load : MemArg -> Instr
F64Load : MemArg -> Instr
I32Load8S : MemArg -> Instr
I32Load8U : MemArg -> Instr
I32Load16S : MemArg -> Instr
I32Load16U : MemArg -> Instr
I64Load8S : MemArg -> Instr
I64Load8U : MemArg -> Instr
I64Load16S : MemArg -> Instr
I64Load16U : MemArg -> Instr
I64Load32S : MemArg -> Instr
I64Load32U : MemArg -> Instr
I32Store : MemArg -> Instr
I64Store : MemArg -> Instr
F32Store : MemArg -> Instr
F64Store : MemArg -> Instr
I32Store8 : MemArg -> Instr
I32Store16 : MemArg -> Instr
I64Store8 : MemArg -> Instr
I64Store16 : MemArg -> Instr
I64Store32 : MemArg -> Instr
MemorySize : Instr
MemoryGrow : Instr
I32Const : I32 -> Instr
I64Const : I64 -> Instr
F32Const : F32 -> Instr
F64Const : F64 -> Instr
I32Eqz : Instr
I32Eq : Instr
I32Ne : Instr
I32LtS : Instr
I32LtU : Instr
I32GtS : Instr
I32GtU : Instr
I32LeS : Instr
I32LeU : Instr
I32GeS : Instr
I32GeU : Instr
I64Eqz : Instr
I64Eq : Instr
I64Ne : Instr
I64LtS : Instr
I64LtU : Instr
I64GtS : Instr
I64GtU : Instr
I64LeS : Instr
I64LeU : Instr
I64GeS : Instr
I64GeU : Instr
F32Eq : Instr
F32Ne : Instr
F32Lt : Instr
F32Gt : Instr
F32Le : Instr
F32Ge : Instr
F64Eq : Instr
F64Ne : Instr
F64Lt : Instr
F64Gt : Instr
F64Le : Instr
F64Ge : Instr
I32Clz : Instr
I32Ctz : Instr
I32Popcnt : Instr
I32Add : Instr
I32Sub : Instr
I32Mul : Instr
I32DivS : Instr
I32DivU : Instr
I32RemS : Instr
I32RemU : Instr
I32And : Instr
I32Or : Instr
I32Xor : Instr
I32Shl : Instr
I32ShrS : Instr
I32ShrU : Instr
I32Rotl : Instr
I32Rotr : Instr
I64Clz : Instr
I64Ctz : Instr
I64Popcnt : Instr
I64Add : Instr
I64Sub : Instr
I64Mul : Instr
I64DivS : Instr
I64DivU : Instr
I64RemS : Instr
I64RemU : Instr
I64And : Instr
I64Or : Instr
I64Xor : Instr
I64Shl : Instr
I64ShrS : Instr
I64ShrU : Instr
I64Rotl : Instr
I64Rotr : Instr
F32Abs : Instr
F32Neg : Instr
F32Ceil : Instr
F32Floor : Instr
F32Trunc : Instr
F32Nearest : Instr
F32Sqrt : Instr
F32Add : Instr
F32Sub : Instr
F32Mul : Instr
F32Div : Instr
F32Min : Instr
F32Max : Instr
F32Copysign : Instr
F64Abs : Instr
F64Neg : Instr
F64Ceil : Instr
F64Floor : Instr
F64Trunc : Instr
F64Nearest : Instr
F64Sqrt : Instr
F64Add : Instr
F64Sub : Instr
F64Mul : Instr
F64Div : Instr
F64Min : Instr
F64Max : Instr
F64Copysign : Instr
I32WrapI64 : Instr
I32TruncF32S : Instr
I32TruncF32U : Instr
I32TruncF64S : Instr
I32TruncF64U : Instr
I64ExtendI32S : Instr
I64ExtendI32U : Instr
I64TruncF32S : Instr
I64TruncF32U : Instr
I64TruncF64S : Instr
I64TruncF64U : Instr
F32ConvertI32S : Instr
F32ConvertI32U : Instr
F32ConvertI64S : Instr
F32ConvertI64U : Instr
F32DemoteF64 : Instr
F64ConvertI32S : Instr
F64ConvertI32U : Instr
F64ConvertI64S : Instr
F64ConvertI64U : Instr
F64PromoteF32 : Instr
I32ReinterpretF32 : Instr
I64ReinterpretF64 : Instr
F32ReinterpretI32 : Instr
F64ReinterpretI64 : Instr
I32Extend8S : Instr
I32Extend16S : Instr
I64Extend8S : Instr
I64Extend16S : Instr
I64Extend32S : Instr
I32TruncSatF32S : Instr
I32TruncSatF32U : Instr
I32TruncSatF64S : Instr
I32TruncSatF64U : Instr
I64TruncSatF32S : Instr
I64TruncSatF32U : Instr
I64TruncSatF64S : Instr
I64TruncSatF64U : Instr
public export
Expr : Type
Expr = List Instr
|
[STATEMENT]
lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a"
for x :: real
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>0 < a; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> x powr a < y powr a
[PROOF STEP]
by (simp add: powr_def) |
From Coq Require Import Program.
From MetaCoq.TypedExtraction Require Import Utils.
From MetaCoq.TypedExtraction Require Import ExAst.
From Equations Require Import Equations.
From MetaCoq.Erasure Require Import EArities.
From MetaCoq.Erasure Require Import EAstUtils.
From MetaCoq.Erasure Require ErasureFunction.
From MetaCoq.PCUIC Require Import PCUICArities.
From MetaCoq.PCUIC Require Import PCUICAstUtils.
From MetaCoq.PCUIC Require Import PCUICCanonicity.
From MetaCoq.PCUIC Require Import PCUICConfluence.
From MetaCoq.PCUIC Require Import PCUICContextConversion.
From MetaCoq.PCUIC Require Import PCUICContexts.
From MetaCoq.PCUIC Require Import PCUICConversion.
From MetaCoq.PCUIC Require Import PCUICInductiveInversion.
From MetaCoq.PCUIC Require Import PCUICInversion.
From MetaCoq.PCUIC Require Import PCUICLiftSubst.
From MetaCoq.PCUIC Require Import PCUICNormal.
From MetaCoq.PCUIC Require Import PCUICSN.
From MetaCoq.PCUIC Require Import PCUICSR.
From MetaCoq.PCUIC Require Import PCUICSafeLemmata.
From MetaCoq.PCUIC Require Import PCUICSubstitution.
From MetaCoq.PCUIC Require Import PCUICTyping.
From MetaCoq.PCUIC Require Import PCUICValidity.
From MetaCoq.PCUIC Require Import PCUICWellScopedCumulativity.
From MetaCoq.PCUIC Require Import PCUICCumulativity.
From MetaCoq.SafeChecker Require Import PCUICSafeReduce.
From MetaCoq.SafeChecker Require Import PCUICSafeRetyping.
From MetaCoq.SafeChecker Require Import PCUICWfEnv.
From MetaCoq.SafeChecker Require Import PCUICWfEnvImpl.
From MetaCoq.Template Require Import Kernames.
From MetaCoq.Template Require Import config.
Import PCUICAst.PCUICEnvTyping.
Import PCUICErrors.
Import PCUICReduction.
Import VectorDef.VectorNotations.
Set Equations Transparent.
Module P := PCUICAst.
Module Ex := ExAst.
Import PCUICAst.
Implicit Types (cf : checker_flags).
Local Existing Instance extraction_checker_flags.
Local Obligation Tactic := simpl in *; program_simplify; CoreTactics.equations_simpl;
try program_solve_wf.
Section FixSigmaExt.
Context {X_type : abstract_env_impl}
{X : X_type.Ο2.Ο1}.
Local Definition heΞ£ Ξ£ (wfΞ£ : abstract_env_ext_rel X Ξ£) :
β₯ wf_ext Ξ£ β₯ := abstract_env_ext_wf _ wfΞ£.
Local Definition HΞ£ Ξ£ (wfΞ£ : abstract_env_ext_rel X Ξ£) :
β₯ wf Ξ£ β₯ := abstract_env_ext_sq_wf _ _ _ wfΞ£.
Opaque ErasureFunction.wf_reduction.
Opaque reduce_term.
Lemma sq_red_transitivity {Ξ£} {Ξ A} B {C} :
β₯red Ξ£ Ξ A Bβ₯ ->
β₯red Ξ£ Ξ B Cβ₯ ->
β₯red Ξ£ Ξ A Cβ₯.
Proof.
intros.
sq.
now transitivity B.
Qed.
Lemma isArity_red (Ξ£ : global_env_ext) Ξ u v :
isArity u ->
red Ξ£ Ξ u v ->
isArity v.
Proof.
intros arity_u r.
induction r using red_rect'; [easy|].
eapply isArity_red1; eassumption.
Qed.
Lemma isType_red_sq Ξ£0 (wf : β₯ wf_ext Ξ£0 β₯) Ξ t t' :
β₯isType Ξ£0 Ξ tβ₯ ->
β₯red Ξ£0 Ξ t t'β₯ ->
β₯isType Ξ£0 Ξ t'β₯.
Proof.
intros [(s & typ)] [r].
sq.
exists s. eapply subject_reduction; eauto.
Qed.
Hint Resolve isType_red_sq : erase.
Lemma isType_prod_dom Ξ£0 (wf : β₯ wf_ext Ξ£0 β₯) Ξ na A B :
β₯isType Ξ£0 Ξ (tProd na A B)β₯ ->
β₯isType Ξ£0 Ξ Aβ₯.
Proof.
intros [(s & typ)].
sq.
apply inversion_Prod in typ as (s' & ? & ? & ? & ?); [|now eauto].
now exists s'.
Qed.
Hint Resolve isType_prod_dom : erase.
Lemma isType_prod_cod Ξ£0 (wf : β₯ wf_ext Ξ£0 β₯) Ξ na A B :
β₯isType Ξ£0 Ξ (tProd na A B)β₯ ->
β₯isType Ξ£0 (Ξ,, vass na A) Bβ₯.
Proof.
intros [(s & typ)].
sq.
apply inversion_Prod in typ as (s' & ? & ? & ? & ?); [|now eauto].
now exists x.
Qed.
Hint Resolve isType_prod_cod : erase.
Hint Resolve Is_conv_to_Arity_red : erase.
Hint Resolve reduce_term_sound sq existT pair : erase.
Definition is_prod_or_sort (t : term) : bool :=
match t with
| tProd _ _ _
| tSort _ => true
| _ => false
end.
Import ErasureFunction.
Lemma not_prod_or_sort_hnf {Ξ£} {wfΞ£ : abstract_env_ext_rel X Ξ£} {Ξ : context} {t : term}
{h : forall Ξ£ : global_env_ext, abstract_env_ext_rel X Ξ£ -> welltyped Ξ£ Ξ t} :
negb (is_prod_or_sort (hnf (X_type := X_type) Ξ t h)) ->
~Is_conv_to_Arity Ξ£ Ξ t.
Proof.
intros nar car.
unfold hnf in nar.
specialize_Ξ£ wfΞ£. pose proof (h _ wfΞ£) as [C hc].
pose proof (reduce_term_sound RedFlags.default _ _ Ξ t h Ξ£ wfΞ£) as [r].
apply PCUICWellScopedCumulativity.closed_red_red in r as r''.
pose proof (reduce_term_complete RedFlags.default _ _ Ξ£ wfΞ£ Ξ t h) as wh.
assert (H : β₯ wf Ξ£ β₯) by now apply HΞ£. destruct H.
apply Is_conv_to_Arity_inv in car as [(?&?&?&[r'])|(?&[r'])]; auto.
- eapply closed_red_confluence in r' as (?&r1&r2); eauto.
apply invert_red_prod in r2 as [? [? [? ?]]]; subst; auto.
destruct wh as [wh].
eapply whnf_red_inv in wh; eauto.
depelim wh.
rewrite H in nar.
now cbn in nar.
- eapply closed_red_confluence in r' as (?&r1&r2); eauto.
apply invert_red_sort in r2 as ->; auto.
destruct wh as [wh].
eapply whnf_red_inv in wh; eauto.
depelim wh.
rewrite H in nar.
now cbn in nar.
Qed.
Inductive term_sub_ctx : context * term -> context * term -> Prop :=
| sub_prod_dom Ξ na A B : term_sub_ctx (Ξ, A) (Ξ, tProd na A B)
| sub_prod_cod Ξ na A B : term_sub_ctx (Ξ,, vass na A, B) (Ξ, tProd na A B)
| sub_app_arg Ξ arg hd arg1 :
In arg (decompose_app (tApp hd arg1)).2 ->
term_sub_ctx (Ξ, arg) (Ξ, tApp hd arg1).
Derive Signature for term_sub_ctx.
Lemma In_app_inv {Y} (x : Y) xs ys :
In x (xs ++ ys) ->
In x xs \/ In x ys.
Proof.
intros isin.
induction xs; [easy|].
cbn in *.
destruct isin as [->|]; [easy|].
apply IHxs in H.
now destruct H.
Qed.
Lemma well_founded_term_sub_ctx : well_founded term_sub_ctx.
Proof.
intros (Ξ & t).
induction t in Ξ |- *; constructor; intros (Ξs & ts) rel; try solve [inversion rel].
- now depelim rel.
- depelim rel.
destruct (mkApps_elim t1 []).
cbn in *.
rewrite -> decompose_app_rec_mkApps, atom_decompose_app in H by assumption.
cbn in *.
apply In_app_inv in H.
destruct H as [|[|]]; cbn in *; subst; [|easy|easy].
apply (IHt1 Ξ).
destruct (firstn n l) using List.rev_ind; [easy|].
rewrite mkApps_app.
constructor.
cbn.
now rewrite -> decompose_app_rec_mkApps, atom_decompose_app.
Qed.
Context (rΞ£ : global_env_ext)
(wfrΞ£ : rΞ£ βΌ_ext X).
Definition erase_rel : Relation_Definitions.relation (β Ξ t, welltyped rΞ£ Ξ t) :=
fun '(Ξs; ts; wfs) '(Ξl; tl; wfl) =>
β₯βm, red rΞ£ Ξl tl m Γ term_sub_ctx (Ξs, ts) (Ξl, m)β₯.
Lemma cored_prod_l (Ξ£ : global_env_ext) Ξ na A A' B :
cored Ξ£ Ξ A A' ->
cored Ξ£ Ξ (tProd na A B) (tProd na A' B).
Proof.
intros cor.
depelim cor.
- eapply cored_red_trans; [easy|].
now constructor.
- apply cored_red in cor as [cor].
eapply cored_red_trans.
2: now apply prod_red_l.
now apply red_prod_l.
Qed.
Lemma cored_prod_r (Ξ£ : global_env_ext) Ξ na A B B' :
cored Ξ£ (Ξ,, vass na A) B B' ->
cored Ξ£ Ξ (tProd na A B) (tProd na A B').
Proof.
intros cor.
depelim cor.
- eapply cored_red_trans; [easy|].
now constructor.
- apply cored_red in cor as [cor].
eapply cored_red_trans.
2: now apply prod_red_r.
now apply red_prod_r.
Qed.
Lemma well_founded_erase_rel : well_founded erase_rel.
Proof.
intros (Ξl & l & wfl).
assert (w : β₯ wf_ext rΞ£ β₯) by now apply heΞ£. sq.
induction (normalisation _ w Ξl l wfl) as [l _ IH].
remember (Ξl, l) as p.
revert wfl IH.
replace Ξl with (fst p) by (now subst).
replace l with (snd p) by (now subst).
clear Ξl l Heqp.
intros wfl IH.
induction (well_founded_term_sub_ctx p) as [p _ IH'] in p, wfl, IH |- *.
constructor.
intros (Ξs & s & wfs) [(m & mred & msub)].
inversion msub; subst; clear msub.
- eapply Relation_Properties.clos_rt_rtn1 in mred. inversion mred; subst.
+ rewrite H0 in wfl,mred,IH.
apply (IH' (p.1, s)).
{ replace p with (p.1, tProd na s B) by (now destruct p; cbn in *; congruence).
cbn.
constructor. }
intros y cor wfly.
cbn in *.
assert (Hred_prod : β₯rΞ£;;; p.1 |- tProd na s B β* tProd na y B β₯).
{ eapply cored_red in cor as [cor].
constructor.
now apply red_prod_l. }
destruct Hred_prod.
unshelve eapply (IH (tProd na y B)).
3: now repeat econstructor.
1: { eapply red_welltyped in wfl; eauto. }
now apply cored_prod_l.
+ apply Relation_Properties.clos_rtn1_rt in X1.
unshelve eapply (IH (tProd na s B)).
3: now repeat econstructor.
1: { eapply red_welltyped in wfl; eauto.
now apply Relation_Properties.clos_rtn1_rt in mred. }
eapply red_neq_cored.
{ now apply Relation_Properties.clos_rtn1_rt in mred. }
intros eq.
destruct p as [Ξ t];cbn in *;subst.
eapply cored_red_trans in X0; eauto.
eapply ErasureFunction.Acc_no_loop in X0; [easy|].
eapply @normalisation; eauto.
- eapply Relation_Properties.clos_rt_rtn1 in mred; inversion mred; subst.
+ apply (IH' (p.1,, vass na A, s)).
{ replace p with (p.1, tProd na A s) by (destruct p; cbn in *; congruence).
cbn.
now constructor. }
intros y cor wfly.
cbn in *.
eapply cored_red in cor as Hcored.
destruct Hcored.
unshelve eapply IH.
4: { constructor.
eexists.
split; try easy.
constructor. }
1: { eapply red_welltyped; eauto.
rewrite H0.
now apply red_prod. }
rewrite H0.
now apply cored_prod_r.
+ apply Relation_Properties.clos_rtn1_rt in X1.
unshelve eapply IH.
4: { constructor.
eexists.
split; try easy.
constructor. }
1: { eapply red_welltyped in wfl; eauto.
now apply Relation_Properties.clos_rtn1_rt in mred. }
eapply red_neq_cored.
{ now apply Relation_Properties.clos_rtn1_rt in mred. }
intros eq.
destruct p as [Ξ t];cbn in *;subst.
eapply cored_red_trans in X0; eauto.
eapply ErasureFunction.Acc_no_loop in X0; [easy|].
eapply @normalisation; eauto.
- eapply Relation_Properties.clos_rt_rtn1 in mred; inversion mred; subst.
+ apply (IH' (p.1, s)).
{ replace p with (p.1, tApp hd arg1) by (destruct p; cbn in *; congruence).
now constructor. }
intros y cor wfly.
destruct (mkApps_elim hd []).
cbn in *.
rewrite -> decompose_app_rec_mkApps, atom_decompose_app in H0 by assumption.
change (tApp ?hd ?arg) with (mkApps hd [arg]) in *.
rewrite <- mkApps_app in *.
set (args := (firstn n l ++ [arg1])%list) in *.
clearbody args.
cbn in *.
assert (cor1 : cored rΞ£ p.1 y s) by easy.
eapply cored_red in cor as [cor].
apply In_split in H0 as (args_pref & args_suf & ->).
unshelve eapply (IH (mkApps f (args_pref ++ y :: args_suf))).
3: { constructor.
econstructor.
split; [easy|].
destruct args_suf using rev_ind.
- rewrite mkApps_app.
constructor.
cbn.
rewrite -> decompose_app_rec_mkApps, atom_decompose_app by auto.
cbn.
now apply in_or_app; right; left.
- rewrite <- app_tip_assoc, app_assoc.
rewrite mkApps_app.
constructor.
cbn.
rewrite -> decompose_app_rec_mkApps, atom_decompose_app by auto.
cbn.
apply in_or_app; left.
apply in_or_app; left.
now apply in_or_app; right; left. }
1: { eapply red_welltyped in wfl; eauto.
rewrite H1.
apply red_mkApps; [easy|].
apply All2_app; [now apply All2_refl|].
constructor; [easy|].
now apply All2_refl. }
depelim cor1; cycle 1.
* eapply cored_red in cor1 as [cor1].
eapply cored_red_trans.
2: apply PCUICReduction.red1_mkApps_r.
2: eapply OnOne2_app.
2: now constructor.
rewrite H1.
apply red_mkApps; [easy|].
apply All2_app; [now apply All2_refl|].
constructor; [easy|].
now apply All2_refl.
* rewrite H1.
constructor.
apply PCUICReduction.red1_mkApps_r.
eapply OnOne2_app.
now constructor.
+ apply Relation_Properties.clos_rtn1_rt in X1.
unshelve eapply (IH (tApp hd arg1)).
3: { constructor.
eexists.
split; try easy.
now constructor. }
1: { eapply red_welltyped in wfl; eauto.
etransitivity; [exact X1|].
now constructor. }
apply red_neq_cored.
{ etransitivity; [exact X1|].
now constructor. }
intros eq.
destruct p;cbn in *;subst.
eapply cored_red_trans in X0; eauto.
eapply ErasureFunction.Acc_no_loop in X0; [easy|].
eapply @normalisation; eauto.
Qed.
Instance WellFounded_erase_rel : WellFounded erase_rel :=
Wf.Acc_intro_generator 1000 well_founded_erase_rel.
Opaque WellFounded_erase_rel.
Hint Constructors term_sub_ctx : erase.
Inductive fot_view : term -> Type :=
| fot_view_prod na A B : fot_view (tProd na A B)
| fot_view_sort univ : fot_view (tSort univ)
| fot_view_other t : negb (is_prod_or_sort t) -> fot_view t.
Equations fot_viewc (t : term) : fot_view t :=
fot_viewc (tProd na A B) := fot_view_prod na A B;
fot_viewc (tSort univ) := fot_view_sort univ;
fot_viewc t := fot_view_other t _.
Lemma tywt {Ξ T Ξ£0} (isT : β₯isType Ξ£0 Ξ Tβ₯) : welltyped Ξ£0 Ξ T.
Proof. destruct isT. now apply isType_welltyped. Qed.
(** Definition of normalized arities *)
Definition arity_ass := aname * term.
Fixpoint mkNormalArity (l : list arity_ass) (s : Universe.t) : term :=
match l with
| [] => tSort s
| (na, A) :: l => tProd na A (mkNormalArity l s)
end.
Lemma isArity_mkNormalArity l s :
isArity (mkNormalArity l s).
Proof.
induction l as [|(na & A) l IH]; cbn; auto.
Qed.
Record conv_arity {Ξ£ Ξ T} : Type := build_conv_arity {
conv_ar_context : list arity_ass;
conv_ar_univ : Universe.t;
conv_ar_red : β₯closed_red Ξ£ Ξ T (mkNormalArity conv_ar_context conv_ar_univ)β₯
}.
Global Arguments conv_arity : clear implicits.
Definition conv_arity_or_not Ξ£ Ξ T : Type :=
(conv_arity Ξ£ Ξ T) + (~β₯conv_arity Ξ£ Ξ Tβ₯).
Definition Is_conv_to_Sort Ξ£ Ξ T : Prop :=
exists univ, β₯red Ξ£ Ξ T (tSort univ)β₯.
Definition is_sort {Ξ£ Ξ T} (c : conv_arity_or_not Ξ£ Ξ T) : option (Is_conv_to_Sort Ξ£ Ξ T).
Proof.
destruct c as [c|not_conv_ar].
- destruct c as [[|(na & A) ctx] univ r].
+ apply Some.
sq. destruct r.
eexists. easy.
+ exact None.
- exact None.
Defined.
Import PCUICSigmaCalculus.
Lemma red_it_mkProd_or_LetIn_smash_context Ξ£ Ξ Ξ t :
red Ξ£ Ξ
(it_mkProd_or_LetIn Ξ t)
(it_mkProd_or_LetIn (smash_context [] Ξ) (expand_lets Ξ t)).
Proof.
induction Ξ in Ξ, t |- * using PCUICInduction.ctx_length_rev_ind; cbn.
- now rewrite expand_lets_nil.
- change (Ξ0 ++ [d]) with ([d],,, Ξ0).
rewrite smash_context_app_expand.
destruct d as [na [b|] ty]; cbn.
+ unfold app_context.
rewrite -> expand_lets_vdef, it_mkProd_or_LetIn_app, app_nil_r.
cbn.
rewrite -> lift0_context, lift0_id, subst_empty.
rewrite subst_context_smash_context.
cbn.
etransitivity.
{ apply red1_red.
apply red_zeta. }
unfold subst1.
rewrite subst_it_mkProd_or_LetIn.
rewrite Nat.add_0_r.
apply X0.
now rewrite subst_context_length.
+ unfold app_context.
rewrite -> expand_lets_vass, !it_mkProd_or_LetIn_app.
cbn.
apply red_prod_r.
rewrite -> subst_context_lift_id by lia.
rewrite lift0_context.
now apply X0.
Qed.
Global Arguments conv_arity : clear implicits.
Lemma conv_arity_Is_conv_to_Arity {Ξ£ Ξ T} :
conv_arity Ξ£ Ξ T ->
Is_conv_to_Arity Ξ£ Ξ T.
Proof.
intros [asses univ r].
eexists.
split; [sq|];tea.
apply isArity_mkNormalArity.
Qed.
Lemma Is_conv_to_Arity_conv_arity {Ξ£ : global_env_ext} {Ξ T} :
β₯ wf Ξ£ β₯ ->
Is_conv_to_Arity Ξ£ Ξ T ->
β₯conv_arity Ξ£ Ξ Tβ₯.
Proof.
intros wfΞ£ (t & [r] & isar).
sq.
destruct (destArity [] t) as [(ctx & univ)|] eqn:dar.
+ set (ctx' := rev_map (fun d => (decl_name d, decl_type d)) (smash_context [] ctx)).
apply (build_conv_arity _ _ _ ctx' univ).
apply PCUICWellScopedCumulativity.closed_red_red in r as r'.
sq.
transitivity t;tea.
apply PCUICLiftSubst.destArity_spec_Some in dar.
cbn in dar.
subst.
replace (mkNormalArity ctx' univ) with (it_mkProd_or_LetIn (smash_context [] ctx) (tSort univ)).
{ destruct r.
assert (PCUICOnFreeVars.on_free_vars (PCUICOnFreeVars.shiftnP #|Ξ| (fun _ : nat => false)) (it_mkProd_or_LetIn ctx (tSort univ))) by (eapply PCUICClosedTyp.red_on_free_vars;eauto;
now rewrite PCUICOnFreeVars.on_free_vars_ctx_on_ctx_free_vars).
constructor;auto.
apply red_it_mkProd_or_LetIn_smash_context. }
subst ctx'.
pose proof (@smash_context_assumption_context [] ctx assumption_context_nil).
clear -H.
induction (smash_context [] ctx) using List.rev_ind; [easy|].
rewrite -> it_mkProd_or_LetIn_app in *.
rewrite rev_map_app.
cbn.
apply assumption_context_app in H as (? & ass_x).
depelim ass_x.
cbn.
f_equal.
now apply IHc.
+ exfalso.
clear -isar dar.
revert dar.
generalize ([] : context).
induction t; intros ctx; cbn in *; eauto; try congruence.
Qed.
Definition is_arity {Ξ£ : global_env_ext} {Ξ T} (wfΞ£ : β₯ wf Ξ£ β₯) (c : conv_arity_or_not Ξ£ Ξ T) :
{Is_conv_to_Arity Ξ£ Ξ T} + {~Is_conv_to_Arity Ξ£ Ξ T}.
Proof.
destruct c; [left|right].
- eapply conv_arity_Is_conv_to_Arity;eauto.
- abstract (intros conv; apply Is_conv_to_Arity_conv_arity in conv; tauto).
Defined.
(** type_flag of a term indexed by the term's type. For example, for
t : T
eq_refl : 5 = 5 : Prop
we would pass T to flag_of_type below, and it would give
is_logical = true, conv_ar = right _. On the other hand, for
(fun (X : Type) => X) : Type -> Type
we would pass Type -> Type and get is_logical = false, conv_ar = left _.
*)
Record type_flag {Ξ£ Ξ T} :=
build_flag
{ (** Type is proposition when fully applied, i.e. either
(T : Prop, or T a0 .. an : Prop). If this is an arity,
indicates whether this is a logical arity (i.e. into Prop). *)
is_logical : bool;
(** Arity that this type is convertible to *)
conv_ar : conv_arity_or_not Ξ£ Ξ T;
}.
Global Arguments type_flag : clear implicits.
Import PCUICSN.
Existing Instance extraction_normalizing.
Hint Resolve abstract_env_wf : erase.
Definition isTT Ξ T :=
forall Ξ£0 (wfΞ£ : abstract_env_ext_rel X Ξ£0), β₯isType Ξ£0 Ξ Tβ₯.
Equations(noeqns) flag_of_type (Ξ : context) (T : term)
(isT : forall Ξ£0 (wfΞ£ : abstract_env_ext_rel X Ξ£0), β₯isType Ξ£0 Ξ Tβ₯)
: type_flag rΞ£ Ξ T
by wf ((Ξ;T; (tywt (isT rΞ£ wfrΞ£))) : (β Ξ t, welltyped rΞ£ Ξ t)) erase_rel :=
flag_of_type Ξ T isT with inspect (hnf (X_type := X_type) Ξ T (fun Ξ£ h => (tywt (isT Ξ£ h)))) :=
| exist T0 is_hnf with fot_viewc T0 := {
| fot_view_prod na A B with flag_of_type (Ξ,, vass na A) B _ := {
| flag_cod := {| is_logical := is_logical flag_cod;
conv_ar := match conv_ar flag_cod with
| inl car =>
inl {| conv_ar_context := (na, A) :: conv_ar_context car;
conv_ar_univ := conv_ar_univ car |}
| inr notar => inr _
end
|}
};
| fot_view_sort univ := {| is_logical := Universe.is_prop univ;
conv_ar := inl {| conv_ar_context := [];
conv_ar_univ := univ; |} |} ;
| fot_view_other T0 discr
with infer X_type X Ξ _ T0 _ :=
{
| @existT K princK with inspect (reduce_to_sort Ξ K _) := {
| exist (Checked_comp (existT _ univ red_univ)) eq :=
{| is_logical := Universe.is_prop univ;
conv_ar := inr _ |};
| exist (TypeError_comp t) eq := !
};
} ;
}.
Ltac reduce_term_sound :=
unfold hnf in *;
match goal with
| [H : reduce_term ?flags _ _ ?Ξ ?t ?wft = ?a |- _] =>
let r := fresh "r" in
pose proof (@reduce_term_sound _ flags _ _ Ξ t wft _ (ltac:(eassumption))) as [r];
rewrite -> H in r
end.
Next Obligation.
assert ( β₯ wf_ext Ξ£0 β₯) by now apply heΞ£.
reduce_term_sound.
apply PCUICWellScopedCumulativity.closed_red_red in r;eauto with erase.
Qed.
Next Obligation.
reduce_term_sound.
apply PCUICWellScopedCumulativity.closed_red_red in r;eauto with erase.
Qed.
Next Obligation.
reduce_term_sound.
destruct car as [ctx univ [r']].
cbn.
constructor.
transitivity (tProd na A B). auto.
now apply closed_red_prod_codom.
Qed.
Next Obligation.
reduce_term_sound.
contradiction notar.
assert (Hwf : β₯ wf rΞ£ β₯) by now apply HΞ£.
apply Is_conv_to_Arity_conv_arity;sq;auto.
specialize r as r'.
destruct r'.
assert (prod_conv : Is_conv_to_Arity rΞ£ Ξ (tProd na A B)).
{ eapply Is_conv_to_Arity_red;eauto.
apply conv_arity_Is_conv_to_Arity;assumption. }
destruct prod_conv as [tm [[redtm] ar]].
apply invert_red_prod in redtm.
destruct redtm as [A' [B' [-> [redAA' redBB']]]].
exists B'; easy.
Qed.
Next Obligation.
remember (hnf _ _ _) as b. symmetry in Heqb.
reduce_term_sound; eauto with erase.
Qed.
Next Obligation.
specialize (isT Ξ£ wfΞ£) as isT'.
sq. destruct isT' as [u Htype].
now eapply typing_wf_local.
Qed.
Next Obligation.
apply well_sorted_wellinferred.
assert (Hwf : β₯ wf_ext Ξ£ β₯) by now apply heΞ£.
destruct Hwf.
remember (hnf Ξ T _) as nf. symmetry in Heqnf.
reduce_term_sound.
assert (tyT : β₯ isType Ξ£ Ξ T β₯) by eauto.
assert (tyNf : β₯ isType Ξ£ Ξ nf β₯).
{ sq. eapply isType_red;eauto.
now apply PCUICWellScopedCumulativity.closed_red_red. }
sq.
now apply BDFromPCUIC.isType_infering_sort.
Defined.
Next Obligation. eauto. Defined.
Next Obligation.
remember (hnf Ξ T _) as nf. symmetry in Heqnf.
reduce_term_sound.
specialize (princK Ξ£ wfΞ£) as HH.
assert (β₯ wf Ξ£ β₯) by now apply HΞ£.
assert (tyT : β₯ isType Ξ£ Ξ T β₯) by eauto.
assert (tyNf : β₯ isType Ξ£ Ξ nf β₯).
{ sq. eapply isType_red;eauto.
now apply PCUICWellScopedCumulativity.closed_red_red. }
sq.
destruct tyT as [u tyT].
apply typing_wf_local in tyT.
apply BDToPCUIC.infering_typing in HH;sq;eauto.
eapply isType_welltyped.
eapply validity; eauto.
Qed.
Next Obligation.
clear eq.
specialize (@not_prod_or_sort_hnf rΞ£ wfrΞ£ _ _ _ discr) as d.
clear red_univ.
remember (hnf Ξ T _) as nf. symmetry in Heqnf.
reduce_term_sound.
destruct r.
destruct H as [car].
apply conv_arity_Is_conv_to_Arity in car;eauto.
Qed.
Next Obligation.
pose proof (PCUICSafeReduce.reduce_to_sort_complete _ wfrΞ£ _ _ eq).
clear eq.
apply (@not_prod_or_sort_hnf rΞ£ wfrΞ£) in discr.
remember (hnf Ξ T _) as nf. symmetry in Heqnf.
reduce_term_sound.
destruct r.
specialize (princK rΞ£ wfrΞ£) as HH.
assert (β₯ wf rΞ£ β₯) by now apply HΞ£.
assert (β₯ wf_ext rΞ£ β₯) by now apply heΞ£.
assert (tyT : β₯ isType rΞ£ Ξ T β₯) by eauto.
assert (tyNf : β₯ isType rΞ£ Ξ nf β₯).
{ sq. eapply isType_red;eauto. }
sq.
destruct tyT as [u tyT].
destruct tyNf as [v tyNf].
apply typing_wf_local in tyT.
apply BDToPCUIC.infering_typing in HH;sq;eauto.
specialize (PCUICPrincipality.common_typing _ _ HH tyNf) as [x[x_le_K[x_le_sort?]]].
apply ws_cumul_pb_Sort_r_inv in x_le_sort as (? & x_red & ?).
specialize (ws_cumul_pb_red_l_inv _ x_le_K x_red) as K_ge_sort.
apply ws_cumul_pb_Sort_l_inv in K_ge_sort as (? & K_red & ?).
exact (H _ K_red).
Qed.
Equations erase_type_discr (t : term) : Prop := {
| tRel _ := False;
| tSort _ := False;
| tProd _ _ _ := False;
| tApp _ _ := False;
| tConst _ _ := False;
| tInd _ _ := False;
| _ := True
}.
Inductive erase_type_view : term -> Type :=
| et_view_rel i : erase_type_view (tRel i)
| et_view_sort u : erase_type_view (tSort u)
| et_view_prod na A B : erase_type_view (tProd na A B)
| et_view_app hd arg : erase_type_view (tApp hd arg)
| et_view_const kn u : erase_type_view (tConst kn u)
| et_view_ind ind u : erase_type_view (tInd ind u)
| et_view_other t : erase_type_discr t -> erase_type_view t.
Equations erase_type_viewc (t : term) : erase_type_view t := {
| tRel i := et_view_rel i;
| tSort u := et_view_sort u;
| tProd na A B := et_view_prod na A B;
| tApp hd arg := et_view_app hd arg;
| tConst kn u := et_view_const kn u;
| tInd ind u := et_view_ind ind u;
| t := et_view_other t _
}.
Inductive tRel_kind :=
(** tRel refers to type variable n in the list of type vars *)
| RelTypeVar (n : nat)
(** tRel refers to an inductive type (used in constructors of inductives) *)
| RelInductive (ind : inductive)
(** tRel refers to something else, for example something logical or a value *)
| RelOther.
Equations(noeqns) erase_type_aux
(Ξ : context)
(erΞ : Vector.t tRel_kind #|Ξ|)
(t : term)
(isT : forall Ξ£ (wfΞ£ : PCUICWfEnv.abstract_env_ext_rel X Ξ£), β₯isType Ξ£ Ξ tβ₯)
(** The index of the next type variable that is being
produced, or None if no more type variables should be
produced (when not at the top level). For example, in
Type -> nat we should erase to nat with one type var,
while in (Type -> nat) -> nat we should erase to (TBox ->
nat) -> nat with no type vars. *)
(next_tvar : option nat) : list name Γ box_type
by wf ((Ξ; t; (tywt (isT rΞ£ wfrΞ£))) : (β Ξ t, welltyped rΞ£ Ξ t)) erase_rel :=
erase_type_aux Ξ erΞ t isT next_tvar
with inspect (reduce_term RedFlags.nodelta X_type X Ξ t (fun Ξ£ h => (tywt (isT Ξ£ h)))) :=
| exist t0 eq_hnf with is_logical (flag_of_type Ξ t0 _) := {
| true := ([], TBox);
| false with erase_type_viewc t0 := {
| et_view_rel i with @Vector.nth_order _ _ erΞ i _ := {
| RelTypeVar n := ([], TVar n);
| RelInductive ind := ([], TInd ind);
| RelOther := ([], TAny)
};
| et_view_sort _ := ([], TBox);
| et_view_prod na A B with flag_of_type Ξ A _ := {
(** For logical things we just box and proceed *)
| {| is_logical := true |} =>
on_snd
(TArr TBox)
(erase_type_aux (Ξ,, vass na A) (RelOther :: erΞ)%vector B _ next_tvar);
(** If the type isn't an arity now, then the domain is a "normal" type like nat. *)
| {| conv_ar := inr _ |} :=
let '(_, dom) := erase_type_aux Ξ erΞ A _ None in
on_snd
(TArr dom)
(erase_type_aux (Ξ,, vass na A) (RelOther :: erΞ)%vector B _ next_tvar);
(** Ok, so it is an arity. We add type variables for all
arities (even non-sorts) because more things are typable without
coercions this way. In particular, type schemes only used
in contravariant positions extract to something typable
even without higher-kinded types. For example:
Definition test (T : Type -> Type) (x : T nat) (y : T bool) : nat := 0.
Definition bar := test option None None.
Here [bar] is perfectly extractable without coercions if T becomes a type
variable. *)
| _ =>
let var :=
match next_tvar with
| Some i => RelTypeVar i
| None => RelOther
end in
let '(tvars, cod) :=
erase_type_aux
(Ξ,, vass na A) (var :: erΞ)%vector
B _
(option_map S next_tvar) in
(if next_tvar then binder_name na :: tvars else tvars, TArr TBox cod)
};
| et_view_app orig_hd orig_arg with inspect (decompose_app (tApp orig_hd orig_arg)) := {
| exist (hd, decomp_args) eq_decomp :=
let hdbt :=
match hd as h return h = hd -> _ with
| tRel i =>
fun _ =>
match @Vector.nth_order _ _ erΞ i _ with
| RelInductive ind => TInd ind
| RelTypeVar i => TVar i
| RelOther => TAny
end
| tConst kn _ => fun _ => TConst kn
| tInd ind _ => fun _ => TInd ind
| _ => fun _ => TAny
end eq_refl in
(** Now for heads that can take args, add args.
Otherwise drop all args. *)
if can_have_args hdbt then
let erase_arg (a : term) (i : In a decomp_args) : box_type :=
let (aT, princaT) := infer X_type X Ξ _ a _ in
match flag_of_type Ξ aT _ with
| {| is_logical := true |} => TBox
| {| conv_ar := car |} =>
match is_sort car with
| Some conv_sort => snd (erase_type_aux Ξ erΞ a _ None)
| None => TAny (* non-sort arity or value *)
end
end in
([], mkTApps hdbt (map_In decomp_args erase_arg))
else
([], hdbt)
};
| et_view_const kn _ := ([], TConst kn);
| et_view_ind ind _ := ([], TInd ind);
| et_view_other t0 _ := ([], TAny)
}
}.
Ltac wfAbstractEnv :=
match goal with
| [H : ?Ξ£ βΌ_ext ?X |- _] =>
pose proof (@abstract_env_ext_wf _ _ _ _ _ X Ξ£ H)
end.
Solve All Obligations with
Tactics.program_simplify;
CoreTactics.equations_simpl; try (reduce_term_sound; destruct r); wfAbstractEnv; eauto with erase.
Next Obligation.
remember (reduce_term _ _ _ _ _ _) as _b;symmetry in Heq_b.
reduce_term_sound; destruct r; eauto using abstract_env_ext_wf with erase.
Qed.
Next Obligation.
reduce_term_sound.
destruct (isT _ wfrΞ£) as [(? & typ)].
assert (β₯ wf rΞ£ β₯) by now apply HΞ£. sq.
destruct r.
eapply subject_reduction with (u := tRel i) in typ; eauto.
apply inversion_Rel in typ as (? & _ & ? & _);[|easy].
now apply nth_error_Some.
Qed.
Next Obligation.
destruct (isT _ wfrΞ£) as [(? & typ)].
assert (β₯ wf rΞ£ β₯) by eauto using HΞ£;sq.
reduce_term_sound;destruct r.
eapply subject_reduction in typ; eauto.
replace (tApp orig_hd orig_arg) with (mkApps (tRel i) decomp_args) in typ; cycle 1.
{ symmetry. apply decompose_app_inv.
now rewrite <- eq_decomp. }
apply inversion_mkApps in typ.
destruct typ as (rel_type & rel_typed & spine).
apply inversion_Rel in rel_typed; [|easy].
apply nth_error_Some.
destruct rel_typed as (? & _ & ? & _).
congruence.
Qed.
Next Obligation.
specialize (isT _ wfΞ£) as isT'.
sq. destruct isT' as [u Htype].
now eapply typing_wf_local.
Qed.
Next Obligation.
destruct (isT _ wfΞ£) as [(? & typ)].
assert (w : β₯ wf Ξ£ β₯) by eauto using HΞ£;sq.
reduce_term_sound;destruct r.
eapply subject_reduction in typ; eauto.
replace (tApp orig_hd orig_arg) with (mkApps hd decomp_args) in typ; cycle 1.
{ symmetry. apply decompose_app_inv.
now rewrite <- eq_decomp. }
apply inversion_mkApps in typ.
destruct typ as (? & ? & spine).
sq.
clear -spine i w.
induction spine; [easy|].
destruct i.
+ subst a.
eapply BDFromPCUIC.typing_infering in t.
destruct t as (? & ? & ?).
econstructor;eauto.
+ easy.
Qed.
Next Obligation.
clear eq_hnf.
destruct (princaT _ wfΞ£) as [inf_aT].
assert (HH : β₯ wf_ext Ξ£0 β₯) by now apply heΞ£.
destruct HH.
specialize (isT _ wfΞ£) as [[? Hty]].
apply typing_wf_local in Hty.
apply BDToPCUIC.infering_typing in inf_aT;eauto with erase.
sq.
now eapply validity.
Qed.
Next Obligation.
clear eq_hnf.
assert (rΞ£ = Ξ£).
{ eapply abstract_env_ext_irr;eauto. }
subst.
destruct (princaT _ wfΞ£) as [inf_aT].
assert (HH : β₯ wf_ext Ξ£ β₯) by now apply heΞ£.
destruct HH.
specialize (isT _ wfΞ£) as [[? Hty]].
apply typing_wf_local in Hty.
apply BDToPCUIC.infering_typing in inf_aT;eauto with erase.
destruct conv_sort as (univ & reduniv).
sq.
exists univ.
eapply type_reduction;eauto.
Qed.
Next Obligation.
reduce_term_sound;destruct r.
sq.
exists (tApp orig_hd orig_arg).
split; [easy|].
constructor.
rewrite eq_decomp.
easy.
Qed.
Definition erase_type (t : term) (isT :forall Ξ£ (wfΞ£ : PCUICWfEnv.abstract_env_ext_rel X Ξ£), β₯isType Ξ£ [] tβ₯) : list name Γ box_type :=
erase_type_aux [] []%vector t isT (Some 0).
Lemma typwt {Ξ t T} Ξ£0 :
β₯Ξ£0 ;;; Ξ |- t : Tβ₯ ->
welltyped Ξ£0 Ξ t.
Proof.
intros [typ].
econstructor; eauto.
Qed.
Inductive erase_type_scheme_view : term -> Type :=
| erase_type_scheme_view_lam na A B : erase_type_scheme_view (tLambda na A B)
| erase_type_scheme_view_other t : negb (isLambda t) -> erase_type_scheme_view t.
Equations erase_type_scheme_viewc (t : term) : erase_type_scheme_view t :=
erase_type_scheme_viewc (tLambda na A B) := erase_type_scheme_view_lam na A B;
erase_type_scheme_viewc t := erase_type_scheme_view_other t _.
Definition type_var_info_of_flag (na : aname) {Ξ£ : global_env_ext} {Ξ t} (w : β₯ wf Ξ£ β₯) (f : type_flag Ξ£ Ξ t) : type_var_info :=
{| tvar_name := binder_name na;
tvar_is_logical := is_logical f;
tvar_is_arity := if is_arity w (conv_ar f) then true else false;
tvar_is_sort := if is_sort (conv_ar f) then true else false; |}.
(** For a non-lambda type scheme, i.e.
t : T1 -> T2 -> ... -> Tn -> Type
where t is not a lambda, finish erasing it as a type scheme
by repeatedly eta expanding it *)
Equations (noeqns) erase_type_scheme_eta
(Ξ : context)
(erΞ : Vector.t tRel_kind #|Ξ|)
(t : term)
(ar_ctx : list arity_ass)
(ar_univ : Universe.t)
(typ : β₯rΞ£;;; Ξ |- t : mkNormalArity ar_ctx ar_univβ₯)
(next_tvar : nat) : list type_var_info Γ box_type :=
erase_type_scheme_eta Ξ erΞ t [] univ typ next_tvar => ([], (erase_type_aux Ξ erΞ t _ None).2);
erase_type_scheme_eta Ξ erΞ t ((na, A) :: ar_ctx) univ typ next_tvar =>
let inf := type_var_info_of_flag na (HΞ£ _ wfrΞ£) (flag_of_type Ξ A _) in
let (kind, new_next_tvar) :=
if tvar_is_arity inf then
(RelTypeVar next_tvar, S next_tvar)
else
(RelOther, next_tvar) in
let '(infs, bt) := erase_type_scheme_eta
(Ξ,, vass na A)
(kind :: erΞ)%vector
(tApp (lift0 1 t) (tRel 0))
ar_ctx univ _
new_next_tvar in
(inf :: infs, bt).
Next Obligation.
assert (H : rΞ£ = Ξ£).
{ eapply abstract_env_ext_irr;eauto. }
rewrite <- H.
destruct typ.
assert (wf_local rΞ£ Ξ) by (eapply typing_wf_local; eauto).
assert (β₯ wf rΞ£ β₯) by now apply HΞ£ .
constructor; eexists;eassumption.
Qed.
Next Obligation.
destruct typ as [typ].
assert (H : rΞ£ = Ξ£0).
{ eapply abstract_env_ext_irr;eauto. }
rewrite <- H.
assert (wf_local rΞ£ Ξ) by (eapply typing_wf_local; eauto).
assert (β₯ wf rΞ£ β₯) by now apply HΞ£ . sq.
apply validity in typ; auto.
apply isType_tProd in typ; auto.
exact (fst typ).
Qed.
Next Obligation.
assert (β₯ wf rΞ£ β₯) by now apply HΞ£ . sq.
apply typing_wf_local in typ as wfl.
assert (wflext : wf_local rΞ£ (Ξ,, vass na A)).
{ apply validity in typ; auto.
apply isType_tProd in typ as (_ & typ); auto.
eapply isType_wf_local; eauto. }
rewrite <- (PCUICSpine.subst_rel0_lift_id 0 (mkNormalArity ar_ctx univ)).
eapply validity in typ as typ_valid;auto.
destruct typ_valid as [u Hty].
eapply type_App.
+ eapply validity in typ as typ;auto.
eapply (PCUICWeakeningTyp.weakening _ _ [_] _ _ _ wflext Hty).
+ eapply (PCUICWeakeningTyp.weakening _ _ [_] _ _ _ wflext typ).
+ fold lift.
eapply (type_Rel _ _ _ (vass na A)); auto.
Qed.
Equations? (noeqns) erase_type_scheme
(Ξ : context)
(erΞ : Vector.t tRel_kind #|Ξ|)
(t : term)
(ar_ctx : list arity_ass)
(ar_univ : Universe.t)
(typ : forall Ξ£0 (wfΞ£ : PCUICWfEnv.abstract_env_ext_rel X Ξ£0), β₯Ξ£0;;; Ξ |- t : mkNormalArity ar_ctx ar_univβ₯)
(next_tvar : nat) : list type_var_info Γ box_type :=
erase_type_scheme Ξ erΞ t [] univ typ next_tvar => ([], (erase_type_aux Ξ erΞ t _ None).2);
erase_type_scheme Ξ erΞ t ((na', A') :: ar_ctx) univ typ next_tvar
with inspect (reduce_term RedFlags.nodelta X_type X Ξ t (fun Ξ£0 h => (typwt _ (typ Ξ£0 h)))) := {
| exist thnf eq_hnf with erase_type_scheme_viewc thnf := {
| erase_type_scheme_view_lam na A body =>
let inf := type_var_info_of_flag na (HΞ£ _ wfrΞ£) (flag_of_type Ξ A _) in
let (kind, new_next_tvar) :=
if tvar_is_arity inf then
(RelTypeVar next_tvar, S next_tvar)
else
(RelOther, next_tvar) in
let '(infs, bt) := erase_type_scheme
(Ξ,, vass na A) (kind :: erΞ)%vector
body ar_ctx univ _ new_next_tvar in
(inf :: infs, bt);
| erase_type_scheme_view_other thnf _ =>
erase_type_scheme_eta Ξ erΞ t ((na', A') :: ar_ctx) univ (typ _ wfrΞ£) next_tvar
}
}.
Proof.
- destruct (typ _ wfΞ£).
constructor; eexists; eauto.
- destruct (typ _ wfΞ£) as [typ0].
reduce_term_sound.
assert (β₯ wf Ξ£0 β₯) by now apply HΞ£.
sq.
destruct r as [?? r].
eapply subject_reduction in r; eauto.
apply inversion_Lambda in r as (?&?&?&?&?); auto.
eexists; eassumption.
- clear inf.
destruct (typ _ wfΞ£) as [typ0].
reduce_term_sound.
assert (β₯ wf Ξ£0 β₯) by now apply HΞ£.
sq.
destruct r as [?? r].
assert (rΞ£ = Ξ£0).
{ eapply abstract_env_ext_irr;eauto. }
subst.
eapply subject_reduction in r; eauto.
apply inversion_Lambda in r as (?&?&?&?&c); auto.
assert (wf_local Ξ£0 Ξ) by (eapply typing_wf_local; eauto).
apply ws_cumul_pb_Prod_Prod_inv_l in c as [???]; auto.
eapply validity in typ0 as typ0; auto.
apply isType_tProd in typ0 as (_ & (u&?)); auto.
assert (PCUICCumulativity.conv_context cumulAlgo_gen Ξ£0 (Ξ,, vass na' A') (Ξ,, vass na A)).
{ constructor; [reflexivity|].
constructor. now symmetry.
apply ws_cumul_pb_forget_conv. now symmetry.
}
eapply type_Cumul.
+ eassumption.
+ eapply PCUICContextConversionTyp.context_conversion; eauto.
eapply typing_wf_local; eassumption.
+ now apply cumulAlgo_cumulSpec.
Qed.
Import ExAst.
Equations? erase_arity
(cst : PCUICEnvironment.constant_body)
(car : conv_arity rΞ£ [] (PCUICEnvironment.cst_type cst))
(wt : β₯on_constant_decl (lift_typing typing) rΞ£ cstβ₯)
: option (list type_var_info Γ box_type) :=
erase_arity cst car wt with inspect (PCUICEnvironment.cst_body cst) := {
| exist (Some body) body_eq =>
Some (erase_type_scheme [] []%vector body (conv_ar_context car) (conv_ar_univ car) _ 0);
| exist None _ => None
}.
Proof.
unfold on_constant_decl in wt.
rewrite body_eq in wt.
cbn in *.
assert (rΞ£ = Ξ£0).
{ eapply abstract_env_ext_irr;eauto. }
subst.
assert (β₯ wf Ξ£0 β₯) by now apply HΞ£.
destruct car as [ctx univ r].
sq.
eapply type_reduction in wt; eauto;cbn.
now destruct r.
Qed.
Equations? erase_constant_decl
(cst : PCUICEnvironment.constant_body)
(wt : β₯on_constant_decl (lift_typing typing) rΞ£ cstβ₯)
: constant_body + option (list type_var_info Γ box_type) :=
erase_constant_decl cst wt with flag_of_type [] (PCUICEnvironment.cst_type cst) _ := {
| {| conv_ar := inl car |} =>
inr (erase_arity cst car wt)
| {| conv_ar := inr notar |} =>
let erased_body := erase_constant_body X_type X cst _ in
inl {| cst_type := erase_type (PCUICEnvironment.cst_type cst) _; cst_body := EAst.cst_body (fst erased_body) |}
}.
Proof.
- assert (rΞ£ = Ξ£0).
{ eapply abstract_env_ext_irr;eauto. }
subst.
assert (β₯ wf Ξ£0 β₯) by now apply HΞ£.
unfold on_constant_decl in wt.
destruct (PCUICEnvironment.cst_body cst); cbn in *.
+ sq;eapply validity;eauto.
+ destruct wt.
eexists; eassumption.
- assert (rΞ£ = Ξ£).
{ eapply abstract_env_ext_irr;eauto. }
easy.
- assert (rΞ£ = Ξ£).
{ eapply abstract_env_ext_irr;eauto. }
subst.
assert (β₯ wf Ξ£ β₯) by now apply HΞ£.
unfold on_constant_decl in wt.
destruct (PCUICEnvironment.cst_body cst).
+ sq.
now eapply validity in wt.
+ sq.
cbn in wt.
destruct wt as (s & ?).
now exists s.
Qed.
Import P.
Equations? (noeqns) erase_ind_arity
(Ξ : context)
(t : term)
(isT : forall Ξ£ (wfΞ£ : PCUICWfEnv.abstract_env_ext_rel X Ξ£), β₯isType Ξ£ Ξ tβ₯) : list type_var_info
by wf ((Ξ; t; tywt (isT _ wfrΞ£)) : (β Ξ t, welltyped rΞ£ Ξ t)) erase_rel :=
erase_ind_arity Ξ t isT with inspect (hnf (X_type := X_type) Ξ t (fun Ξ£ h => (tywt (isT Ξ£ h)))) := {
| exist (tProd na A B) hnf_eq =>
let hd := type_var_info_of_flag na (HΞ£ _ wfrΞ£)(flag_of_type Ξ A _) in
let tl := erase_ind_arity (Ξ,, vass na A) B _ in
hd :: tl;
| exist _ _ := []
}.
Proof.
all: specialize (isT _ wfrΞ£) as typ.
- assert (β₯ wf_ext Ξ£0 β₯) by (now apply heΞ£);
assert (rΞ£ = Ξ£0) by (eapply abstract_env_ext_irr;eauto);subst;
reduce_term_sound; destruct r; eauto with erase.
- assert (β₯ wf_ext Ξ£ β₯) by (now apply heΞ£);
assert (rΞ£ = Ξ£) by (eapply abstract_env_ext_irr;eauto);subst;
reduce_term_sound; destruct r; eauto with erase.
- reduce_term_sound; destruct r; eauto with erase.
Qed.
Definition ind_aname (oib : PCUICEnvironment.one_inductive_body) :=
{| binder_name := nNamed (PCUICEnvironment.ind_name oib);
binder_relevance := PCUICEnvironment.ind_relevance oib |}.
Definition arities_contexts
(mind : kername)
(oibs : list PCUICEnvironment.one_inductive_body) : βΞ, Vector.t tRel_kind #|Ξ| :=
(fix f (oibs : list PCUICEnvironment.one_inductive_body)
(i : nat)
(Ξ : context) (erΞ : Vector.t tRel_kind #|Ξ|) :=
match oibs with
| [] => (Ξ; erΞ)
| oib :: oibs =>
f oibs
(S i)
(Ξ,, vass (ind_aname oib) (PCUICEnvironment.ind_type oib))
(RelInductive {| inductive_mind := mind;
inductive_ind := i |} :: erΞ)%vector
end) oibs 0 [] []%vector.
Lemma arities_contexts_cons_1 mind oib oibs :
(arities_contexts mind (oib :: oibs)).Ο1 =
(arities_contexts mind oibs).Ο1 ++ [vass (ind_aname oib) (PCUICEnvironment.ind_type oib)].
Proof.
unfold arities_contexts.
match goal with
| |- (?f' _ _ _ _).Ο1 = _ => set (f := f')
end.
assert (H : forall oibs n Ξ erΞ, (f oibs n Ξ erΞ).Ο1 = (f oibs 0 [] []%vector).Ο1 ++ Ξ).
{ clear.
intros oibs.
induction oibs as [|oib oibs IH]; [easy|].
intros n Ξ erΞ.
cbn.
rewrite IH; symmetry; rewrite IH.
now rewrite <- List.app_assoc. }
now rewrite H.
Qed.
Lemma arities_contexts_1 mind oibs :
(arities_contexts mind oibs).Ο1 = arities_context oibs.
Proof.
induction oibs as [|oib oibs IH]; [easy|].
rewrite arities_contexts_cons_1.
unfold arities_context.
rewrite rev_map_cons.
f_equal.
apply IH.
Qed.
Inductive view_prod : term -> Type :=
| view_prod_prod na A B : view_prod (tProd na A B)
| view_prod_other t : negb (isProd t) -> view_prod t.
Equations view_prodc (t : term) : view_prod t :=
| tProd na A B => view_prod_prod na A B;
| t => view_prod_other t _.
(** Constructors are treated slightly differently to types as we always
generate type variables for parameters *)
Equations? (noeqns) erase_ind_ctor
(Ξ : context)
(erΞ : Vector.t tRel_kind #|Ξ|)
(t : term)
(isT : forall Ξ£ : global_env_ext, abstract_env_ext_rel X Ξ£ -> β₯isType Ξ£ Ξ tβ₯)
(next_par : nat)
(tvars : list type_var_info) : box_type
by struct tvars :=
erase_ind_ctor Ξ erΞ t isT next_par [] := (erase_type_aux Ξ erΞ t isT None).2;
erase_ind_ctor Ξ erΞ t isT next_par (tvar :: tvars)
with inspect (reduce_term RedFlags.nodelta X_type X Ξ t (fun Ξ£0 h => (tywt (isT Ξ£0 h)))) :=
| exist t0 eq_red with view_prodc t0 := {
| view_prod_prod na A B =>
let rel_kind := if tvar_is_arity tvar then RelTypeVar next_par else RelOther in
let '(_, dom) := erase_type_aux Ξ erΞ A _ None in
let cod := erase_ind_ctor (Ξ,, vass na A) (rel_kind :: erΞ)%vector B _ (S next_par) tvars in
TArr dom cod;
| view_prod_other _ _ => TAny (* unreachable *)
}.
Proof.
1-2: assert (rΞ£ = Ξ£) by (eapply abstract_env_ext_irr;eauto);subst.
all: assert (β₯ wf_ext Ξ£ β₯) by (now apply heΞ£).
all: specialize (isT _ wfrΞ£) as typ.
all: reduce_term_sound; destruct r; subst;eauto with erase.
Qed.
Import ExAst.
Definition erase_ind_body
(mind : kername)
(mib : PCUICEnvironment.mutual_inductive_body)
(oib : PCUICEnvironment.one_inductive_body)
(wt : β₯βi, on_ind_body cumulSpec0 (lift_typing typing) rΞ£ mind mib i oibβ₯) : one_inductive_body.
Proof.
unshelve refine (
let is_propositional :=
match destArity [] (ind_type oib) with
| Some (_, u) => is_propositional u
| None => false
end in
let oib_tvars := erase_ind_arity [] (PCUICEnvironment.ind_type oib) _ in
let ctx := inspect (arities_contexts mind (PCUICEnvironment.ind_bodies mib)) in
let ind_params := firstn (PCUICEnvironment.ind_npars mib) oib_tvars in
let erase_ind_ctor (p : PCUICEnvironment.constructor_body) (is_in : In p (PCUICEnvironment.ind_ctors oib)) :=
let bt := erase_ind_ctor (proj1_sig ctx).Ο1 (proj1_sig ctx).Ο2 p.(PCUICEnvironment.cstr_type) _ 0 ind_params in
let '(ctor_args, _) := decompose_arr bt in
let fix decomp_names ty :=
match ty with
| P.tProd na A B => binder_name na :: decomp_names B
| P.tLetIn na a A b => decomp_names b
| _ => []
end in
(p.(PCUICEnvironment.cstr_name), combine (decomp_names p.(PCUICEnvironment.cstr_type)) ctor_args, p.(PCUICEnvironment.cstr_arity)) in
let ctors := map_In (PCUICEnvironment.ind_ctors oib) erase_ind_ctor in
let erase_ind_proj (p : PCUICEnvironment.projection_body) (is_in : In p (PCUICEnvironment.ind_projs oib)) :=
(p.(PCUICEnvironment.proj_name), TBox) (* TODO *) in
let projs := map_In (PCUICEnvironment.ind_projs oib) erase_ind_proj in
{| ind_name := PCUICEnvironment.ind_name oib;
ind_propositional := is_propositional;
ind_kelim := PCUICEnvironment.ind_kelim oib;
ind_type_vars := oib_tvars;
ind_ctors := ctors;
ind_projs := projs |}).
all: intros;assert (rΞ£ = Ξ£) by (eapply abstract_env_ext_irr;eauto);subst.
- abstract (
destruct wt as [wt];sq;
exact (onArity wt.Ο2)).
- abstract (
destruct p;
cbn in *;
destruct wt as [[ind_index wt]];
pose proof (onConstructors wt) as on_ctors;
unfold on_constructors in *;
induction on_ctors; [easy|];
destruct is_in as [->|later]; [|easy];
constructor;
destruct (on_ctype r) as (s & typ);
rewrite <- (arities_contexts_1 mind) in typ;
cbn in *;
now exists s).
Defined.
Program Definition erase_ind
(kn : kername)
(mib : PCUICEnvironment.mutual_inductive_body)
(wt : β₯on_inductive cumulSpec0 (lift_typing typing) rΞ£ kn mibβ₯) : mutual_inductive_body :=
let inds := map_In (PCUICEnvironment.ind_bodies mib) (fun oib is_in => erase_ind_body kn mib oib _) in
{| ind_npars := PCUICEnvironment.ind_npars mib; ind_bodies := inds; ind_finite := PCUICEnvironment.ind_finite mib |}.
Next Obligation.
apply In_nth_error in is_in.
destruct is_in as (i & nth_some).
destruct wt as [wt].
constructor.
exists i.
specialize (onInductives wt).
change i with (0 + i).
generalize 0 as n.
revert i nth_some.
induction (PCUICEnvironment.ind_bodies mib) as [|? oibs IH]; intros i nth_some n inds_wt.
- now rewrite nth_error_nil in nth_some.
- inversion inds_wt; subst; clear inds_wt.
destruct i; cbn in *.
+ replace a with oib in * by congruence.
now rewrite Nat.add_0_r.
+ specialize (IH _ nth_some (S n)).
now rewrite Nat.add_succ_r.
Qed.
End FixSigmaExt.
Section EraseEnv.
Local Existing Instance extraction_checker_flags.
Import ExAst.
Definition fake_guard_impl : FixCoFix -> global_env_ext -> PCUICEnvironment.context
-> BasicAst.mfixpoint PCUICAst.term -> bool :=
fun fix_cofix Ξ£ Ξ mfix => true.
Axiom fake_guard_correct : forall (fix_cofix : FixCoFix) (Ξ£ : global_env_ext)
(Ξ : PCUICEnvironment.context) (mfix : BasicAst.mfixpoint PCUICAst.term),
guard fix_cofix Ξ£ Ξ mfix <-> fake_guard_impl fix_cofix Ξ£ Ξ mfix.
Instance fake_guard_impl_instance : abstract_guard_impl :=
{| guard_impl := fake_guard_impl;
guard_correct := fake_guard_correct |}.
Program Definition erase_global_decl
(Ξ£ext : global_env_ext)
(wfΞ£ext : β₯ wf_ext Ξ£ext β₯)
(kn : kername)
(decl : PCUICEnvironment.global_decl)
(wt : β₯on_global_decl cumulSpec0 (lift_typing typing) Ξ£ext kn declβ₯)
: global_decl :=
match decl with
| PCUICEnvironment.ConstantDecl cst =>
match @erase_constant_decl canonical_abstract_env_impl _ Ξ£ext _ cst _ with
| inl cst => ConstantDecl cst
| inr ta => TypeAliasDecl ta
end
| PCUICEnvironment.InductiveDecl mib => InductiveDecl (@erase_ind canonical_abstract_env_impl _ Ξ£ext _ kn mib _)
end.
Solve Obligations with now unshelve econstructor;eauto.
Fixpoint box_type_deps (t : box_type) : KernameSet.t :=
match t with
| TBox
| TAny
| TVar _ => KernameSet.empty
| TArr t1 t2
| TApp t1 t2 => KernameSet.union (box_type_deps t1) (box_type_deps t2)
| TInd ind => KernameSet.singleton (inductive_mind ind)
| TConst kn => KernameSet.singleton kn
end.
Definition decl_deps (decl : global_decl) : KernameSet.t :=
match decl with
| ConstantDecl body =>
let seen :=
match cst_body body with
| Some body => term_global_deps body
| None => KernameSet.empty
end in
KernameSet.union (box_type_deps (cst_type body).2) seen
| InductiveDecl mib =>
let one_inductive_body_deps oib :=
let seen := fold_left (fun seen '(_, bt) => KernameSet.union seen (box_type_deps bt))
(flat_map (compose snd fst) (ind_ctors oib))
KernameSet.empty in
fold_left (fun seen bt => KernameSet.union seen (box_type_deps bt))
(map snd (ind_projs oib)) seen in
fold_left (fun seen oib => KernameSet.union seen (one_inductive_body_deps oib))
(ind_bodies mib)
KernameSet.empty
| TypeAliasDecl (Some (nms, ty)) => box_type_deps ty
| _ => KernameSet.empty
end.
(** Erase the global declarations by the specified names and their
non-erased dependencies recursively. Ignore dependencies for which
[ignore_deps] returnes [true] *)
Program Fixpoint erase_global_decls_deps_recursive
(Ξ£ : PCUICEnvironment.global_declarations)
(universes : ContextSet.t)
(retroknowledge : Retroknowledge.t)
(wfΞ£ : β₯wf (mk_global_env universes Ξ£ retroknowledge)β₯)
(include : KernameSet.t)
(ignore_deps : kername -> bool) : global_env :=
match Ξ£ with
| [] => []
| (kn, decl) :: Ξ£ =>
let Ξ£ext := (Ξ£, universes_decl_of_decl decl) in
if KernameSet.mem kn include then
(** We still erase ignored inductives and constants for two reasons:
- For inductives, we want to allow pattern matches on them and we need
information about them to print names.
- For constants, we use their type to do deboxing. *)
let decl := erase_global_decl ((mk_global_env universes Ξ£ retroknowledge), PCUICLookup.universes_decl_of_decl decl) _ kn decl _ in
let with_deps := negb (ignore_deps kn) in
let new_deps := if with_deps then decl_deps decl else KernameSet.empty in
let Ξ£er := erase_global_decls_deps_recursive
Ξ£ universes retroknowledge _
(KernameSet.union new_deps include) ignore_deps in
(kn, with_deps, decl) :: Ξ£er
else
erase_global_decls_deps_recursive Ξ£ universes retroknowledge _ include ignore_deps
end.
Ltac invert_wf :=
match goal with
| [H : β₯ wf _ β₯ |- _] => sq; inversion H;subst;clear H;cbn in *
| [H : on_global_decls _ _ _ _ (_ :: _) |- _] => inversion H;subst;clear H;cbn in *
| [H : on_global_decls_data _ _ _ _ _ _ _ |- _] => inversion H; subst; clear H; cbn in *
end.
Next Obligation.
repeat invert_wf;split;auto;split;auto.
Qed.
Next Obligation.
repeat invert_wf.
destruct decl;cbn in *;auto.
Qed.
Next Obligation.
repeat invert_wf;split;auto;split;auto.
Qed.
Next Obligation.
repeat invert_wf;split;auto;split;auto.
Qed.
End EraseEnv.
Global Arguments is_logical {_ _ _}.
Global Arguments conv_ar {_ _ _}.
Global Arguments is_sort {_ _ _}.
Global Arguments is_arity {_ _ _}.
|
function skelVargplvmModify(handle, yVal, chan, skel, padding, zeroIndices)
% SKELMODIFY Update visualisation of skeleton data.
% FORMAT
% DESC updates a skeleton representation in a 3-D plot.
% ARG handle : a vector of handles to the structure to be updated.
% ARG channels : the channels to update the skeleton with.
% ARG skel : the skeleton structure.
%
% SEEALSO : skelVisualise
%
% COPYRIGHT : Neil D. Lawrence, 2005, 2006
% MODIFICATIONS: Andreas C. Damianou, 2014
% SHEFFIELDML
if nargin<5
padding = 0;
end
channels = demCmu35VargplvmLoadChannels(yVal,skel);
channels = [channels zeros(1, padding)];
vals = skel2xyz(skel, channels);
connect = skelConnectionMatrix(skel);
indices = find(connect);
[I, J] = ind2sub(size(connect), indices);
set(handle(1), 'Xdata', vals(:, 1), 'Ydata', vals(:, 3), 'Zdata', ...
vals(:, 2));
%/~
%set(handle(1), 'visible', 'on')
%~/
for i = 1:length(indices)
set(handle(i+1), 'Xdata', [vals(I(i), 1) vals(J(i), 1)], ...
'Ydata', [vals(I(i), 3) vals(J(i), 3)], ...
'Zdata', [vals(I(i), 2) vals(J(i), 2)]);
end
function [vals, connect] = wrapAround(vals, lim, connect);
quot = lim(2) - lim(1);
vals = rem(vals, quot)+lim(1);
nVals = floor(vals/quot);
for i = 1:size(connect, 1)
for j = find(connect(i, :))
if nVals(i) ~= nVals(j)
connect(i, j) = 0;
end
end
end
|
@testset "Orderings" begin
import KnuthBendix.set_inversion!
import Base.Order.lt
A = Alphabet(['a', 'b', 'c', 'd'])
set_inversion!(A, 'a', 'b')
set_inversion!(A, 'c', 'd')
lenlexord = LenLex(A)
@test lenlexord isa Base.Order.Ordering
u1 = Word([1,2])
u3 = Word([1,3])
u4 = Word([1,2,3])
u5 = Word([1,4,2])
@test lt(lenlexord, u1, u3) == true
@test lt(lenlexord, u3, u1) == false
@test lt(lenlexord, u3, u4) == true
@test lt(lenlexord, u4, u5) == true
@test lt(lenlexord, u5, u4) == false
@test lt(lenlexord, u1, u1) == false
wo = WreathOrder(A)
@test wo isa KnuthBendix.WordOrdering
w1 = Word([3,1,4,2,3])
w5 = Word([1,2,3,2,4,1,2,4])
w4 = Word([1,3,1,2,4,2,1])
w6 = Word([1,2,3,2,1,4,2,4])
w2 = Word([1,3,4,3,2])
w3 = Word([1,3,2,1,4,1,2])
Ξ΅ = one(w1)
a = [w1, w5, w4, w6, w2, w3, Ξ΅]
@test sort(a, order = wo) == [Ξ΅, w1, w2, w3, w4, w5, w6]
rpo = RecursivePathOrder(A)
w1 = Word([1])
w2 = Word([2])
w14 = Word([1,4])
w214 = Word([2,1,4])
w41 = Word([4,1])
w241 = Word([2,4,1])
w1224 = Word([1,2,3,2,1,4,2,4])
w32141 = Word([3,2,1,4,1])
a = [w14, w1, w1224, w214, Ξ΅]
b = [w32141, w214, w1, Ξ΅, w241, w2, w41]
@test sort(a, order = rpo) == [Ξ΅, w1, w14, w214, w1224]
@test sort(b, order = rpo) == [Ξ΅, w1, w2, w214, w41, w241, w32141]
end
|
If $S$ is a locally connected space and $c$ is a component of $S$, then $c$ is locally connected. |
section \<open>Generalised Cylindric Kleene Lattices\<close>
text \<open>Using this mathematical component requires downloading the Archive of Formal Proofs.\<close>
theory GCKL
imports CKL2
begin
primrec list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"list_inter [] ys = []" |
"list_inter (x # xs) ys = (let zs = list_inter xs ys in (if x \<in> set ys then x # zs else zs))"
lemma list_set_inter: "set (list_inter ks ls) = set ks \<inter> set ls"
by (induct ks, simp_all, metis (full_types) Int_insert_left list.simps(15))
context cylindric_l_monoid_zerol
begin
primrec cyl_list :: "('a::linorder) list \<Rightarrow> 'b \<Rightarrow> 'b" where
"cyl_list [] x = x" |
"cyl_list (i # is) x = cyl i (cyl_list is x)"
lemma cyl_list_nil_fun [simp]: "cyl_list [] = id"
unfolding fun_eq_iff by simp
lemma cyl_list_append: "cyl_list ks (cyl_list ls x) = cyl_list (ks @ ls) x"
by (induct ks, simp_all)
lemma cyl_list_cons_fun: "cyl_list (l # ls) = cyl l \<circ> cyl_list ls"
unfolding fun_eq_iff comp_def by simp
lemma setmem_cyl: "l \<in> set ls \<Longrightarrow> cyl_list (l # ls) x = cyl_list ls x"
by (induct ls, simp_all, metis cyl_comm cyl_fix_im)
lemma cyl_list_remdups: "cyl_list (remdups ls) x = cyl_list ls x"
apply (induct ls)
using setmem_cyl by auto
lemma cyl_list_el_comm: "cyl_list ls (cyl i x) = cyl i (cyl_list ls x)"
by (induct ls, simp_all add: cyl_comm)
lemma cyl_sorted_aux: "sorted ls \<Longrightarrow> cyl i (cyl_list ls x) = cyl_list (insort i ls) x"
by (induct ls, simp_all add: cyl_comm)
lemma cyl_sorted: "cyl i (cyl_list (sort ls) x) = cyl_list (insort i (sort ls)) x"
by (simp add: cyl_sorted_aux)
lemma cyl_list_sort: "cyl_list ls x = cyl_list (sort ls) x"
by (induct ls, simp_all add: cyl_sorted)
definition cyl_set :: "('a::linorder) set \<Rightarrow> 'b \<Rightarrow> 'b" where
"cyl_set X x = cyl_list (sorted_list_of_set X) x"
lemma cyl_list_set: "cyl_list ls x = cyl_set (set ls) x"
unfolding cyl_set_def by (metis cyl_list_remdups cyl_list_sort sorted_list_of_set_sort_remdups)
lemma cyl_list_set_eq: "set ks = set ls \<Longrightarrow> cyl_list ks x = cyl_list ls x"
by (simp add: cyl_list_set)
lemma cyl_list_iso_aux: "cyl_list ls x \<le> cyl_list (ks @ ls) x"
by (induct ks, simp_all, meson cyl_ext order_trans)
lemma cyl_list_iso: "set ks \<subseteq> set ls \<Longrightarrow> cyl_list ks x \<le> cyl_list ls x"
by (metis UnCI Un_Int_distrib cyl_list_set cylindric_l_monoid_zerol.cyl_list_iso_aux cylindric_l_monoid_zerol_axioms inf.commute inf.orderE set_append subsetI sup.idem)
lemma "cyl_list ks x \<le> cyl_list ls x \<Longrightarrow> set ks \<subseteq> set ls"
(*nitpick*)
oops
lemma cyl_list_idem [simp]: "cyl_list ls (cyl_list ls x) = cyl_list ls x"
using cyl_list_append cyl_list_set by auto
lemma cyl_list_one_cons: "cyl i 1 \<cdot> cyl_list ls 1 = cyl_list (i # ls) 1"
apply (induct ls)
apply simp_all
by (smt cyl_1_unl cyl_comm cyl_id_seq_eq cyl_oplax cyl_rel_prop_var1 join.le_iff_sup join.sup.order_iff mult.assoc)
lemma cyl_list_one_app: "cyl_list ks 1 \<cdot> cyl_list ls 1 = cyl_list (ks @ ls) 1"
by (induct ks, simp_all, metis cyl_list.simps(2) cyl_list_one_cons mult.assoc)
lemma "i \<noteq> j \<Longrightarrow> cyl_list (ls @ [i]) 1 \<sqinter> cyl_list (ls @ [j]) 1 = cyl_list ls 1"
oops
lemma "i \<noteq> j \<Longrightarrow> cyl_list ([i] @ ls) 1 \<sqinter> cyl_list ([j] @ ls) 1 = cyl_list ls 1"
oops
lemma rect_id: "cyl_list ks 1 \<sqinter> cyl_list ls 1 = cyl_list (list_inter ks ls) 1"
oops
lemma c1_list [simp]: "cyl_list ls 0 = 0"
by (induct ls, simp_all)
lemma c2_list: "x \<le> cyl_list ls x"
apply (induct ls)
apply simp_all
using cyl_ext dual_order.trans by blast
lemma c3_list: "cyl_list ls (x + y) = cyl_list ls x + cyl_list ls y"
by (induct ls, simp_all add: cyl_sup_add)
lemma c4_list: "\<forall>x y. cyl_list ls (x \<sqinter> cyl_list ls y) = cyl_list ls x \<sqinter> cyl_list ls y"
by (induct ls, simp_all, metis cyl_inf cyl_list_el_comm)
lemma c5_list: "cyl_list ks (cyl_list ls x) = cyl_list ls (cyl_list ks x)"
by (induct ls, simp_all add: cyl_list_el_comm)
lemma c5_list_var: "cyl_list (ks @ ls) x = cyl_list (ls @ ks) x"
using c5_list cyl_list_append by auto
lemma c6_list: "\<forall>x y. cyl_list ls (x \<cdot> cyl_list ls y) = cyl_list ls x \<cdot> cyl_list ls y"
by (induct ls, simp_all, metis cyl_list_el_comm cyl_multr)
lemma c7_list: "\<forall>x y. cyl_list ls (cyl_list ls x \<cdot> y) = cyl_list ls x \<cdot> cyl_list ls y"
by (induct ls, simp_all, metis cyl_list_el_comm cyl_multl)
lemma c8_list:
assumes "cyl_list ks 1 \<sqinter> cyl_list ls 1 = cyl_list (list_inter ks ls) 1"
shows "list_inter ks ls = [] \<Longrightarrow> cyl_list ks 1 \<sqinter> cyl_list ls 1 = 1"
by (simp add: assms)
lemma c9_list:
assumes "\<forall>ks ls. cyl_list ks 1 \<sqinter> cyl_list ls 1 = cyl_list (list_inter ks ls) 1"
shows"(cyl_list ks 1 \<cdot> cyl_list ls 1) \<sqinter> (cyl_list ks 1 \<cdot> cyl_list ms 1) = cyl_list ks (cyl_list ls 1 \<sqinter> cyl_list ms 1)"
by (metis Un_Int_distrib cyl_list_append cyl_list_one_app cylindric_l_monoid_zerol.cyl_list_set assms cylindric_l_monoid_zerol_axioms list_set_inter set_append)
lemma c10_list: "cyl_list ks (cyl_list ls 1) = cyl_list ks 1 \<cdot> cyl_list ls 1"
by (simp add: cyl_list_append cyl_list_one_app)
lemma cyl_list_id_absorb1:
assumes "\<forall>ks ls. cyl_list ks 1 \<sqinter> cyl_list ls 1 = cyl_list (list_inter ks ls) 1"
shows "cyl_list ks 1 \<cdot> (cyl_list ks 1 \<sqinter> cyl_list ls 1) = cyl_list ks 1"
by (metis (no_types, lifting) assms c10_list c2_list c4_list c5_list inf.commute inf_absorb2)
lemma cyl_list_id_absorb2:
assumes "\<forall>ks ls. cyl_list ks 1 \<sqinter> cyl_list ls 1 = cyl_list (list_inter ks ls) 1"
shows "cyl_list ks 1 \<sqinter> (cyl_list ks 1 \<cdot> cyl_list ls 1) = cyl_list ks 1"
by (metis (no_types, lifting) assms c10_list c2_list c4_list c5_list inf.commute inf_absorb2)
end
context cylindric_kleene_lattice_zerol
begin
lemma c11_list: "cyl_list ls (x\<^sup>\<oplus>) \<le> (cyl_list ls x::'b::{l_monoid_zerol,kleene_algebra_zerol})\<^sup>\<oplus>"
by (induct ls, simp_all, metis cyl_iso cyl_kplus cyl_list.simps(2) order_trans)
lemma cyl_id_star_fix [simp]: "(cyl_list ls 1)\<^sup>\<star> = cyl_list ls 1"
by (metis c2_list c7_list cyl_list_idem mult.right_neutral star_inductr_var_eq2 sup_id_star1)
lemma cyl_id_plus_fix [simp]: "(cyl_list ls 1)\<^sup>\<oplus> = cyl_list ls 1"
by (metis c7_list cyl_list_idem kplus_def mult_oner star_inductr_var_eq2)
end
context cylindric_l_monoid_zerol
begin
text \<open>We define programs with frames\<close>
definition fprog :: "'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
"fprog ls x = x \<sqinter> cyl_list ls 1"
lemma fprog1: "fprog ls x \<le> x"
by (simp add: fprog_def)
lemma fprog2: "set ks \<subseteq> set ls \<Longrightarrow> fprog ks x \<le> fprog ls x"
unfolding fprog_def using cyl_list_iso inf_mono by blast
lemma fprog3: "x \<le> y \<Longrightarrow> fprog ls x \<le> fprog ls y"
unfolding fprog_def using inf_mono by blast
lemma fprog4: "fprog ls x + fprog ls y \<le> fprog ls (x + y)"
by (simp add: fprog3)
lemma fprog4_eq:
assumes "\<forall>x y z. (x + y) \<sqinter> (z::'b) = (x \<sqinter> y) + (x \<sqinter> z)"
shows "fprog ls (x + y) = fprog ls x + fprog ls y"
by (metis (no_types, lifting) add_idem' assms lmon_lat.inf_sup_absorb)
lemma fprog5: "fprog ls x \<cdot> fprog ls y \<le> fprog ls (x \<cdot> y)"
unfolding fprog_def
by (metis (no_types, lifting) cyl_list_idem cylindric_l_monoid_zerol.c10_list cylindric_l_monoid_zerol_axioms inf_le1 inf_le2 le_inf_iff mult_isol_var)
lemma fprog7: "x \<le> 1 \<Longrightarrow> fprog ls x = x"
by (metis c2_list dual_order.trans fprog_def inf.orderE)
end
context cylindric_kleene_lattice_zerol
begin
lemma fprog6: "(fprog ls x)\<^sup>\<star> \<le> fprog ls (x\<^sup>\<star>)"
by (metis cyl_id_star_fix fprog1 fprog_def inf_le2 le_inf_iff star_iso)
lemma refine_skip: "1 \<le> x \<Longrightarrow> fprog ls 1 \<le> fprog ls x"
by (simp add: fprog3)
lemma refine4: "y \<le> x \<Longrightarrow> fprog ls y \<le> x"
using dual_order.trans fprog1 by blast
lemma refine_seq: "x \<cdot> y \<le> z \<Longrightarrow> fprog ls x \<cdot> fprog ls y \<le> fprog ls z"
by (meson fprog3 fprog5 order_subst1)
lemma refine_cond: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> p \<cdot> x + q \<cdot> y \<le> z \<Longrightarrow> p \<cdot> fprog ls x + q \<cdot> fprog ls y \<le> fprog ls z"
using fprog7 refine_seq by fastforce
lemma refine_loop: "p \<le> 1 \<Longrightarrow> q \<le> 1 \<Longrightarrow> (p \<cdot> x)\<^sup>\<star> \<cdot> q \<le> y \<Longrightarrow> (p \<cdot> fprog ls x)\<^sup>\<star> \<cdot> q \<le> fprog ls y"
proof-
assume h1: "p \<le> 1"
and h2: "q \<le> 1"
and h3: "(p \<cdot> x)\<^sup>\<star> \<cdot> q \<le> y"
hence "(p \<cdot> fprog ls x)\<^sup>\<star> \<cdot> q \<le> (fprog ls (p \<cdot> x)\<^sup>\<star>) \<cdot> q"
by (metis fprog5 fprog7 h1 mult_isor star_iso)
also have "... \<le> fprog ls ((p \<cdot> x)\<^sup>\<star> \<cdot> q)"
by (smt cyl_id_star_fix cyl_list_idem cylindric_l_monoid_zerol.c10_list cylindric_l_monoid_zerol.fprog1 cylindric_l_monoid_zerol.fprog7 cylindric_l_monoid_zerol_axioms fprog_def h2 inf.absorb_iff2 inf_le2 le_inf_iff mult_isol_var star_iso)
also have "... \<le> fprog ls y"
by (simp add: fprog3 h3)
finally show ?thesis.
qed
end
end |
module System.File.Buffer
import public System.File.Error
import System.File.Handle
import System.File.Meta
import System.File.Mode
import System.File.ReadWrite
import System.File.Support
import public System.File.Types
import Data.Buffer
%default total
%foreign support "idris2_readBufferData"
"node:lambda:(f,b,l,m) => require('fs').readSync(f.fd,b,l,m)"
prim__readBufferData : FilePtr -> Buffer -> (offset : Int) -> (maxbytes : Int) -> PrimIO Int
%foreign support "idris2_writeBufferData"
"node:lambda:(f,b,l,m) => require('fs').writeSync(f.fd,b,l,m)"
prim__writeBufferData : FilePtr -> Buffer -> (offset : Int) -> (maxbytes : Int) -> PrimIO Int
||| Read the data from the file into the given buffer.
|||
||| @ fh the file handle to read from
||| @ buf the buffer to read the data into
||| @ offset the position in buffer to start adding
||| @ maxbytes the maximum size to read; must not exceed buffer length
export
readBufferData : HasIO io => (fh : File) -> (buf : Buffer) ->
(offset : Int) ->
(maxbytes : Int) ->
io (Either FileError ())
readBufferData (FHandle h) buf offset max
= do read <- primIO (prim__readBufferData h buf offset max)
if read >= 0
then pure (Right ())
else pure (Left FileReadError)
||| Write the data from the buffer to the given `File`.
||| (If you do not have a `File` and just want to write to a file at a known
||| path, you probably want to use `writeBufferToFile`.)
|||
||| @ fh the file handle to write to
||| @ buf the buffer from which to get the data to write
||| @ offset the position in buffer to write from
||| @ maxbytes the maximum size to write; must not exceed buffer length
export
writeBufferData : HasIO io => (fh : File) -> (buf : Buffer) ->
(offset : Int) ->
(maxbytes : Int) ->
io (Either FileError ())
writeBufferData (FHandle h) buf offset max
= do written <- primIO (prim__writeBufferData h buf offset max)
if written >= 0
then pure (Right ())
else pure (Left FileWriteError)
||| Attempt to write the data from the buffer to the file at the specified file
||| name.
|||
||| @ fn the file name to write to
||| @ buf the buffer from which to get the data to write
||| @ max the maximum size to write; must not exceed buffer length
export
writeBufferToFile : HasIO io => (fn : String) -> (buf : Buffer) -> (max : Int) ->
io (Either FileError ())
writeBufferToFile fn buf max
= do Right f <- openFile fn WriteTruncate
| Left err => pure (Left err)
Right ok <- writeBufferData f buf 0 max
| Left err => pure (Left err)
closeFile f
pure (Right ok)
||| Create a new buffer by opening a file and reading its contents into a new
||| buffer whose size matches the file size.
|||
||| @ fn the name of the file to read
export
createBufferFromFile : HasIO io => (fn : String) -> io (Either FileError Buffer)
createBufferFromFile fn
= do Right f <- openFile fn Read
| Left err => pure (Left err)
Right size <- fileSize f
| Left err => pure (Left err)
Just buf <- newBuffer size
| Nothing => pure (Left FileReadError)
Right ok <- readBufferData f buf 0 size
| Left err => pure (Left err)
closeFile f
pure (Right buf)
|
[STATEMENT]
lemma addfunsetD_range : "f \<in> addfunset A M \<Longrightarrow> range f \<subseteq> M"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. f \<in> addfunset A M \<Longrightarrow> range f \<subseteq> M
[PROOF STEP]
unfolding addfunset_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. f \<in> {f. supp f \<subseteq> A \<and> range f \<subseteq> M \<and> (\<forall>x\<in>A. \<forall>y\<in>A. f (x + y) = f x + f y)} \<Longrightarrow> range f \<subseteq> M
[PROOF STEP]
by fast |
import computability.regular_expressions
import computability.language
open list
universes u
variables {Ξ± Ξ² Ξ³ : Type*}
namespace regular_expressions
def prod : list (regular_expression Ξ±) β regular_expression Ξ±
| [] := 1
| (x :: xs) := x * prod xs
@[simp]
lemma prod_nil : prod ([] : list (regular_expression Ξ±)) = 1 := by refl
@[simp]
lemma prod_cons {x : regular_expression Ξ±} {xs} : prod (x :: xs) = x * prod xs := by refl
lemma mathces_prod {x : list (regular_expression Ξ±)} :
(prod x).matches = list.prod (map (Ξ» x : regular_expression Ξ±, x.matches) x) :=
begin
induction x,
simp,
simp,
rw x_ih,
end
@[simp]
lemma matches_prod_cons {x : regular_expression Ξ±} {xs} :
(prod (x :: xs)).matches = x.matches * (prod xs).matches := by simp
@[simp]
lemma matches_prod_append {x y : list (regular_expression Ξ±)} :
(prod (x ++ y)).matches = (prod x).matches * (prod y).matches :=
begin
induction x,
simp,
simp,
rw [x_ih, β mul_assoc],
end
@[simp]
lemma map_prod {f : Ξ± β Ξ²} {x : list (regular_expression Ξ±)} :
regular_expression.map f (prod x) = prod (map (Ξ» x, regular_expression.map f x) x) :=
begin
induction x,
simp,
simp,
rw x_ih,
end
end regular_expressions |
[GOAL]
Ξ± : Type u_1
a b : β
hβ : delta * a < b
hβ : delta * b < a
β’ 0 < delta
[PROOFSTEP]
decide
[GOAL]
Ξ± : Type u_1
a b : β
hβ : delta * a < b
hβ : delta * b < a
β’ a β€ delta * (delta * a)
[PROOFSTEP]
simpa [mul_assoc] using Nat.mul_le_mul_right a (by decide : 1 β€ delta * delta)
[GOAL]
Ξ± : Type u_1
a b : β
hβ : delta * a < b
hβ : delta * b < a
β’ 1 β€ delta * delta
[PROOFSTEP]
decide
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Sized (node s l x r)
β’ node s l x r = Ordnode.node' l x r
[PROOFSTEP]
rw [h.1]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
hl : Sized t
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
β’ C t
[PROOFSTEP]
induction t with
| nil => exact H0
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
hl : Sized t
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
β’ C t
[PROOFSTEP]
induction t with
| nil => exact H0
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
[GOAL]
case nil
Ξ± : Type u_1
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
hl : Sized nil
β’ C nil
[PROOFSTEP]
| nil => exact H0
[GOAL]
case nil
Ξ± : Type u_1
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
hl : Sized nil
β’ C nil
[PROOFSTEP]
exact H0
[GOAL]
case node
Ξ± : Type u_1
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l : Sized lβ β C lβ
t_ih_r : Sized rβ β C rβ
hl : Sized (node sizeβ lβ xβ rβ)
β’ C (node sizeβ lβ xβ rβ)
[PROOFSTEP]
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
[GOAL]
case node
Ξ± : Type u_1
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l : Sized lβ β C lβ
t_ih_r : Sized rβ β C rβ
hl : Sized (node sizeβ lβ xβ rβ)
β’ C (node sizeβ lβ xβ rβ)
[PROOFSTEP]
rw [hl.eq_node']
[GOAL]
case node
Ξ± : Type u_1
C : Ordnode Ξ± β Prop
H0 : C nil
H1 : β (l : Ordnode Ξ±) (x : Ξ±) (r : Ordnode Ξ±), C l β C r β C (Ordnode.node' l x r)
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l : Sized lβ β C lβ
t_ih_r : Sized rβ β C rβ
hl : Sized (node sizeβ lβ xβ rβ)
β’ C (Ordnode.node' lβ xβ rβ)
[PROOFSTEP]
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβ : s = size l + size r + 1
hβ : Sized l
hβ : Sized r
β’ size (node s l x r) = realSize (node s l x r)
[PROOFSTEP]
rw [size, hβ, size_eq_realSize hβ, size_eq_realSize hβ]
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβ : s = size l + size r + 1
hβ : Sized l
hβ : Sized r
β’ realSize l + realSize r + 1 = realSize (node (realSize l + realSize r + 1) l x r)
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
ht : Sized t
β’ size t = 0 β t = nil
[PROOFSTEP]
cases t <;> [simp; simp [ht.1]]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
ht : Sized t
β’ size t = 0 β t = nil
[PROOFSTEP]
cases t
[GOAL]
case nil
Ξ± : Type u_1
ht : Sized nil
β’ size nil = 0 β nil = nil
[PROOFSTEP]
simp
[GOAL]
case node
Ξ± : Type u_1
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
ht : Sized (node sizeβ lβ xβ rβ)
β’ size (node sizeβ lβ xβ rβ) = 0 β node sizeβ lβ xβ rβ = nil
[PROOFSTEP]
simp [ht.1]
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Sized (node s l x r)
β’ 0 < s
[PROOFSTEP]
rw [h.1]
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Sized (node s l x r)
β’ 0 < size l + size r + 1
[PROOFSTEP]
apply Nat.le_add_left
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (dual (node s l x r)) = node s l x r
[PROOFSTEP]
rw [dual, dual, dual_dual l, dual_dual r]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
β’ size (dual t) = size t
[PROOFSTEP]
cases t
[GOAL]
case nil
Ξ± : Type u_1
β’ size (dual nil) = size nil
[PROOFSTEP]
rfl
[GOAL]
case node
Ξ± : Type u_1
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ size (dual (node sizeβ lβ xβ rβ)) = size (node sizeβ lβ xβ rβ)
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
β’ Decidable (Balanced nil)
[PROOFSTEP]
unfold Balanced
[GOAL]
Ξ± : Type u_1
β’ Decidable True
[PROOFSTEP]
infer_instance
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
β’ Decidable (Balanced (node sizeβ l xβ r))
[PROOFSTEP]
unfold Balanced
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
β’ Decidable (BalancedSz (size l) (size r) β§ Balanced l β§ Balanced r)
[PROOFSTEP]
haveI := Balanced.dec l
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
this : Decidable (Balanced l)
β’ Decidable (BalancedSz (size l) (size r) β§ Balanced l β§ Balanced r)
[PROOFSTEP]
haveI := Balanced.dec r
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
thisβ : Decidable (Balanced l)
this : Decidable (Balanced r)
β’ Decidable (BalancedSz (size l) (size r) β§ Balanced l β§ Balanced r)
[PROOFSTEP]
infer_instance
[GOAL]
Ξ± : Type u_1
l r : β
β’ l + r β€ 1 β r + l β€ 1
[PROOFSTEP]
rw [add_comm]
[GOAL]
Ξ± : Type u_1
l r : β
β’ r + l β€ 1 β r + l β€ 1
[PROOFSTEP]
exact id
[GOAL]
Ξ± : Type u_1
l : β
β’ BalancedSz l 0 β l β€ 1
[PROOFSTEP]
simp (config := { contextual := true }) [BalancedSz]
[GOAL]
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
H : BalancedSz l rβ
β’ BalancedSz l rβ
[PROOFSTEP]
refine' or_iff_not_imp_left.2 fun h => _
[GOAL]
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
H : BalancedSz l rβ
h : Β¬l + rβ β€ 1
β’ l β€ delta * rβ β§ rβ β€ delta * l
[PROOFSTEP]
refine' β¨_, hβ.resolve_left hβ©
[GOAL]
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
H : BalancedSz l rβ
h : Β¬l + rβ β€ 1
β’ l β€ delta * rβ
[PROOFSTEP]
cases H with
| inl H =>
cases rβ
Β· cases h (le_trans (Nat.add_le_add_left (Nat.zero_le _) _) H)
Β· exact le_trans (le_trans (Nat.le_add_right _ _) H) (Nat.le_add_left 1 _)
| inr H => exact le_trans H.1 (Nat.mul_le_mul_left _ hβ)
[GOAL]
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
H : BalancedSz l rβ
h : Β¬l + rβ β€ 1
β’ l β€ delta * rβ
[PROOFSTEP]
cases H with
| inl H =>
cases rβ
Β· cases h (le_trans (Nat.add_le_add_left (Nat.zero_le _) _) H)
Β· exact le_trans (le_trans (Nat.le_add_right _ _) H) (Nat.le_add_left 1 _)
| inr H => exact le_trans H.1 (Nat.mul_le_mul_left _ hβ)
[GOAL]
case inl
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
h : Β¬l + rβ β€ 1
H : l + rβ β€ 1
β’ l β€ delta * rβ
[PROOFSTEP]
| inl H =>
cases rβ
Β· cases h (le_trans (Nat.add_le_add_left (Nat.zero_le _) _) H)
Β· exact le_trans (le_trans (Nat.le_add_right _ _) H) (Nat.le_add_left 1 _)
[GOAL]
case inl
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
h : Β¬l + rβ β€ 1
H : l + rβ β€ 1
β’ l β€ delta * rβ
[PROOFSTEP]
cases rβ
[GOAL]
case inl.zero
Ξ± : Type u_1
l rβ : β
H : l + rβ β€ 1
hβ : rβ β€ Nat.zero
hβ : l + Nat.zero β€ 1 β¨ Nat.zero β€ delta * l
h : Β¬l + Nat.zero β€ 1
β’ l β€ delta * Nat.zero
[PROOFSTEP]
cases h (le_trans (Nat.add_le_add_left (Nat.zero_le _) _) H)
[GOAL]
case inl.succ
Ξ± : Type u_1
l rβ : β
H : l + rβ β€ 1
nβ : β
hβ : rβ β€ Nat.succ nβ
hβ : l + Nat.succ nβ β€ 1 β¨ Nat.succ nβ β€ delta * l
h : Β¬l + Nat.succ nβ β€ 1
β’ l β€ delta * Nat.succ nβ
[PROOFSTEP]
exact le_trans (le_trans (Nat.le_add_right _ _) H) (Nat.le_add_left 1 _)
[GOAL]
case inr
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
h : Β¬l + rβ β€ 1
H : l β€ delta * rβ β§ rβ β€ delta * l
β’ l β€ delta * rβ
[PROOFSTEP]
| inr H => exact le_trans H.1 (Nat.mul_le_mul_left _ hβ)
[GOAL]
case inr
Ξ± : Type u_1
l rβ rβ : β
hβ : rβ β€ rβ
hβ : l + rβ β€ 1 β¨ rβ β€ delta * l
h : Β¬l + rβ β€ 1
H : l β€ delta * rβ β§ rβ β€ delta * l
β’ l β€ delta * rβ
[PROOFSTEP]
exact le_trans H.1 (Nat.mul_le_mul_left _ hβ)
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
b : BalancedSz (size l) (size r)
bl : Balanced l
br : Balanced r
β’ BalancedSz (size (Ordnode.dual r)) (size (Ordnode.dual l))
[PROOFSTEP]
rw [size_dual, size_dual]
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
b : BalancedSz (size l) (size r)
bl : Balanced l
br : Balanced r
β’ BalancedSz (size r) (size l)
[PROOFSTEP]
exact b.symm
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (node' l x r) = node' (dual r) x (dual l)
[PROOFSTEP]
simp [node', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ dual (node3L l x m y r) = node3R (dual r) y (dual m) x (dual l)
[PROOFSTEP]
simp [node3L, node3R, dual_node', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ dual (node3R l x m y r) = node3L (dual r) y (dual m) x (dual l)
[PROOFSTEP]
simp [node3L, node3R, dual_node', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ dual (node4L l x m y r) = node4R (dual r) y (dual m) x (dual l)
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
β’ dual (node4L l x nil y r) = node4R (dual r) y (dual nil) x (dual l)
[PROOFSTEP]
simp [node4L, node4R, node3R, dual_node3L, dual_node', add_comm]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ dual (node4L l x (node sizeβ lβ xβ rβ) y r) = node4R (dual r) y (dual (node sizeβ lβ xβ rβ)) x (dual l)
[PROOFSTEP]
simp [node4L, node4R, node3R, dual_node3L, dual_node', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ dual (node4R l x m y r) = node4L (dual r) y (dual m) x (dual l)
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
β’ dual (node4R l x nil y r) = node4L (dual r) y (dual nil) x (dual l)
[PROOFSTEP]
simp [node4L, node4R, node3L, dual_node3R, dual_node', add_comm]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ dual (node4R l x (node sizeβ lβ xβ rβ) y r) = node4L (dual r) y (dual (node sizeβ lβ xβ rβ)) x (dual l)
[PROOFSTEP]
simp [node4L, node4R, node3L, dual_node3R, dual_node', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (rotateL l x r) = rotateR (dual r) x (dual l)
[PROOFSTEP]
cases r
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
β’ dual (rotateL l x nil) = rotateR (dual nil) x (dual l)
[PROOFSTEP]
simp [rotateL, rotateR, dual_node']
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ dual (rotateL l x (node sizeβ lβ xβ rβ)) = rotateR (dual (node sizeβ lβ xβ rβ)) x (dual l)
[PROOFSTEP]
simp [rotateL, rotateR, dual_node']
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ dual (if size lβ < ratio * size rβ then node3L l x lβ xβ rβ else node4L l x lβ xβ rβ) =
if size lβ < ratio * size rβ then node3R (dual rβ) xβ (dual lβ) x (dual l)
else node4R (dual rβ) xβ (dual lβ) x (dual l)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hβ : size lβ < ratio * size rβ
β’ dual (node3L l x lβ xβ rβ) = node3R (dual rβ) xβ (dual lβ) x (dual l)
[PROOFSTEP]
simp [dual_node3L, dual_node4L, node3R, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hβ : Β¬size lβ < ratio * size rβ
β’ dual (node4L l x lβ xβ rβ) = node4R (dual rβ) xβ (dual lβ) x (dual l)
[PROOFSTEP]
simp [dual_node3L, dual_node4L, node3R, add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (rotateR l x r) = rotateL (dual r) x (dual l)
[PROOFSTEP]
rw [β dual_dual (rotateL _ _ _), dual_rotateL, dual_dual, dual_dual]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (balance' l x r) = balance' (dual r) x (dual l)
[PROOFSTEP]
simp [balance', add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual
(if size l + size r β€ 1 then node' l x r
else
if delta * size l < size r then rotateL l x r
else if delta * size r < size l then rotateR l x r else node' l x r) =
if size l + size r β€ 1 then node' (dual r) x (dual l)
else
if delta * size r < size l then rotateL (dual r) x (dual l)
else if delta * size l < size r then rotateR (dual r) x (dual l) else node' (dual r) x (dual l)
[PROOFSTEP]
split_ifs with h h_1 h_2
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : size l + size r β€ 1
β’ dual (node' l x r) = node' (dual r) x (dual l)
[PROOFSTEP]
simp [dual_node', dual_rotateL, dual_rotateR, add_comm]
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Β¬size l + size r β€ 1
h_1 : delta * size l < size r
h_2 : delta * size r < size l
β’ dual (rotateL l x r) = rotateL (dual r) x (dual l)
[PROOFSTEP]
simp [dual_node', dual_rotateL, dual_rotateR, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Β¬size l + size r β€ 1
h_1 : delta * size l < size r
h_2 : Β¬delta * size r < size l
β’ dual (rotateL l x r) = rotateR (dual r) x (dual l)
[PROOFSTEP]
simp [dual_node', dual_rotateL, dual_rotateR, add_comm]
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Β¬size l + size r β€ 1
h_1 : Β¬delta * size l < size r
hβ : delta * size r < size l
β’ dual (rotateR l x r) = rotateL (dual r) x (dual l)
[PROOFSTEP]
simp [dual_node', dual_rotateL, dual_rotateR, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Β¬size l + size r β€ 1
h_1 : Β¬delta * size l < size r
hβ : Β¬delta * size r < size l
β’ dual (node' l x r) = node' (dual r) x (dual l)
[PROOFSTEP]
simp [dual_node', dual_rotateL, dual_rotateR, add_comm]
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Β¬size l + size r β€ 1
h_1 : delta * size l < size r
h_2 : delta * size r < size l
β’ rotateR (dual r) x (dual l) = rotateL (dual r) x (dual l)
[PROOFSTEP]
cases delta_lt_false h_1 h_2
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (balanceL l x r) = balanceR (dual r) x (dual l)
[PROOFSTEP]
unfold balanceL balanceR
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id r = t β Ordnode Ξ±) (id r)
(fun h =>
(_ : nil = id r) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ Ordnode.singleton x)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 l x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else
node (ls + 1) (node (lls + size lrl + 1) ll lx lrl) lrx (node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id l = id l))
(fun rs l_1 x_1 r_1 h =>
(_ : node rs l_1 x_1 r_1 = id r) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ node (rs + 1) nil x r)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r_2 h =>
(_ : node lls l x_2 r_2 = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x r)
else
node (ls + rs + 1) (node (lls + size lrl + 1) ll lx lrl) lrx
(node (size lrr + rs + 1) lrr x r))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls + rs + 1) l x r)
(_ : id l = id l))
(_ : id r = id r)) =
Ordnode.casesOn (motive := fun t => id (dual r) = t β Ordnode Ξ±) (id (dual r))
(fun h =>
(_ : nil = id (dual r)) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual l)) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual l))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual l) = id (dual l)))
(fun ls l_1 x_1 r_1 h =>
(_ : node ls l_1 x_1 r_1 = id (dual r)) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ node (ls + 1) (dual r) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual l)) βΈ
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r_2 h =>
(_ : node rrs l x_2 r_2 = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) (dual r) x rl) rx rr
else
node (ls + rs + 1) (node (ls + size rll + 1) (dual r) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls + rs + 1) (dual r) x (dual l))
(_ : id (dual l) = id (dual l)))
(_ : id (dual r) = id (dual r))
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ Ordnode.singleton x)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 l x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else
node (ls + 1) (node (lls + size lrl + 1) ll lx lrl) lrx (node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id l = id l))
(fun rs l_1 x_1 r h =>
(_ : node rs l_1 x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + size lrl + 1) ll lx lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls + rs + 1) l x nil)
(_ : id l = id l))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual l)) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual l))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual l) = id (dual l)))
(fun ls l_1 x_1 r h =>
(_ : node ls l_1 x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ node (ls + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual l)) βΈ
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) (dual nil) x rl) rx rr
else
node (ls + rs + 1) (node (ls + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls + rs + 1) (dual nil) x (dual l))
(_ : id (dual l) = id (dual l)))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
cases' l with ls ll lx lr
[GOAL]
case nil.nil
Ξ± : Type u_1
x : Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h => (_ : nil = id nil) βΈ Ordnode.singleton x)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id nil) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 nil x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else
node (ls + 1) (node (lls + size lrl + 1) ll lx lrl) lrx (node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id nil = id nil))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h => (_ : nil = id nil) βΈ node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id nil) βΈ
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + size lrl + 1) ll lx lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls + rs + 1) nil x nil)
(_ : id nil = id nil))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h => (_ : nil = id (dual nil)) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual nil))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual nil) = id (dual nil)))
(fun ls l x_1 r h =>
(_ : node ls l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h => (_ : nil = id (dual nil)) βΈ node (ls + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual nil)) βΈ
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) (dual nil) x rl) rx rr
else
node (ls + rs + 1) (node (ls + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls + rs + 1) (dual nil) x (dual nil))
(_ : id (dual nil) = id (dual nil)))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
rfl
[GOAL]
case nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t β Ordnode Ξ±) (id (node ls ll lx lr))
(fun h => (_ : nil = id (node ls ll lx lr)) βΈ Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
(_ : node ls_1 ll_1 lx_1 lr_1 = id (node ls ll lx lr)) βΈ
Ordnode.casesOn (motive := fun t => id ll_1 = t β Ordnode Ξ±) (id ll_1)
(fun h =>
(_ : nil = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => lr_1 = t β Ordnode Ξ±) lr_1
(fun h => (_ : nil = lr_1) βΈ node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr_1) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr_1 = lr_1))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll_1 lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll_1 = id ll_1))
(_ : id (node ls ll lx lr) = id (node ls ll lx lr)))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t β Ordnode Ξ±) (id (node ls ll lx lr))
(fun h => (_ : nil = id (node ls ll lx lr)) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll_1 lx_1 lr_1 h =>
(_ : node ls_1 ll_1 lx_1 lr_1 = id (node ls ll lx lr)) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll_1 = t β Ordnode Ξ±) (id ll_1)
(fun h => (_ : nil = id ll_1) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then
node (ls_1 + rs + 1) ll_1 lx_1 (node (rs + lrs + 1) lr_1 x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll_1 lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll_1 = id ll_1)
else node (ls_1 + rs + 1) (node ls ll lx lr) x nil)
(_ : id (node ls ll lx lr) = id (node ls ll lx lr)))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls ll lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls ll lx lr))) (fun h => (_ : nil = id (dual (node ls ll lx lr))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls ll lx lr))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls ll lx lr)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls ll lx lr)) = id (dual (node ls ll lx lr))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls ll lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls ll lx lr)))
(fun h => (_ : nil = id (dual (node ls ll lx lr))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls ll lx lr))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls ll lx lr)))
(_ : id (dual (node ls ll lx lr)) = id (dual (node ls ll lx lr))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
cases' ll with lls lll llx llr
[GOAL]
case nil.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx lr) = t β Ordnode Ξ±) (id (node ls nil lx lr))
(fun h => (_ : nil = id (node ls nil lx lr)) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr_1 h =>
(_ : node ls_1 ll lx_1 lr_1 = id (node ls nil lx lr)) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr_1 = t β Ordnode Ξ±) lr_1
(fun h => (_ : nil = lr_1) βΈ node 2 (node ls nil lx lr) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr_1) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr_1 = lr_1))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll = id ll))
(_ : id (node ls nil lx lr) = id (node ls nil lx lr)))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx lr) = t β Ordnode Ξ±) (id (node ls nil lx lr))
(fun h => (_ : nil = id (node ls nil lx lr)) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr_1 h =>
(_ : node ls_1 ll lx_1 lr_1 = id (node ls nil lx lr)) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr_1 x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls nil lx lr) x nil)
(_ : id (node ls nil lx lr) = id (node ls nil lx lr)))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls nil lx lr))) (fun h => (_ : nil = id (dual (node ls nil lx lr))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx lr))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls nil lx lr)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls nil lx lr)) = id (dual (node ls nil lx lr))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls nil lx lr)))
(fun h => (_ : nil = id (dual (node ls nil lx lr))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx lr))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls nil lx lr)))
(_ : id (dual (node ls nil lx lr)) = id (dual (node ls nil lx lr))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls (node lls lll llx llr) lx lr) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx lr))
(fun h => (_ : nil = id (node ls (node lls lll llx llr) lx lr)) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr_1 h =>
(_ : node ls_1 ll lx_1 lr_1 = id (node ls (node lls lll llx llr) lx lr)) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr_1 = t β Ordnode Ξ±) lr_1
(fun h => (_ : nil = lr_1) βΈ node 2 (node ls (node lls lll llx llr) lx lr) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr_1) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr_1 = lr_1))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll = id ll))
(_ : id (node ls (node lls lll llx llr) lx lr) = id (node ls (node lls lll llx llr) lx lr)))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls (node lls lll llx llr) lx lr) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx lr))
(fun h => (_ : nil = id (node ls (node lls lll llx llr) lx lr)) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr_1 h =>
(_ : node ls_1 ll lx_1 lr_1 = id (node ls (node lls lll llx llr) lx lr)) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr_1 x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls (node lls lll llx llr) lx lr) x nil)
(_ : id (node ls (node lls lll llx llr) lx lr) = id (node ls (node lls lll llx llr) lx lr)))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls (node lls lll llx llr) lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx lr)))
(fun h => (_ : nil = id (dual (node ls (node lls lll llx llr) lx lr))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx lr))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls (node lls lll llx llr) lx lr)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls (node lls lll llx llr) lx lr)) = id (dual (node ls (node lls lll llx llr) lx lr))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls (node lls lll llx llr) lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx lr)))
(fun h => (_ : nil = id (dual (node ls (node lls lll llx llr) lx lr))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx lr))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls (node lls lll llx llr) lx lr)))
(_ : id (dual (node ls (node lls lll llx llr) lx lr)) = id (dual (node ls (node lls lll llx llr) lx lr))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case nil.node.nil.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx nil) = t β Ordnode Ξ±) (id (node ls nil lx nil))
(fun h => (_ : nil = id (node ls nil lx nil)) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls nil lx nil)) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 (node ls nil lx nil) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id (node ls nil lx nil) = id (node ls nil lx nil)))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx nil) = t β Ordnode Ξ±) (id (node ls nil lx nil))
(fun h => (_ : nil = id (node ls nil lx nil)) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls nil lx nil)) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls nil lx nil) x nil)
(_ : id (node ls nil lx nil) = id (node ls nil lx nil)))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx nil)) = t β Ordnode Ξ±)
(id (dual (node ls nil lx nil))) (fun h => (_ : nil = id (dual (node ls nil lx nil))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx nil))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls nil lx nil)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls nil lx nil)) = id (dual (node ls nil lx nil))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx nil)) = t β Ordnode Ξ±)
(id (dual (node ls nil lx nil)))
(fun h => (_ : nil = id (dual (node ls nil lx nil))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx nil))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls nil lx nil)))
(_ : id (dual (node ls nil lx nil)) = id (dual (node ls nil lx nil))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx (node lrs lrl lrx lrr)) = t β Ordnode Ξ±)
(id (node ls nil lx (node lrs lrl lrx lrr)))
(fun h => (_ : nil = id (node ls nil lx (node lrs lrl lrx lrr))) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls nil lx (node lrs lrl lrx lrr))) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 (node ls nil lx (node lrs lrl lrx lrr)) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id (node ls nil lx (node lrs lrl lrx lrr)) = id (node ls nil lx (node lrs lrl lrx lrr))))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls nil lx (node lrs lrl lrx lrr)) = t β Ordnode Ξ±)
(id (node ls nil lx (node lrs lrl lrx lrr)))
(fun h => (_ : nil = id (node ls nil lx (node lrs lrl lrx lrr))) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls nil lx (node lrs lrl lrx lrr))) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls nil lx (node lrs lrl lrx lrr)) x nil)
(_ : id (node ls nil lx (node lrs lrl lrx lrr)) = id (node ls nil lx (node lrs lrl lrx lrr))))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx (node lrs lrl lrx lrr))) = t β Ordnode Ξ±)
(id (dual (node ls nil lx (node lrs lrl lrx lrr))))
(fun h => (_ : nil = id (dual (node ls nil lx (node lrs lrl lrx lrr)))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx (node lrs lrl lrx lrr)))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls nil lx (node lrs lrl lrx lrr))))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls nil lx (node lrs lrl lrx lrr))) = id (dual (node ls nil lx (node lrs lrl lrx lrr)))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls nil lx (node lrs lrl lrx lrr))) = t β Ordnode Ξ±)
(id (dual (node ls nil lx (node lrs lrl lrx lrr))))
(fun h => (_ : nil = id (dual (node ls nil lx (node lrs lrl lrx lrr)))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls nil lx (node lrs lrl lrx lrr)))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls nil lx (node lrs lrl lrx lrr))))
(_ : id (dual (node ls nil lx (node lrs lrl lrx lrr))) = id (dual (node ls nil lx (node lrs lrl lrx lrr)))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls (node lls lll llx llr) lx nil) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx nil))
(fun h => (_ : nil = id (node ls (node lls lll llx llr) lx nil)) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls (node lls lll llx llr) lx nil)) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 (node ls (node lls lll llx llr) lx nil) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id (node ls (node lls lll llx llr) lx nil) = id (node ls (node lls lll llx llr) lx nil)))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t => id (node ls (node lls lll llx llr) lx nil) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx nil))
(fun h => (_ : nil = id (node ls (node lls lll llx llr) lx nil)) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls (node lls lll llx llr) lx nil)) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls (node lls lll llx llr) lx nil) x nil)
(_ : id (node ls (node lls lll llx llr) lx nil) = id (node ls (node lls lll llx llr) lx nil)))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls (node lls lll llx llr) lx nil)) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx nil)))
(fun h => (_ : nil = id (dual (node ls (node lls lll llx llr) lx nil))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx nil))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls (node lls lll llx llr) lx nil)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls (node lls lll llx llr) lx nil)) = id (dual (node ls (node lls lll llx llr) lx nil))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls (node lls lll llx llr) lx nil)) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx nil)))
(fun h => (_ : nil = id (dual (node ls (node lls lll llx llr) lx nil))) βΈ node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx nil))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else node (ls_1 + rs + 1) (dual nil) x (dual (node ls (node lls lll llx llr) lx nil)))
(_ : id (dual (node ls (node lls lll llx llr) lx nil)) = id (dual (node ls (node lls lll llx llr) lx nil))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h =>
(_ : nil = id nil) βΈ
Ordnode.casesOn (motive := fun t =>
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))
(fun h => (_ : nil = id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) βΈ Ordnode.singleton x)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h =>
(_ : nil = lr) βΈ node 2 (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll lx_1 (node (lrs + 1) lr x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ :
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) =
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(fun rs l x_1 r h =>
(_ : node rs l x_1 r = id nil) βΈ
Ordnode.casesOn (motive := fun t =>
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) = t β Ordnode Ξ±)
(id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))
(fun h =>
(_ : nil = id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) βΈ node (rs + 1) nil x nil)
(fun ls_1 ll lx_1 lr h =>
(_ : node ls_1 ll lx_1 lr = id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) βΈ
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls_1 + rs + 1) ll lx_1 (node (rs + lrs + 1) lr x nil)
else
node (ls_1 + rs + 1) (node (lls + size lrl + 1) ll lx_1 lrl) lrx
(node (size lrr + rs + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls_1 + rs + 1) (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil)
(_ :
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) =
id (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(_ : id nil = id nil)) =
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h =>
(_ : nil = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t =>
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(fun h =>
(_ : nil = id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h =>
(_ : nil = rl) βΈ
node 2 nil x (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ :
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) =
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t =>
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) = t β Ordnode Ξ±)
(id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(fun h =>
(_ : nil = id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))) βΈ
node (ls_1 + 1) (dual nil) x nil)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))) βΈ
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr) (fun h => (_ : nil = id rr) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then
node (ls_1 + rs + 1) (node (ls_1 + rls + 1) (dual nil) x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + size rll + 1) (dual nil) x rll) rlx
(node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr)
else
node (ls_1 + rs + 1) (dual nil) x (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))))
(_ :
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))) =
id (dual (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)))))
(_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ node 3 (node 1 nil x nil) lrx (node 1 nil lx nil) = node 3 (Ordnode.singleton x) lrx (Ordnode.singleton lx)
[PROOFSTEP]
try rfl
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ node 3 (node 1 nil x nil) lrx (node 1 nil lx nil) = node 3 (Ordnode.singleton x) lrx (Ordnode.singleton lx)
[PROOFSTEP]
rfl
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ node 3 (node 1 nil x nil) lx (node lls (dual llr) llx (dual lll)) =
node 3 (Ordnode.singleton x) lx (node lls (dual llr) llx (dual lll))
[PROOFSTEP]
try rfl
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ node 3 (node 1 nil x nil) lx (node lls (dual llr) llx (dual lll)) =
node 3 (Ordnode.singleton x) lx (node lls (dual llr) llx (dual lll))
[PROOFSTEP]
rfl
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(if lrs < ratio * lls then node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (ls + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx (node (size lrr + 1) lrr x nil)) =
if lrs < ratio * lls then
node (ls + 1) (node (lrs + 1) nil x (node lrs (dual lrr) lrx (dual lrl))) lx (node lls (dual llr) llx (dual lll))
else
node (ls + 1) (node (size (dual lrr) + 1) nil x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
try rfl
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(if lrs < ratio * lls then node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (ls + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx (node (size lrr + 1) lrr x nil)) =
if lrs < ratio * lls then
node (ls + 1) (node (lrs + 1) nil x (node lrs (dual lrr) lrx (dual lrl))) lx (node lls (dual llr) llx (dual lll))
else
node (ls + 1) (node (size (dual lrr) + 1) nil x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
rfl
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(if lrs < ratio * lls then node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (ls + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx (node (size lrr + 1) lrr x nil)) =
if lrs < ratio * lls then
node (ls + 1) (node (lrs + 1) nil x (node lrs (dual lrr) lrx (dual lrl))) lx (node lls (dual llr) llx (dual lll))
else
node (ls + 1) (node (size (dual lrr) + 1) nil x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : lrs < ratio * lls
β’ dual (node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)) =
node (ls + 1) (node (lrs + 1) nil x (node lrs (dual lrr) lrx (dual lrl))) lx (node lls (dual llr) llx (dual lll))
[PROOFSTEP]
repeat simp [h, add_comm]
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : lrs < ratio * lls
β’ dual (node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)) =
node (ls + 1) (node (lrs + 1) nil x (node lrs (dual lrr) lrx (dual lrl))) lx (node lls (dual llr) llx (dual lll))
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : Β¬lrs < ratio * lls
β’ dual (node (ls + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx (node (size lrr + 1) lrr x nil)) =
node (ls + 1) (node (size (dual lrr) + 1) nil x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
repeat simp [h, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : Β¬lrs < ratio * lls
β’ dual (node (ls + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx (node (size lrr + 1) lrr x nil)) =
node (ls + 1) (node (size (dual lrr) + 1) nil x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t β Ordnode Ξ±) (id (node rs rl rx rr))
(fun h =>
(_ : nil = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ Ordnode.singleton x)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 l x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else
node (ls + 1) (node (lls + size lrl + 1) ll lx lrl) lrx (node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id l = id l))
(fun rs_1 l_1 x_1 r h =>
(_ : node rs_1 l_1 x_1 r = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id l = t β Ordnode Ξ±) (id l)
(fun h => (_ : nil = id l) βΈ node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id l) βΈ
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + size lrl + 1) ll lx lrl) lrx
(node (size lrr + rs_1 + 1) lrr x (node rs rl rx rr)))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls + rs_1 + 1) l x (node rs rl rx rr))
(_ : id l = id l))
(_ : id (node rs rl rx rr) = id (node rs rl rx rr))) =
Ordnode.casesOn (motive := fun t => id (dual (node rs rl rx rr)) = t β Ordnode Ξ±) (id (dual (node rs rl rx rr)))
(fun h =>
(_ : nil = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual l)) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual l))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual l) = id (dual l)))
(fun ls l_1 x_1 r h =>
(_ : node ls l_1 x_1 r = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual l) = t β Ordnode Ξ±) (id (dual l))
(fun h => (_ : nil = id (dual l)) βΈ node (ls + 1) (dual (node rs rl rx rr)) x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
(_ : node rs_1 rl_1 rx_1 rr_1 = id (dual l)) βΈ
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t β Ordnode Ξ±) (id rr_1)
(fun h => (_ : nil = id rr_1) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr_1) βΈ
Ordnode.casesOn (motive := fun t => id rl_1 = t β Ordnode Ξ±) (id rl_1)
(fun h => (_ : nil = id rl_1) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl_1) βΈ
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (dual (node rs rl rx rr)) x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + size rll + 1) (dual (node rs rl rx rr)) x rll) rlx
(node (size rlr + rrs + 1) rlr rx_1 rr_1))
(_ : id rl_1 = id rl_1))
(_ : id rr_1 = id rr_1)
else node (ls + rs_1 + 1) (dual (node rs rl rx rr)) x (dual l))
(_ : id (dual l) = id (dual l)))
(_ : id (dual (node rs rl rx rr)) = id (dual (node rs rl rx rr)))
[PROOFSTEP]
cases' l with ls ll lx lr
[GOAL]
case node.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t β Ordnode Ξ±) (id (node rs rl rx rr))
(fun h =>
(_ : nil = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h => (_ : nil = id nil) βΈ Ordnode.singleton x)
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id nil) βΈ
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h =>
(_ : nil = id ll) βΈ
Ordnode.casesOn (motive := fun t => lr = t β Ordnode Ξ±) lr
(fun h => (_ : nil = lr) βΈ node 2 nil x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr) βΈ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
(_ : lr = lr))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else
node (ls + 1) (node (lls + size lrl + 1) ll lx lrl) lrx (node (size lrr + 1) lrr x nil))
(_ : id lr = id lr))
(_ : id ll = id ll))
(_ : id nil = id nil))
(fun rs_1 l x_1 r h =>
(_ : node rs_1 l x_1 r = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id nil = t β Ordnode Ξ±) (id nil)
(fun h => (_ : nil = id nil) βΈ node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
(_ : node ls ll lx lr = id nil) βΈ
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t β Ordnode Ξ±) (id ll)
(fun h => (_ : nil = id ll) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll) βΈ
Ordnode.casesOn (motive := fun t => id lr = t β Ordnode Ξ±) (id lr)
(fun h => (_ : nil = id lr) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr) βΈ
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + size lrl + 1) ll lx lrl) lrx
(node (size lrr + rs_1 + 1) lrr x (node rs rl rx rr)))
(_ : id lr = id lr))
(_ : id ll = id ll)
else node (ls + rs_1 + 1) nil x (node rs rl rx rr))
(_ : id nil = id nil))
(_ : id (node rs rl rx rr) = id (node rs rl rx rr))) =
Ordnode.casesOn (motive := fun t => id (dual (node rs rl rx rr)) = t β Ordnode Ξ±) (id (dual (node rs rl rx rr)))
(fun h =>
(_ : nil = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h => (_ : nil = id (dual nil)) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual nil)) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual nil))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual nil) = id (dual nil)))
(fun ls l x_1 r h =>
(_ : node ls l x_1 r = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual nil) = t β Ordnode Ξ±) (id (dual nil))
(fun h => (_ : nil = id (dual nil)) βΈ node (ls + 1) (dual (node rs rl rx rr)) x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
(_ : node rs_1 rl_1 rx_1 rr_1 = id (dual nil)) βΈ
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t β Ordnode Ξ±) (id rr_1)
(fun h => (_ : nil = id rr_1) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr_1) βΈ
Ordnode.casesOn (motive := fun t => id rl_1 = t β Ordnode Ξ±) (id rl_1)
(fun h => (_ : nil = id rl_1) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl_1) βΈ
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (dual (node rs rl rx rr)) x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + size rll + 1) (dual (node rs rl rx rr)) x rll) rlx
(node (size rlr + rrs + 1) rlr rx_1 rr_1))
(_ : id rl_1 = id rl_1))
(_ : id rr_1 = id rr_1)
else node (ls + rs_1 + 1) (dual (node rs rl rx rr)) x (dual nil))
(_ : id (dual nil) = id (dual nil)))
(_ : id (dual (node rs rl rx rr)) = id (dual (node rs rl rx rr)))
[PROOFSTEP]
rfl
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
β’ dual
(Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t β Ordnode Ξ±) (id (node rs rl rx rr))
(fun h =>
(_ : nil = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t β Ordnode Ξ±) (id (node ls ll lx lr))
(fun h => (_ : nil = id (node ls ll lx lr)) βΈ Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
(_ : node ls_1 ll_1 lx_1 lr_1 = id (node ls ll lx lr)) βΈ
Ordnode.casesOn (motive := fun t => id ll_1 = t β Ordnode Ξ±) (id ll_1)
(fun h =>
(_ : nil = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => lr_1 = t β Ordnode Ξ±) lr_1
(fun h => (_ : nil = lr_1) βΈ node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h =>
(_ : node size l lrx r = lr_1) βΈ node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x))
(_ : lr_1 = lr_1))
(fun lls l x_1 r h =>
(_ : node lls l x_1 r = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + size lrl + 1) ll_1 lx_1 lrl) lrx
(node (size lrr + 1) lrr x nil))
(_ : id lr_1 = id lr_1))
(_ : id ll_1 = id ll_1))
(_ : id (node ls ll lx lr) = id (node ls ll lx lr)))
(fun rs_1 l x_1 r h =>
(_ : node rs_1 l x_1 r = id (node rs rl rx rr)) βΈ
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t β Ordnode Ξ±) (id (node ls ll lx lr))
(fun h => (_ : nil = id (node ls ll lx lr)) βΈ node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls_1 ll_1 lx_1 lr_1 h =>
(_ : node ls_1 ll_1 lx_1 lr_1 = id (node ls ll lx lr)) βΈ
if ls_1 > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll_1 = t β Ordnode Ξ±) (id ll_1)
(fun h => (_ : nil = id ll_1) βΈ nil)
(fun lls l x_2 r h =>
(_ : node lls l x_2 r = id ll_1) βΈ
Ordnode.casesOn (motive := fun t => id lr_1 = t β Ordnode Ξ±) (id lr_1)
(fun h => (_ : nil = id lr_1) βΈ nil)
(fun lrs lrl lrx lrr h =>
(_ : node lrs lrl lrx lrr = id lr_1) βΈ
if lrs < ratio * lls then
node (ls_1 + rs_1 + 1) ll_1 lx_1 (node (rs_1 + lrs + 1) lr_1 x (node rs rl rx rr))
else
node (ls_1 + rs_1 + 1) (node (lls + size lrl + 1) ll_1 lx_1 lrl) lrx
(node (size lrr + rs_1 + 1) lrr x (node rs rl rx rr)))
(_ : id lr_1 = id lr_1))
(_ : id ll_1 = id ll_1)
else node (ls_1 + rs_1 + 1) (node ls ll lx lr) x (node rs rl rx rr))
(_ : id (node ls ll lx lr) = id (node ls ll lx lr)))
(_ : id (node rs rl rx rr) = id (node rs rl rx rr))) =
Ordnode.casesOn (motive := fun t => id (dual (node rs rl rx rr)) = t β Ordnode Ξ±) (id (dual (node rs rl rx rr)))
(fun h =>
(_ : nil = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls ll lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls ll lx lr))) (fun h => (_ : nil = id (dual (node ls ll lx lr))) βΈ Ordnode.singleton x)
(fun rs rl rx rr h =>
(_ : node rs rl rx rr = id (dual (node ls ll lx lr))) βΈ
Ordnode.casesOn (motive := fun t => id rr = t β Ordnode Ξ±) (id rr)
(fun h =>
(_ : nil = id rr) βΈ
Ordnode.casesOn (motive := fun t => rl = t β Ordnode Ξ±) rl
(fun h => (_ : nil = rl) βΈ node 2 nil x (dual (node ls ll lx lr)))
(fun size l rlx r h =>
(_ : node size l rlx r = rl) βΈ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
(_ : rl = rl))
(fun rrs l x_1 r h =>
(_ : node rrs l x_1 r = id rr) βΈ
Ordnode.casesOn (motive := fun t => id rl = t β Ordnode Ξ±) (id rl)
(fun h => (_ : nil = id rl) βΈ node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl) βΈ
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else
node (rs + 1) (node (size rll + 1) nil x rll) rlx (node (size rlr + rrs + 1) rlr rx rr))
(_ : id rl = id rl))
(_ : id rr = id rr))
(_ : id (dual (node ls ll lx lr)) = id (dual (node ls ll lx lr))))
(fun ls_1 l x_1 r h =>
(_ : node ls_1 l x_1 r = id (dual (node rs rl rx rr))) βΈ
Ordnode.casesOn (motive := fun t => id (dual (node ls ll lx lr)) = t β Ordnode Ξ±)
(id (dual (node ls ll lx lr)))
(fun h => (_ : nil = id (dual (node ls ll lx lr))) βΈ node (ls_1 + 1) (dual (node rs rl rx rr)) x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
(_ : node rs_1 rl_1 rx_1 rr_1 = id (dual (node ls ll lx lr))) βΈ
if rs_1 > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr_1 = t β Ordnode Ξ±) (id rr_1)
(fun h => (_ : nil = id rr_1) βΈ nil)
(fun rrs l x_2 r h =>
(_ : node rrs l x_2 r = id rr_1) βΈ
Ordnode.casesOn (motive := fun t => id rl_1 = t β Ordnode Ξ±) (id rl_1)
(fun h => (_ : nil = id rl_1) βΈ nil)
(fun rls rll rlx rlr h =>
(_ : node rls rll rlx rlr = id rl_1) βΈ
if rls < ratio * rrs then
node (ls_1 + rs_1 + 1) (node (ls_1 + rls + 1) (dual (node rs rl rx rr)) x rl_1) rx_1
rr_1
else
node (ls_1 + rs_1 + 1) (node (ls_1 + size rll + 1) (dual (node rs rl rx rr)) x rll) rlx
(node (size rlr + rrs + 1) rlr rx_1 rr_1))
(_ : id rl_1 = id rl_1))
(_ : id rr_1 = id rr_1)
else node (ls_1 + rs_1 + 1) (dual (node rs rl rx rr)) x (dual (node ls ll lx lr)))
(_ : id (dual (node ls ll lx lr)) = id (dual (node ls ll lx lr))))
(_ : id (dual (node rs rl rx rr)) = id (dual (node rs rl rx rr)))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
β’ dual
(if ls > delta * rs then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => (_ : nil = id ll) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id ll) βΈ
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => (_ : nil = id lr) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id lr) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)) =
if ls > delta * rs then
rec (motive := fun t => dual ll = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual ll)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual ll)) βΈ
rec (motive := fun t => dual lr = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual lr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual lr)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual lr)) lx
(dual ll)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual ll)))
(dual lr) (_ : id (dual lr) = id (dual lr)))
(dual ll) (_ : id (dual ll) = id (dual ll))
else node (rs + ls + 1) (node rs (dual rr) rx (dual rl)) x (node ls (dual lr) lx (dual ll))
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hβ : ls > delta * rs
β’ dual
(rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => (_ : nil = id ll) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id ll) βΈ
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => (_ : nil = id lr) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id lr) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)) =
rec (motive := fun t => dual ll = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual ll)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual ll)) βΈ
rec (motive := fun t => dual lr = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual lr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual lr)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual lr)) lx (dual ll)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual ll)))
(dual lr) (_ : id (dual lr) = id (dual lr)))
(dual ll) (_ : id (dual ll) = id (dual ll))
case neg
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hβ : Β¬ls > delta * rs
β’ dual (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)) =
node (rs + ls + 1) (node rs (dual rr) rx (dual rl)) x (node ls (dual lr) lx (dual ll))
[PROOFSTEP]
swap
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hβ : Β¬ls > delta * rs
β’ dual (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)) =
node (rs + ls + 1) (node rs (dual rr) rx (dual rl)) x (node ls (dual lr) lx (dual ll))
[PROOFSTEP]
simp [add_comm]
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hβ : ls > delta * rs
β’ dual
(rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => (_ : nil = id ll) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id ll) βΈ
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => (_ : nil = id lr) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id lr) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)) =
rec (motive := fun t => dual ll = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual ll)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual ll)) βΈ
rec (motive := fun t => dual lr = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual lr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual lr)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual lr)) lx (dual ll)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual ll)))
(dual lr) (_ : id (dual lr) = id (dual lr)))
(dual ll) (_ : id (dual ll) = id (dual ll))
[PROOFSTEP]
cases' ll with lls lll llx llr
[GOAL]
case pos.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
hβ : ls > delta * rs
β’ dual
(rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id nil) βΈ
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => (_ : nil = id lr) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id lr) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) nil lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
nil (_ : id nil = id nil)) =
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual nil)) βΈ
rec (motive := fun t => dual lr = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual lr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual lr)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual lr)) lx (dual nil)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual nil)))
(dual lr) (_ : id (dual lr) = id (dual lr)))
(dual nil) (_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case pos.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => (_ : nil = id lr) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id lr) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual lr = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual lr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual lr)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual lr)) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual lr) (_ : id (dual lr) = id (dual lr)))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case pos.nil.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
β’ dual
(rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id nil) βΈ
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id nil) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) nil lx (node (rs + size_1 + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
nil (_ : id nil = id nil))
nil (_ : id nil = id nil)) =
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual nil)) βΈ
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual nil)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual nil)) lx
(dual nil)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual nil)))
(dual nil) (_ : id (dual nil) = id (dual nil)))
(dual nil) (_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
try rfl
[GOAL]
case pos.nil.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
β’ dual
(rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id nil) βΈ
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id nil) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) nil lx (node (rs + size_1 + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
nil (_ : id nil = id nil))
nil (_ : id nil = id nil)) =
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual nil)) βΈ
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual nil)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual nil)) lx
(dual nil)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual nil)))
(dual nil) (_ : id (dual nil) = id (dual nil)))
(dual nil) (_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
rfl
[GOAL]
case pos.nil.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id nil) βΈ
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lrs lrl lrx lrr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (node lrs lrl lrx lrr)) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) nil lx (node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
nil (_ : id nil = id nil)) =
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual nil)) βΈ
rec (motive := fun t => dual (node lrs lrl lrx lrr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lrs lrl lrx lrr))) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual (node lrs lrl lrx lrr))) βΈ
if size_1 < ratio * size then
node (rs + ls + 1)
(node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual (node lrs lrl lrx lrr))) lx
(dual nil)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual nil)))
(dual (node lrs lrl lrx lrr)) (_ : id (dual (node lrs lrl lrx lrr)) = id (dual (node lrs lrl lrx lrr))))
(dual nil) (_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
try rfl
[GOAL]
case pos.nil.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id nil) βΈ
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lrs lrl lrx lrr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (node lrs lrl lrx lrr)) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) nil lx (node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
nil (_ : id nil = id nil)) =
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual nil)) βΈ
rec (motive := fun t => dual (node lrs lrl lrx lrr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lrs lrl lrx lrr))) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual (node lrs lrl lrx lrr))) βΈ
if size_1 < ratio * size then
node (rs + ls + 1)
(node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual (node lrs lrl lrx lrr))) lx
(dual nil)
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual nil)))
(dual (node lrs lrl lrx lrr)) (_ : id (dual (node lrs lrl lrx lrr)) = id (dual (node lrs lrl lrx lrr))))
(dual nil) (_ : id (dual nil) = id (dual nil))
[PROOFSTEP]
rfl
[GOAL]
case pos.node.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id nil) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + size_1 + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
nil (_ : id nil = id nil))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual nil)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual nil)) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual nil) (_ : id (dual nil) = id (dual nil)))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
try rfl
[GOAL]
case pos.node.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => (_ : nil = id nil) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id nil) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + size_1 + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
nil (_ : id nil = id nil))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual nil = t β Ordnode Ξ±) (fun h => (_ : nil = id (dual nil)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual nil)) βΈ
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual nil)) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual nil) (_ : id (dual nil) = id (dual nil)))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
rfl
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lrs lrl lrx lrr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (node lrs lrl lrx lrr)) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual (node lrs lrl lrx lrr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lrs lrl lrx lrr))) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual (node lrs lrl lrx lrr))) βΈ
if size_1 < ratio * size then
node (rs + ls + 1)
(node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual (node lrs lrl lrx lrr))) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual (node lrs lrl lrx lrr)) (_ : id (dual (node lrs lrl lrx lrr)) = id (dual (node lrs lrl lrx lrr))))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
try rfl
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lrs lrl lrx lrr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (node lrs lrl lrx lrr)) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual (node lrs lrl lrx lrr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lrs lrl lrx lrr))) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual (node lrs lrl lrx lrr))) βΈ
if size_1 < ratio * size then
node (rs + ls + 1)
(node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual (node lrs lrl lrx lrr))) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual (node lrs lrl lrx lrr)) (_ : id (dual (node lrs lrl lrx lrr)) = id (dual (node lrs lrl lrx lrr))))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
rfl
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lls lll llx llr)) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (node lls lll llx llr)) βΈ
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±)
(fun h => (_ : nil = id (node lrs lrl lrx lrr)) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (node lrs lrl lrx lrr)) βΈ
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))) =
rec (motive := fun t => dual (node lls lll llx llr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lls lll llx llr))) βΈ nil)
(fun size l x_1 r l_ih r_ih h =>
(_ : node size l x_1 r = id (dual (node lls lll llx llr))) βΈ
rec (motive := fun t => dual (node lrs lrl lrx lrr) = t β Ordnode Ξ±)
(fun h => (_ : nil = id (dual (node lrs lrl lrx lrr))) βΈ nil)
(fun size_1 l x_2 r l_ih r_ih h =>
(_ : node size_1 l x_2 r = id (dual (node lrs lrl lrx lrr))) βΈ
if size_1 < ratio * size then
node (rs + ls + 1)
(node (rs + size_1 + 1) (node rs (dual rr) rx (dual rl)) x (dual (node lrs lrl lrx lrr))) lx
(dual (node lls lll llx llr))
else
node (rs + ls + 1) (node (rs + Ordnode.size l + 1) (node rs (dual rr) rx (dual rl)) x l) x_2
(node (Ordnode.size r + size + 1) r lx (dual (node lls lll llx llr))))
(dual (node lrs lrl lrx lrr)) (_ : id (dual (node lrs lrl lrx lrr)) = id (dual (node lrs lrl lrx lrr))))
(dual (node lls lll llx llr)) (_ : id (dual (node lls lll llx llr)) = id (dual (node lls lll llx llr)))
[PROOFSTEP]
dsimp only [dual, id]
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
β’ dual
(if lrs < ratio * lls then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + lrs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + rs + 1) lrr x (node rs rl rx rr))) =
if lrs < ratio * lls then
node (rs + ls + 1) (node (rs + lrs + 1) (node rs (dual rr) rx (dual rl)) x (node lrs (dual lrr) lrx (dual lrl)))
lx (node lls (dual llr) llx (dual lll))
else
node (rs + ls + 1) (node (rs + size (dual lrr) + 1) (node rs (dual rr) rx (dual rl)) x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : lrs < ratio * lls
β’ dual
(node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + lrs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))) =
node (rs + ls + 1) (node (rs + lrs + 1) (node rs (dual rr) rx (dual rl)) x (node lrs (dual lrr) lrx (dual lrl))) lx
(node lls (dual llr) llx (dual lll))
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
ls : β
lx : Ξ±
hβ : ls > delta * rs
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
h : Β¬lrs < ratio * lls
β’ dual
(node (ls + rs + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + rs + 1) lrr x (node rs rl rx rr))) =
node (rs + ls + 1) (node (rs + size (dual lrr) + 1) (node rs (dual rr) rx (dual rl)) x (dual lrr)) lrx
(node (size (dual lrl) + lls + 1) (dual lrl) lx (node lls (dual llr) llx (dual lll)))
[PROOFSTEP]
simp [h, add_comm]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (balanceR l x r) = balanceL (dual r) x (dual l)
[PROOFSTEP]
rw [β dual_dual (balanceL _ _ _), dual_balanceL, dual_dual, dual_dual]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
hl : Sized l
hm : Sized m
hr : Sized r
β’ Sized (Ordnode.node4L l x m y r)
[PROOFSTEP]
cases m <;> [exact (hl.node' hm).node' hr; exact (hl.node' hm.2.1).node' (hm.2.2.node' hr)]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
hl : Sized l
hm : Sized m
hr : Sized r
β’ Sized (Ordnode.node4L l x m y r)
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hm : Sized nil
β’ Sized (Ordnode.node4L l x nil y r)
[PROOFSTEP]
exact (hl.node' hm).node' hr
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ Sized (Ordnode.node4L l x (node sizeβ lβ xβ rβ) y r)
[PROOFSTEP]
exact (hl.node' hm.2.1).node' (hm.2.2.node' hr)
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ size (node3L l x m y r) = size l + size m + size r + 2
[PROOFSTEP]
dsimp [node3L, node', size]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (((match l with
| nil => 0
| node sz l x r => sz) +
match m with
| nil => 0
| node sz l x r => sz) +
1 +
match r with
| nil => 0
| node sz l x r => sz) +
1 =
(((match l with
| nil => 0
| node sz l x r => sz) +
match m with
| nil => 0
| node sz l x r => sz) +
match r with
| nil => 0
| node sz l x r => sz) +
2
[PROOFSTEP]
rw [add_right_comm _ 1]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ size (node3R l x m y r) = size l + size m + size r + 2
[PROOFSTEP]
dsimp [node3R, node', size]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match l with
| nil => 0
| node sz l x r => sz) +
(((match m with
| nil => 0
| node sz l x r => sz) +
match r with
| nil => 0
| node sz l x r => sz) +
1) +
1 =
(((match l with
| nil => 0
| node sz l x r => sz) +
match m with
| nil => 0
| node sz l x r => sz) +
match r with
| nil => 0
| node sz l x r => sz) +
2
[PROOFSTEP]
rw [β add_assoc, β add_assoc]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
hm : Sized m
β’ size (node4L l x m y r) = size l + size m + size r + 2
[PROOFSTEP]
cases m <;> simp [node4L, node3L, node'] <;> [skip; simp [size, hm.1]]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
hm : Sized m
β’ size (node4L l x m y r) = size l + size m + size r + 2
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hm : Sized nil
β’ size (node4L l x nil y r) = size l + size nil + size r + 2
[PROOFSTEP]
simp [node4L, node3L, node']
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size (node4L l x (node sizeβ lβ xβ rβ) y r) = size l + size (node sizeβ lβ xβ rβ) + size r + 2
[PROOFSTEP]
simp [node4L, node3L, node']
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hm : Sized nil
β’ size l + 1 + size r = size l + size r + 1
[PROOFSTEP]
skip
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size l + size lβ + 1 + (size rβ + size r + 1) = size l + sizeβ + size r + 1
[PROOFSTEP]
simp [size, hm.1]
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hm : Sized nil
β’ size l + 1 + size r = size l + size r + 1
[PROOFSTEP]
abel
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
hm : Sized nil
β’ size l + 1 + size r = size l + size r + 1
[PROOFSTEP]
abel
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size l + size lβ + 1 + (size rβ + size r + 1) = size l + (size lβ + size rβ + 1) + size r + 1
[PROOFSTEP]
abel
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size l + size lβ + 1 + (size rβ + size r + 1) = size l + (size lβ + size rβ + 1) + size r + 1
[PROOFSTEP]
abel
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
sl : Sized l
sr : Sized r
β’ size l + size r + 1 = size (Ordnode.dual r) + size (Ordnode.dual l) + 1
[PROOFSTEP]
simp [size_dual, add_comm]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
h : Sized (Ordnode.dual t)
β’ Sized t
[PROOFSTEP]
rw [β dual_dual t]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
h : Sized (Ordnode.dual t)
β’ Sized (Ordnode.dual (Ordnode.dual t))
[PROOFSTEP]
exact h.dual
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ Sized (Ordnode.rotateL l x r)
[PROOFSTEP]
cases r
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hl : Sized l
hr : Sized nil
β’ Sized (Ordnode.rotateL l x nil)
[PROOFSTEP]
exact hl.node' hr
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hl : Sized l
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hr : Sized (node sizeβ lβ xβ rβ)
β’ Sized (Ordnode.rotateL l x (node sizeβ lβ xβ rβ))
[PROOFSTEP]
rw [rotateL]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hl : Sized l
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hr : Sized (node sizeβ lβ xβ rβ)
β’ Sized (if size lβ < ratio * size rβ then Ordnode.node3L l x lβ xβ rβ else Ordnode.node4L l x lβ xβ rβ)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hl : Sized l
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hr : Sized (node sizeβ lβ xβ rβ)
hβ : size lβ < ratio * size rβ
β’ Sized (Ordnode.node3L l x lβ xβ rβ)
[PROOFSTEP]
exact hl.node3L hr.2.1 hr.2.2
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hl : Sized l
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hr : Sized (node sizeβ lβ xβ rβ)
hβ : Β¬size lβ < ratio * size rβ
β’ Sized (Ordnode.node4L l x lβ xβ rβ)
[PROOFSTEP]
exact hl.node4L hr.2.1 hr.2.2
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ Sized (Ordnode.dual (Ordnode.rotateR l x r))
[PROOFSTEP]
rw [dual_rotateR]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ Sized (Ordnode.rotateL (Ordnode.dual r) x (Ordnode.dual l))
[PROOFSTEP]
exact hr.dual.rotateL hl.dual
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hm : Sized r
β’ size (Ordnode.rotateL l x r) = size l + size r + 1
[PROOFSTEP]
cases r
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
hm : Sized nil
β’ size (Ordnode.rotateL l x nil) = size l + size nil + 1
[PROOFSTEP]
simp [Ordnode.rotateL]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size (Ordnode.rotateL l x (node sizeβ lβ xβ rβ)) = size l + size (node sizeβ lβ xβ rβ) + 1
[PROOFSTEP]
simp [Ordnode.rotateL]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size (if size lβ < ratio * size rβ then Ordnode.node3L l x lβ xβ rβ else Ordnode.node4L l x lβ xβ rβ) =
size l + sizeβ + 1
[PROOFSTEP]
simp [size, hm.1]
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
β’ size (if size lβ < ratio * size rβ then Ordnode.node3L l x lβ xβ rβ else Ordnode.node4L l x lβ xβ rβ) =
size l + (size lβ + size rβ + 1) + 1
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : size lβ < ratio * size rβ
β’ size (Ordnode.node3L l x lβ xβ rβ) = size l + (size lβ + size rβ + 1) + 1
[PROOFSTEP]
simp [node3L_size, node4L_size hm.2.1]
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : Β¬size lβ < ratio * size rβ
β’ size (Ordnode.node4L l x lβ xβ rβ) = size l + (size lβ + size rβ + 1) + 1
[PROOFSTEP]
simp [node3L_size, node4L_size hm.2.1]
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : size lβ < ratio * size rβ
β’ size l + size lβ + size rβ + 1 = size l + (size lβ + size rβ + 1)
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : size lβ < ratio * size rβ
β’ size l + size lβ + size rβ + 1 = size l + (size lβ + size rβ + 1)
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : Β¬size lβ < ratio * size rβ
β’ size l + size lβ + size rβ + 1 = size l + (size lβ + size rβ + 1)
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hm : Sized (node sizeβ lβ xβ rβ)
hβ : Β¬size lβ < ratio * size rβ
β’ size l + size lβ + size rβ + 1 = size l + (size lβ + size rβ + 1)
[PROOFSTEP]
abel
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
β’ size (Ordnode.rotateR l x r) = size l + size r + 1
[PROOFSTEP]
rw [β size_dual, dual_rotateR, hl.dual.rotateL_size, size_dual, size_dual, add_comm (size l)]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ Sized (Ordnode.balance' l x r)
[PROOFSTEP]
unfold balance'
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ Sized
(if size l + size r β€ 1 then Ordnode.node' l x r
else
if size r > delta * size l then Ordnode.rotateL l x r
else if size l > delta * size r then Ordnode.rotateR l x r else Ordnode.node' l x r)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβ : size l + size r β€ 1
β’ Sized (Ordnode.node' l x r)
[PROOFSTEP]
exact hl.node' hr
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒΉ : Β¬size l + size r β€ 1
hβ : size r > delta * size l
β’ Sized (Ordnode.rotateL l x r)
[PROOFSTEP]
exact hl.rotateL hr
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : size l > delta * size r
β’ Sized (Ordnode.rotateR l x r)
[PROOFSTEP]
exact hl.rotateR hr
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : Β¬size l > delta * size r
β’ Sized (Ordnode.node' l x r)
[PROOFSTEP]
exact hl.node' hr
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ size (balance' l x r) = size l + size r + 1
[PROOFSTEP]
unfold balance'
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
β’ size
(if size l + size r β€ 1 then node' l x r
else
if size r > delta * size l then rotateL l x r
else if size l > delta * size r then rotateR l x r else node' l x r) =
size l + size r + 1
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβ : size l + size r β€ 1
β’ size (node' l x r) = size l + size r + 1
[PROOFSTEP]
rfl
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒΉ : Β¬size l + size r β€ 1
hβ : size r > delta * size l
β’ size (rotateL l x r) = size l + size r + 1
[PROOFSTEP]
exact hr.rotateL_size
[GOAL]
case pos
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : size l > delta * size r
β’ size (rotateR l x r) = size l + size r + 1
[PROOFSTEP]
exact hl.rotateR_size
[GOAL]
case neg
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Sized l
hr : Sized r
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : Β¬size l > delta * size r
β’ size (node' l x r) = size l + size r + 1
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
x : Ξ±
β’ Any P {x} β P x
[PROOFSTEP]
rintro (β¨β¨β©β© | h | β¨β¨β©β©)
[GOAL]
case inr.inl
Ξ± : Type u_1
P : Ξ± β Prop
x : Ξ±
h : P x
β’ P x
[PROOFSTEP]
exact h
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
β’ β (x : Ξ±), Emem x nil β P x
[PROOFSTEP]
rintro _ β¨β©
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P (node sizeβ l x r) β β (x_1 : Ξ±), Emem x_1 (node sizeβ l x r) β P x_1
[PROOFSTEP]
simp [All, Emem, all_iff_forall, Any, or_imp, forall_and]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
β’ Any P nil β β x, Emem x nil β§ P x
[PROOFSTEP]
rintro β¨β©
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
β’ (β x, Emem x nil β§ P x) β Any P nil
[PROOFSTEP]
rintro β¨_, β¨β©, _β©
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ Any P (node sizeβ l x r) β β x_1, Emem x_1 (node sizeβ l x r) β§ P x_1
[PROOFSTEP]
simp only [Emem]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ Any P (node sizeβ l x r) β β x_1, Any (Eq x_1) (node sizeβ l x r) β§ P x_1
[PROOFSTEP]
simp [Any, any_iff_exists, or_and_right, exists_or]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ All P (node3L l x m y r) β All P l β§ P x β§ All P m β§ P y β§ All P r
[PROOFSTEP]
simp [node3L, all_node', and_assoc]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ All P (node4L l x m y r) β All P l β§ P x β§ All P m β§ P y β§ All P r
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
β’ All P (node4L l x nil y r) β All P l β§ P x β§ All P nil β§ P y β§ All P r
[PROOFSTEP]
simp [node4L, all_node', All, all_node3L, and_assoc]
[GOAL]
case node
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ All P (node4L l x (node sizeβ lβ xβ rβ) y r) β All P l β§ P x β§ All P (node sizeβ lβ xβ rβ) β§ P y β§ All P r
[PROOFSTEP]
simp [node4L, all_node', All, all_node3L, and_assoc]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ All P (node4R l x m y r) β All P l β§ P x β§ All P m β§ P y β§ All P r
[PROOFSTEP]
cases m
[GOAL]
case nil
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
β’ All P (node4R l x nil y r) β All P l β§ P x β§ All P nil β§ P y β§ All P r
[PROOFSTEP]
simp [node4R, all_node', All, all_node3R, and_assoc]
[GOAL]
case node
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ All P (node4R l x (node sizeβ lβ xβ rβ) y r) β All P l β§ P x β§ All P (node sizeβ lβ xβ rβ) β§ P y β§ All P r
[PROOFSTEP]
simp [node4R, all_node', All, all_node3R, and_assoc]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P (rotateL l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
cases r
[GOAL]
case nil
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
β’ All P (rotateL l x nil) β All P l β§ P x β§ All P nil
[PROOFSTEP]
simp [rotateL, all_node']
[GOAL]
case node
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ All P (rotateL l x (node sizeβ lβ xβ rβ)) β All P l β§ P x β§ All P (node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [rotateL, all_node']
[GOAL]
case node
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ All P (if size lβ < ratio * size rβ then node3L l x lβ xβ rβ else node4L l x lβ xβ rβ) β
All P l β§ P x β§ All P (node sizeβ lβ xβ rβ)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hβ : size lβ < ratio * size rβ
β’ All P (node3L l x lβ xβ rβ) β All P l β§ P x β§ All P (node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [all_node3L, all_node4L, All, and_assoc]
[GOAL]
case neg
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
hβ : Β¬size lβ < ratio * size rβ
β’ All P (node4L l x lβ xβ rβ) β All P l β§ P x β§ All P (node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [all_node3L, all_node4L, All, and_assoc]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P (rotateR l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
rw [β all_dual, dual_rotateR, all_rotateL]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P (dual r) β§ P x β§ All P (dual l) β All P l β§ P x β§ All P r
[PROOFSTEP]
simp [all_dual, and_comm, and_left_comm, and_assoc]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P (balance' l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
rw [balance']
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ All P
(if size l + size r β€ 1 then node' l x r
else
if size r > delta * size l then rotateL l x r
else if size l > delta * size r then rotateR l x r else node' l x r) β
All P l β§ P x β§ All P r
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβ : size l + size r β€ 1
β’ All P (node' l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
simp [all_node', all_rotateL, all_rotateR]
[GOAL]
case pos
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβΒΉ : Β¬size l + size r β€ 1
hβ : size r > delta * size l
β’ All P (rotateL l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
simp [all_node', all_rotateL, all_rotateR]
[GOAL]
case pos
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : size l > delta * size r
β’ All P (rotateR l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
simp [all_node', all_rotateL, all_rotateR]
[GOAL]
case neg
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hβΒ² : Β¬size l + size r β€ 1
hβΒΉ : Β¬size r > delta * size l
hβ : Β¬size l > delta * size r
β’ All P (node' l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
simp [all_node', all_rotateL, all_rotateR]
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
r' : List Ξ±
β’ foldr List.cons (node sizeβ l x r) r' = toList (node sizeβ l x r) ++ r'
[PROOFSTEP]
rw [foldr, foldr_cons_eq_toList l, foldr_cons_eq_toList r, β List.cons_append, β List.append_assoc, β
foldr_cons_eq_toList l]
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
r' : List Ξ±
β’ foldr List.cons l (x :: toList r) ++ r' = toList (node sizeβ l x r) ++ r'
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ toList (node s l x r) = toList l ++ x :: toList r
[PROOFSTEP]
rw [toList, foldr, foldr_cons_eq_toList]
[GOAL]
Ξ± : Type u_1
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ toList l ++ x :: foldr List.cons r [] = toList l ++ x :: toList r
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
x : Ξ±
t : Ordnode Ξ±
β’ Emem x t β x β toList t
[PROOFSTEP]
unfold Emem
[GOAL]
Ξ± : Type u_1
x : Ξ±
t : Ordnode Ξ±
β’ Any (Eq x) t β x β toList t
[PROOFSTEP]
induction t
[GOAL]
case nil
Ξ± : Type u_1
x : Ξ±
β’ Any (Eq x) nil β x β toList nil
[PROOFSTEP]
simp [Any, *, or_assoc]
[GOAL]
case node
Ξ± : Type u_1
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
l_ihβ : Any (Eq x) lβ β x β toList lβ
r_ihβ : Any (Eq x) rβ β x β toList rβ
β’ Any (Eq x) (node sizeβ lβ xβ rβ) β x β toList (node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [Any, *, or_assoc]
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
β’ List.length (toList (node sizeβ l xβ r)) = realSize (node sizeβ l xβ r)
[PROOFSTEP]
rw [toList_node, List.length_append, List.length_cons, length_toList' l, length_toList' r]
[GOAL]
Ξ± : Type u_1
sizeβ : β
l : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
β’ realSize l + Nat.succ (realSize r) = realSize (node sizeβ l xβ r)
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
h : Sized t
β’ List.length (toList t) = size t
[PROOFSTEP]
rw [length_toList', size_eq_realSize h]
[GOAL]
Ξ± : Type u_1
tβ tβ : Ordnode Ξ±
hβ : Sized tβ
hβ : Sized tβ
h : toList tβ = toList tβ
β’ size tβ = size tβ
[PROOFSTEP]
rw [β length_toList hβ, h, length_toList hβ]
[GOAL]
Ξ± : Type u_1
instβΒΉ : LE Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
h : Sized t
h_mem : x β t
β’ 0 < size t
[PROOFSTEP]
cases t
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : LE Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
h : Sized nil
h_mem : x β nil
β’ 0 < size nil
[PROOFSTEP]
{contradiction
}
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : LE Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
h : Sized nil
h_mem : x β nil
β’ 0 < size nil
[PROOFSTEP]
contradiction
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : LE Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
h : Sized (node sizeβ lβ xβ rβ)
h_mem : x β node sizeβ lβ xβ rβ
β’ 0 < size (node sizeβ lβ xβ rβ)
[PROOFSTEP]
{simp [h.1]
}
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : LE Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
h : Sized (node sizeβ lβ xβ rβ)
h_mem : x β node sizeβ lβ xβ rβ
β’ 0 < size (node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [h.1]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
x : Ξ±
β’ findMax' x (dual t) = findMin' t x
[PROOFSTEP]
rw [β findMin'_dual, dual_dual]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
β’ findMax (dual t) = findMin t
[PROOFSTEP]
rw [β findMin_dual, dual_dual]
[GOAL]
Ξ± : Type u_1
sizeβ sz : β
l' : Ordnode Ξ±
y : Ξ±
r' : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ dual (eraseMin (node sizeβ (node sz l' y r') x r)) = eraseMax (dual (node sizeβ (node sz l' y r') x r))
[PROOFSTEP]
rw [eraseMin, dual_balanceR, dual_eraseMin (node sz l' y r'), dual, dual, dual, eraseMax]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
β’ dual (eraseMax t) = eraseMin (dual t)
[PROOFSTEP]
rw [β dual_dual (eraseMin _), dual_eraseMin, dual_dual]
[GOAL]
Ξ± : Type u_1
xβ ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
β’ splitMin' (node ls ll lx lr) x r = (findMin' (node ls ll lx lr) x, eraseMin (node xβ (node ls ll lx lr) x r))
[PROOFSTEP]
rw [splitMin', splitMin_eq ls ll lx lr, findMin', eraseMin]
[GOAL]
Ξ± : Type u_1
xβ : β
l : Ordnode Ξ±
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
β’ splitMax' l x (node ls ll lx lr) = (eraseMax (node xβ l x (node ls ll lx lr)), findMax' x (node ls ll lx lr))
[PROOFSTEP]
rw [splitMax', splitMax_eq ls ll lx lr, findMax', eraseMax]
[GOAL]
Ξ± : Type u_1
t : Ordnode Ξ±
β’ merge t nil = t
[PROOFSTEP]
cases t
[GOAL]
case nil
Ξ± : Type u_1
β’ merge nil nil = nil
[PROOFSTEP]
rfl
[GOAL]
case node
Ξ± : Type u_1
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
β’ merge (node sizeβ lβ xβ rβ) nil = node sizeβ lβ xβ rβ
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ dual (Ordnode.insert x (node sizeβ l y r)) = Ordnode.insert x (dual (node sizeβ l y r))
[PROOFSTEP]
have : @cmpLE Ξ±α΅α΅ _ _ x y = cmpLE y x := rfl
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
this : cmpLE x y = cmpLE y x
β’ dual (Ordnode.insert x (node sizeβ l y r)) = Ordnode.insert x (dual (node sizeβ l y r))
[PROOFSTEP]
rw [Ordnode.insert, dual, Ordnode.insert, this, β cmpLE_swap x y]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
this : cmpLE x y = cmpLE y x
β’ dual
(match cmpLE x y with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.swap (cmpLE x y) with
| Ordering.lt => balanceL (Ordnode.insert x (dual r)) y (dual l)
| Ordering.eq => node sizeβ (dual r) x (dual l)
| Ordering.gt => balanceR (dual r) y (Ordnode.insert x (dual l))
[PROOFSTEP]
cases cmpLE x y
[GOAL]
case lt
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
this : cmpLE x y = cmpLE y x
β’ dual
(match Ordering.lt with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.swap Ordering.lt with
| Ordering.lt => balanceL (Ordnode.insert x (dual r)) y (dual l)
| Ordering.eq => node sizeβ (dual r) x (dual l)
| Ordering.gt => balanceR (dual r) y (Ordnode.insert x (dual l))
[PROOFSTEP]
simp [Ordering.swap, Ordnode.insert, dual_balanceL, dual_balanceR, dual_insert]
[GOAL]
case eq
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
this : cmpLE x y = cmpLE y x
β’ dual
(match Ordering.eq with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.swap Ordering.eq with
| Ordering.lt => balanceL (Ordnode.insert x (dual r)) y (dual l)
| Ordering.eq => node sizeβ (dual r) x (dual l)
| Ordering.gt => balanceR (dual r) y (Ordnode.insert x (dual l))
[PROOFSTEP]
simp [Ordering.swap, Ordnode.insert, dual_balanceL, dual_balanceR, dual_insert]
[GOAL]
case gt
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
this : cmpLE x y = cmpLE y x
β’ dual
(match Ordering.gt with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.swap Ordering.gt with
| Ordering.lt => balanceL (Ordnode.insert x (dual r)) y (dual l)
| Ordering.eq => node sizeβ (dual r) x (dual l)
| Ordering.gt => balanceR (dual r) y (Ordnode.insert x (dual l))
[PROOFSTEP]
simp [Ordering.swap, Ordnode.insert, dual_balanceL, dual_balanceR, dual_insert]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
β’ balance l x r = balance' l x r
[PROOFSTEP]
cases' l with ls ll lx lr
[GOAL]
case nil
Ξ± : Type u_1
x : Ξ±
r : Ordnode Ξ±
hr : Balanced r
sr : Sized r
hl : Balanced nil
sl : Sized nil
β’ balance nil x r = balance' nil x r
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case nil.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
hr : Balanced nil
sr : Sized nil
β’ balance nil x nil = balance' nil x nil
[PROOFSTEP]
rfl
[GOAL]
case nil.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ balance nil x (node rs rl rx rr) = balance' nil x (node rs rl rx rr)
[PROOFSTEP]
rw [sr.eq_node'] at hr β’
[GOAL]
case nil.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node' rl rx rr)
sr : Sized (node rs rl rx rr)
β’ balance nil x (node' rl rx rr) = balance' nil x (node' rl rx rr)
[PROOFSTEP]
cases' rl with rls rll rlx rlr
[GOAL]
case nil.node.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node' nil rx rr)
sr : Sized (node rs nil rx rr)
β’ balance nil x (node' nil rx rr) = balance' nil x (node' nil rx rr)
[PROOFSTEP]
cases' rr with rrs rrl rrx rrr
[GOAL]
case nil.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rr : Ordnode Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx rr)
sr : Sized (node rs (node rls rll rlx rlr) rx rr)
β’ balance nil x (node' (node rls rll rlx rlr) rx rr) = balance' nil x (node' (node rls rll rlx rlr) rx rr)
[PROOFSTEP]
cases' rr with rrs rrl rrx rrr
[GOAL]
case nil.node.nil.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
hr : Balanced (node' nil rx nil)
sr : Sized (node rs nil rx nil)
β’ balance nil x (node' nil rx nil) = balance' nil x (node' nil rx nil)
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs rrl rrx rrr))
sr : Sized (node rs nil rx (node rrs rrl rrx rrr))
β’ balance nil x (node' nil rx (node rrs rrl rrx rrr)) = balance' nil x (node' nil rx (node rrs rrl rrx rrr))
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
β’ balance nil x (node' (node rls rll rlx rlr) rx nil) = balance' nil x (node' (node rls rll rlx rlr) rx nil)
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ balance nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
balance' nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case nil.node.nil.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
hr : Balanced (node' nil rx nil)
sr : Sized (node rs nil rx nil)
β’ node 2 nil x (node' nil rx nil) =
if 0 + (0 + 1) β€ 1 then node' nil x (node' nil rx nil)
else
if 0 + 1 > 0 then rotateL nil x (node' nil rx nil)
else if 0 > delta * (0 + 1) then rotateR nil x (node' nil rx nil) else node' nil x (node' nil rx nil)
[PROOFSTEP]
rfl
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs rrl rrx rrr))
sr : Sized (node rs nil rx (node rrs rrl rrx rrr))
β’ node 3 (Ordnode.singleton x) rx (node rrs rrl rrx rrr) =
if 0 + (0 + rrs + 1) β€ 1 then node' nil x (node' nil rx (node rrs rrl rrx rrr))
else
if 0 + rrs + 1 > 0 then rotateL nil x (node' nil rx (node rrs rrl rrx rrr))
else
if 0 > delta * (0 + rrs + 1) then rotateR nil x (node' nil rx (node rrs rrl rrx rrr))
else node' nil x (node' nil rx (node rrs rrl rrx rrr))
[PROOFSTEP]
have : size rrl = 0 β§ size rrr = 0 := by
have := balancedSz_zero.1 hr.1.symm
rwa [size, sr.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs rrl rrx rrr))
sr : Sized (node rs nil rx (node rrs rrl rrx rrr))
β’ size rrl = 0 β§ size rrr = 0
[PROOFSTEP]
have := balancedSz_zero.1 hr.1.symm
[GOAL]
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs rrl rrx rrr))
sr : Sized (node rs nil rx (node rrs rrl rrx rrr))
this : size (node rrs rrl rrx rrr) β€ 1
β’ size rrl = 0 β§ size rrr = 0
[PROOFSTEP]
rwa [size, sr.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
case nil.node.nil.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs rrl rrx rrr))
sr : Sized (node rs nil rx (node rrs rrl rrx rrr))
this : size rrl = 0 β§ size rrr = 0
β’ node 3 (Ordnode.singleton x) rx (node rrs rrl rrx rrr) =
if 0 + (0 + rrs + 1) β€ 1 then node' nil x (node' nil rx (node rrs rrl rrx rrr))
else
if 0 + rrs + 1 > 0 then rotateL nil x (node' nil rx (node rrs rrl rrx rrr))
else
if 0 > delta * (0 + rrs + 1) then rotateR nil x (node' nil rx (node rrs rrl rrx rrr))
else node' nil x (node' nil rx (node rrs rrl rrx rrr))
[PROOFSTEP]
cases sr.2.2.2.1.size_eq_zero.1 this.1
[GOAL]
case nil.node.nil.node.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' nil rx (node rrs nil rrx rrr))
sr : Sized (node rs nil rx (node rrs nil rrx rrr))
this : size nil = 0 β§ size rrr = 0
β’ node 3 (Ordnode.singleton x) rx (node rrs nil rrx rrr) =
if 0 + (0 + rrs + 1) β€ 1 then node' nil x (node' nil rx (node rrs nil rrx rrr))
else
if 0 + rrs + 1 > 0 then rotateL nil x (node' nil rx (node rrs nil rrx rrr))
else
if 0 > delta * (0 + rrs + 1) then rotateR nil x (node' nil rx (node rrs nil rrx rrr))
else node' nil x (node' nil rx (node rrs nil rrx rrr))
[PROOFSTEP]
cases sr.2.2.2.2.size_eq_zero.1 this.2
[GOAL]
case nil.node.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rrs : β
rrx : Ξ±
hr : Balanced (node' nil rx (node rrs nil rrx nil))
sr : Sized (node rs nil rx (node rrs nil rrx nil))
this : size nil = 0 β§ size nil = 0
β’ node 3 (Ordnode.singleton x) rx (node rrs nil rrx nil) =
if 0 + (0 + rrs + 1) β€ 1 then node' nil x (node' nil rx (node rrs nil rrx nil))
else
if 0 + rrs + 1 > 0 then rotateL nil x (node' nil rx (node rrs nil rrx nil))
else
if 0 > delta * (0 + rrs + 1) then rotateR nil x (node' nil rx (node rrs nil rrx nil))
else node' nil x (node' nil rx (node rrs nil rrx nil))
[PROOFSTEP]
obtain rfl : rrs = 1 := sr.2.2.1
[GOAL]
case nil.node.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ node 3 (Ordnode.singleton x) rx (node 1 nil rrx nil) =
if 0 + (0 + 1 + 1) β€ 1 then node' nil x (node' nil rx (node 1 nil rrx nil))
else
if 0 + 1 + 1 > 0 then rotateL nil x (node' nil rx (node 1 nil rrx nil))
else
if 0 > delta * (0 + 1 + 1) then rotateR nil x (node' nil rx (node 1 nil rrx nil))
else node' nil x (node' nil rx (node 1 nil rrx nil))
[PROOFSTEP]
rw [if_neg, if_pos, rotateL, if_pos]
[GOAL]
case nil.node.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ node 3 (Ordnode.singleton x) rx (node 1 nil rrx nil) = node3L nil x nil rx (node 1 nil rrx nil)
[PROOFSTEP]
rfl
[GOAL]
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ size nil < ratio * size (node 1 nil rrx nil)
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ 0 + 1 + 1 > 0
case nil.node.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ Β¬0 + (0 + 1 + 1) β€ 1
[PROOFSTEP]
all_goals dsimp only [size]; decide
[GOAL]
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ size nil < ratio * size (node 1 nil rrx nil)
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ 0 < ratio * 1
[PROOFSTEP]
decide
[GOAL]
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ 0 + 1 + 1 > 0
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ 0 + 1 + 1 > 0
[PROOFSTEP]
decide
[GOAL]
case nil.node.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ Β¬0 + (0 + 1 + 1) β€ 1
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rrx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' nil rx (node 1 nil rrx nil))
sr : Sized (node rs nil rx (node 1 nil rrx nil))
β’ Β¬0 + (0 + 1 + 1) β€ 1
[PROOFSTEP]
decide
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) =
if 0 + (rls + 1) β€ 1 then node' nil x (node' (node rls rll rlx rlr) rx nil)
else
if rls + 1 > 0 then rotateL nil x (node' (node rls rll rlx rlr) rx nil)
else
if 0 > delta * (rls + 1) then rotateR nil x (node' (node rls rll rlx rlr) rx nil)
else node' nil x (node' (node rls rll rlx rlr) rx nil)
[PROOFSTEP]
have : size rll = 0 β§ size rlr = 0 := by
have := balancedSz_zero.1 hr.1
rwa [size, sr.2.1.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
β’ size rll = 0 β§ size rlr = 0
[PROOFSTEP]
have := balancedSz_zero.1 hr.1
[GOAL]
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
this : size (node rls rll rlx rlr) β€ 1
β’ size rll = 0 β§ size rlr = 0
[PROOFSTEP]
rwa [size, sr.2.1.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
case nil.node.node.nil
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
this : size rll = 0 β§ size rlr = 0
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) =
if 0 + (rls + 1) β€ 1 then node' nil x (node' (node rls rll rlx rlr) rx nil)
else
if rls + 1 > 0 then rotateL nil x (node' (node rls rll rlx rlr) rx nil)
else
if 0 > delta * (rls + 1) then rotateR nil x (node' (node rls rll rlx rlr) rx nil)
else node' nil x (node' (node rls rll rlx rlr) rx nil)
[PROOFSTEP]
cases sr.2.1.2.1.size_eq_zero.1 this.1
[GOAL]
case nil.node.node.nil.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node' (node rls nil rlx rlr) rx nil)
sr : Sized (node rs (node rls nil rlx rlr) rx nil)
this : size nil = 0 β§ size rlr = 0
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) =
if 0 + (rls + 1) β€ 1 then node' nil x (node' (node rls nil rlx rlr) rx nil)
else
if rls + 1 > 0 then rotateL nil x (node' (node rls nil rlx rlr) rx nil)
else
if 0 > delta * (rls + 1) then rotateR nil x (node' (node rls nil rlx rlr) rx nil)
else node' nil x (node' (node rls nil rlx rlr) rx nil)
[PROOFSTEP]
cases sr.2.1.2.2.size_eq_zero.1 this.2
[GOAL]
case nil.node.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rlx : Ξ±
hr : Balanced (node' (node rls nil rlx nil) rx nil)
sr : Sized (node rs (node rls nil rlx nil) rx nil)
this : size nil = 0 β§ size nil = 0
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) =
if 0 + (rls + 1) β€ 1 then node' nil x (node' (node rls nil rlx nil) rx nil)
else
if rls + 1 > 0 then rotateL nil x (node' (node rls nil rlx nil) rx nil)
else
if 0 > delta * (rls + 1) then rotateR nil x (node' (node rls nil rlx nil) rx nil)
else node' nil x (node' (node rls nil rlx nil) rx nil)
[PROOFSTEP]
obtain rfl : rls = 1 := sr.2.1.1
[GOAL]
case nil.node.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) =
if 0 + (1 + 1) β€ 1 then node' nil x (node' (node 1 nil rlx nil) rx nil)
else
if 1 + 1 > 0 then rotateL nil x (node' (node 1 nil rlx nil) rx nil)
else
if 0 > delta * (1 + 1) then rotateR nil x (node' (node 1 nil rlx nil) rx nil)
else node' nil x (node' (node 1 nil rlx nil) rx nil)
[PROOFSTEP]
rw [if_neg, if_pos, rotateL, if_neg]
[GOAL]
case nil.node.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx) = node4L nil x (node 1 nil rlx nil) rx nil
[PROOFSTEP]
rfl
[GOAL]
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬size (node 1 nil rlx nil) < ratio * size nil
case nil.node.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ 1 + 1 > 0
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬0 + (1 + 1) β€ 1
[PROOFSTEP]
all_goals dsimp only [size]; decide
[GOAL]
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬size (node 1 nil rlx nil) < ratio * size nil
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬1 < ratio * 0
[PROOFSTEP]
decide
[GOAL]
case nil.node.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ 1 + 1 > 0
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ 1 + 1 > 0
[PROOFSTEP]
decide
[GOAL]
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬0 + (1 + 1) β€ 1
[PROOFSTEP]
dsimp only [size]
[GOAL]
case nil.node.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx rlx : Ξ±
this : size nil = 0 β§ size nil = 0
hr : Balanced (node' (node 1 nil rlx nil) rx nil)
sr : Sized (node rs (node 1 nil rlx nil) rx nil)
β’ Β¬0 + (1 + 1) β€ 1
[PROOFSTEP]
decide
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ (if rls < ratio * rrs then
node (rls + rrs + 1 + 1) (node (rls + 1) nil x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
else
node (rls + rrs + 1 + 1) (node (size rll + 1) nil x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))) =
if 0 + (rls + rrs + 1) β€ 1 then node' nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else
if rls + rrs + 1 > 0 then rotateL nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else
if 0 > delta * (rls + rrs + 1) then rotateR nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else node' nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
[PROOFSTEP]
symm
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ (if 0 + (rls + rrs + 1) β€ 1 then node' nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else
if rls + rrs + 1 > 0 then rotateL nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else
if 0 > delta * (rls + rrs + 1) then rotateR nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
else node' nil x (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))) =
if rls < ratio * rrs then
node (rls + rrs + 1 + 1) (node (rls + 1) nil x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
else
node (rls + rrs + 1 + 1) (node (size rll + 1) nil x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
rw [zero_add, if_neg, if_pos, rotateL]
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ (if size (node rls rll rlx rlr) < ratio * size (node rrs rrl rrx rrr) then
node3L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)
else node4L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
if rls < ratio * rrs then
node (rls + rrs + 1 + 1) (node (rls + 1) nil x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
else
node (rls + rrs + 1 + 1) (node (size rll + 1) nil x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
dsimp only [size_node]
[GOAL]
case nil.node.node.node
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ (if rls < ratio * rrs then node3L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)
else node4L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
if rls < ratio * rrs then
node (rls + rrs + 1 + 1) (node (rls + 1) nil x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
else
node (rls + rrs + 1 + 1) (node (size rll + 1) nil x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : rls < ratio * rrs
β’ node3L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr) =
node (rls + rrs + 1 + 1) (node (rls + 1) nil x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
[PROOFSTEP]
simp [node3L, node']
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : rls < ratio * rrs
β’ rls + 1 + rrs = rls + rrs + 1
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : rls < ratio * rrs
β’ rls + 1 + rrs = rls + rrs + 1
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : Β¬rls < ratio * rrs
β’ node4L nil x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr) =
node (rls + rrs + 1 + 1) (node (size rll + 1) nil x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
simp [node4L, node', sr.2.1.1]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : Β¬rls < ratio * rrs
β’ size rll + 1 + (size rlr + rrs + 1) = size rll + size rlr + 1 + rrs + 1
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
hβ : Β¬rls < ratio * rrs
β’ size rll + 1 + (size rlr + rrs + 1) = size rll + size rlr + 1 + rrs + 1
[PROOFSTEP]
abel
[GOAL]
case nil.node.node.node.hc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ rls + rrs + 1 > 0
[PROOFSTEP]
apply Nat.zero_lt_succ
[GOAL]
case nil.node.node.node.hnc
Ξ± : Type u_1
x : Ξ±
hl : Balanced nil
sl : Sized nil
rs : β
rx : Ξ±
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node' (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
β’ Β¬rls + rrs + 1 β€ 1
[PROOFSTEP]
exact not_le_of_gt (Nat.succ_lt_succ (add_pos sr.2.1.pos sr.2.2.pos))
[GOAL]
case node
Ξ± : Type u_1
x : Ξ±
r : Ordnode Ξ±
hr : Balanced r
sr : Sized r
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
β’ balance (node ls ll lx lr) x r = balance' (node ls ll lx lr) x r
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
hr : Balanced nil
sr : Sized nil
β’ balance (node ls ll lx lr) x nil = balance' (node ls ll lx lr) x nil
[PROOFSTEP]
rw [sl.eq_node'] at hl β’
[GOAL]
case node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node' ll lx lr)
sl : Sized (node ls ll lx lr)
hr : Balanced nil
sr : Sized nil
β’ balance (node' ll lx lr) x nil = balance' (node' ll lx lr) x nil
[PROOFSTEP]
cases' ll with lls lll llx llr
[GOAL]
case node.nil.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
hr : Balanced nil
sr : Sized nil
hl : Balanced (node' nil lx lr)
sl : Sized (node ls nil lx lr)
β’ balance (node' nil lx lr) x nil = balance' (node' nil lx lr) x nil
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case node.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx lr)
sl : Sized (node ls (node lls lll llx llr) lx lr)
β’ balance (node' (node lls lll llx llr) lx lr) x nil = balance' (node' (node lls lll llx llr) lx lr) x nil
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case node.nil.nil.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
hl : Balanced (node' nil lx nil)
sl : Sized (node ls nil lx nil)
β’ balance (node' nil lx nil) x nil = balance' (node' nil lx nil) x nil
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case node.nil.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
β’ balance (node' nil lx (node lrs lrl lrx lrr)) x nil = balance' (node' nil lx (node lrs lrl lrx lrr)) x nil
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case node.nil.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
β’ balance (node' (node lls lll llx llr) lx nil) x nil = balance' (node' (node lls lll llx llr) lx nil) x nil
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case node.nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ balance (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil =
balance' (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
[PROOFSTEP]
dsimp [balance, balance']
[GOAL]
case node.nil.nil.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
hl : Balanced (node' nil lx nil)
sl : Sized (node ls nil lx nil)
β’ node 2 (node' nil lx nil) x nil =
if 1 β€ 1 then node' (node' nil lx nil) x nil
else
if 0 > delta * (0 + 1) then rotateL (node' nil lx nil) x nil
else if 0 + 1 > 0 then rotateR (node' nil lx nil) x nil else node' (node' nil lx nil) x nil
[PROOFSTEP]
rfl
[GOAL]
case node.nil.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
if 0 + lrs + 1 β€ 1 then node' (node' nil lx (node lrs lrl lrx lrr)) x nil
else
if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs lrl lrx lrr)) x nil
else
if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs lrl lrx lrr)) x nil
else node' (node' nil lx (node lrs lrl lrx lrr)) x nil
[PROOFSTEP]
have : size lrl = 0 β§ size lrr = 0 := by
have := balancedSz_zero.1 hl.1.symm
rwa [size, sl.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
β’ size lrl = 0 β§ size lrr = 0
[PROOFSTEP]
have := balancedSz_zero.1 hl.1.symm
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
this : size (node lrs lrl lrx lrr) β€ 1
β’ size lrl = 0 β§ size lrr = 0
[PROOFSTEP]
rwa [size, sl.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
case node.nil.nil.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
this : size lrl = 0 β§ size lrr = 0
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
if 0 + lrs + 1 β€ 1 then node' (node' nil lx (node lrs lrl lrx lrr)) x nil
else
if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs lrl lrx lrr)) x nil
else
if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs lrl lrx lrr)) x nil
else node' (node' nil lx (node lrs lrl lrx lrr)) x nil
[PROOFSTEP]
cases sl.2.2.2.1.size_eq_zero.1 this.1
[GOAL]
case node.nil.nil.node.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' nil lx (node lrs nil lrx lrr))
sl : Sized (node ls nil lx (node lrs nil lrx lrr))
this : size nil = 0 β§ size lrr = 0
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
if 0 + lrs + 1 β€ 1 then node' (node' nil lx (node lrs nil lrx lrr)) x nil
else
if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs nil lrx lrr)) x nil
else
if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs nil lrx lrr)) x nil
else node' (node' nil lx (node lrs nil lrx lrr)) x nil
[PROOFSTEP]
cases sl.2.2.2.2.size_eq_zero.1 this.2
[GOAL]
case node.nil.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrs : β
lrx : Ξ±
hl : Balanced (node' nil lx (node lrs nil lrx nil))
sl : Sized (node ls nil lx (node lrs nil lrx nil))
this : size nil = 0 β§ size nil = 0
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
if 0 + lrs + 1 β€ 1 then node' (node' nil lx (node lrs nil lrx nil)) x nil
else
if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs nil lrx nil)) x nil
else
if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs nil lrx nil)) x nil
else node' (node' nil lx (node lrs nil lrx nil)) x nil
[PROOFSTEP]
obtain rfl : lrs = 1 := sl.2.2.1
[GOAL]
case node.nil.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
if 0 + 1 + 1 β€ 1 then node' (node' nil lx (node 1 nil lrx nil)) x nil
else
if 0 > delta * (0 + 1 + 1) then rotateL (node' nil lx (node 1 nil lrx nil)) x nil
else
if 0 + 1 + 1 > 0 then rotateR (node' nil lx (node 1 nil lrx nil)) x nil
else node' (node' nil lx (node 1 nil lrx nil)) x nil
[PROOFSTEP]
rw [if_neg, if_neg, if_pos, rotateR, if_neg]
[GOAL]
case node.nil.nil.node.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) = node4R nil lx (node 1 nil lrx nil) x nil
[PROOFSTEP]
rfl
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬size (node 1 nil lrx nil) < ratio * size nil
case node.nil.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ 0 + 1 + 1 > 0
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 > delta * (0 + 1 + 1)
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 + 1 + 1 β€ 1
[PROOFSTEP]
all_goals dsimp only [size]; decide
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬size (node 1 nil lrx nil) < ratio * size nil
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬1 < ratio * 0
[PROOFSTEP]
decide
[GOAL]
case node.nil.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ 0 + 1 + 1 > 0
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.nil.node.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ 0 + 1 + 1 > 0
[PROOFSTEP]
decide
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 > delta * (0 + 1 + 1)
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 > delta * (0 + 1 + 1)
[PROOFSTEP]
decide
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 + 1 + 1 β€ 1
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.nil.node.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lrx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
β’ Β¬0 + 1 + 1 β€ 1
[PROOFSTEP]
decide
[GOAL]
case node.nil.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
β’ node 3 (node lls lll llx llr) lx (Ordnode.singleton x) =
if lls + 1 β€ 1 then node' (node' (node lls lll llx llr) lx nil) x nil
else
if 0 > delta * (lls + 1) then rotateL (node' (node lls lll llx llr) lx nil) x nil
else
if lls + 1 > 0 then rotateR (node' (node lls lll llx llr) lx nil) x nil
else node' (node' (node lls lll llx llr) lx nil) x nil
[PROOFSTEP]
have : size lll = 0 β§ size llr = 0 := by
have := balancedSz_zero.1 hl.1
rwa [size, sl.2.1.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
β’ size lll = 0 β§ size llr = 0
[PROOFSTEP]
have := balancedSz_zero.1 hl.1
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
this : size (node lls lll llx llr) β€ 1
β’ size lll = 0 β§ size llr = 0
[PROOFSTEP]
rwa [size, sl.2.1.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
case node.nil.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
this : size lll = 0 β§ size llr = 0
β’ node 3 (node lls lll llx llr) lx (Ordnode.singleton x) =
if lls + 1 β€ 1 then node' (node' (node lls lll llx llr) lx nil) x nil
else
if 0 > delta * (lls + 1) then rotateL (node' (node lls lll llx llr) lx nil) x nil
else
if lls + 1 > 0 then rotateR (node' (node lls lll llx llr) lx nil) x nil
else node' (node' (node lls lll llx llr) lx nil) x nil
[PROOFSTEP]
cases sl.2.1.2.1.size_eq_zero.1 this.1
[GOAL]
case node.nil.node.nil.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node' (node lls nil llx llr) lx nil)
sl : Sized (node ls (node lls nil llx llr) lx nil)
this : size nil = 0 β§ size llr = 0
β’ node 3 (node lls nil llx llr) lx (Ordnode.singleton x) =
if lls + 1 β€ 1 then node' (node' (node lls nil llx llr) lx nil) x nil
else
if 0 > delta * (lls + 1) then rotateL (node' (node lls nil llx llr) lx nil) x nil
else
if lls + 1 > 0 then rotateR (node' (node lls nil llx llr) lx nil) x nil
else node' (node' (node lls nil llx llr) lx nil) x nil
[PROOFSTEP]
cases sl.2.1.2.2.size_eq_zero.1 this.2
[GOAL]
case node.nil.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
llx : Ξ±
hl : Balanced (node' (node lls nil llx nil) lx nil)
sl : Sized (node ls (node lls nil llx nil) lx nil)
this : size nil = 0 β§ size nil = 0
β’ node 3 (node lls nil llx nil) lx (Ordnode.singleton x) =
if lls + 1 β€ 1 then node' (node' (node lls nil llx nil) lx nil) x nil
else
if 0 > delta * (lls + 1) then rotateL (node' (node lls nil llx nil) lx nil) x nil
else
if lls + 1 > 0 then rotateR (node' (node lls nil llx nil) lx nil) x nil
else node' (node' (node lls nil llx nil) lx nil) x nil
[PROOFSTEP]
obtain rfl : lls = 1 := sl.2.1.1
[GOAL]
case node.nil.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ node 3 (node 1 nil llx nil) lx (Ordnode.singleton x) =
if 1 + 1 β€ 1 then node' (node' (node 1 nil llx nil) lx nil) x nil
else
if 0 > delta * (1 + 1) then rotateL (node' (node 1 nil llx nil) lx nil) x nil
else
if 1 + 1 > 0 then rotateR (node' (node 1 nil llx nil) lx nil) x nil
else node' (node' (node 1 nil llx nil) lx nil) x nil
[PROOFSTEP]
rw [if_neg, if_neg, if_pos, rotateR, if_pos]
[GOAL]
case node.nil.node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ node 3 (node 1 nil llx nil) lx (Ordnode.singleton x) = node3R (node 1 nil llx nil) lx nil x nil
[PROOFSTEP]
rfl
[GOAL]
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ size nil < ratio * size (node 1 nil llx nil)
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ 1 + 1 > 0
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬0 > delta * (1 + 1)
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬1 + 1 β€ 1
[PROOFSTEP]
all_goals dsimp only [size]; decide
[GOAL]
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ size nil < ratio * size (node 1 nil llx nil)
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ 0 < ratio * 1
[PROOFSTEP]
decide
[GOAL]
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ 1 + 1 > 0
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.node.nil.refl.refl.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ 1 + 1 > 0
[PROOFSTEP]
decide
[GOAL]
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬0 > delta * (1 + 1)
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬0 > delta * (1 + 1)
[PROOFSTEP]
decide
[GOAL]
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬1 + 1 β€ 1
[PROOFSTEP]
dsimp only [size]
[GOAL]
case node.nil.node.nil.refl.refl.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
llx : Ξ±
this : size nil = 0 β§ size nil = 0
hl : Balanced (node' (node 1 nil llx nil) lx nil)
sl : Sized (node ls (node 1 nil llx nil) lx nil)
β’ Β¬1 + 1 β€ 1
[PROOFSTEP]
decide
[GOAL]
case node.nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ (if lrs < ratio * lls then
node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (lls + lrs + 1 + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + 1) lrr x nil)) =
if lls + lrs + 1 β€ 1 then node' (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else
if 0 > delta * (lls + lrs + 1) then rotateL (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else
if lls + lrs + 1 > 0 then rotateR (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else node' (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
[PROOFSTEP]
symm
[GOAL]
case node.nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ (if lls + lrs + 1 β€ 1 then node' (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else
if 0 > delta * (lls + lrs + 1) then rotateL (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else
if lls + lrs + 1 > 0 then rotateR (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil
else node' (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x nil) =
if lrs < ratio * lls then
node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (lls + lrs + 1 + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + 1) lrr x nil)
[PROOFSTEP]
rw [if_neg, if_neg, if_pos, rotateR]
[GOAL]
case node.nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ (if size (node lrs lrl lrx lrr) < ratio * size (node lls lll llx llr) then
node3R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil
else node4R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil) =
if lrs < ratio * lls then
node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (lls + lrs + 1 + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + 1) lrr x nil)
[PROOFSTEP]
dsimp only [size_node]
[GOAL]
case node.nil.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ (if lrs < ratio * lls then node3R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil
else node4R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil) =
if lrs < ratio * lls then
node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (lls + lrs + 1 + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + 1) lrr x nil)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : lrs < ratio * lls
β’ node3R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil =
node (lls + lrs + 1 + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
[PROOFSTEP]
simp [node3R, node']
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : lrs < ratio * lls
β’ lls + (lrs + 1) = lls + lrs + 1
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : lrs < ratio * lls
β’ lls + (lrs + 1) = lls + lrs + 1
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : Β¬lrs < ratio * lls
β’ node4R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x nil =
node (lls + lrs + 1 + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + 1) lrr x nil)
[PROOFSTEP]
simp [node4R, node', sl.2.2.1]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : Β¬lrs < ratio * lls
β’ lls + size lrl + 1 + (size lrr + 1) = lls + (size lrl + size lrr + 1) + 1
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
hβ : Β¬lrs < ratio * lls
β’ lls + size lrl + 1 + (size lrr + 1) = lls + (size lrl + size lrr + 1) + 1
[PROOFSTEP]
abel
[GOAL]
case node.nil.node.node.hc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ lls + lrs + 1 > 0
[PROOFSTEP]
apply Nat.zero_lt_succ
[GOAL]
case node.nil.node.node.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ Β¬0 > delta * (lls + lrs + 1)
[PROOFSTEP]
apply Nat.not_lt_zero
[GOAL]
case node.nil.node.node.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
hr : Balanced nil
sr : Sized nil
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node' (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
β’ Β¬lls + lrs + 1 β€ 1
[PROOFSTEP]
exact not_le_of_gt (Nat.succ_lt_succ (add_pos sl.2.1.pos sl.2.2.pos))
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ balance (node ls ll lx lr) x (node rs rl rx rr) = balance' (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
simp [balance, balance']
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ (if delta * ls < rs then
rec (motive := fun t => rl = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x rl) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
rl (_ : id rl = id rl)
else
if delta * rs < ls then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)) =
if ls + rs β€ 1 then node' (node ls ll lx lr) x (node rs rl rx rr)
else
if delta * ls < rs then rotateL (node ls ll lx lr) x (node rs rl rx rr)
else
if delta * rs < ls then rotateR (node ls ll lx lr) x (node rs rl rx rr)
else node' (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
symm
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ (if ls + rs β€ 1 then node' (node ls ll lx lr) x (node rs rl rx rr)
else
if delta * ls < rs then rotateL (node ls ll lx lr) x (node rs rl rx rr)
else
if delta * rs < ls then rotateR (node ls ll lx lr) x (node rs rl rx rr)
else node' (node ls ll lx lr) x (node rs rl rx rr)) =
if delta * ls < rs then
rec (motive := fun t => rl = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x rl) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
rl (_ : id rl = id rl)
else
if delta * rs < ls then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
rw [if_neg]
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ (if delta * ls < rs then rotateL (node ls ll lx lr) x (node rs rl rx rr)
else
if delta * rs < ls then rotateR (node ls ll lx lr) x (node rs rl rx rr)
else node' (node ls ll lx lr) x (node rs rl rx rr)) =
if delta * ls < rs then
rec (motive := fun t => rl = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x rl) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
rl (_ : id rl = id rl)
else
if delta * rs < ls then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
split_ifs with h h_1
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : delta * ls < rs
β’ rotateL (node ls ll lx lr) x (node rs rl rx rr) =
rec (motive := fun t => rl = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x rl) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
rl (_ : id rl = id rl)
[PROOFSTEP]
have rd : delta β€ size rl + size rr :=
by
have := lt_of_le_of_lt (Nat.mul_le_mul_left _ sl.pos) h
rwa [sr.1, Nat.lt_succ_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : delta * ls < rs
β’ delta β€ size rl + size rr
[PROOFSTEP]
have := lt_of_le_of_lt (Nat.mul_le_mul_left _ sl.pos) h
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : delta * ls < rs
this : delta * Nat.succ 0 < rs
β’ delta β€ size rl + size rr
[PROOFSTEP]
rwa [sr.1, Nat.lt_succ_iff] at this
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : delta * ls < rs
rd : delta β€ size rl + size rr
β’ rotateL (node ls ll lx lr) x (node rs rl rx rr) =
rec (motive := fun t => rl = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x rl) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
rl (_ : id rl = id rl)
[PROOFSTEP]
cases' rl with rls rll rlx rlr
[GOAL]
case pos.nil
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
rr : Ordnode Ξ±
h : delta * ls < rs
hr : Balanced (node rs nil rx rr)
sr : Sized (node rs nil rx rr)
rd : delta β€ size nil + size rr
β’ rotateL (node ls ll lx lr) x (node rs nil rx rr) =
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x nil) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
nil (_ : id nil = id nil)
[PROOFSTEP]
rw [size, zero_add] at rd
[GOAL]
case pos.nil
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
rr : Ordnode Ξ±
h : delta * ls < rs
hr : Balanced (node rs nil rx rr)
sr : Sized (node rs nil rx rr)
rd : delta β€ size rr
β’ rotateL (node ls ll lx lr) x (node rs nil rx rr) =
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x nil) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
nil (_ : id nil = id nil)
[PROOFSTEP]
exact absurd (le_trans rd (balancedSz_zero.1 hr.1.symm)) (by decide)
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
rr : Ordnode Ξ±
h : delta * ls < rs
hr : Balanced (node rs nil rx rr)
sr : Sized (node rs nil rx rr)
rd : delta β€ size rr
β’ Β¬delta β€ 1
[PROOFSTEP]
decide
[GOAL]
case pos.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
rr : Ordnode Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx rr)
sr : Sized (node rs (node rls rll rlx rlr) rx rr)
rd : delta β€ size (node rls rll rlx rlr) + size rr
β’ rotateL (node ls ll lx lr) x (node rs (node rls rll rlx rlr) rx rr) =
rec (motive := fun t => node rls rll rlx rlr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => rr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then
node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx rr
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx rr))
rr (_ : id rr = id rr))
(node rls rll rlx rlr) (_ : id (node rls rll rlx rlr) = id (node rls rll rlx rlr))
[PROOFSTEP]
cases' rr with rrs rrl rrx rrr
[GOAL]
case pos.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
rd : delta β€ size (node rls rll rlx rlr) + size nil
β’ rotateL (node ls ll lx lr) x (node rs (node rls rll rlx rlr) rx nil) =
rec (motive := fun t => node rls rll rlx rlr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then
node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx nil
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx nil))
nil (_ : id nil = id nil))
(node rls rll rlx rlr) (_ : id (node rls rll rlx rlr) = id (node rls rll rlx rlr))
[PROOFSTEP]
exact absurd (le_trans rd (balancedSz_zero.1 hr.1)) (by decide)
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx nil)
sr : Sized (node rs (node rls rll rlx rlr) rx nil)
rd : delta β€ size (node rls rll rlx rlr) + size nil
β’ Β¬delta β€ 1
[PROOFSTEP]
decide
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
β’ rotateL (node ls ll lx lr) x (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
rec (motive := fun t => node rls rll rlx rlr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => node rrs rrl rrx rrr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l_1 x_2 r_1 l_ih r_ih h =>
if size < ratio * size_1 then
node (ls + rs + 1) (node (ls + size + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx
(node rrs rrl rrx rrr)
else
node (ls + rs + 1) (node (ls + Ordnode.size l + 1) (node ls ll lx lr) x l) x_1
(node (Ordnode.size r + size_1 + 1) r rx (node rrs rrl rrx rrr)))
(node rrs rrl rrx rrr) (_ : id (node rrs rrl rrx rrr) = id (node rrs rrl rrx rrr)))
(node rls rll rlx rlr) (_ : id (node rls rll rlx rlr) = id (node rls rll rlx rlr))
[PROOFSTEP]
dsimp [rotateL]
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
β’ (if rls < ratio * rrs then node3L (node ls ll lx lr) x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)
else node4L (node ls ll lx lr) x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)) =
if rls < ratio * rrs then
node (ls + rs + 1) (node (ls + rls + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
else
node (ls + rs + 1) (node (ls + size rll + 1) (node ls ll lx lr) x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : rls < ratio * rrs
β’ node3L (node ls ll lx lr) x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr) =
node (ls + rs + 1) (node (ls + rls + 1) (node ls ll lx lr) x (node rls rll rlx rlr)) rx (node rrs rrl rrx rrr)
[PROOFSTEP]
simp [node3L, node', sr.1]
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : rls < ratio * rrs
β’ ls + rls + 1 + rrs = ls + (rls + rrs + 1)
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : rls < ratio * rrs
β’ ls + rls + 1 + rrs = ls + (rls + rrs + 1)
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : Β¬rls < ratio * rrs
β’ node4L (node ls ll lx lr) x (node rls rll rlx rlr) rx (node rrs rrl rrx rrr) =
node (ls + rs + 1) (node (ls + size rll + 1) (node ls ll lx lr) x rll) rlx
(node (size rlr + rrs + 1) rlr rx (node rrs rrl rrx rrr))
[PROOFSTEP]
simp [node4L, node', sr.1, sr.2.1.1]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : Β¬rls < ratio * rrs
β’ ls + size rll + 1 + (size rlr + rrs + 1) = ls + (size rll + size rlr + 1 + rrs + 1)
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rx : Ξ±
h : delta * ls < rs
rls : β
rll : Ordnode Ξ±
rlx : Ξ±
rlr : Ordnode Ξ±
rrs : β
rrl : Ordnode Ξ±
rrx : Ξ±
rrr : Ordnode Ξ±
hr : Balanced (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
sr : Sized (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr))
rd : delta β€ size (node rls rll rlx rlr) + size (node rrs rrl rrx rrr)
hβ : Β¬rls < ratio * rrs
β’ ls + size rll + 1 + (size rlr + rrs + 1) = ls + (size rll + size rlr + 1 + rrs + 1)
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
β’ rotateR (node ls ll lx lr) x (node rs rl rx rr) =
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
[PROOFSTEP]
have ld : delta β€ size ll + size lr :=
by
have := lt_of_le_of_lt (Nat.mul_le_mul_left _ sr.pos) h_1
rwa [sl.1, Nat.lt_succ_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
β’ delta β€ size ll + size lr
[PROOFSTEP]
have := lt_of_le_of_lt (Nat.mul_le_mul_left _ sr.pos) h_1
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
this : delta * Nat.succ 0 < ls
β’ delta β€ size ll + size lr
[PROOFSTEP]
rwa [sl.1, Nat.lt_succ_iff] at this
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
ld : delta β€ size ll + size lr
β’ rotateR (node ls ll lx lr) x (node rs rl rx rr) =
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
[PROOFSTEP]
cases' ll with lls lll llx llr
[GOAL]
case pos.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
hl : Balanced (node ls nil lx lr)
sl : Sized (node ls nil lx lr)
ld : delta β€ size nil + size lr
β’ rotateR (node ls nil lx lr) x (node rs rl rx rr) =
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) nil lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
nil (_ : id nil = id nil)
[PROOFSTEP]
rw [size, zero_add] at ld
[GOAL]
case pos.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
hl : Balanced (node ls nil lx lr)
sl : Sized (node ls nil lx lr)
ld : delta β€ size lr
β’ rotateR (node ls nil lx lr) x (node rs rl rx rr) =
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) nil lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) nil lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
nil (_ : id nil = id nil)
[PROOFSTEP]
exact absurd (le_trans ld (balancedSz_zero.1 hl.1.symm)) (by decide)
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
hl : Balanced (node ls nil lx lr)
sl : Sized (node ls nil lx lr)
ld : delta β€ size lr
β’ Β¬delta β€ 1
[PROOFSTEP]
decide
[GOAL]
case pos.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx lr)
sl : Sized (node ls (node lls lll llx llr) lx lr)
ld : delta β€ size (node lls lll llx llr) + size lr
β’ rotateR (node ls (node lls lll llx llr) lx lr) x (node rs rl rx rr) =
rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))
[PROOFSTEP]
cases' lr with lrs lrl lrx lrr
[GOAL]
case pos.node.nil
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
ld : delta β€ size (node lls lll llx llr) + size nil
β’ rotateR (node ls (node lls lll llx llr) lx nil) x (node rs rl rx rr) =
rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => nil = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (size_1 + rs + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
nil (_ : id nil = id nil))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))
[PROOFSTEP]
exact absurd (le_trans ld (balancedSz_zero.1 hl.1)) (by decide)
[GOAL]
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx nil)
sl : Sized (node ls (node lls lll llx llr) lx nil)
ld : delta β€ size (node lls lll llx llr) + size nil
β’ Β¬delta β€ 1
[PROOFSTEP]
decide
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
β’ rotateR (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr)) x (node rs rl rx rr) =
rec (motive := fun t => node lls lll llx llr = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => node lrs lrl lrx lrr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (size_1 + rs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) (node lls lll llx llr) lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) (_ : id (node lrs lrl lrx lrr) = id (node lrs lrl lrx lrr)))
(node lls lll llx llr) (_ : id (node lls lll llx llr) = id (node lls lll llx llr))
[PROOFSTEP]
dsimp [rotateR]
[GOAL]
case pos.node.node
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
β’ (if lrs < ratio * lls then node3R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x (node rs rl rx rr)
else node4R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x (node rs rl rx rr)) =
if lrs < ratio * lls then
node (ls + rs + 1) (node lls lll llx llr) lx (node (lrs + rs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + rs + 1) lrr x (node rs rl rx rr))
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : lrs < ratio * lls
β’ node3R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x (node rs rl rx rr) =
node (ls + rs + 1) (node lls lll llx llr) lx (node (lrs + rs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
[PROOFSTEP]
simp [node3R, node', sl.1]
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : lrs < ratio * lls
β’ lls + (lrs + rs + 1) = lls + lrs + 1 + rs
[PROOFSTEP]
abel
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : lrs < ratio * lls
β’ lls + (lrs + rs + 1) = lls + lrs + 1 + rs
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : Β¬lrs < ratio * lls
β’ node4R (node lls lll llx llr) lx (node lrs lrl lrx lrr) x (node rs rl rx rr) =
node (ls + rs + 1) (node (lls + size lrl + 1) (node lls lll llx llr) lx lrl) lrx
(node (size lrr + rs + 1) lrr x (node rs rl rx rr))
[PROOFSTEP]
simp [node4R, node', sl.1, sl.2.2.1]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : Β¬lrs < ratio * lls
β’ lls + size lrl + 1 + (size lrr + rs + 1) = lls + (size lrl + size lrr + 1) + 1 + rs
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
lx : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
lls : β
lll : Ordnode Ξ±
llx : Ξ±
llr : Ordnode Ξ±
lrs : β
lrl : Ordnode Ξ±
lrx : Ξ±
lrr : Ordnode Ξ±
hl : Balanced (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
sl : Sized (node ls (node lls lll llx llr) lx (node lrs lrl lrx lrr))
ld : delta β€ size (node lls lll llx llr) + size (node lrs lrl lrx lrr)
hβ : Β¬lrs < ratio * lls
β’ lls + size lrl + 1 + (size lrr + rs + 1) = lls + (size lrl + size lrr + 1) + 1 + rs
[PROOFSTEP]
abel
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
h : Β¬delta * ls < rs
h_1 : Β¬delta * rs < ls
β’ node' (node ls ll lx lr) x (node rs rl rx rr) = node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
simp [node']
[GOAL]
case node.node.hnc
Ξ± : Type u_1
x : Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Balanced (node ls ll lx lr)
sl : Sized (node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Balanced (node rs rl rx rr)
sr : Sized (node rs rl rx rr)
β’ Β¬ls + rs β€ 1
[PROOFSTEP]
exact not_le_of_gt (add_le_add (Nat.succ_le_of_lt sl.pos) (Nat.succ_le_of_lt sr.pos))
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
sl : Sized l
sr : Sized r
H1 : size l = 0 β size r β€ 1
H2 : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
β’ balanceL l x r = balance l x r
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case nil
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sl : Sized l
sr : Sized nil
H1 : size l = 0 β size nil β€ 1
H2 : 1 β€ size l β 1 β€ size nil β size nil β€ delta * size l
β’ balanceL l x nil = balance l x nil
[PROOFSTEP]
rfl
[GOAL]
case node
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
sl : Sized l
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
H1 : size l = 0 β size (node rs rl rx rr) β€ 1
H2 : 1 β€ size l β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size l
β’ balanceL l x (node rs rl rx rr) = balance l x (node rs rl rx rr)
[PROOFSTEP]
cases' l with ls ll lx lr
[GOAL]
case node.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
sl : Sized nil
H1 : size nil = 0 β size (node rs rl rx rr) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size nil
β’ balanceL nil x (node rs rl rx rr) = balance nil x (node rs rl rx rr)
[PROOFSTEP]
have : size rl = 0 β§ size rr = 0 := by
have := H1 rfl
rwa [size, sr.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
sl : Sized nil
H1 : size nil = 0 β size (node rs rl rx rr) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size nil
β’ size rl = 0 β§ size rr = 0
[PROOFSTEP]
have := H1 rfl
[GOAL]
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
sl : Sized nil
H1 : size nil = 0 β size (node rs rl rx rr) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size nil
this : size (node rs rl rx rr) β€ 1
β’ size rl = 0 β§ size rr = 0
[PROOFSTEP]
rwa [size, sr.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
[GOAL]
case node.nil
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
sl : Sized nil
H1 : size nil = 0 β size (node rs rl rx rr) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size nil
this : size rl = 0 β§ size rr = 0
β’ balanceL nil x (node rs rl rx rr) = balance nil x (node rs rl rx rr)
[PROOFSTEP]
cases sr.2.1.size_eq_zero.1 this.1
[GOAL]
case node.nil.refl
Ξ± : Type u_1
x : Ξ±
rs : β
rx : Ξ±
rr : Ordnode Ξ±
sl : Sized nil
sr : Sized (node rs nil rx rr)
H1 : size nil = 0 β size (node rs nil rx rr) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs nil rx rr) β size (node rs nil rx rr) β€ delta * size nil
this : size nil = 0 β§ size rr = 0
β’ balanceL nil x (node rs nil rx rr) = balance nil x (node rs nil rx rr)
[PROOFSTEP]
cases sr.2.2.size_eq_zero.1 this.2
[GOAL]
case node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
rs : β
rx : Ξ±
sl : Sized nil
sr : Sized (node rs nil rx nil)
H1 : size nil = 0 β size (node rs nil rx nil) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs nil rx nil) β size (node rs nil rx nil) β€ delta * size nil
this : size nil = 0 β§ size nil = 0
β’ balanceL nil x (node rs nil rx nil) = balance nil x (node rs nil rx nil)
[PROOFSTEP]
rw [sr.eq_node']
[GOAL]
case node.nil.refl.refl
Ξ± : Type u_1
x : Ξ±
rs : β
rx : Ξ±
sl : Sized nil
sr : Sized (node rs nil rx nil)
H1 : size nil = 0 β size (node rs nil rx nil) β€ 1
H2 : 1 β€ size nil β 1 β€ size (node rs nil rx nil) β size (node rs nil rx nil) β€ delta * size nil
this : size nil = 0 β§ size nil = 0
β’ balanceL nil x (node' nil rx nil) = balance nil x (node' nil rx nil)
[PROOFSTEP]
rfl
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
sl : Sized (node ls ll lx lr)
H1 : size (node ls ll lx lr) = 0 β size (node rs rl rx rr) β€ 1
H2 :
1 β€ size (node ls ll lx lr) β 1 β€ size (node rs rl rx rr) β size (node rs rl rx rr) β€ delta * size (node ls ll lx lr)
β’ balanceL (node ls ll lx lr) x (node rs rl rx rr) = balance (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
replace H2 : Β¬rs > delta * ls := not_lt_of_le (H2 sl.pos sr.pos)
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
sl : Sized (node ls ll lx lr)
H1 : size (node ls ll lx lr) = 0 β size (node rs rl rx rr) β€ 1
H2 : Β¬rs > delta * ls
β’ balanceL (node ls ll lx lr) x (node rs rl rx rr) = balance (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
simp [balanceL, balance, H2]
[GOAL]
case node.node
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
sl : Sized (node ls ll lx lr)
H1 : size (node ls ll lx lr) = 0 β size (node rs rl rx rr) β€ 1
H2 : Β¬rs > delta * ls
β’ (if delta * rs < ls then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)) =
if delta * rs < ls then
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
sl : Sized (node ls ll lx lr)
H1 : size (node ls ll lx lr) = 0 β size (node rs rl rx rr) β€ 1
H2 : Β¬rs > delta * ls
hβ : delta * rs < ls
β’ rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll) =
rec (motive := fun t => ll = t β Ordnode Ξ±) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t β Ordnode Ξ±) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (size_1 + rs + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + Ordnode.size l + 1) ll lx l) x_2
(node (Ordnode.size r + rs + 1) r x (node rs rl rx rr)))
lr (_ : id lr = id lr))
ll (_ : id ll = id ll)
[PROOFSTEP]
simp [add_comm]
[GOAL]
case neg
Ξ± : Type u_1
x : Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
sr : Sized (node rs rl rx rr)
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
sl : Sized (node ls ll lx lr)
H1 : size (node ls ll lx lr) = 0 β size (node rs rl rx rr) β€ 1
H2 : Β¬rs > delta * ls
hβ : Β¬delta * rs < ls
β’ node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr) =
node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)
[PROOFSTEP]
simp [add_comm]
[GOAL]
Ξ± : Type u_1
n m : β
β’ Raised n m β n β€ m β§ m β€ n + 1
[PROOFSTEP]
constructor
[GOAL]
case mp
Ξ± : Type u_1
n m : β
β’ Raised n m β n β€ m β§ m β€ n + 1
case mpr Ξ± : Type u_1 n m : β β’ n β€ m β§ m β€ n + 1 β Raised n m
[PROOFSTEP]
rintro (rfl | rfl)
[GOAL]
case mp.inl
Ξ± : Type u_1
m : β
β’ m β€ m β§ m β€ m + 1
[PROOFSTEP]
exact β¨le_rfl, Nat.le_succ _β©
[GOAL]
case mp.inr
Ξ± : Type u_1
n : β
β’ n β€ n + 1 β§ n + 1 β€ n + 1
[PROOFSTEP]
exact β¨Nat.le_succ _, le_rflβ©
[GOAL]
case mpr
Ξ± : Type u_1
n m : β
β’ n β€ m β§ m β€ n + 1 β Raised n m
[PROOFSTEP]
rintro β¨hβ, hββ©
[GOAL]
case mpr.intro
Ξ± : Type u_1
n m : β
hβ : n β€ m
hβ : m β€ n + 1
β’ Raised n m
[PROOFSTEP]
rcases eq_or_lt_of_le hβ with (rfl | hβ)
[GOAL]
case mpr.intro.inl
Ξ± : Type u_1
n : β
hβ : n β€ n
hβ : n β€ n + 1
β’ Raised n n
[PROOFSTEP]
exact Or.inl rfl
[GOAL]
case mpr.intro.inr
Ξ± : Type u_1
n m : β
hββ : n β€ m
hβ : m β€ n + 1
hβ : n < m
β’ Raised n m
[PROOFSTEP]
exact Or.inr (le_antisymm hβ hβ)
[GOAL]
Ξ± : Type u_1
n m : β
H : Raised n m
β’ Nat.dist n m β€ 1
[PROOFSTEP]
cases' raised_iff.1 H with H1 H2
[GOAL]
case intro
Ξ± : Type u_1
n m : β
H : Raised n m
H1 : n β€ m
H2 : m β€ n + 1
β’ Nat.dist n m β€ 1
[PROOFSTEP]
rwa [Nat.dist_eq_sub_of_le H1, tsub_le_iff_left]
[GOAL]
Ξ± : Type u_1
n m : β
H : Raised n m
β’ Nat.dist m n β€ 1
[PROOFSTEP]
rw [Nat.dist_comm]
[GOAL]
Ξ± : Type u_1
n m : β
H : Raised n m
β’ Nat.dist n m β€ 1
[PROOFSTEP]
exact H.dist_le
[GOAL]
Ξ± : Type u_1
k n m : β
H : Raised n m
β’ Raised (k + n) (k + m)
[PROOFSTEP]
rcases H with (rfl | rfl)
[GOAL]
case inl
Ξ± : Type u_1
k m : β
β’ Raised (k + m) (k + m)
[PROOFSTEP]
exact Or.inl rfl
[GOAL]
case inr
Ξ± : Type u_1
k n : β
β’ Raised (k + n) (k + (n + 1))
[PROOFSTEP]
exact Or.inr rfl
[GOAL]
Ξ± : Type u_1
k n m : β
H : Raised n m
β’ Raised (n + k) (m + k)
[PROOFSTEP]
rw [add_comm, add_comm m]
[GOAL]
Ξ± : Type u_1
k n m : β
H : Raised n m
β’ Raised (k + n) (k + m)
[PROOFSTEP]
exact H.add_left _
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
xβ xβ : Ξ±
rβ rβ : Ordnode Ξ±
H : Raised (size rβ) (size rβ)
β’ Raised (size (node' l xβ rβ)) (size (node' l xβ rβ))
[PROOFSTEP]
rw [node', size, size]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
xβ xβ : Ξ±
rβ rβ : Ordnode Ξ±
H : Raised (size rβ) (size rβ)
β’ Raised (size l + size rβ + 1) (size l + size rβ + 1)
[PROOFSTEP]
generalize size rβ = m at H β’
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
xβ xβ : Ξ±
rβ rβ : Ordnode Ξ±
m : β
H : Raised (size rβ) m
β’ Raised (size l + size rβ + 1) (size l + m + 1)
[PROOFSTEP]
rcases H with (rfl | rfl)
[GOAL]
case inl
Ξ± : Type u_1
l : Ordnode Ξ±
xβ xβ : Ξ±
rβ rβ : Ordnode Ξ±
β’ Raised (size l + size rβ + 1) (size l + size rβ + 1)
[PROOFSTEP]
exact Or.inl rfl
[GOAL]
case inr
Ξ± : Type u_1
l : Ordnode Ξ±
xβ xβ : Ξ±
rβ rβ : Ordnode Ξ±
β’ Raised (size l + size rβ + 1) (size l + (size rβ + 1) + 1)
[PROOFSTEP]
exact Or.inr rfl
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ balanceL l x r = balance' l x r
[PROOFSTEP]
rw [β balance_eq_balance' hl hr sl sr, balanceL_eq_balance sl sr]
[GOAL]
case H1
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ size l = 0 β size r β€ 1
[PROOFSTEP]
intro l0
[GOAL]
case H1
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
l0 : size l = 0
β’ size r β€ 1
[PROOFSTEP]
rw [l0] at H
[GOAL]
case H1
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' 0 β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz 0 r'
l0 : size l = 0
β’ size r β€ 1
[PROOFSTEP]
rcases H with (β¨_, β¨β¨β©β© | β¨β¨β©β©, Hβ© | β¨r', e, Hβ©)
[GOAL]
case H1.inl.intro.intro.inl.refl
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l0 : size l = 0
H : BalancedSz 0 (size r)
β’ size r β€ 1
[PROOFSTEP]
exact balancedSz_zero.1 H.symm
[GOAL]
case H1.inr.intro.intro
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l0 : size l = 0
r' : β
e : Raised (size r) r'
H : BalancedSz 0 r'
β’ size r β€ 1
[PROOFSTEP]
exact le_trans (raised_iff.1 e).1 (balancedSz_zero.1 H.symm)
[GOAL]
case H2
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ 1 β€ size l β 1 β€ size r β size r β€ delta * size l
[PROOFSTEP]
intro l1 _
[GOAL]
case H2
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
l1 : 1 β€ size l
aβ : 1 β€ size r
β’ size r β€ delta * size l
[PROOFSTEP]
rcases H with (β¨l', e, H | β¨_, Hββ©β© | β¨r', e, H | β¨_, Hββ©β©)
[GOAL]
case H2.inl.intro.intro.inl
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
l' : β
e : Raised l' (size l)
H : l' + size r β€ 1
β’ size r β€ delta * size l
[PROOFSTEP]
exact le_trans (le_trans (Nat.le_add_left _ _) H) (mul_pos (by decide) l1 : (0 : β) < _)
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
l' : β
e : Raised l' (size l)
H : l' + size r β€ 1
β’ 0 < delta
[PROOFSTEP]
decide
[GOAL]
case H2.inl.intro.intro.inr.intro
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
l' : β
e : Raised l' (size l)
leftβ : l' β€ delta * size r
Hβ : size r β€ delta * l'
β’ size r β€ delta * size l
[PROOFSTEP]
exact le_trans Hβ (Nat.mul_le_mul_left _ (raised_iff.1 e).1)
[GOAL]
case H2.inr.intro.intro.inl
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
r' : β
e : Raised (size r) r'
H : size l + r' β€ 1
β’ size r β€ delta * size l
[PROOFSTEP]
cases raised_iff.1 e
[GOAL]
case H2.inr.intro.intro.inl.intro
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
r' : β
e : Raised (size r) r'
H : size l + r' β€ 1
leftβ : size r β€ r'
rightβ : r' β€ size r + 1
β’ size r β€ delta * size l
[PROOFSTEP]
unfold delta
[GOAL]
case H2.inr.intro.intro.inl.intro
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
r' : β
e : Raised (size r) r'
H : size l + r' β€ 1
leftβ : size r β€ r'
rightβ : r' β€ size r + 1
β’ size r β€ 3 * size l
[PROOFSTEP]
linarith
[GOAL]
case H2.inr.intro.intro.inr.intro
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
l1 : 1 β€ size l
aβ : 1 β€ size r
r' : β
e : Raised (size r) r'
leftβ : size l β€ delta * r'
Hβ : r' β€ delta * size l
β’ size r β€ delta * size l
[PROOFSTEP]
exact le_trans (raised_iff.1 e).1 Hβ
[GOAL]
Ξ± : Type u_1
l r : Ordnode Ξ±
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ (β l', Raised l' (size (dual r)) β§ BalancedSz l' (size (dual l))) β¨
β r', Raised (size (dual l)) r' β§ BalancedSz (size (dual r)) r'
[PROOFSTEP]
rw [size_dual, size_dual]
[GOAL]
Ξ± : Type u_1
l r : Ordnode Ξ±
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ (β l', Raised l' (size r) β§ BalancedSz l' (size l)) β¨ β r', Raised (size l) r' β§ BalancedSz (size r) r'
[PROOFSTEP]
exact H.symm.imp (Exists.imp fun _ => And.imp_right BalancedSz.symm) (Exists.imp fun _ => And.imp_right BalancedSz.symm)
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ size (balanceL l x r) = size l + size r + 1
[PROOFSTEP]
rw [balanceL_eq_balance' hl hr sl sr H, size_balance' sl sr]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ All P (balanceL l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
rw [balanceL_eq_balance' hl hr sl sr H, all_balance']
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ balanceR l x r = balance' l x r
[PROOFSTEP]
rw [β dual_dual (balanceR l x r), dual_balanceR,
balanceL_eq_balance' hr.dual hl.dual sr.dual sl.dual (balance_sz_dual H), β dual_balance', dual_dual]
[GOAL]
Ξ± : Type u_1
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ size (balanceR l x r) = size l + size r + 1
[PROOFSTEP]
rw [balanceR_eq_balance' hl hr sl sr H, size_balance' sl sr]
[GOAL]
Ξ± : Type u_1
P : Ξ± β Prop
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
hl : Balanced l
hr : Balanced r
sl : Sized l
sr : Sized r
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ All P (balanceR l x r) β All P l β§ P x β§ All P r
[PROOFSTEP]
rw [balanceR_eq_balance' hl hr sl sr H, all_balance']
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Bounded nil oβ oβ
β’ Bounded (Ordnode.dual nil) oβ oβ
[PROOFSTEP]
cases oβ
[GOAL]
case none
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithTop Ξ±
h : Bounded nil none oβ
β’ Bounded (Ordnode.dual nil) oβ none
[PROOFSTEP]
cases oβ
[GOAL]
case some
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithTop Ξ±
valβ : Ξ±
h : Bounded nil (some valβ) oβ
β’ Bounded (Ordnode.dual nil) oβ (some valβ)
[PROOFSTEP]
cases oβ
[GOAL]
case none.none
Ξ± : Type u_1
instβ : Preorder Ξ±
h : Bounded nil none none
β’ Bounded (Ordnode.dual nil) none none
[PROOFSTEP]
trivial
[GOAL]
case none.some
Ξ± : Type u_1
instβ : Preorder Ξ±
valβ : Ξ±
h : Bounded nil none (some valβ)
β’ Bounded (Ordnode.dual nil) (some valβ) none
[PROOFSTEP]
trivial
[GOAL]
case some.none
Ξ± : Type u_1
instβ : Preorder Ξ±
valβ : Ξ±
h : Bounded nil (some valβ) none
β’ Bounded (Ordnode.dual nil) none (some valβ)
[PROOFSTEP]
trivial
[GOAL]
case some.some
Ξ± : Type u_1
instβ : Preorder Ξ±
valβΒΉ valβ : Ξ±
h : Bounded nil (some valβΒΉ) (some valβ)
β’ Bounded (Ordnode.dual nil) (some valβ) (some valβΒΉ)
[PROOFSTEP]
trivial
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Bounded (Ordnode.dual t) oβ oβ
β’ Bounded t oβ oβ
[PROOFSTEP]
have := Bounded.dual h
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Bounded (Ordnode.dual t) oβ oβ
this : Bounded (Ordnode.dual (Ordnode.dual t)) oβ oβ
β’ Bounded t oβ oβ
[PROOFSTEP]
rwa [dual_dual, OrderDual.Preorder.dual_dual] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Bounded nil oβ oβ
β’ Bounded nil β₯ oβ
[PROOFSTEP]
cases oβ
[GOAL]
case none
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
h : Bounded nil oβ none
β’ Bounded nil β₯ none
[PROOFSTEP]
trivial
[GOAL]
case some
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
valβ : Ξ±
h : Bounded nil oβ (some valβ)
β’ Bounded nil β₯ (some valβ)
[PROOFSTEP]
trivial
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Bounded nil oβ oβ
β’ Bounded nil oβ β€
[PROOFSTEP]
cases oβ
[GOAL]
case none
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithTop Ξ±
h : Bounded nil none oβ
β’ Bounded nil none β€
[PROOFSTEP]
trivial
[GOAL]
case some
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithTop Ξ±
valβ : Ξ±
h : Bounded nil (some valβ) oβ
β’ Bounded nil (some valβ) β€
[PROOFSTEP]
trivial
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
tβ tβ : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
x : Ξ±
hβ : Bounded tβ oβ βx
hβ : Bounded tβ (βx) oβ
β’ All (fun y => All (fun z => y < z) tβ) tβ
[PROOFSTEP]
refine hβ.mem_lt.imp fun y yx => ?_
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
tβ tβ : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
x : Ξ±
hβ : Bounded tβ oβ βx
hβ : Bounded tβ (βx) oβ
y : Ξ±
yx : y < x
β’ All (fun z => y < z) tβ
[PROOFSTEP]
exact hβ.mem_gt.imp fun z xz => lt_trans yx xz
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ol : Bounded l oβ βx
Or : Bounded r (βx) oβ
sl : Sized l
sr : Sized r
b : BalancedSz (size l) (size r)
bl : Balanced l
br : Balanced r
ol' : Bounded (Ordnode.dual l) (βx) oβ
sl' : Sized (Ordnode.dual l)
bl' : Balanced (Ordnode.dual l)
or' : Bounded (Ordnode.dual r) oβ βx
sr' : Sized (Ordnode.dual r)
br' : Balanced (Ordnode.dual r)
β’ size l + size r + 1 = size (Ordnode.dual r) + size (Ordnode.dual l) + 1
[PROOFSTEP]
simp [size_dual, add_comm]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ol : Bounded l oβ βx
Or : Bounded r (βx) oβ
sl : Sized l
sr : Sized r
b : BalancedSz (size l) (size r)
bl : Balanced l
br : Balanced r
ol' : Bounded (Ordnode.dual l) (βx) oβ
sl' : Sized (Ordnode.dual l)
bl' : Balanced (Ordnode.dual l)
or' : Bounded (Ordnode.dual r) oβ βx
sr' : Sized (Ordnode.dual r)
br' : Balanced (Ordnode.dual r)
β’ BalancedSz (size (Ordnode.dual r)) (size (Ordnode.dual l))
[PROOFSTEP]
rw [size_dual, size_dual]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ol : Bounded l oβ βx
Or : Bounded r (βx) oβ
sl : Sized l
sr : Sized r
b : BalancedSz (size l) (size r)
bl : Balanced l
br : Balanced r
ol' : Bounded (Ordnode.dual l) (βx) oβ
sl' : Sized (Ordnode.dual l)
bl' : Balanced (Ordnode.dual l)
or' : Bounded (Ordnode.dual r) oβ βx
sr' : Sized (Ordnode.dual r)
br' : Balanced (Ordnode.dual r)
β’ BalancedSz (size r) (size l)
[PROOFSTEP]
exact b.symm
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (Ordnode.dual t) oβ
β’ Valid' oβ t oβ
[PROOFSTEP]
have := Valid'.dual h
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (Ordnode.dual t) oβ
this : Valid' oβ (Ordnode.dual (Ordnode.dual t)) oβ
β’ Valid' oβ t oβ
[PROOFSTEP]
rwa [dual_dual, OrderDual.Preorder.dual_dual] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c d : β
lrβ : 3 * (b + c + 1 + d) β€ 16 * a + 9
mrβ : b + c + 1 β€ 3 * d
mmβ : b β€ 3 * c
β’ b < 3 * a + 1
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
b c d : β
mrβ : b + c + 1 β€ 3 * d
β’ c β€ 3 * d
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
b c d : β
mrβ : 2 * d β€ b + c + 1
mmβ : b β€ 3 * c
β’ d β€ 3 * c
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c d : β
lrβ : 3 * a β€ b + c + 1 + d
mrβ : b + c + 1 β€ 3 * d
mmβ : b β€ 3 * c
β’ a + b + 1 β€ 3 * (c + d + 1)
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c d : β
lrβ : 3 * (b + c + 1 + d) β€ 16 * a + 9
mrβ : 2 * d β€ b + c + 1
mmβ : c β€ 3 * b
β’ c + d + 1 β€ 3 * (a + b + 1)
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
m : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hm : Valid' (βx) m βy
hr : Valid' (βy) r oβ
Hm : 0 < size m
H :
size l = 0 β§ size m = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size m β§
delta * size l β€ size m + size r β§ 3 * (size m + size r) β€ 16 * size l + 9 β§ size m β€ delta * size r
β’ Valid' oβ (Ordnode.node4L l x m y r) oβ
[PROOFSTEP]
cases' m with s ml z mr
[GOAL]
case nil
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
hm : Valid' (βx) nil βy
Hm : 0 < size nil
H :
size l = 0 β§ size nil = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size nil β§
delta * size l β€ size nil + size r β§ 3 * (size nil + size r) β€ 16 * size l + 9 β§ size nil β€ delta * size r
β’ Valid' oβ (Ordnode.node4L l x nil y r) oβ
[PROOFSTEP]
cases Hm
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
H :
size l = 0 β§ size (Ordnode.node s ml z mr) = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size (Ordnode.node s ml z mr) β§
delta * size l β€ size (Ordnode.node s ml z mr) + size r β§
3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9 β§
size (Ordnode.node s ml z mr) β€ delta * size r
β’ Valid' oβ (Ordnode.node4L l x (Ordnode.node s ml z mr) y r) oβ
[PROOFSTEP]
suffices :
BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
H :
size l = 0 β§ size (Ordnode.node s ml z mr) = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size (Ordnode.node s ml z mr) β§
delta * size l β€ size (Ordnode.node s ml z mr) + size r β§
3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9 β§
size (Ordnode.node s ml z mr) β€ delta * size r
this :
BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
β’ Valid' oβ (Ordnode.node4L l x (Ordnode.node s ml z mr) y r) oβ
case this
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
H :
size l = 0 β§ size (Ordnode.node s ml z mr) = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size (Ordnode.node s ml z mr) β§
delta * size l β€ size (Ordnode.node s ml z mr) + size r β§
3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9 β§
size (Ordnode.node s ml z mr) β€ delta * size r
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
exact Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
[GOAL]
case this
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
H :
size l = 0 β§ size (Ordnode.node s ml z mr) = 1 β§ size r β€ 1 β¨
0 < size l β§
ratio * size r β€ size (Ordnode.node s ml z mr) β§
delta * size l β€ size (Ordnode.node s ml z mr) + size r β§
3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9 β§
size (Ordnode.node s ml z mr) β€ delta * size r
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rcases H with (β¨l0, m1, r0β© | β¨l0, mrβ, lrβ, lrβ, mrββ©)
[GOAL]
case this.inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size (Ordnode.node s ml z mr) = 1
r0 : size r β€ 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
[GOAL]
case this.inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
r0 : size r β€ 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [l0, m1.1, m1.2]
[GOAL]
case this.inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
r0 : size r β€ 1
β’ BalancedSz 0 0 β§ BalancedSz 0 (size r) β§ BalancedSz (0 + 0 + 1) (0 + size r + 1)
[PROOFSTEP]
revert r0
[GOAL]
case this.inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
β’ size r β€ 1 β BalancedSz 0 0 β§ BalancedSz 0 (size r) β§ BalancedSz (0 + 0 + 1) (0 + size r + 1)
[PROOFSTEP]
rcases size r with (_ | _ | _) <;> [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
[GOAL]
case this.inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
β’ size r β€ 1 β BalancedSz 0 0 β§ BalancedSz 0 (size r) β§ BalancedSz (0 + 0 + 1) (0 + size r + 1)
[PROOFSTEP]
rcases size r with (_ | _ | _)
[GOAL]
case this.inl.intro.intro.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
β’ Nat.zero β€ 1 β BalancedSz 0 0 β§ BalancedSz 0 Nat.zero β§ BalancedSz (0 + 0 + 1) (0 + Nat.zero + 1)
[PROOFSTEP]
decide
[GOAL]
case this.inl.intro.intro.succ.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
β’ Nat.succ Nat.zero β€ 1 β
BalancedSz 0 0 β§ BalancedSz 0 (Nat.succ Nat.zero) β§ BalancedSz (0 + 0 + 1) (0 + Nat.succ Nat.zero + 1)
[PROOFSTEP]
decide
[GOAL]
case this.inl.intro.intro.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
nβ : β
β’ Nat.succ (Nat.succ nβ) β€ 1 β
BalancedSz 0 0 β§ BalancedSz 0 (Nat.succ (Nat.succ nβ)) β§ BalancedSz (0 + 0 + 1) (0 + Nat.succ (Nat.succ nβ) + 1)
[PROOFSTEP]
intro r0
[GOAL]
case this.inl.intro.intro.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
nβ : β
r0 : Nat.succ (Nat.succ nβ) β€ 1
β’ BalancedSz 0 0 β§ BalancedSz 0 (Nat.succ (Nat.succ nβ)) β§ BalancedSz (0 + 0 + 1) (0 + Nat.succ (Nat.succ nβ) + 1)
[PROOFSTEP]
unfold BalancedSz delta
[GOAL]
case this.inl.intro.intro.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : size l = 0
m1 : size ml = 0 β§ size mr = 0
nβ : β
r0 : Nat.succ (Nat.succ nβ) β€ 1
β’ (0 + 0 β€ 1 β¨ 0 β€ 3 * 0 β§ 0 β€ 3 * 0) β§
(0 + Nat.succ (Nat.succ nβ) β€ 1 β¨ 0 β€ 3 * Nat.succ (Nat.succ nβ) β§ Nat.succ (Nat.succ nβ) β€ 3 * 0) β§
(0 + 0 + 1 + (0 + Nat.succ (Nat.succ nβ) + 1) β€ 1 β¨
0 + 0 + 1 β€ 3 * (0 + Nat.succ (Nat.succ nβ) + 1) β§ 0 + Nat.succ (Nat.succ nβ) + 1 β€ 3 * (0 + 0 + 1))
[PROOFSTEP]
linarith
[GOAL]
case this.inr.intro.intro.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size (Ordnode.node s ml z mr)
lrβ : delta * size l β€ size (Ordnode.node s ml z mr) + size r
lrβ : 3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9
mrβ : size (Ordnode.node s ml z mr) β€ delta * size r
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size r) with r0 r0
[GOAL]
case this.inr.intro.intro.intro.intro.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size (Ordnode.node s ml z mr)
lrβ : delta * size l β€ size (Ordnode.node s ml z mr) + size r
lrβ : 3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9
mrβ : size (Ordnode.node s ml z mr) β€ delta * size r
r0 : size r = 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [r0] at mrβ
[GOAL]
case this.inr.intro.intro.intro.intro.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size (Ordnode.node s ml z mr)
lrβ : delta * size l β€ size (Ordnode.node s ml z mr) + size r
lrβ : 3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9
mrβ : size (Ordnode.node s ml z mr) β€ delta * 0
r0 : size r = 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
cases not_le_of_lt Hm mrβ
[GOAL]
case this.inr.intro.intro.intro.intro.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size (Ordnode.node s ml z mr)
lrβ : delta * size l β€ size (Ordnode.node s ml z mr) + size r
lrβ : 3 * (size (Ordnode.node s ml z mr) + size r) β€ 16 * size l + 9
mrβ : size (Ordnode.node s ml z mr) β€ delta * size r
r0 : size r > 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [hm.2.size_eq] at lrβ lrβ mrβ mrβ
[GOAL]
case this.inr.intro.intro.intro.intro.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
by_cases mm : size ml + size mr β€ 1
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
have r1 := le_antisymm ((mul_le_mul_left (by decide)).1 (le_trans mrβ (Nat.succ_le_succ mm) : _ β€ ratio * 1)) r0
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
β’ 0 < ratio
[PROOFSTEP]
decide
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
r1 : size r = 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [r1, add_assoc] at lrβ
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
r1 : size r = 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
have l1 := le_antisymm ((mul_le_mul_left (by decide)).1 (le_trans lrβ (add_le_add_right mm 2) : _ β€ delta * 1)) l0
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
r1 : size r = 1
β’ 0 < delta
[PROOFSTEP]
decide
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
r1 : size r = 1
l1 : size l = 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [l1, r1]
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : size ml + size mr β€ 1
r1 : size r = 1
l1 : size l = 1
β’ BalancedSz 1 (size ml) β§ BalancedSz (size mr) 1 β§ BalancedSz (1 + size ml + 1) (size mr + 1 + 1)
[PROOFSTEP]
revert mm
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
β’ size ml + size mr β€ 1 β
BalancedSz 1 (size ml) β§ BalancedSz (size mr) 1 β§ BalancedSz (1 + size ml + 1) (size mr + 1 + 1)
[PROOFSTEP]
cases size ml
[GOAL]
case pos.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
β’ Nat.zero + size mr β€ 1 β
BalancedSz 1 Nat.zero β§ BalancedSz (size mr) 1 β§ BalancedSz (1 + Nat.zero + 1) (size mr + 1 + 1)
[PROOFSTEP]
cases size mr
[GOAL]
case pos.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
β’ Nat.succ nβ + size mr β€ 1 β
BalancedSz 1 (Nat.succ nβ) β§ BalancedSz (size mr) 1 β§ BalancedSz (1 + Nat.succ nβ + 1) (size mr + 1 + 1)
[PROOFSTEP]
cases size mr
[GOAL]
case pos.zero.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
β’ Nat.zero + Nat.zero β€ 1 β
BalancedSz 1 Nat.zero β§ BalancedSz Nat.zero 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.zero + 1 + 1)
[PROOFSTEP]
intro mm
[GOAL]
case pos.zero.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
β’ Nat.zero + Nat.succ nβ β€ 1 β
BalancedSz 1 Nat.zero β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
intro mm
[GOAL]
case pos.succ.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
β’ Nat.succ nβ + Nat.zero β€ 1 β
BalancedSz 1 (Nat.succ nβ) β§ BalancedSz Nat.zero 1 β§ BalancedSz (1 + Nat.succ nβ + 1) (Nat.zero + 1 + 1)
[PROOFSTEP]
intro mm
[GOAL]
case pos.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβΒΉ nβ : β
β’ Nat.succ nβΒΉ + Nat.succ nβ β€ 1 β
BalancedSz 1 (Nat.succ nβΒΉ) β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.succ nβΒΉ + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
intro mm
[GOAL]
case pos.zero.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
mm : Nat.zero + Nat.zero β€ 1
β’ BalancedSz 1 Nat.zero β§ BalancedSz Nat.zero 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.zero + 1 + 1)
[PROOFSTEP]
decide
[GOAL]
case pos.zero.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
mm : Nat.zero + Nat.succ nβ β€ 1
β’ BalancedSz 1 Nat.zero β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
rw [Nat.zero_eq, zero_add] at mm
[GOAL]
case pos.zero.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
mm : Nat.succ nβ β€ 1
β’ BalancedSz 1 Nat.zero β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
rcases mm with (_ | β¨β¨β©β©)
[GOAL]
case pos.zero.succ.refl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
β’ BalancedSz 1 Nat.zero β§ BalancedSz (Nat.succ 0) 1 β§ BalancedSz (1 + Nat.zero + 1) (Nat.succ 0 + 1 + 1)
[PROOFSTEP]
decide
[GOAL]
case pos.succ.zero
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβ : β
mm : Nat.succ nβ + Nat.zero β€ 1
β’ BalancedSz 1 (Nat.succ nβ) β§ BalancedSz Nat.zero 1 β§ BalancedSz (1 + Nat.succ nβ + 1) (Nat.zero + 1 + 1)
[PROOFSTEP]
rcases mm with (_ | β¨β¨β©β©)
[GOAL]
case pos.succ.zero.refl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
β’ BalancedSz 1 (Nat.succ 0) β§ BalancedSz Nat.zero 1 β§ BalancedSz (1 + Nat.succ 0 + 1) (Nat.zero + 1 + 1)
[PROOFSTEP]
decide
[GOAL]
case pos.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβΒΉ nβ : β
mm : Nat.succ nβΒΉ + Nat.succ nβ β€ 1
β’ BalancedSz 1 (Nat.succ nβΒΉ) β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.succ nβΒΉ + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
rw [Nat.succ_add] at mm
[GOAL]
case pos.succ.succ
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + (1 + 1)
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
r1 : size r = 1
l1 : size l = 1
nβΒΉ nβ : β
mm : Nat.succ (nβΒΉ + Nat.succ nβ) β€ 1
β’ BalancedSz 1 (Nat.succ nβΒΉ) β§ BalancedSz (Nat.succ nβ) 1 β§ BalancedSz (1 + Nat.succ nβΒΉ + 1) (Nat.succ nβ + 1 + 1)
[PROOFSTEP]
rcases mm with (_ | β¨β¨β©β©)
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rcases hm.3.1.resolve_left mm with β¨mmβ, mmββ©
[GOAL]
case neg.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size ml) with ml0 ml0
[GOAL]
case neg.intro.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml = 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [ml0, mul_zero, le_zero_iff] at mmβ
[GOAL]
case neg.intro.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr = 0
ml0 : size ml = 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
rw [ml0, mmβ] at mm
[GOAL]
case neg.intro.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬0 + 0 β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr = 0
ml0 : size ml = 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
cases mm (by decide)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬0 + 0 β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr = 0
ml0 : size ml = 0
β’ 0 + 0 β€ 1
[PROOFSTEP]
decide
[GOAL]
case neg.intro.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
have : 2 * size l β€ size ml + size mr + 1 :=
by
have := Nat.mul_le_mul_left ratio lrβ
rw [mul_left_comm, mul_add] at this
have := le_trans this (add_le_add_left mrβ _)
rw [β Nat.succ_mul] at this
exact (mul_le_mul_left (by decide)).1 this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
β’ 2 * size l β€ size ml + size mr + 1
[PROOFSTEP]
have := Nat.mul_le_mul_left ratio lrβ
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : ratio * (delta * size l) β€ ratio * (size ml + size mr + 1 + size r)
β’ 2 * size l β€ size ml + size mr + 1
[PROOFSTEP]
rw [mul_left_comm, mul_add] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : delta * (ratio * size l) β€ ratio * (size ml + size mr + 1) + ratio * size r
β’ 2 * size l β€ size ml + size mr + 1
[PROOFSTEP]
have := le_trans this (add_le_add_left mrβ _)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
thisβ : delta * (ratio * size l) β€ ratio * (size ml + size mr + 1) + ratio * size r
this : delta * (ratio * size l) β€ ratio * (size ml + size mr + 1) + (size ml + size mr + 1)
β’ 2 * size l β€ size ml + size mr + 1
[PROOFSTEP]
rw [β Nat.succ_mul] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
thisβ : delta * (ratio * size l) β€ ratio * (size ml + size mr + 1) + ratio * size r
this : delta * (ratio * size l) β€ Nat.succ ratio * (size ml + size mr + 1)
β’ 2 * size l β€ size ml + size mr + 1
[PROOFSTEP]
exact (mul_le_mul_left (by decide)).1 this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
thisβ : delta * (ratio * size l) β€ ratio * (size ml + size mr + 1) + ratio * size r
this : delta * (ratio * size l) β€ Nat.succ ratio * (size ml + size mr + 1)
β’ 0 < delta
[PROOFSTEP]
decide
[GOAL]
case neg.intro.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ BalancedSz (size l) (size ml) β§
BalancedSz (size mr) (size r) β§ BalancedSz (size l + size ml + 1) (size mr + size r + 1)
[PROOFSTEP]
refine' β¨Or.inr β¨_, _β©, Or.inr β¨_, _β©, Or.inr β¨_, _β©β©
[GOAL]
case neg.intro.inr.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size l β€ delta * size ml
[PROOFSTEP]
refine' (mul_le_mul_left (by decide)).1 (le_trans this _)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ 0 < 2
[PROOFSTEP]
decide
[GOAL]
case neg.intro.inr.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size ml + size mr + 1 β€ 2 * (delta * size ml)
[PROOFSTEP]
rw [two_mul, Nat.succ_le_iff]
[GOAL]
case neg.intro.inr.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size ml + size mr < delta * size ml + delta * size ml
[PROOFSTEP]
refine' add_lt_add_of_lt_of_le _ mmβ
[GOAL]
case neg.intro.inr.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size ml < delta * size ml
[PROOFSTEP]
simpa using (mul_lt_mul_right ml0).2 (by decide : 1 < 3)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ 1 < 3
[PROOFSTEP]
decide
[GOAL]
case neg.intro.inr.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size ml β€ delta * size l
[PROOFSTEP]
exact Nat.le_of_lt_succ (Valid'.node4L_lemmaβ lrβ mrβ mmβ)
[GOAL]
case neg.intro.inr.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size mr β€ delta * size r
[PROOFSTEP]
exact Valid'.node4L_lemmaβ mrβ
[GOAL]
case neg.intro.inr.refine'_4
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size r β€ delta * size mr
[PROOFSTEP]
exact Valid'.node4L_lemmaβ mrβ mmβ
[GOAL]
case neg.intro.inr.refine'_5
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size l + size ml + 1 β€ delta * (size mr + size r + 1)
[PROOFSTEP]
exact Valid'.node4L_lemmaβ lrβ mrβ mmβ
[GOAL]
case neg.intro.inr.refine'_6
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βy) r oβ
s : β
ml : Ordnode Ξ±
z : Ξ±
mr : Ordnode Ξ±
hm : Valid' (βx) (Ordnode.node s ml z mr) βy
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mrβ : ratio * size r β€ size ml + size mr + 1
lrβ : delta * size l β€ size ml + size mr + 1 + size r
lrβ : 3 * (size ml + size mr + 1 + size r) β€ 16 * size l + 9
mrβ : size ml + size mr + 1 β€ delta * size r
r0 : size r > 0
mm : Β¬size ml + size mr β€ 1
mmβ : size ml β€ delta * size mr
mmβ : size mr β€ delta * size ml
ml0 : size ml > 0
this : 2 * size l β€ size ml + size mr + 1
β’ size mr + size r + 1 β€ delta * (size l + size ml + 1)
[PROOFSTEP]
exact Valid'.node4L_lemmaβ
lrβ mrβ mmβ
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c : β
H2 : 3 * a β€ b + c
hbβ : c β€ 3 * b
β’ a β€ 3 * b
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c : β
H3 : 2 * (b + c) β€ 9 * a + 3
h : b < 2 * c
β’ b < 3 * a + 1
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c : β
H2 : 3 * a β€ b + c
h : b < 2 * c
β’ a + b < 3 * c
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b : β
H3 : 2 * b β€ 9 * a + 3
β’ 3 * b β€ 16 * a + 9
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size l < size r
H3 : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
β’ Valid' oβ (Ordnode.rotateL l x r) oβ
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case nil
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) nil oβ
H1 : Β¬size l + size nil β€ 1
H2 : delta * size l < size nil
H3 : 2 * size nil β€ 9 * size l + 5 β¨ size nil β€ 3
β’ Valid' oβ (Ordnode.rotateL l x nil) oβ
[PROOFSTEP]
cases H2
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l < size (Ordnode.node rs rl rx rr)
H3 : 2 * size (Ordnode.node rs rl rx rr) β€ 9 * size l + 5 β¨ size (Ordnode.node rs rl rx rr) β€ 3
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * size (Ordnode.node rs rl rx rr) β€ 9 * size l + 5 β¨ size (Ordnode.node rs rl rx rr) β€ 3
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
rw [hr.2.size_eq] at H3
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr + 1) β€ 9 * size l + 5 β¨ size rl + size rr + 1 β€ 3
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
replace H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2 :=
H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
have H3_0 : size l = 0 β size rl + size rr β€ 2 := by
intro l0; rw [l0] at H3
exact (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
β’ size l = 0 β size rl + size rr β€ 2
[PROOFSTEP]
intro l0
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
l0 : size l = 0
β’ size rl + size rr β€ 2
[PROOFSTEP]
rw [l0] at H3
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * 0 + 3 β¨ size rl + size rr β€ 2
l0 : size l = 0
β’ size rl + size rr β€ 2
[PROOFSTEP]
exact (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * 0 + 3 β¨ size rl + size rr β€ 2
l0 : size l = 0
h : 2 * (size rl + size rr) β€ 9 * 0 + 3
β’ 0 < 2
[PROOFSTEP]
decide
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * 0 + 3 β¨ size rl + size rr β€ 2
l0 : size l = 0
h : 2 * (size rl + size rr) β€ 9 * 0 + 3
β’ 9 * 0 + 3 β€ 2 * 2
[PROOFSTEP]
decide
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
have H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3 := fun l0 : 1 β€ size l =>
(or_iff_left_of_imp <| by intro; linarith).1 H3
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
l0 : 1 β€ size l
β’ size rl + size rr β€ 2 β 2 * (size rl + size rr) β€ 9 * size l + 3
[PROOFSTEP]
intro
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
l0 : 1 β€ size l
aβ : size rl + size rr β€ 2
β’ 2 * (size rl + size rr) β€ 9 * size l + 3
[PROOFSTEP]
linarith
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
have ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1 := by intros; linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
β’ β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
[PROOFSTEP]
intros
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
aβΒ² bβ : β
aβΒΉ : 1 β€ aβΒ²
aβ : aβΒ² + bβ β€ 2
β’ bβ β€ 1
[PROOFSTEP]
linarith
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
have hlp : size l > 0 β Β¬size rl + size rr β€ 1 := fun l0 hb =>
absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by decide)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
l0 : size l > 0
hb : size rl + size rr β€ 1
β’ Β¬delta * Nat.succ 0 β€ 1
[PROOFSTEP]
decide
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
β’ Valid' oβ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) oβ
[PROOFSTEP]
rw [rotateL]
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
β’ Valid' oβ (if size rl < ratio * size rr then Ordnode.node3L l x rl rx rr else Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
split_ifs with h
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
β’ Valid' oβ (Ordnode.node3L l x rl rx rr) oβ
[PROOFSTEP]
have rr0 : size rr > 0 := (mul_lt_mul_left (by decide)).1 (lt_of_le_of_lt (Nat.zero_le _) h : ratio * 0 < _)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
β’ 0 < ratio
[PROOFSTEP]
decide
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
β’ Valid' oβ (Ordnode.node3L l x rl rx rr) oβ
[PROOFSTEP]
suffices BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr) by
exact hl.node3L hr.left hr.right this.1 this.2
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
this : BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
β’ Valid' oβ (Ordnode.node3L l x rl rx rr) oβ
[PROOFSTEP]
exact hl.node3L hr.left hr.right this.1 this.2
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
β’ BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size l) with l0 l0
[GOAL]
case pos.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
β’ BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
[PROOFSTEP]
rw [l0]
[GOAL]
case pos.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
replace H3 := H3_0 l0
[GOAL]
case pos.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
have := hr.3.1
[GOAL]
case pos.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz (size rl) (size rr)
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size rl) with rl0 rl0
[GOAL]
case pos.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz (size rl) (size rr)
rl0 : size rl = 0
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
rw [rl0] at this β’
[GOAL]
case pos.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz 0 (size rr)
rl0 : size rl = 0
β’ BalancedSz 0 0 β§ BalancedSz (0 + 0 + 1) (size rr)
[PROOFSTEP]
rw [le_antisymm (balancedSz_zero.1 this.symm) rr0]
[GOAL]
case pos.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz 0 (size rr)
rl0 : size rl = 0
β’ BalancedSz 0 0 β§ BalancedSz (0 + 0 + 1) 1
[PROOFSTEP]
decide
[GOAL]
case pos.inl.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz (size rl) (size rr)
rl0 : size rl > 0
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
have rr1 : size rr = 1 := le_antisymm (ablem rl0 H3) rr0
[GOAL]
case pos.inl.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
this : BalancedSz (size rl) (size rr)
rl0 : size rl > 0
rr1 : size rr = 1
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
rw [add_comm] at H3
[GOAL]
case pos.inl.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rr + size rl β€ 2
this : BalancedSz (size rl) (size rr)
rl0 : size rl > 0
rr1 : size rr = 1
β’ BalancedSz 0 (size rl) β§ BalancedSz (0 + size rl + 1) (size rr)
[PROOFSTEP]
rw [rr1, show size rl = 1 from le_antisymm (ablem rr0 H3) rl0]
[GOAL]
case pos.inl.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l = 0
H3 : size rr + size rl β€ 2
this : BalancedSz (size rl) (size rr)
rl0 : size rl > 0
rr1 : size rr = 1
β’ BalancedSz 0 1 β§ BalancedSz (0 + 1 + 1) 1
[PROOFSTEP]
decide
[GOAL]
case pos.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
β’ BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
[PROOFSTEP]
replace H3 := H3p l0
[GOAL]
case pos.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
β’ BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
[PROOFSTEP]
rcases hr.3.1.resolve_left (hlp l0) with β¨_, hbββ©
[GOAL]
case pos.inr.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
leftβ : size rl β€ delta * size rr
hbβ : size rr β€ delta * size rl
β’ BalancedSz (size l) (size rl) β§ BalancedSz (size l + size rl + 1) (size rr)
[PROOFSTEP]
refine' β¨Or.inr β¨_, _β©, Or.inr β¨_, _β©β©
[GOAL]
case pos.inr.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
leftβ : size rl β€ delta * size rr
hbβ : size rr β€ delta * size rl
β’ size l β€ delta * size rl
[PROOFSTEP]
exact Valid'.rotateL_lemmaβ H2 hbβ
[GOAL]
case pos.inr.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
leftβ : size rl β€ delta * size rr
hbβ : size rr β€ delta * size rl
β’ size rl β€ delta * size l
[PROOFSTEP]
exact Nat.le_of_lt_succ (Valid'.rotateL_lemmaβ H3 h)
[GOAL]
case pos.inr.intro.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
leftβ : size rl β€ delta * size rr
hbβ : size rr β€ delta * size rl
β’ size l + size rl + 1 β€ delta * size rr
[PROOFSTEP]
exact Valid'.rotateL_lemmaβ H2 h
[GOAL]
case pos.inr.intro.refine'_4
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : size rl < ratio * size rr
rr0 : size rr > 0
l0 : size l > 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3
leftβ : size rl β€ delta * size rr
hbβ : size rr β€ delta * size rl
β’ size rr β€ delta * (size l + size rl + 1)
[PROOFSTEP]
exact le_trans hbβ (Nat.mul_le_mul_left _ <| le_trans (Nat.le_add_left _ _) (Nat.le_add_right _ _))
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size rl) with rl0 rl0
[GOAL]
case neg.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl = 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
rw [rl0, not_lt, le_zero_iff, Nat.mul_eq_zero] at h
[GOAL]
case neg.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : ratio = 0 β¨ size rr = 0
rl0 : size rl = 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
replace h := h.resolve_left (by decide)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : ratio = 0 β¨ size rr = 0
rl0 : size rl = 0
β’ Β¬ratio = 0
[PROOFSTEP]
decide
[GOAL]
case neg.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
rl0 : size rl = 0
h : size rr = 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
erw [rl0, h, le_zero_iff, Nat.mul_eq_zero] at H2
[GOAL]
case neg.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta = 0 β¨ size l = 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
rl0 : size rl = 0
h : size rr = 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
rw [hr.2.size_eq, rl0, h, H2.resolve_left (by decide)] at H1
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + (0 + 0 + 1) β€ 1
H2 : delta = 0 β¨ size l = 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
rl0 : size rl = 0
h : size rr = 0
β’ Β¬delta = 0
[PROOFSTEP]
decide
[GOAL]
case neg.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬0 + (0 + 0 + 1) β€ 1
H2 : delta = 0 β¨ size l = 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
rl0 : size rl = 0
h : size rr = 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
cases H1 (by decide)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬0 + (0 + 0 + 1) β€ 1
H2 : delta = 0 β¨ size l = 0
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
rl0 : size rl = 0
h : size rr = 0
β’ 0 + (0 + 0 + 1) β€ 1
[PROOFSTEP]
decide
[GOAL]
case neg.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
β’ Valid' oβ (Ordnode.node4L l x rl rx rr) oβ
[PROOFSTEP]
refine' hl.node4L hr.left hr.right rl0 _
[GOAL]
case neg.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size l) with l0 l0
[GOAL]
case neg.inr.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
replace H3 := H3_0 l0
[GOAL]
case neg.inr.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size rr) with rr0 rr0
[GOAL]
case neg.inr.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
rr0 : size rr = 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
have := hr.3.1
[GOAL]
case neg.inr.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
rr0 : size rr = 0
this : BalancedSz (size rl) (size rr)
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
rw [rr0] at this
[GOAL]
case neg.inr.inl.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
rr0 : size rr = 0
this : BalancedSz (size rl) 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
exact Or.inl β¨l0, le_antisymm (balancedSz_zero.1 this) rl0, rr0.symm βΈ zero_le_oneβ©
[GOAL]
case neg.inr.inl.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
rr0 : size rr > 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
exact Or.inl β¨l0, le_antisymm (ablem rr0 <| by rwa [add_comm]) rl0, ablem rl0 H3β©
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l = 0
H3 : size rl + size rr β€ 2
rr0 : size rr > 0
β’ size rr + size rl β€ 2
[PROOFSTEP]
rwa [add_comm]
[GOAL]
case neg.inr.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' (βx) (Ordnode.node rs rl rx rr) oβ
H1 : Β¬size l + size (Ordnode.node rs rl rx rr) β€ 1
H2 : delta * size l β€ size rl + size rr
H3 : 2 * (size rl + size rr) β€ 9 * size l + 3 β¨ size rl + size rr β€ 2
H3_0 : size l = 0 β size rl + size rr β€ 2
H3p : size l > 0 β 2 * (size rl + size rr) β€ 9 * size l + 3
ablem : β {a b : β}, 1 β€ a β a + b β€ 2 β b β€ 1
hlp : size l > 0 β Β¬size rl + size rr β€ 1
h : Β¬size rl < ratio * size rr
rl0 : size rl > 0
l0 : size l > 0
β’ size l = 0 β§ size rl = 1 β§ size rr β€ 1 β¨
0 < size l β§
ratio * size rr β€ size rl β§
delta * size l β€ size rl + size rr β§ 3 * (size rl + size rr) β€ 16 * size l + 9 β§ size rl β€ delta * size rr
[PROOFSTEP]
exact Or.inr β¨l0, not_lt.1 h, H2, Valid'.rotateL_lemmaβ (H3p l0), (hr.3.1.resolve_left (hlp l0)).1β©
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (Ordnode.rotateR l x r) oβ
[PROOFSTEP]
refine' Valid'.dual_iff.2 _
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (Ordnode.dual (Ordnode.rotateR l x r)) oβ
[PROOFSTEP]
rw [dual_rotateR]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (Ordnode.rotateL (Ordnode.dual r) x (Ordnode.dual l)) oβ
[PROOFSTEP]
refine' hr.dual.rotateL hl.dual _ _ _
[GOAL]
case refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Β¬size (Ordnode.dual r) + size (Ordnode.dual l) β€ 1
[PROOFSTEP]
rwa [size_dual, size_dual, add_comm]
[GOAL]
case refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ delta * size (Ordnode.dual r) < size (Ordnode.dual l)
[PROOFSTEP]
rwa [size_dual, size_dual]
[GOAL]
case refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H1 : Β¬size l + size r β€ 1
H2 : delta * size r < size l
H3 : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ 2 * size (Ordnode.dual l) β€ 9 * size (Ordnode.dual r) + 5 β¨ size (Ordnode.dual l) β€ 3
[PROOFSTEP]
rwa [size_dual, size_dual]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (balance' l x r) oβ
[PROOFSTEP]
rw [balance']
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ
(if size l + size r β€ 1 then Ordnode.node' l x r
else
if size r > delta * size l then Ordnode.rotateL l x r
else if size l > delta * size r then Ordnode.rotateR l x r else Ordnode.node' l x r)
oβ
[PROOFSTEP]
split_ifs with h h_1 h_2
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
h : size l + size r β€ 1
β’ Valid' oβ (Ordnode.node' l x r) oβ
[PROOFSTEP]
exact hl.node' hr (Or.inl h)
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
h : Β¬size l + size r β€ 1
h_1 : size r > delta * size l
β’ Valid' oβ (Ordnode.rotateL l x r) oβ
[PROOFSTEP]
exact hl.rotateL hr h h_1 Hβ
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
h : Β¬size l + size r β€ 1
h_1 : Β¬size r > delta * size l
h_2 : size l > delta * size r
β’ Valid' oβ (Ordnode.rotateR l x r) oβ
[PROOFSTEP]
exact hl.rotateR hr h h_2 Hβ
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
h : Β¬size l + size r β€ 1
h_1 : Β¬size r > delta * size l
h_2 : Β¬size l > delta * size r
β’ Valid' oβ (Ordnode.node' l x r) oβ
[PROOFSTEP]
exact hl.node' hr (Or.inr β¨not_lt.1 h_2, not_lt.1 h_1β©)
[GOAL]
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
β’ 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
[PROOFSTEP]
suffices @size Ξ± r β€ 3 * (size l + 1)
by
cases' Nat.eq_zero_or_pos (size l) with l0 l0
Β· apply Or.inr; rwa [l0] at this
change 1 β€ _ at l0 ; apply Or.inl; linarith
[GOAL]
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
β’ 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size l) with l0 l0
[GOAL]
case inl
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
l0 : size l = 0
β’ 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
[PROOFSTEP]
apply Or.inr
[GOAL]
case inl.h
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
l0 : size l = 0
β’ size r β€ 3
[PROOFSTEP]
rwa [l0] at this
[GOAL]
case inr
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
l0 : size l > 0
β’ 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
[PROOFSTEP]
change 1 β€ _ at l0
[GOAL]
case inr
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
l0 : 1 β€ size l
β’ 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
[PROOFSTEP]
apply Or.inl
[GOAL]
case inr.h
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
this : size r β€ 3 * (size l + 1)
l0 : 1 β€ size l
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
linarith
[GOAL]
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
r' : β
H1 : BalancedSz l' r'
H2 : Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l'
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
rcases H2 with (β¨hl, rflβ© | β¨hr, rflβ©)
[GOAL]
case inl.intro
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
hl : Nat.dist (size l) l' β€ 1
H1 : BalancedSz l' (size r)
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
rcases H1 with (h | β¨_, hββ©)
[GOAL]
case inr.intro
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
H1 : BalancedSz (size l) r'
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
rcases H1 with (h | β¨_, hββ©)
[GOAL]
case inl.intro.inl
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
hl : Nat.dist (size l) l' β€ 1
h : l' + size r β€ 1
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
exact le_trans (Nat.le_add_left _ _) (le_trans h (Nat.le_add_left _ _))
[GOAL]
case inl.intro.inr.intro
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l : Ordnode Ξ±
l' : β
r : Ordnode Ξ±
hl : Nat.dist (size l) l' β€ 1
leftβ : l' β€ delta * size r
hβ : size r β€ delta * l'
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
exact le_trans hβ (Nat.mul_le_mul_left _ <| le_trans (Nat.dist_tri_right _ _) (Nat.add_le_add_left hl _))
[GOAL]
case inr.intro.inl
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
h : size l + r' β€ 1
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
exact le_trans (Nat.dist_tri_left' _ _) (le_trans (add_le_add hr (le_trans (Nat.le_add_left _ _) h)) (by linarith))
[GOAL]
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
h : size l + r' β€ 1
β’ 1 + 1 β€ 3 * (size l + 1)
[PROOFSTEP]
linarith
[GOAL]
case inr.intro.inr.intro
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
leftβ : size l β€ delta * r'
hβ : r' β€ delta * size l
β’ size r β€ 3 * (size l + 1)
[PROOFSTEP]
rw [Nat.mul_succ]
[GOAL]
case inr.intro.inr.intro
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
leftβ : size l β€ delta * r'
hβ : r' β€ delta * size l
β’ size r β€ 3 * size l + 3
[PROOFSTEP]
exact le_trans (Nat.dist_tri_right' _ _) (add_le_add hβ (le_trans hr (by decide)))
[GOAL]
Ξ±β : Type u_1
instβ : Preorder Ξ±β
Ξ± : Type u_2
l r : Ordnode Ξ±
r' : β
hr : Nat.dist (size r) r' β€ 1
leftβ : size l β€ delta * r'
hβ : r' β€ delta * size l
β’ 1 β€ 3
[PROOFSTEP]
decide
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : β l' r', BalancedSz l' r' β§ (Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l')
β’ Valid' oβ (Ordnode.balance l x r) oβ
[PROOFSTEP]
rw [balance_eq_balance' hl.3 hr.3 hl.2 hr.2]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : β l' r', BalancedSz l' r' β§ (Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l')
β’ Valid' oβ (Ordnode.balance' l x r) oβ
[PROOFSTEP]
exact hl.balance' hr H
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (balanceL l x r) oβ
[PROOFSTEP]
rw [balanceL_eq_balance hl.2 hr.2 Hβ Hβ, balance_eq_balance' hl.3 hr.3 hl.2 hr.2]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ Valid' oβ (Ordnode.balance' l x r) oβ
[PROOFSTEP]
refine' hl.balance'_aux hr (Or.inl _) Hβ
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size r) with r0 r0
[GOAL]
case inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r = 0
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
rw [r0]
[GOAL]
case inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r = 0
β’ 2 * 0 β€ 9 * size l + 5
[PROOFSTEP]
exact Nat.zero_le _
[GOAL]
case inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
cases' Nat.eq_zero_or_pos (size l) with l0 l0
[GOAL]
case inr.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
l0 : size l = 0
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
rw [l0]
[GOAL]
case inr.inl
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
l0 : size l = 0
β’ 2 * size r β€ 9 * 0 + 5
[PROOFSTEP]
exact le_trans (Nat.mul_le_mul_left _ (Hβ l0)) (by decide)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
l0 : size l = 0
β’ 2 * 1 β€ 9 * 0 + 5
[PROOFSTEP]
decide
[GOAL]
case inr.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 1 β€ size l β 1 β€ size r β size r β€ delta * size l
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
l0 : size l > 0
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
replace Hβ : _ β€ 3 * _ := Hβ l0 r0
[GOAL]
case inr.inr
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size l = 0 β size r β€ 1
Hβ : 2 * size l β€ 9 * size r + 5 β¨ size l β€ 3
r0 : size r > 0
l0 : size l > 0
Hβ : size r β€ 3 * size l
β’ 2 * size r β€ 9 * size l + 5
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ Valid' oβ (Ordnode.balanceL l x r) oβ
[PROOFSTEP]
rw [balanceL_eq_balance' hl.3 hr.3 hl.2 hr.2 H]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ Valid' oβ (Ordnode.balance' l x r) oβ
[PROOFSTEP]
refine' hl.balance' hr _
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : (β l', Raised l' (size l) β§ BalancedSz l' (size r)) β¨ β r', Raised (size r) r' β§ BalancedSz (size l) r'
β’ β l' r', BalancedSz l' r' β§ (Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l')
[PROOFSTEP]
rcases H with (β¨l', e, Hβ© | β¨r', e, Hβ©)
[GOAL]
case inl.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
l' : β
e : Raised l' (size l)
H : BalancedSz l' (size r)
β’ β l' r', BalancedSz l' r' β§ (Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l')
[PROOFSTEP]
exact β¨_, _, H, Or.inl β¨e.dist_le', rflβ©β©
[GOAL]
case inr.intro.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
r' : β
e : Raised (size r) r'
H : BalancedSz (size l) r'
β’ β l' r', BalancedSz l' r' β§ (Nat.dist (size l) l' β€ 1 β§ size r = r' β¨ Nat.dist (size r) r' β€ 1 β§ size l = l')
[PROOFSTEP]
exact β¨_, _, H, Or.inr β¨e.dist_le, rflβ©β©
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size r = 0 β size l β€ 1
Hβ : 1 β€ size r β 1 β€ size l β size l β€ delta * size r
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
β’ Valid' oβ (balanceR l x r) oβ
[PROOFSTEP]
rw [Valid'.dual_iff, dual_balanceR]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size r = 0 β size l β€ 1
Hβ : 1 β€ size r β 1 β€ size l β size l β€ delta * size r
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
β’ Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
[PROOFSTEP]
have := hr.dual.balanceL_aux hl.dual
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size r = 0 β size l β€ 1
Hβ : 1 β€ size r β 1 β€ size l β size l β€ delta * size r
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
this :
(size (Ordnode.dual r) = 0 β size (Ordnode.dual l) β€ 1) β
(1 β€ size (Ordnode.dual r) β 1 β€ size (Ordnode.dual l) β size (Ordnode.dual l) β€ delta * size (Ordnode.dual r)) β
2 * size (Ordnode.dual r) β€ 9 * size (Ordnode.dual l) + 5 β¨ size (Ordnode.dual r) β€ 3 β
Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
β’ Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
[PROOFSTEP]
rw [size_dual, size_dual] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
Hβ : size r = 0 β size l β€ 1
Hβ : 1 β€ size r β 1 β€ size l β size l β€ delta * size r
Hβ : 2 * size r β€ 9 * size l + 5 β¨ size r β€ 3
this :
(size r = 0 β size l β€ 1) β
(1 β€ size r β 1 β€ size l β size l β€ delta * size r) β
2 * size r β€ 9 * size l + 5 β¨ size r β€ 3 β Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
β’ Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
[PROOFSTEP]
exact this Hβ Hβ Hβ
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ Valid' oβ (Ordnode.balanceR l x r) oβ
[PROOFSTEP]
rw [Valid'.dual_iff, dual_balanceR]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l βx
hr : Valid' (βx) r oβ
H : (β l', Raised (size l) l' β§ BalancedSz l' (size r)) β¨ β r', Raised r' (size r) β§ BalancedSz (size l) r'
β’ Valid' oβ (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) oβ
[PROOFSTEP]
exact hr.dual.balanceL hl.dual (balance_sz_dual H)
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node s l x r) oβ
β’ Valid' oβ (eraseMax (Ordnode.node' l x r)) β(findMax' x r) β§
size (Ordnode.node' l x r) = size (eraseMax (Ordnode.node' l x r)) + 1
[PROOFSTEP]
have := H.2.eq_node'
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node s l x r) oβ
this : Ordnode.node s l x r = Ordnode.node' l x r
β’ Valid' oβ (eraseMax (Ordnode.node' l x r)) β(findMax' x r) β§
size (Ordnode.node' l x r) = size (eraseMax (Ordnode.node' l x r)) + 1
[PROOFSTEP]
rw [this] at H
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node' l x r) oβ
this : Ordnode.node s l x r = Ordnode.node' l x r
β’ Valid' oβ (eraseMax (Ordnode.node' l x r)) β(findMax' x r) β§
size (Ordnode.node' l x r) = size (eraseMax (Ordnode.node' l x r)) + 1
[PROOFSTEP]
clear this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node' l x r) oβ
β’ Valid' oβ (eraseMax (Ordnode.node' l x r)) β(findMax' x r) β§
size (Ordnode.node' l x r) = size (eraseMax (Ordnode.node' l x r)) + 1
[PROOFSTEP]
induction' r with rs rl rx rr _ IHrr generalizing l x oβ
[GOAL]
case nil
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x nil) oβ
β’ Valid' oβ (eraseMax (Ordnode.node' l x nil)) β(findMax' x nil) β§
size (Ordnode.node' l x nil) = size (eraseMax (Ordnode.node' l x nil)) + 1
[PROOFSTEP]
exact β¨H.left, rflβ©
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node rs rl rx rr)) oβ
β’ Valid' oβ (eraseMax (Ordnode.node' l x (Ordnode.node rs rl rx rr))) β(findMax' x (Ordnode.node rs rl rx rr)) β§
size (Ordnode.node' l x (Ordnode.node rs rl rx rr)) =
size (eraseMax (Ordnode.node' l x (Ordnode.node rs rl rx rr))) + 1
[PROOFSTEP]
have := H.2.2.2.eq_node'
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node rs rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
β’ Valid' oβ (eraseMax (Ordnode.node' l x (Ordnode.node rs rl rx rr))) β(findMax' x (Ordnode.node rs rl rx rr)) β§
size (Ordnode.node' l x (Ordnode.node rs rl rx rr)) =
size (eraseMax (Ordnode.node' l x (Ordnode.node rs rl rx rr))) + 1
[PROOFSTEP]
rw [this] at H β’
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node' rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
β’ Valid' oβ (eraseMax (Ordnode.node' l x (Ordnode.node' rl rx rr))) β(findMax' x (Ordnode.node' rl rx rr)) β§
size (Ordnode.node' l x (Ordnode.node' rl rx rr)) = size (eraseMax (Ordnode.node' l x (Ordnode.node' rl rx rr))) + 1
[PROOFSTEP]
rcases IHrr H.right with β¨h, eβ©
[GOAL]
case node.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node' rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
h : Valid' (βx) (eraseMax (Ordnode.node' rl rx rr)) β(findMax' rx rr)
e : size (Ordnode.node' rl rx rr) = size (eraseMax (Ordnode.node' rl rx rr)) + 1
β’ Valid' oβ (eraseMax (Ordnode.node' l x (Ordnode.node' rl rx rr))) β(findMax' x (Ordnode.node' rl rx rr)) β§
size (Ordnode.node' l x (Ordnode.node' rl rx rr)) = size (eraseMax (Ordnode.node' l x (Ordnode.node' rl rx rr))) + 1
[PROOFSTEP]
refine' β¨Valid'.balanceL H.left h (Or.inr β¨_, Or.inr e, H.3.1β©), _β©
[GOAL]
case node.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node' rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
h : Valid' (βx) (eraseMax (Ordnode.node' rl rx rr)) β(findMax' rx rr)
e : size (Ordnode.node' rl rx rr) = size (eraseMax (Ordnode.node' rl rx rr)) + 1
β’ size (Ordnode.node' l x (Ordnode.node' rl rx rr)) = size (eraseMax (Ordnode.node' l x (Ordnode.node' rl rx rr))) + 1
[PROOFSTEP]
rw [eraseMax, size_balanceL H.3.2.1 h.3 H.2.2.1 h.2 (Or.inr β¨_, Or.inr e, H.3.1β©)]
[GOAL]
case node.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node' rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
h : Valid' (βx) (eraseMax (Ordnode.node' rl rx rr)) β(findMax' rx rr)
e : size (Ordnode.node' rl rx rr) = size (eraseMax (Ordnode.node' rl rx rr)) + 1
β’ size (Ordnode.node' l x (Ordnode.node' rl rx rr)) = size l + size (eraseMax (Ordnode.node' rl rx rr)) + 1 + 1
[PROOFSTEP]
rw [size, e]
[GOAL]
case node.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
lβ : Ordnode Ξ±
xβ : Ξ±
r : Ordnode Ξ±
oββ : WithBot Ξ±
oβ : WithTop Ξ±
Hβ : Valid' oββ (Ordnode.node' lβ xβ r) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
l_ihβ :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rl) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rl)) β(findMax' x rl) β§
size (Ordnode.node' l x rl) = size (eraseMax (Ordnode.node' l x rl)) + 1
IHrr :
β {l : Ordnode Ξ±} {x : Ξ±} {oβ : WithBot Ξ±},
Valid' oβ (Ordnode.node' l x rr) oβ β
Valid' oβ (eraseMax (Ordnode.node' l x rr)) β(findMax' x rr) β§
size (Ordnode.node' l x rr) = size (eraseMax (Ordnode.node' l x rr)) + 1
l : Ordnode Ξ±
x : Ξ±
oβ : WithBot Ξ±
H : Valid' oβ (Ordnode.node' l x (Ordnode.node' rl rx rr)) oβ
this : Ordnode.node rs rl rx rr = Ordnode.node' rl rx rr
h : Valid' (βx) (eraseMax (Ordnode.node' rl rx rr)) β(findMax' rx rr)
e : size (Ordnode.node' rl rx rr) = size (eraseMax (Ordnode.node' rl rx rr)) + 1
β’ size l + (size (eraseMax (Ordnode.node' rl rx rr)) + 1) + 1 =
size l + size (eraseMax (Ordnode.node' rl rx rr)) + 1 + 1
[PROOFSTEP]
rfl
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node s l x r) oβ
β’ Valid' (β(findMin' l x)) (eraseMin (Ordnode.node' l x r)) oβ β§
size (Ordnode.node' l x r) = size (eraseMin (Ordnode.node' l x r)) + 1
[PROOFSTEP]
have := H.dual.eraseMax_aux
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
H : Valid' oβ (Ordnode.node s l x r) oβ
this :
Valid' oβ (eraseMax (Ordnode.node' (Ordnode.dual r) x (Ordnode.dual l))) β(findMax' x (Ordnode.dual l)) β§
size (Ordnode.node' (Ordnode.dual r) x (Ordnode.dual l)) =
size (eraseMax (Ordnode.node' (Ordnode.dual r) x (Ordnode.dual l))) + 1
β’ Valid' (β(findMin' l x)) (eraseMin (Ordnode.node' l x r)) oβ β§
size (Ordnode.node' l x r) = size (eraseMin (Ordnode.node' l x r)) + 1
[PROOFSTEP]
rwa [β dual_node', size_dual, β dual_eraseMin, size_dual, β Valid'.dual_iff, findMax'_dual] at this
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Valid (node sizeβ l x r)
β’ Valid (eraseMin (node sizeβ l x r))
[PROOFSTEP]
rw [h.2.eq_node']
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
sizeβ : β
l : Ordnode Ξ±
x : Ξ±
r : Ordnode Ξ±
h : Valid (node sizeβ l x r)
β’ Valid (eraseMin (node' l x r))
[PROOFSTEP]
exact h.eraseMin_aux.1.valid
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (eraseMax t)
[PROOFSTEP]
rw [Valid.dual_iff, dual_eraseMax]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (eraseMin (dual t))
[PROOFSTEP]
exact eraseMin.valid h.dual
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l oβ
hr : Valid' oβ r oβ
sep : All (fun x => All (fun y => x < y) r) l
bal : BalancedSz (size l) (size r)
β’ Valid' oβ (glue l r) oβ β§ size (glue l r) = size l + size r
[PROOFSTEP]
cases' l with ls ll lx lr
[GOAL]
case nil
Ξ± : Type u_1
instβ : Preorder Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hr : Valid' oβ r oβ
hl : Valid' oβ nil oβ
sep : All (fun x => All (fun y => x < y) r) nil
bal : BalancedSz (size nil) (size r)
β’ Valid' oβ (glue nil r) oβ β§ size (glue nil r) = size nil + size r
[PROOFSTEP]
exact β¨hr, (zero_add _).symmβ©
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hr : Valid' oβ r oβ
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
sep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size r)
β’ Valid' oβ (glue (Ordnode.node ls ll lx lr) r) oβ β§
size (glue (Ordnode.node ls ll lx lr) r) = size (Ordnode.node ls ll lx lr) + size r
[PROOFSTEP]
cases' r with rs rl rx rr
[GOAL]
case node.nil
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ nil oβ
sep : All (fun x => All (fun y => x < y) nil) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size nil)
β’ Valid' oβ (glue (Ordnode.node ls ll lx lr) nil) oβ β§
size (glue (Ordnode.node ls ll lx lr) nil) = size (Ordnode.node ls ll lx lr) + size nil
[PROOFSTEP]
exact β¨hl, rflβ©
[GOAL]
case node.node
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
β’ Valid' oβ (glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) oβ β§
size (glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
dsimp [glue]
[GOAL]
case node.node
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
β’ Valid' oβ
(if ls > rs then Ordnode.balanceR (splitMax' ll lx lr).fst (splitMax' ll lx lr).snd (Ordnode.node rs rl rx rr)
else Ordnode.balanceL (Ordnode.node ls ll lx lr) (splitMin' rl rx rr).fst (splitMin' rl rx rr).snd)
oβ β§
size
(if ls > rs then Ordnode.balanceR (splitMax' ll lx lr).fst (splitMax' ll lx lr).snd (Ordnode.node rs rl rx rr)
else Ordnode.balanceL (Ordnode.node ls ll lx lr) (splitMin' rl rx rr).fst (splitMin' rl rx rr).snd) =
ls + rs
[PROOFSTEP]
split_ifs
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
β’ Valid' oβ (Ordnode.balanceR (splitMax' ll lx lr).fst (splitMax' ll lx lr).snd (Ordnode.node rs rl rx rr)) oβ β§
size (Ordnode.balanceR (splitMax' ll lx lr).fst (splitMax' ll lx lr).snd (Ordnode.node rs rl rx rr)) = ls + rs
[PROOFSTEP]
rw [splitMax_eq]
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
β’ Valid' oβ
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr))
oβ β§
size
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr)) =
ls + rs
case pos.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
β’ β
[PROOFSTEP]
cases' Valid'.eraseMax_aux hl with v e
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
β’ Valid' oβ
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr))
oβ β§
size
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr)) =
ls + rs
case pos.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
β’ β
[PROOFSTEP]
suffices H
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H : ?m.354852
β’ Valid' oβ
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr))
oβ β§
size
(Ordnode.balanceR (eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node ?pos.sβ ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr)) =
ls + rs
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
β’ ?m.354852
case pos.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
β’ β
[PROOFSTEP]
refine' β¨Valid'.balanceR v (hr.of_gt _ _) H, _β©
[GOAL]
case pos.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H :
(β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
β’ Bounded nil (β(eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).snd) oβ
[PROOFSTEP]
refine' findMax'_all (P := fun a : Ξ± => Bounded nil (a : WithTop Ξ±) oβ) lx lr hl.1.2.to_nil (sep.2.2.imp _)
[GOAL]
case pos.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H :
(β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
β’ β (a : Ξ±), All (fun y => a < y) (Ordnode.node rs rl rx rr) β Bounded nil (βa) oβ
[PROOFSTEP]
exact fun x h => hr.1.2.to_nil.mono_left (le_of_lt h.2.1)
[GOAL]
case pos.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H :
(β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
β’ All (fun x => x > (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).snd)
(Ordnode.node rs rl rx rr)
[PROOFSTEP]
exact @findMax'_all _ (fun a => All (Β· > a) (.node rs rl rx rr)) lx lr sep.2.1 sep.2.2
[GOAL]
case pos.intro.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H :
(β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
β’ size
(Ordnode.balanceR (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst
(eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).snd (Ordnode.node rs rl rx rr)) =
ls + rs
[PROOFSTEP]
rw [size_balanceR v.3 hr.3 v.2 hr.2 H, add_right_comm, β e, hl.2.1]
[GOAL]
case pos.intro.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
H :
(β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
β’ size (Ordnode.node' ll lx lr) + size (Ordnode.node rs rl rx rr) = size ll + size lr + 1 + rs
[PROOFSTEP]
rfl
[GOAL]
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
β’ (β l',
Raised (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) l' β§
BalancedSz l' (size (Ordnode.node rs rl rx rr))) β¨
β r',
Raised r' (size (Ordnode.node rs rl rx rr)) β§
BalancedSz (size (eraseMax (Ordnode.node (size ll + size lr + 1) ll lx lr), findMax' lx lr).fst) r'
[PROOFSTEP]
refine' Or.inl β¨_, Or.inr e, _β©
[GOAL]
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : ls > rs
v : Valid' oβ (eraseMax (Ordnode.node' ll lx lr)) β(findMax' lx lr)
e : size (Ordnode.node' ll lx lr) = size (eraseMax (Ordnode.node' ll lx lr)) + 1
β’ BalancedSz (size (Ordnode.node' ll lx lr)) (size (Ordnode.node rs rl rx rr))
[PROOFSTEP]
rwa [hl.2.eq_node'] at bal
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
β’ Valid' oβ (Ordnode.balanceL (Ordnode.node ls ll lx lr) (splitMin' rl rx rr).fst (splitMin' rl rx rr).snd) oβ β§
size (Ordnode.balanceL (Ordnode.node ls ll lx lr) (splitMin' rl rx rr).fst (splitMin' rl rx rr).snd) = ls + rs
[PROOFSTEP]
rw [splitMin_eq]
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
β’ Valid' oβ
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd)
oβ β§
size
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd) =
ls + rs
case neg.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
β’ β
[PROOFSTEP]
cases' Valid'.eraseMin_aux hr with v e
[GOAL]
case neg.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
β’ Valid' oβ
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd)
oβ β§
size
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd) =
ls + rs
case neg.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
β’ β
[PROOFSTEP]
suffices H
[GOAL]
case neg.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H : ?m.355955
β’ Valid' oβ
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd)
oβ β§
size
(Ordnode.balanceL (Ordnode.node ls ll lx lr) (findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node ?neg.sβ rl rx rr)).snd) =
ls + rs
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
β’ ?m.355955
case neg.s
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
β’ β
[PROOFSTEP]
refine' β¨Valid'.balanceL (hl.of_lt _ _) v H, _β©
[GOAL]
case neg.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H :
(β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
β’ Bounded nil oβ β(findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).fst
[PROOFSTEP]
refine' @findMin'_all (P := fun a : Ξ± => Bounded nil oβ (a : WithBot Ξ±)) rl rx (sep.2.1.1.imp _) hr.1.1.to_nil
[GOAL]
case neg.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H :
(β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
β’ β (a : Ξ±), lx < a β Bounded nil oβ βa
[PROOFSTEP]
exact fun y h => hl.1.1.to_nil.mono_right (le_of_lt h)
[GOAL]
case neg.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H :
(β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
β’ All (fun x => x < (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).fst)
(Ordnode.node ls ll lx lr)
[PROOFSTEP]
exact
@findMin'_all _ (fun a => All (Β· < a) (.node ls ll lx lr)) rl rx
(all_iff_forall.2 fun x hx => sep.imp fun y hy => all_iff_forall.1 hy.1 _ hx) (sep.imp fun y hy => hy.2.1)
[GOAL]
case neg.intro.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H :
(β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
β’ size
(Ordnode.balanceL (Ordnode.node ls ll lx lr)
(findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).fst
(findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) =
ls + rs
[PROOFSTEP]
rw [size_balanceL hl.3 v.3 hl.2 v.2 H, add_assoc, β e, hr.2.1]
[GOAL]
case neg.intro.refine'_3
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
H :
(β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
β’ size (Ordnode.node ls ll lx lr) + size (Ordnode.node' rl rx rr) = ls + (size rl + size rr + 1)
[PROOFSTEP]
rfl
[GOAL]
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
β’ (β l',
Raised l' (size (Ordnode.node ls ll lx lr)) β§
BalancedSz l' (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd)) β¨
β r',
Raised (size (findMin' rl rx, eraseMin (Ordnode.node (size rl + size rr + 1) rl rx rr)).snd) r' β§
BalancedSz (size (Ordnode.node ls ll lx lr)) r'
[PROOFSTEP]
refine' Or.inr β¨_, Or.inr e, _β©
[GOAL]
case H
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
bal : BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node rs rl rx rr))
hβ : Β¬ls > rs
v : Valid' (β(findMin' rl rx)) (eraseMin (Ordnode.node' rl rx rr)) oβ
e : size (Ordnode.node' rl rx rr) = size (eraseMin (Ordnode.node' rl rx rr)) + 1
β’ BalancedSz (size (Ordnode.node ls ll lx lr)) (size (Ordnode.node' rl rx rr))
[PROOFSTEP]
rwa [hr.2.eq_node'] at bal
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
a b c : β
hβ : 3 * a < b + c + 1
hβ : b β€ 3 * c
β’ 2 * (a + b) β€ 9 * c + 5
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : delta * ls < rs
v : Valid' oβ t βrx
e : size t = ls + size rl
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
rw [hl.2.1] at e
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : delta * ls < rs
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
rw [hl.2.1, hr.2.1, delta] at h
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
rcases hr.3.1 with (H | β¨hrβ, hrββ©)
[GOAL]
case inl
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
H : size rl + size rr β€ 1
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
linarith
[GOAL]
case inr.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
suffices Hβ
[GOAL]
case inr.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : ?m.365245
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
β’ ?m.365245
[PROOFSTEP]
suffices Hβ
[GOAL]
case inr.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : ?m.365245
Hβ : ?m.365252
β’ Valid' oβ (Ordnode.balanceL t rx rr) oβ β§ size (Ordnode.balanceL t rx rr) = ls + rs
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : ?m.365245
β’ ?m.365252
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
β’ ?m.365245
[PROOFSTEP]
refine' β¨Valid'.balanceL_aux v hr.right Hβ Hβ _, _β©
[GOAL]
case inr.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
Hβ : size t = 0 β size rr β€ 1
β’ 2 * size t β€ 9 * size rr + 5 β¨ size t β€ 3
[PROOFSTEP]
rw [e]
[GOAL]
case inr.intro.refine'_1
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
Hβ : size t = 0 β size rr β€ 1
β’ 2 * (size ll + size lr + 1 + size rl) β€ 9 * size rr + 5 β¨ size ll + size lr + 1 + size rl β€ 3
[PROOFSTEP]
exact Or.inl (Valid'.merge_lemma h hrβ)
[GOAL]
case inr.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
Hβ : size t = 0 β size rr β€ 1
β’ size (Ordnode.balanceL t rx rr) = ls + rs
[PROOFSTEP]
rw [balanceL_eq_balance v.2 hr.2.2.2 Hβ Hβ, balance_eq_balance' v.3 hr.3.2.2 v.2 hr.2.2.2, size_balance' v.2 hr.2.2.2,
e, hl.2.1, hr.2.1]
[GOAL]
case inr.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
Hβ : size t = 0 β size rr β€ 1
β’ size ll + size lr + 1 + size rl + size rr + 1 = size ll + size lr + 1 + (size rl + size rr + 1)
[PROOFSTEP]
abel
[GOAL]
case inr.intro.refine'_2
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
Hβ : size t = 0 β size rr β€ 1
β’ size ll + size lr + 1 + size rl + size rr + 1 = size ll + size lr + 1 + (size rl + size rr + 1)
[PROOFSTEP]
abel
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
β’ size t = 0 β size rr β€ 1
[PROOFSTEP]
rw [e, add_right_comm]
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
Hβ : 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
β’ size ll + size lr + size rl + 1 = 0 β size rr β€ 1
[PROOFSTEP]
rintro β¨β©
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
β’ 1 β€ size t β 1 β€ size rr β size rr β€ delta * size t
[PROOFSTEP]
intro _ _
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
aβΒΉ : 1 β€ size t
aβ : 1 β€ size rr
β’ size rr β€ delta * size t
[PROOFSTEP]
rw [e]
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ delta * size rl
aβΒΉ : 1 β€ size t
aβ : 1 β€ size rr
β’ size rr β€ delta * (size ll + size lr + 1 + size rl)
[PROOFSTEP]
unfold delta at hrβ β’
[GOAL]
case Hβ
Ξ± : Type u_1
instβ : Preorder Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr t : Ordnode Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
h : 3 * (size ll + size lr + 1) < size rl + size rr + 1
v : Valid' oβ t βrx
e : size t = size ll + size lr + 1 + size rl
hrβ : size rl β€ delta * size rr
hrβ : size rr β€ 3 * size rl
aβΒΉ : 1 β€ size t
aβ : 1 β€ size rr
β’ size rr β€ 3 * (size ll + size lr + 1 + size rl)
[PROOFSTEP]
linarith
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
l r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ l oβ
hr : Valid' oβ r oβ
sep : All (fun x => All (fun y => x < y) r) l
β’ Valid' oβ (merge l r) oβ β§ size (merge l r) = size l + size r
[PROOFSTEP]
induction' l with ls ll lx lr _ IHlr generalizing oβ oβ r
[GOAL]
case nil
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ l oββ
hrβ : Valid' oββ rβ oββ
sepβ : All (fun x => All (fun y => x < y) rβ) l
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ nil oβ
hr : Valid' oβ r oβ
sep : All (fun x => All (fun y => x < y) r) nil
β’ Valid' oβ (merge nil r) oβ β§ size (merge nil r) = size nil + size r
[PROOFSTEP]
exact β¨hr, (zero_add _).symmβ©
[GOAL]
case node
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ l oββ
hrβ : Valid' oββ rβ oββ
sepβ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ r oβ
sep : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
β’ Valid' oβ (merge (Ordnode.node ls ll lx lr) r) oβ β§
size (merge (Ordnode.node ls ll lx lr) r) = size (Ordnode.node ls ll lx lr) + size r
[PROOFSTEP]
induction' r with rs rl rx rr IHrl _ generalizing oβ oβ
[GOAL]
case node.nil
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ nil oβ
sep : All (fun x => All (fun y => x < y) nil) (Ordnode.node ls ll lx lr)
β’ Valid' oβ (merge (Ordnode.node ls ll lx lr) nil) oβ β§
size (merge (Ordnode.node ls ll lx lr) nil) = size (Ordnode.node ls ll lx lr) + size nil
[PROOFSTEP]
exact β¨hl, rflβ©
[GOAL]
case node.node
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
β’ Valid' oβ (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) oβ β§
size (merge (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
rw [merge_node]
[GOAL]
case node.node
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
β’ Valid' oβ
(if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr
else
if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))
else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr))
oβ β§
size
(if delta * ls < rs then Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr
else
if delta * rs < ls then Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))
else Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
split_ifs with h h_1
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : delta * ls < rs
β’ Valid' oβ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) oβ β§
size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
cases' IHrl (hl.of_lt hr.1.1.to_nil <| sep.imp fun x h => h.2.1) hr.left (sep.imp fun x h => h.1) with v e
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : delta * ls < rs
v : Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) βrx
e : size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
β’ Valid' oβ (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) oβ β§
size (Ordnode.balanceL (merge (Ordnode.node ls ll lx lr) rl) rx rr) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
exact Valid'.merge_auxβ hl hr h v e
[GOAL]
case pos
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
β’ Valid' oβ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) oβ β§
size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
cases' IHlr hl.right (hr.of_gt hl.1.2.to_nil sep.2.1) sep.2.2 with v e
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
v : Valid' (βlx) (merge lr (Ordnode.node rs rl rx rr)) oβ
e : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)
β’ Valid' oβ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) oβ β§
size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
have := Valid'.merge_auxβ hr.dual hl.dual h_1 v.dual
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
v : Valid' (βlx) (merge lr (Ordnode.node rs rl rx rr)) oβ
e : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)
this :
size (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) = rs + size (Ordnode.dual lr) β
Valid' oβ (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) oβ β§
size (Ordnode.balanceL (Ordnode.dual (merge lr (Ordnode.node rs rl rx rr))) lx (Ordnode.dual ll)) = rs + ls
β’ Valid' oβ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) oβ β§
size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
rw [size_dual, add_comm, size_dual, β dual_balanceR, β Valid'.dual_iff, size_dual, add_comm rs] at this
[GOAL]
case pos.intro
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : Β¬delta * ls < rs
h_1 : delta * rs < ls
v : Valid' (βlx) (merge lr (Ordnode.node rs rl rx rr)) oβ
e : size (merge lr (Ordnode.node rs rl rx rr)) = size lr + size (Ordnode.node rs rl rx rr)
this :
size (merge lr (Ordnode.node rs rl rx rr)) = size lr + rs β
Valid' oβ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) oβ β§
size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) = ls + rs
β’ Valid' oβ (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) oβ β§
size (Ordnode.balanceR ll lx (merge lr (Ordnode.node rs rl rx rr))) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
exact this e
[GOAL]
case neg
Ξ± : Type u_1
instβ : Preorder Ξ±
l rβ : Ordnode Ξ±
oββΒΉ : WithBot Ξ±
oββΒΉ : WithTop Ξ±
hlβΒΉ : Valid' oββΒΉ l oββΒΉ
hrβΒΉ : Valid' oββΒΉ rβ oββΒΉ
sepβΒΉ : All (fun x => All (fun y => x < y) rβ) l
ls : β
ll : Ordnode Ξ±
lx : Ξ±
lr : Ordnode Ξ±
l_ihβ :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ ll oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) ll β Valid' oβ (merge ll r) oβ β§ size (merge ll r) = size ll + size r
IHlr :
β {r : Ordnode Ξ±} {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ lr oβ β
Valid' oβ r oβ β
All (fun x => All (fun y => x < y) r) lr β Valid' oβ (merge lr r) oβ β§ size (merge lr r) = size lr + size r
r : Ordnode Ξ±
oββ : WithBot Ξ±
oββ : WithTop Ξ±
hlβ : Valid' oββ (Ordnode.node ls ll lx lr) oββ
hrβ : Valid' oββ r oββ
sepβ : All (fun x => All (fun y => x < y) r) (Ordnode.node ls ll lx lr)
rs : β
rl : Ordnode Ξ±
rx : Ξ±
rr : Ordnode Ξ±
IHrl :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rl oβ β
All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rl) oβ β§
size (merge (Ordnode.node ls ll lx lr) rl) = size (Ordnode.node ls ll lx lr) + size rl
r_ihβ :
β {oβ : WithBot Ξ±} {oβ : WithTop Ξ±},
Valid' oβ (Ordnode.node ls ll lx lr) oβ β
Valid' oβ rr oβ β
All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) β
Valid' oβ (merge (Ordnode.node ls ll lx lr) rr) oβ β§
size (merge (Ordnode.node ls ll lx lr) rr) = size (Ordnode.node ls ll lx lr) + size rr
oβ : WithBot Ξ±
oβ : WithTop Ξ±
hl : Valid' oβ (Ordnode.node ls ll lx lr) oβ
hr : Valid' oβ (Ordnode.node rs rl rx rr) oβ
sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)
h : Β¬delta * ls < rs
h_1 : Β¬delta * rs < ls
β’ Valid' oβ (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) oβ β§
size (Ordnode.glue (Ordnode.node ls ll lx lr) (Ordnode.node rs rl rx rr)) =
size (Ordnode.node ls ll lx lr) + size (Ordnode.node rs rl rx rr)
[PROOFSTEP]
refine' Valid'.glue_aux hl hr sep (Or.inr β¨not_lt.1 h_1, not_lt.1 hβ©)
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
β’ Valid' oβ (insertWith f x (node sz l y r)) oβ β§ Raised (size (node sz l y r)) (size (insertWith f x (node sz l y r)))
[PROOFSTEP]
rw [insertWith, cmpLE]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
β’ Valid' oβ
(match if x β€ y then if y β€ x then Ordering.eq else Ordering.lt else Ordering.gt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r))
oβ β§
Raised (size (node sz l y r))
(size
(match if x β€ y then if y β€ x then Ordering.eq else Ordering.lt else Ordering.gt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r)))
[PROOFSTEP]
split_ifs with h_1 h_2
[GOAL]
case pos
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : y β€ x
β’ Valid' oβ
(match Ordering.eq with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r))
oβ β§
Raised (size (node sz l y r))
(size
(match Ordering.eq with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r)))
[PROOFSTEP]
dsimp only
[GOAL]
case neg
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
β’ Valid' oβ
(match Ordering.lt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r))
oβ β§
Raised (size (node sz l y r))
(size
(match Ordering.lt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r)))
[PROOFSTEP]
dsimp only
[GOAL]
case neg
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
β’ Valid' oβ
(match Ordering.gt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r))
oβ β§
Raised (size (node sz l y r))
(size
(match Ordering.gt with
| Ordering.lt => balanceL (insertWith f x l) y r
| Ordering.eq => node sz l (f y) r
| Ordering.gt => balanceR l y (insertWith f x r)))
[PROOFSTEP]
dsimp only
[GOAL]
case pos
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : y β€ x
β’ Valid' oβ (node sz l (f y) r) oβ β§ Raised (size (node sz l y r)) (size (node sz l (f y) r))
[PROOFSTEP]
rcases h with β¨β¨lx, xrβ©, hs, hbβ©
[GOAL]
case pos.mk.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : y β€ x
hs : Sized (node sz l y r)
hb : Balanced (node sz l y r)
lx : Bounded l oβ βy
xr : Bounded r (βy) oβ
β’ Valid' oβ (node sz l (f y) r) oβ β§ Raised (size (node sz l y r)) (size (node sz l (f y) r))
[PROOFSTEP]
rcases hf _ β¨h_1, h_2β© with β¨xf, fxβ©
[GOAL]
case pos.mk.intro.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : y β€ x
hs : Sized (node sz l y r)
hb : Balanced (node sz l y r)
lx : Bounded l oβ βy
xr : Bounded r (βy) oβ
xf : x β€ f y
fx : f y β€ x
β’ Valid' oβ (node sz l (f y) r) oβ β§ Raised (size (node sz l y r)) (size (node sz l (f y) r))
[PROOFSTEP]
refine' β¨β¨β¨lx.mono_right (le_trans h_2 xf), xr.mono_left (le_trans fx h_1)β©, hs, hbβ©, Or.inl rflβ©
[GOAL]
case neg
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
β’ Valid' oβ (balanceL (insertWith f x l) y r) oβ β§
Raised (size (node sz l y r)) (size (balanceL (insertWith f x l) y r))
[PROOFSTEP]
rcases insertWith.valid_aux f x hf h.left bl (lt_of_le_not_le h_1 h_2) with β¨vl, eβ©
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
vl : Valid' oβ (insertWith f x l) βy
e : Raised (size l) (size (insertWith f x l))
β’ Valid' oβ (balanceL (insertWith f x l) y r) oβ β§
Raised (size (node sz l y r)) (size (balanceL (insertWith f x l) y r))
[PROOFSTEP]
suffices H
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
vl : Valid' oβ (insertWith f x l) βy
e : Raised (size l) (size (insertWith f x l))
H : ?m.378189
β’ Valid' oβ (balanceL (insertWith f x l) y r) oβ β§
Raised (size (node sz l y r)) (size (balanceL (insertWith f x l) y r))
[PROOFSTEP]
refine' β¨vl.balanceL h.right H, _β©
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
vl : Valid' oβ (insertWith f x l) βy
e : Raised (size l) (size (insertWith f x l))
H :
(β l', Raised l' (size (insertWith f x l)) β§ BalancedSz l' (size r)) β¨
β r', Raised (size r) r' β§ BalancedSz (size (insertWith f x l)) r'
β’ Raised (size (node sz l y r)) (size (balanceL (insertWith f x l) y r))
[PROOFSTEP]
rw [size_balanceL vl.3 h.3.2.2 vl.2 h.2.2.2 H, h.2.size_eq]
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
vl : Valid' oβ (insertWith f x l) βy
e : Raised (size l) (size (insertWith f x l))
H :
(β l', Raised l' (size (insertWith f x l)) β§ BalancedSz l' (size r)) β¨
β r', Raised (size r) r' β§ BalancedSz (size (insertWith f x l)) r'
β’ Raised (size l + size r + 1) (size (insertWith f x l) + size r + 1)
[PROOFSTEP]
refine' (e.add_right _).add_right _
[GOAL]
case H
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : x β€ y
h_2 : Β¬y β€ x
vl : Valid' oβ (insertWith f x l) βy
e : Raised (size l) (size (insertWith f x l))
β’ (β l', Raised l' (size (insertWith f x l)) β§ BalancedSz l' (size r)) β¨
β r', Raised (size r) r' β§ BalancedSz (size (insertWith f x l)) r'
[PROOFSTEP]
exact Or.inl β¨_, e, h.3.1β©
[GOAL]
case neg
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
β’ Valid' oβ (balanceR l y (insertWith f x r)) oβ β§
Raised (size (node sz l y r)) (size (balanceR l y (insertWith f x r)))
[PROOFSTEP]
have : y < x := lt_of_le_not_le ((total_of (Β· β€ Β·) _ _).resolve_left h_1) h_1
[GOAL]
case neg
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
β’ Valid' oβ (balanceR l y (insertWith f x r)) oβ β§
Raised (size (node sz l y r)) (size (balanceR l y (insertWith f x r)))
[PROOFSTEP]
rcases insertWith.valid_aux f x hf h.right this br with β¨vr, eβ©
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
vr : Valid' (βy) (insertWith f x r) oβ
e : Raised (size r) (size (insertWith f x r))
β’ Valid' oβ (balanceR l y (insertWith f x r)) oβ β§
Raised (size (node sz l y r)) (size (balanceR l y (insertWith f x r)))
[PROOFSTEP]
suffices H
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
vr : Valid' (βy) (insertWith f x r) oβ
e : Raised (size r) (size (insertWith f x r))
H : ?m.378711
β’ Valid' oβ (balanceR l y (insertWith f x r)) oβ β§
Raised (size (node sz l y r)) (size (balanceR l y (insertWith f x r)))
[PROOFSTEP]
refine' β¨h.left.balanceR vr H, _β©
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
vr : Valid' (βy) (insertWith f x r) oβ
e : Raised (size r) (size (insertWith f x r))
H :
(β l', Raised (size l) l' β§ BalancedSz l' (size (insertWith f x r))) β¨
β r', Raised r' (size (insertWith f x r)) β§ BalancedSz (size l) r'
β’ Raised (size (node sz l y r)) (size (balanceR l y (insertWith f x r)))
[PROOFSTEP]
rw [size_balanceR h.3.2.1 vr.3 h.2.2.1 vr.2 H, h.2.size_eq]
[GOAL]
case neg.intro
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
vr : Valid' (βy) (insertWith f x r) oβ
e : Raised (size r) (size (insertWith f x r))
H :
(β l', Raised (size l) l' β§ BalancedSz l' (size (insertWith f x r))) β¨
β r', Raised r' (size (insertWith f x r)) β§ BalancedSz (size l) r'
β’ Raised (size l + size r + 1) (size l + size (insertWith f x r) + 1)
[PROOFSTEP]
refine' (e.add_left _).add_right _
[GOAL]
case H
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
f : Ξ± β Ξ±
x : Ξ±
hf : β (y : Ξ±), x β€ y β§ y β€ x β x β€ f y β§ f y β€ x
sz : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
oβ : WithBot Ξ±
oβ : WithTop Ξ±
h : Valid' oβ (node sz l y r) oβ
bl : Bounded nil oβ βx
br : Bounded nil (βx) oβ
h_1 : Β¬x β€ y
this : y < x
vr : Valid' (βy) (insertWith f x r) oβ
e : Raised (size r) (size (insertWith f x r))
β’ (β l', Raised (size l) l' β§ BalancedSz l' (size (insertWith f x r))) β¨
β r', Raised r' (size (insertWith f x r)) β§ BalancedSz (size l) r'
[PROOFSTEP]
exact Or.inr β¨_, e, h.3.1β©
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ Ordnode.insert x (node sizeβ l y r) = insertWith (fun x_1 => x) x (node sizeβ l y r)
[PROOFSTEP]
unfold Ordnode.insert insertWith
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match cmpLE x y with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match cmpLE x y with
| Ordering.lt => balanceL (insertWith (fun x_1 => x) x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (insertWith (fun x_1 => x) x r)
[PROOFSTEP]
cases cmpLE x y
[GOAL]
case lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.lt with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.lt with
| Ordering.lt => balanceL (insertWith (fun x_1 => x) x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (insertWith (fun x_1 => x) x r)
[PROOFSTEP]
simp [insert_eq_insertWith]
[GOAL]
case eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.eq with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.eq with
| Ordering.lt => balanceL (insertWith (fun x_1 => x) x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (insertWith (fun x_1 => x) x r)
[PROOFSTEP]
simp [insert_eq_insertWith]
[GOAL]
case gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.gt with
| Ordering.lt => balanceL (Ordnode.insert x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (Ordnode.insert x r)) =
match Ordering.gt with
| Ordering.lt => balanceL (insertWith (fun x_1 => x) x l) y r
| Ordering.eq => node sizeβ l x r
| Ordering.gt => balanceR l y (insertWith (fun x_1 => x) x r)
[PROOFSTEP]
simp [insert_eq_insertWith]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (Ordnode.insert x t)
[PROOFSTEP]
rw [insert_eq_insertWith]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (insertWith (fun x_1 => x) x t)
[PROOFSTEP]
exact insertWith.valid _ _ (fun _ _ => β¨le_rfl, le_rflβ©) h
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ insert' x (node sizeβ l y r) = insertWith id x (node sizeβ l y r)
[PROOFSTEP]
unfold insert' insertWith
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match cmpLE x y with
| Ordering.lt => balanceL (insert' x l) y r
| Ordering.eq => node sizeβ l y r
| Ordering.gt => balanceR l y (insert' x r)) =
match cmpLE x y with
| Ordering.lt => balanceL (insertWith id x l) y r
| Ordering.eq => node sizeβ l (id y) r
| Ordering.gt => balanceR l y (insertWith id x r)
[PROOFSTEP]
cases cmpLE x y
[GOAL]
case lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.lt with
| Ordering.lt => balanceL (insert' x l) y r
| Ordering.eq => node sizeβ l y r
| Ordering.gt => balanceR l y (insert' x r)) =
match Ordering.lt with
| Ordering.lt => balanceL (insertWith id x l) y r
| Ordering.eq => node sizeβ l (id y) r
| Ordering.gt => balanceR l y (insertWith id x r)
[PROOFSTEP]
simp [insert'_eq_insertWith]
[GOAL]
case eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.eq with
| Ordering.lt => balanceL (insert' x l) y r
| Ordering.eq => node sizeβ l y r
| Ordering.gt => balanceR l y (insert' x r)) =
match Ordering.eq with
| Ordering.lt => balanceL (insertWith id x l) y r
| Ordering.eq => node sizeβ l (id y) r
| Ordering.gt => balanceR l y (insertWith id x r)
[PROOFSTEP]
simp [insert'_eq_insertWith]
[GOAL]
case gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
l : Ordnode Ξ±
y : Ξ±
r : Ordnode Ξ±
β’ (match Ordering.gt with
| Ordering.lt => balanceL (insert' x l) y r
| Ordering.eq => node sizeβ l y r
| Ordering.gt => balanceR l y (insert' x r)) =
match Ordering.gt with
| Ordering.lt => balanceL (insertWith id x l) y r
| Ordering.eq => node sizeβ l (id y) r
| Ordering.gt => balanceR l y (insertWith id x r)
[PROOFSTEP]
simp [insert'_eq_insertWith]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (insert' x t)
[PROOFSTEP]
rw [insert'_eq_insertWith]
[GOAL]
Ξ± : Type u_1
instβΒ² : Preorder Ξ±
instβΒΉ : IsTotal Ξ± fun x x_1 => x β€ x_1
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
h : Valid t
β’ Valid (insertWith id x t)
[PROOFSTEP]
exact insertWith.valid _ _ (fun _ => id) h
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
β’ Valid' (Option.map f aβ) (map f t) (Option.map f aβ) β§ size (map f t) = size t
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil =>
simp [map]; apply valid'_nil
cases aβ; Β· trivial
cases aβ; Β· trivial
simp [Bounded]
exact f_strict_mono h.ord
| node _ _ _ _ t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
simp [map]
constructor
Β· exact And.intro t_l_valid.ord t_r_valid.ord
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.sz.1
Β· constructor
Β· exact t_l_valid.sz
Β· exact t_r_valid.sz
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.bal.1
Β· constructor
Β· exact t_l_valid.bal
Β· exact t_r_valid.bal
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
β’ Valid' (Option.map f aβ) (map f t) (Option.map f aβ) β§ size (map f t) = size t
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil =>
simp [map]; apply valid'_nil
cases aβ; Β· trivial
cases aβ; Β· trivial
simp [Bounded]
exact f_strict_mono h.ord
| node _ _ _ _ t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
simp [map]
constructor
Β· exact And.intro t_l_valid.ord t_r_valid.ord
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.sz.1
Β· constructor
Β· exact t_l_valid.sz
Β· exact t_r_valid.sz
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.bal.1
Β· constructor
Β· exact t_l_valid.bal
Β· exact t_r_valid.bal
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' (Option.map f aβ) (map f nil) (Option.map f aβ) β§ size (map f nil) = size nil
[PROOFSTEP]
| nil =>
simp [map]; apply valid'_nil
cases aβ; Β· trivial
cases aβ; Β· trivial
simp [Bounded]
exact f_strict_mono h.ord
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' (Option.map f aβ) (map f nil) (Option.map f aβ) β§ size (map f nil) = size nil
[PROOFSTEP]
simp [map]
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' (Option.map f aβ) nil (Option.map f aβ)
[PROOFSTEP]
apply valid'_nil
[GOAL]
case nil.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Bounded nil (Option.map f aβ) (Option.map f aβ)
[PROOFSTEP]
cases aβ
[GOAL]
case nil.h.none
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithTop Ξ±
h : Valid' none nil aβ
β’ Bounded nil (Option.map f none) (Option.map f aβ)
[PROOFSTEP]
trivial
[GOAL]
case nil.h.some
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
aβ : WithTop Ξ±
valβ : Ξ±
h : Valid' (some valβ) nil aβ
β’ Bounded nil (Option.map f (some valβ)) (Option.map f aβ)
[PROOFSTEP]
cases aβ
[GOAL]
case nil.h.some.none
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
valβ : Ξ±
h : Valid' (some valβ) nil none
β’ Bounded nil (Option.map f (some valβ)) (Option.map f none)
[PROOFSTEP]
trivial
[GOAL]
case nil.h.some.some
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
valβΒΉ valβ : Ξ±
h : Valid' (some valβΒΉ) nil (some valβ)
β’ Bounded nil (Option.map f (some valβΒΉ)) (Option.map f (some valβ))
[PROOFSTEP]
simp [Bounded]
[GOAL]
case nil.h.some.some
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
valβΒΉ valβ : Ξ±
h : Valid' (some valβΒΉ) nil (some valβ)
β’ f valβΒΉ < f valβ
[PROOFSTEP]
exact f_strict_mono h.ord
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ lβ aβ β Valid' (Option.map f aβ) (map f lβ) (Option.map f aβ) β§ size (map f lβ) = size lβ
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ rβ aβ β Valid' (Option.map f aβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
| node _ _ _ _ t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
simp [map]
constructor
Β· exact And.intro t_l_valid.ord t_r_valid.ord
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.sz.1
Β· constructor
Β· exact t_l_valid.sz
Β· exact t_r_valid.sz
Β· constructor
Β· rw [t_l_size, t_r_size]; exact h.bal.1
Β· constructor
Β· exact t_l_valid.bal
Β· exact t_r_valid.bal
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ lβ aβ β Valid' (Option.map f aβ) (map f lβ) (Option.map f aβ) β§ size (map f lβ) = size lβ
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ rβ aβ β Valid' (Option.map f aβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
have t_ih_l' := t_ih_l h.left
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ lβ aβ β Valid' (Option.map f aβ) (map f lβ) (Option.map f aβ) β§ size (map f lβ) = size lβ
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ rβ aβ β Valid' (Option.map f aβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_ih_l' : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ) β§ size (map f lβ) = size lβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
have t_ih_r' := t_ih_r h.right
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ lβ aβ β Valid' (Option.map f aβ) (map f lβ) (Option.map f aβ) β§ size (map f lβ) = size lβ
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ rβ aβ β Valid' (Option.map f aβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_ih_l' : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ) β§ size (map f lβ) = size lβ
t_ih_r' : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
clear t_ih_l t_ih_r
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_ih_l' : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ) β§ size (map f lβ) = size lβ
t_ih_r' : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
cases' t_ih_l' with t_l_valid t_l_size
[GOAL]
case node.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_ih_r' : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ) β§ size (map f rβ) = size rβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
cases' t_ih_r' with t_r_valid t_r_size
[GOAL]
case node.intro.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Valid' (Option.map f aβ) (map f (Ordnode.node sizeβ lβ xβ rβ)) (Option.map f aβ) β§
size (map f (Ordnode.node sizeβ lβ xβ rβ)) = size (Ordnode.node sizeβ lβ xβ rβ)
[PROOFSTEP]
simp [map]
[GOAL]
case node.intro.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Valid' (Option.map f aβ) (Ordnode.node sizeβ (map f lβ) (f xβ) (map f rβ)) (Option.map f aβ)
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.ord
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Bounded (Ordnode.node sizeβ (map f lβ) (f xβ) (map f rβ)) (Option.map f aβ) (Option.map f aβ)
[PROOFSTEP]
exact And.intro t_l_valid.ord t_r_valid.ord
[GOAL]
case node.intro.intro.sz
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Sized (Ordnode.node sizeβ (map f lβ) (f xβ) (map f rβ))
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.sz.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ sizeβ = size (map f lβ) + size (map f rβ) + 1
[PROOFSTEP]
rw [t_l_size, t_r_size]
[GOAL]
case node.intro.intro.sz.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ sizeβ = size lβ + size rβ + 1
[PROOFSTEP]
exact h.sz.1
[GOAL]
case node.intro.intro.sz.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Sized (map f lβ) β§ Sized (map f rβ)
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.sz.right.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Sized (map f lβ)
[PROOFSTEP]
exact t_l_valid.sz
[GOAL]
case node.intro.intro.sz.right.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Sized (map f rβ)
[PROOFSTEP]
exact t_r_valid.sz
[GOAL]
case node.intro.intro.bal
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Balanced (Ordnode.node sizeβ (map f lβ) (f xβ) (map f rβ))
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.bal.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ BalancedSz (size (map f lβ)) (size (map f rβ))
[PROOFSTEP]
rw [t_l_size, t_r_size]
[GOAL]
case node.intro.intro.bal.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ BalancedSz (size lβ) (size rβ)
[PROOFSTEP]
exact h.bal.1
[GOAL]
case node.intro.intro.bal.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Balanced (map f lβ) β§ Balanced (map f rβ)
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.bal.right.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Balanced (map f lβ)
[PROOFSTEP]
exact t_l_valid.bal
[GOAL]
case node.intro.intro.bal.right.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
Ξ² : Type u_2
instβ : Preorder Ξ²
f : Ξ± β Ξ²
f_strict_mono : StrictMono f
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ lβ xβ rβ) aβ
t_l_valid : Valid' (Option.map f aβ) (map f lβ) (Option.map f βxβ)
t_l_size : size (map f lβ) = size lβ
t_r_valid : Valid' (Option.map f βxβ) (map f rβ) (Option.map f aβ)
t_r_size : size (map f rβ) = size rβ
β’ Balanced (map f rβ)
[PROOFSTEP]
exact t_r_valid.bal
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
β’ Valid' aβ (erase x t) aβ β§ Raised (size (erase x t)) (size t)
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil => simp [erase, Raised]; exact h
| node _ t_l t_x t_r t_ih_l t_ih_r =>
simp [erase]
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
cases cmpLE x t_x <;> rw [h.sz.1]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceR t_l_valid h.right h_balanceable
Β· rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz h_balanceable]
repeat apply Raised.add_right
exact t_l_size
Β· left; exists t_l.size; exact And.intro t_l_size h.bal.1
Β· have h_glue := Valid'.glue h.left h.right h.bal.1
cases' h_glue with h_glue_valid h_glue_sized
constructor
Β· exact h_glue_valid
Β· right; rw [h_glue_sized]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceL h.left t_r_valid h_balanceable
Β· rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz h_balanceable]
apply Raised.add_right
apply Raised.add_left
exact t_r_size
Β· right; exists t_r.size; exact And.intro t_r_size h.bal.1
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
β’ Valid' aβ (erase x t) aβ β§ Raised (size (erase x t)) (size t)
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil => simp [erase, Raised]; exact h
| node _ t_l t_x t_r t_ih_l t_ih_r =>
simp [erase]
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
cases cmpLE x t_x <;> rw [h.sz.1]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceR t_l_valid h.right h_balanceable
Β· rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz h_balanceable]
repeat apply Raised.add_right
exact t_l_size
Β· left; exists t_l.size; exact And.intro t_l_size h.bal.1
Β· have h_glue := Valid'.glue h.left h.right h.bal.1
cases' h_glue with h_glue_valid h_glue_sized
constructor
Β· exact h_glue_valid
Β· right; rw [h_glue_sized]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceL h.left t_r_valid h_balanceable
Β· rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz h_balanceable]
apply Raised.add_right
apply Raised.add_left
exact t_r_size
Β· right; exists t_r.size; exact And.intro t_r_size h.bal.1
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' aβ (erase x nil) aβ β§ Raised (size (erase x nil)) (size nil)
[PROOFSTEP]
| nil => simp [erase, Raised]; exact h
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' aβ (erase x nil) aβ β§ Raised (size (erase x nil)) (size nil)
[PROOFSTEP]
simp [erase, Raised]
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
β’ Valid' aβ nil aβ
[PROOFSTEP]
exact h
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_l aβ β Valid' aβ (erase x t_l) aβ β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_r aβ β Valid' aβ (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
β’ Valid' aβ (erase x (Ordnode.node sizeβ t_l t_x t_r)) aβ β§
Raised (size (erase x (Ordnode.node sizeβ t_l t_x t_r))) (size (Ordnode.node sizeβ t_l t_x t_r))
[PROOFSTEP]
| node _ t_l t_x t_r t_ih_l t_ih_r =>
simp [erase]
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
cases' t_ih_l' with t_l_valid t_l_size
cases' t_ih_r' with t_r_valid t_r_size
cases cmpLE x t_x <;> rw [h.sz.1]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceR t_l_valid h.right h_balanceable
Β· rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz h_balanceable]
repeat apply Raised.add_right
exact t_l_size
Β· left; exists t_l.size; exact And.intro t_l_size h.bal.1
Β· have h_glue := Valid'.glue h.left h.right h.bal.1
cases' h_glue with h_glue_valid h_glue_sized
constructor
Β· exact h_glue_valid
Β· right; rw [h_glue_sized]
Β· suffices h_balanceable
constructor
Β· exact Valid'.balanceL h.left t_r_valid h_balanceable
Β· rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz h_balanceable]
apply Raised.add_right
apply Raised.add_left
exact t_r_size
Β· right; exists t_r.size; exact And.intro t_r_size h.bal.1
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_l aβ β Valid' aβ (erase x t_l) aβ β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_r aβ β Valid' aβ (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
β’ Valid' aβ (erase x (Ordnode.node sizeβ t_l t_x t_r)) aβ β§
Raised (size (erase x (Ordnode.node sizeβ t_l t_x t_r))) (size (Ordnode.node sizeβ t_l t_x t_r))
[PROOFSTEP]
simp [erase]
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_l aβ β Valid' aβ (erase x t_l) aβ β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_r aβ β Valid' aβ (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
have t_ih_l' := t_ih_l h.left
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_l aβ β Valid' aβ (erase x t_l) aβ β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_r aβ β Valid' aβ (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_ih_l' : Valid' aβ (erase x t_l) βt_x β§ Raised (size (erase x t_l)) (size t_l)
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
have t_ih_r' := t_ih_r h.right
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_l aβ β Valid' aβ (erase x t_l) aβ β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r :
β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±},
Valid' aβ t_r aβ β Valid' aβ (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_ih_l' : Valid' aβ (erase x t_l) βt_x β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r' : Valid' (βt_x) (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
clear t_ih_l t_ih_r
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_ih_l' : Valid' aβ (erase x t_l) βt_x β§ Raised (size (erase x t_l)) (size t_l)
t_ih_r' : Valid' (βt_x) (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
cases' t_ih_l' with t_l_valid t_l_size
[GOAL]
case node.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_ih_r' : Valid' (βt_x) (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
cases' t_ih_r' with t_r_valid t_r_size
[GOAL]
case node.intro.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match cmpLE x t_x with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
cases cmpLE x t_x
[GOAL]
case node.intro.intro.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
rw [h.sz.1]
[GOAL]
case node.intro.intro.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
rw [h.sz.1]
[GOAL]
case node.intro.intro.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
sizeβ
[PROOFSTEP]
rw [h.sz.1]
[GOAL]
case node.intro.intro.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
suffices h_balanceable
[GOAL]
case node.intro.intro.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable : ?m.391248
β’ Valid' aβ
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
case h_balanceable
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ ?m.391248
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.lt.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable : ?m.391248
β’ Valid' aβ
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ
[PROOFSTEP]
exact Valid'.balanceR t_l_valid h.right h_balanceable
[GOAL]
case node.intro.intro.lt.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised
(size
(match Ordering.lt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz h_balanceable]
[GOAL]
case node.intro.intro.lt.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised (size (erase x t_l) + size t_r + 1) (size t_l + size t_r + 1)
[PROOFSTEP]
repeat apply Raised.add_right
[GOAL]
case node.intro.intro.lt.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised (size (erase x t_l) + size t_r + 1) (size t_l + size t_r + 1)
[PROOFSTEP]
apply Raised.add_right
[GOAL]
case node.intro.intro.lt.right.H
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised (size (erase x t_l) + size t_r) (size t_l + size t_r)
[PROOFSTEP]
apply Raised.add_right
[GOAL]
case node.intro.intro.lt.right.H.H
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised (size (erase x t_l)) (size t_l)
[PROOFSTEP]
apply Raised.add_right
[GOAL]
case node.intro.intro.lt.right.H.H
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
β’ Raised (size (erase x t_l)) (size t_l)
[PROOFSTEP]
exact t_l_size
[GOAL]
case h_balanceable
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ (β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)) β¨
β r', Raised r' (size t_r) β§ BalancedSz (size (erase x t_l)) r'
[PROOFSTEP]
left
[GOAL]
case h_balanceable.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ β l', Raised (size (erase x t_l)) l' β§ BalancedSz l' (size t_r)
[PROOFSTEP]
exists t_l.size
[GOAL]
case h_balanceable.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Raised (size (erase x t_l)) (size t_l) β§ BalancedSz (size t_l) (size t_r)
[PROOFSTEP]
exact And.intro t_l_size h.bal.1
[GOAL]
case node.intro.intro.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
have h_glue := Valid'.glue h.left h.right h.bal.1
[GOAL]
case node.intro.intro.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_glue : Valid' aβ (Ordnode.glue t_l t_r) aβ β§ size (Ordnode.glue t_l t_r) = size t_l + size t_r
β’ Valid' aβ
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
cases' h_glue with h_glue_valid h_glue_sized
[GOAL]
case node.intro.intro.eq.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_glue_valid : Valid' aβ (Ordnode.glue t_l t_r) aβ
h_glue_sized : size (Ordnode.glue t_l t_r) = size t_l + size t_r
β’ Valid' aβ
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.eq.intro.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_glue_valid : Valid' aβ (Ordnode.glue t_l t_r) aβ
h_glue_sized : size (Ordnode.glue t_l t_r) = size t_l + size t_r
β’ Valid' aβ
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ
[PROOFSTEP]
exact h_glue_valid
[GOAL]
case node.intro.intro.eq.intro.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_glue_valid : Valid' aβ (Ordnode.glue t_l t_r) aβ
h_glue_sized : size (Ordnode.glue t_l t_r) = size t_l + size t_r
β’ Raised
(size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
right
[GOAL]
case node.intro.intro.eq.intro.right.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_glue_valid : Valid' aβ (Ordnode.glue t_l t_r) aβ
h_glue_sized : size (Ordnode.glue t_l t_r) = size t_l + size t_r
β’ size t_l + size t_r + 1 =
size
(match Ordering.eq with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)) +
1
[PROOFSTEP]
rw [h_glue_sized]
[GOAL]
case node.intro.intro.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Valid' aβ
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
suffices h_balanceable
[GOAL]
case node.intro.intro.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable : ?m.391865
β’ Valid' aβ
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ β§
Raised
(size
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
case h_balanceable
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ ?m.391865
[PROOFSTEP]
constructor
[GOAL]
case node.intro.intro.gt.left
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable : ?m.391865
β’ Valid' aβ
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r))
aβ
[PROOFSTEP]
exact Valid'.balanceL h.left t_r_valid h_balanceable
[GOAL]
case node.intro.intro.gt.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised l' (size t_l) β§ BalancedSz l' (size (erase x t_r))) β¨
β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
β’ Raised
(size
(match Ordering.gt with
| Ordering.lt => Ordnode.balanceR (erase x t_l) t_x t_r
| Ordering.eq => Ordnode.glue t_l t_r
| Ordering.gt => Ordnode.balanceL t_l t_x (erase x t_r)))
(size t_l + size t_r + 1)
[PROOFSTEP]
rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz h_balanceable]
[GOAL]
case node.intro.intro.gt.right
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised l' (size t_l) β§ BalancedSz l' (size (erase x t_r))) β¨
β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
β’ Raised (size t_l + size (erase x t_r) + 1) (size t_l + size t_r + 1)
[PROOFSTEP]
apply Raised.add_right
[GOAL]
case node.intro.intro.gt.right.H
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised l' (size t_l) β§ BalancedSz l' (size (erase x t_r))) β¨
β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
β’ Raised (size t_l + size (erase x t_r)) (size t_l + size t_r)
[PROOFSTEP]
apply Raised.add_left
[GOAL]
case node.intro.intro.gt.right.H.H
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_balanceable :
(β l', Raised l' (size t_l) β§ BalancedSz l' (size (erase x t_r))) β¨
β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
β’ Raised (size (erase x t_r)) (size t_r)
[PROOFSTEP]
exact t_r_size
[GOAL]
case h_balanceable
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ (β l', Raised l' (size t_l) β§ BalancedSz l' (size (erase x t_r))) β¨
β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
[PROOFSTEP]
right
[GOAL]
case h_balanceable.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ β r', Raised (size (erase x t_r)) r' β§ BalancedSz (size t_l) r'
[PROOFSTEP]
exists t_r.size
[GOAL]
case h_balanceable.h
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (Ordnode.node sizeβ t_l t_x t_r) aβ
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ Raised (size (erase x t_r)) (size t_r) β§ BalancedSz (size t_l) (size t_r)
[PROOFSTEP]
exact And.intro t_r_size h.bal.1
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
h_mem : x β t
β’ size (erase x t) = size t - 1
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil => contradiction
| node _ t_l t_x t_r t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
dsimp only [Membership.mem, mem] at h_mem
unfold erase
revert h_mem; cases cmpLE x t_x <;> intro h_mem <;> dsimp only at h_mem β’
Β· have t_ih_l := t_ih_l' h_mem
clear t_ih_l' t_ih_r'
have t_l_h := Valid'.erase_aux x h.left
cases' t_l_h with t_l_valid t_l_size
rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz
(Or.inl (Exists.intro t_l.size (And.intro t_l_size h.bal.1)))]
rw [t_ih_l, h.sz.1]
have h_pos_t_l_size := pos_size_of_mem h.left.sz h_mem
revert h_pos_t_l_size; cases' t_l.size with t_l_size <;> intro h_pos_t_l_size
Β· cases h_pos_t_l_size
Β· simp [Nat.succ_add]
Β· rw [(Valid'.glue h.left h.right h.bal.1).2, h.sz.1]; rfl
Β· have t_ih_r := t_ih_r' h_mem
clear t_ih_l' t_ih_r'
have t_r_h := Valid'.erase_aux x h.right
cases' t_r_h with t_r_valid t_r_size
rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz
(Or.inr (Exists.intro t_r.size (And.intro t_r_size h.bal.1)))]
rw [t_ih_r, h.sz.1]
have h_pos_t_r_size := pos_size_of_mem h.right.sz h_mem
revert h_pos_t_r_size; cases' t_r.size with t_r_size <;> intro h_pos_t_r_size
Β· cases h_pos_t_r_size
Β· simp [Nat.succ_add, Nat.add_succ]
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ t aβ
h_mem : x β t
β’ size (erase x t) = size t - 1
[PROOFSTEP]
induction t generalizing aβ aβ with
| nil => contradiction
| node _ t_l t_x t_r t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
dsimp only [Membership.mem, mem] at h_mem
unfold erase
revert h_mem; cases cmpLE x t_x <;> intro h_mem <;> dsimp only at h_mem β’
Β· have t_ih_l := t_ih_l' h_mem
clear t_ih_l' t_ih_r'
have t_l_h := Valid'.erase_aux x h.left
cases' t_l_h with t_l_valid t_l_size
rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz
(Or.inl (Exists.intro t_l.size (And.intro t_l_size h.bal.1)))]
rw [t_ih_l, h.sz.1]
have h_pos_t_l_size := pos_size_of_mem h.left.sz h_mem
revert h_pos_t_l_size; cases' t_l.size with t_l_size <;> intro h_pos_t_l_size
Β· cases h_pos_t_l_size
Β· simp [Nat.succ_add]
Β· rw [(Valid'.glue h.left h.right h.bal.1).2, h.sz.1]; rfl
Β· have t_ih_r := t_ih_r' h_mem
clear t_ih_l' t_ih_r'
have t_r_h := Valid'.erase_aux x h.right
cases' t_r_h with t_r_valid t_r_size
rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz
(Or.inr (Exists.intro t_r.size (And.intro t_r_size h.bal.1)))]
rw [t_ih_r, h.sz.1]
have h_pos_t_r_size := pos_size_of_mem h.right.sz h_mem
revert h_pos_t_r_size; cases' t_r.size with t_r_size <;> intro h_pos_t_r_size
Β· cases h_pos_t_r_size
Β· simp [Nat.succ_add, Nat.add_succ]
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
h_mem : x β nil
β’ size (erase x nil) = size nil - 1
[PROOFSTEP]
| nil => contradiction
[GOAL]
case nil
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ nil aβ
h_mem : x β nil
β’ size (erase x nil) = size nil - 1
[PROOFSTEP]
contradiction
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_l aβ β x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_r aβ β x β t_r β size (erase x t_r) = size t_r - 1
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : x β node sizeβ t_l t_x t_r
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
| node _ t_l t_x t_r t_ih_l t_ih_r =>
have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
clear t_ih_l t_ih_r
dsimp only [Membership.mem, mem] at h_mem
unfold erase
revert h_mem; cases cmpLE x t_x <;> intro h_mem <;> dsimp only at h_mem β’
Β· have t_ih_l := t_ih_l' h_mem
clear t_ih_l' t_ih_r'
have t_l_h := Valid'.erase_aux x h.left
cases' t_l_h with t_l_valid t_l_size
rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz
(Or.inl (Exists.intro t_l.size (And.intro t_l_size h.bal.1)))]
rw [t_ih_l, h.sz.1]
have h_pos_t_l_size := pos_size_of_mem h.left.sz h_mem
revert h_pos_t_l_size; cases' t_l.size with t_l_size <;> intro h_pos_t_l_size
Β· cases h_pos_t_l_size
Β· simp [Nat.succ_add]
Β· rw [(Valid'.glue h.left h.right h.bal.1).2, h.sz.1]; rfl
Β· have t_ih_r := t_ih_r' h_mem
clear t_ih_l' t_ih_r'
have t_r_h := Valid'.erase_aux x h.right
cases' t_r_h with t_r_valid t_r_size
rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz
(Or.inr (Exists.intro t_r.size (And.intro t_r_size h.bal.1)))]
rw [t_ih_r, h.sz.1]
have h_pos_t_r_size := pos_size_of_mem h.right.sz h_mem
revert h_pos_t_r_size; cases' t_r.size with t_r_size <;> intro h_pos_t_r_size
Β· cases h_pos_t_r_size
Β· simp [Nat.succ_add, Nat.add_succ]
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_l aβ β x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_r aβ β x β t_r β size (erase x t_r) = size t_r - 1
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : x β node sizeβ t_l t_x t_r
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_ih_l' := t_ih_l h.left
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_l aβ β x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_r aβ β x β t_r β size (erase x t_r) = size t_r - 1
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : x β node sizeβ t_l t_x t_r
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_ih_r' := t_ih_r h.right
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
t_ih_l : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_l aβ β x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r : β {aβ : WithBot Ξ±} {aβ : WithTop Ξ±}, Valid' aβ t_r aβ β x β t_r β size (erase x t_r) = size t_r - 1
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : x β node sizeβ t_l t_x t_r
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
clear t_ih_l t_ih_r
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : x β node sizeβ t_l t_x t_r
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
dsimp only [Membership.mem, mem] at h_mem
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem :
(match cmpLE x t_x with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ size (erase x (node sizeβ t_l t_x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
unfold erase
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem :
(match cmpLE x t_x with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ size
(match cmpLE x t_x with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
revert h_mem
[GOAL]
case node
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ (match cmpLE x t_x with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true β
size
(match cmpLE x t_x with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
cases cmpLE x t_x
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ (match Ordering.lt with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true β
size
(match Ordering.lt with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
intro h_mem
[GOAL]
case node.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ (match Ordering.eq with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true β
size
(match Ordering.eq with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
intro h_mem
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
β’ (match Ordering.gt with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true β
size
(match Ordering.gt with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
intro h_mem
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem :
(match Ordering.lt with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true
β’ size
(match Ordering.lt with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
dsimp only at h_mem β’
[GOAL]
case node.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem :
(match Ordering.eq with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true
β’ size
(match Ordering.eq with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
dsimp only at h_mem β’
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem :
(match Ordering.gt with
| Ordering.lt => mem x t_l
| Ordering.eq => true
| Ordering.gt => mem x t_r) =
true
β’ size
(match Ordering.gt with
| Ordering.lt => balanceR (erase x t_l) t_x t_r
| Ordering.eq => glue t_l t_r
| Ordering.gt => balanceL t_l t_x (erase x t_r)) =
size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
dsimp only at h_mem β’
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : mem x t_l = true
β’ size (balanceR (erase x t_l) t_x t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_ih_l := t_ih_l' h_mem
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
β’ size (balanceR (erase x t_l) t_x t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
clear t_ih_l' t_ih_r'
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
β’ size (balanceR (erase x t_l) t_x t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_l_h := Valid'.erase_aux x h.left
[GOAL]
case node.lt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_h : Valid' aβ (erase x t_l) βt_x β§ Raised (size (erase x t_l)) (size t_l)
β’ size (balanceR (erase x t_l) t_x t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
cases' t_l_h with t_l_valid t_l_size
[GOAL]
case node.lt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ size (balanceR (erase x t_l) t_x t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
rw [size_balanceR t_l_valid.bal h.right.bal t_l_valid.sz h.right.sz
(Or.inl (Exists.intro t_l.size (And.intro t_l_size h.bal.1)))]
[GOAL]
case node.lt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ size (erase x t_l) + size t_r + 1 = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
rw [t_ih_l, h.sz.1]
[GOAL]
case node.lt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ size t_l - 1 + size t_r + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
have h_pos_t_l_size := pos_size_of_mem h.left.sz h_mem
[GOAL]
case node.lt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
h_pos_t_l_size : 0 < size t_l
β’ size t_l - 1 + size t_r + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
revert h_pos_t_l_size
[GOAL]
case node.lt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ 0 < size t_l β size t_l - 1 + size t_r + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
cases' t_l.size with t_l_size
[GOAL]
case node.lt.intro.zero
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
β’ 0 < Nat.zero β Nat.zero - 1 + size t_r + 1 = size (node (Nat.zero + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
intro h_pos_t_l_size
[GOAL]
case node.lt.intro.succ
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_sizeβ : Raised (size (erase x t_l)) (size t_l)
t_l_size : β
β’ 0 < Nat.succ t_l_size β
Nat.succ t_l_size - 1 + size t_r + 1 = size (node (Nat.succ t_l_size + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
intro h_pos_t_l_size
[GOAL]
case node.lt.intro.zero
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_size : Raised (size (erase x t_l)) (size t_l)
h_pos_t_l_size : 0 < Nat.zero
β’ Nat.zero - 1 + size t_r + 1 = size (node (Nat.zero + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
cases h_pos_t_l_size
[GOAL]
case node.lt.intro.succ
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_l = true
t_ih_l : size (erase x t_l) = size t_l - 1
t_l_valid : Valid' aβ (erase x t_l) βt_x
t_l_sizeβ : Raised (size (erase x t_l)) (size t_l)
t_l_size : β
h_pos_t_l_size : 0 < Nat.succ t_l_size
β’ Nat.succ t_l_size - 1 + size t_r + 1 = size (node (Nat.succ t_l_size + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
simp [Nat.succ_add]
[GOAL]
case node.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : true = true
β’ size (glue t_l t_r) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
rw [(Valid'.glue h.left h.right h.bal.1).2, h.sz.1]
[GOAL]
case node.eq
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : true = true
β’ size t_l + size t_r = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
rfl
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : mem x t_r = true
β’ size (balanceL t_l t_x (erase x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_ih_r := t_ih_r' h_mem
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
t_ih_l' : x β t_l β size (erase x t_l) = size t_l - 1
t_ih_r' : x β t_r β size (erase x t_r) = size t_r - 1
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
β’ size (balanceL t_l t_x (erase x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
clear t_ih_l' t_ih_r'
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
β’ size (balanceL t_l t_x (erase x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
have t_r_h := Valid'.erase_aux x h.right
[GOAL]
case node.gt
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_h : Valid' (βt_x) (erase x t_r) aβ β§ Raised (size (erase x t_r)) (size t_r)
β’ size (balanceL t_l t_x (erase x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
cases' t_r_h with t_r_valid t_r_size
[GOAL]
case node.gt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ size (balanceL t_l t_x (erase x t_r)) = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
rw [size_balanceL h.left.bal t_r_valid.bal h.left.sz t_r_valid.sz
(Or.inr (Exists.intro t_r.size (And.intro t_r_size h.bal.1)))]
[GOAL]
case node.gt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ size t_l + size (erase x t_r) + 1 = size (node sizeβ t_l t_x t_r) - 1
[PROOFSTEP]
rw [t_ih_r, h.sz.1]
[GOAL]
case node.gt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ size t_l + (size t_r - 1) + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
have h_pos_t_r_size := pos_size_of_mem h.right.sz h_mem
[GOAL]
case node.gt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_pos_t_r_size : 0 < size t_r
β’ size t_l + (size t_r - 1) + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
revert h_pos_t_r_size
[GOAL]
case node.gt.intro
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ 0 < size t_r β size t_l + (size t_r - 1) + 1 = size (node (size t_l + size t_r + 1) t_l t_x t_r) - 1
[PROOFSTEP]
cases' t_r.size with t_r_size
[GOAL]
case node.gt.intro.zero
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
β’ 0 < Nat.zero β size t_l + (Nat.zero - 1) + 1 = size (node (size t_l + Nat.zero + 1) t_l t_x t_r) - 1
[PROOFSTEP]
intro h_pos_t_r_size
[GOAL]
case node.gt.intro.succ
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_sizeβ : Raised (size (erase x t_r)) (size t_r)
t_r_size : β
β’ 0 < Nat.succ t_r_size β
size t_l + (Nat.succ t_r_size - 1) + 1 = size (node (size t_l + Nat.succ t_r_size + 1) t_l t_x t_r) - 1
[PROOFSTEP]
intro h_pos_t_r_size
[GOAL]
case node.gt.intro.zero
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_size : Raised (size (erase x t_r)) (size t_r)
h_pos_t_r_size : 0 < Nat.zero
β’ size t_l + (Nat.zero - 1) + 1 = size (node (size t_l + Nat.zero + 1) t_l t_x t_r) - 1
[PROOFSTEP]
cases h_pos_t_r_size
[GOAL]
case node.gt.intro.succ
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
sizeβ : β
t_l : Ordnode Ξ±
t_x : Ξ±
t_r : Ordnode Ξ±
aβ : WithBot Ξ±
aβ : WithTop Ξ±
h : Valid' aβ (node sizeβ t_l t_x t_r) aβ
h_mem : mem x t_r = true
t_ih_r : size (erase x t_r) = size t_r - 1
t_r_valid : Valid' (βt_x) (erase x t_r) aβ
t_r_sizeβ : Raised (size (erase x t_r)) (size t_r)
t_r_size : β
h_pos_t_r_size : 0 < Nat.succ t_r_size
β’ size t_l + (Nat.succ t_r_size - 1) + 1 = size (node (size t_l + Nat.succ t_r_size + 1) t_l t_x t_r) - 1
[PROOFSTEP]
simp [Nat.succ_add, Nat.add_succ]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : Ordset Ξ±
h : s = β
β’ empty βs = true
[PROOFSTEP]
cases h
[GOAL]
case refl
Ξ± : Type u_1
instβ : Preorder Ξ±
β’ empty ββ
= true
[PROOFSTEP]
exact rfl
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : Ordset Ξ±
h : empty βs = true
β’ s = β
[PROOFSTEP]
cases s with
| mk s_val _ => cases s_val <;> [rfl; cases h]
[GOAL]
Ξ± : Type u_1
instβ : Preorder Ξ±
s : Ordset Ξ±
h : empty βs = true
β’ s = β
[PROOFSTEP]
cases s with
| mk s_val _ => cases s_val <;> [rfl; cases h]
[GOAL]
case mk
Ξ± : Type u_1
instβ : Preorder Ξ±
s_val : Ordnode Ξ±
propertyβ : Valid s_val
h : empty β{ val := s_val, property := propertyβ } = true
β’ { val := s_val, property := propertyβ } = β
[PROOFSTEP]
| mk s_val _ => cases s_val <;> [rfl; cases h]
[GOAL]
case mk
Ξ± : Type u_1
instβ : Preorder Ξ±
s_val : Ordnode Ξ±
propertyβ : Valid s_val
h : empty β{ val := s_val, property := propertyβ } = true
β’ { val := s_val, property := propertyβ } = β
[PROOFSTEP]
cases s_val <;> [rfl; cases h]
[GOAL]
case mk
Ξ± : Type u_1
instβ : Preorder Ξ±
s_val : Ordnode Ξ±
propertyβ : Valid s_val
h : empty β{ val := s_val, property := propertyβ } = true
β’ { val := s_val, property := propertyβ } = β
[PROOFSTEP]
cases s_val
[GOAL]
case mk.nil
Ξ± : Type u_1
instβ : Preorder Ξ±
propertyβ : Valid Ordnode.nil
h : empty β{ val := Ordnode.nil, property := propertyβ } = true
β’ { val := Ordnode.nil, property := propertyβ } = β
[PROOFSTEP]
rfl
[GOAL]
case mk.node
Ξ± : Type u_1
instβ : Preorder Ξ±
sizeβ : β
lβ : Ordnode Ξ±
xβ : Ξ±
rβ : Ordnode Ξ±
propertyβ : Valid (node sizeβ lβ xβ rβ)
h : empty β{ val := node sizeβ lβ xβ rβ, property := propertyβ } = true
β’ { val := node sizeβ lβ xβ rβ, property := propertyβ } = β
[PROOFSTEP]
cases h
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordset Ξ±
h_mem : x β t
β’ 0 < size t
[PROOFSTEP]
simp [Membership.mem, mem] at h_mem
[GOAL]
Ξ± : Type u_1
instβΒΉ : Preorder Ξ±
instβ : DecidableRel fun x x_1 => x β€ x_1
x : Ξ±
t : Ordset Ξ±
h_mem : Ordnode.mem x βt = true
β’ 0 < size t
[PROOFSTEP]
apply Ordnode.pos_size_of_mem t.property.sz h_mem
|
Theorem Modus_Ponens: forall P Q:Prop, (P->Q)->P->Q.
Proof.
intros.
apply H.
apply H0.
Qed.
|
module Data.Range
%access public export
%default total
||| Range is a generic data type over the `Ord` type class. It describes a value
||| between two bounds.
data Range : Ord a => a -> a -> Type where
MkRange : Ord a => (x,y,z : a) ->
{auto prf : (x >= y && x <= z = True)} ->
Range y z
|
//==================================================================================================
/**
Copyright 2016 NumScale SAS
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
**/
//==================================================================================================
#ifndef BOOST_SIMD_ARCH_COMMON_SIMD_FUNCTION_RSQRT_HPP_INCLUDED
#define BOOST_SIMD_ARCH_COMMON_SIMD_FUNCTION_RSQRT_HPP_INCLUDED
#include <boost/simd/function/raw.hpp>
#include <boost/simd/function/rec.hpp>
#include <boost/simd/function/sqrt.hpp>
#include <boost/simd/detail/overload.hpp>
#include <boost/simd/detail/traits.hpp>
namespace boost { namespace simd { namespace ext
{
namespace bd = boost::dispatch;
namespace bs = boost::simd;
BOOST_DISPATCH_OVERLOAD_IF( rsqrt_
, (typename A0, typename X)
, (detail::is_native<X>)
, bd::cpu_
, bs::pack_< bd::floating_<A0>, X >
)
{
BOOST_FORCEINLINE A0 operator() ( A0 const& a0) const BOOST_NOEXCEPT
{
return bs::rec(bs::sqrt(a0));
}
};
BOOST_DISPATCH_OVERLOAD_IF( rsqrt_
, (typename A0, typename X)
, (detail::is_native<X>)
, bd::cpu_
, boost::simd::raw_tag
, bs::pack_< bd::floating_<A0>, X >
)
{
BOOST_FORCEINLINE A0 operator() (const raw_tag &, A0 const& a0) const BOOST_NOEXCEPT
{
return bs::rec(bs::raw_(bs::sqrt)(a0));
}
};
} } }
#endif
|
lemma LIM_zero_iff: "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F" for f :: "'a \<Rightarrow> 'b::real_normed_vector" |
State Before: Ξ± : Type u_1
Ξ±' : Type ?u.4489185
Ξ² : Type u_2
Ξ²' : Type ?u.4489191
Ξ³ : Type ?u.4489194
E : Type ?u.4489197
instββ· : MeasurableSpace Ξ±
instββΆ : MeasurableSpace Ξ±'
instββ΅ : MeasurableSpace Ξ²
instββ΄ : MeasurableSpace Ξ²'
instβΒ³ : MeasurableSpace Ξ³
ΞΌ ΞΌ' : Measure Ξ±
Ξ½ Ξ½' : Measure Ξ²
Ο : Measure Ξ³
instβΒ² : NormedAddCommGroup E
instβΒΉ : SigmaFinite Ξ½
instβ : SigmaFinite ΞΌ
y : Ξ²
β’ Measure.prod ΞΌ (dirac y) = map (fun x => (x, y)) ΞΌ State After: Ξ± : Type u_1
Ξ±' : Type ?u.4489185
Ξ² : Type u_2
Ξ²' : Type ?u.4489191
Ξ³ : Type ?u.4489194
E : Type ?u.4489197
instββ· : MeasurableSpace Ξ±
instββΆ : MeasurableSpace Ξ±'
instββ΅ : MeasurableSpace Ξ²
instββ΄ : MeasurableSpace Ξ²'
instβΒ³ : MeasurableSpace Ξ³
ΞΌ ΞΌ' : Measure Ξ±
Ξ½ Ξ½' : Measure Ξ²
Ο : Measure Ξ³
instβΒ² : NormedAddCommGroup E
instβΒΉ : SigmaFinite Ξ½
instβ : SigmaFinite ΞΌ
y : Ξ²
s : Set Ξ±
t : Set Ξ²
hs : MeasurableSet s
ht : MeasurableSet t
β’ ββ(map (fun x => (x, y)) ΞΌ) (s ΓΛ’ t) = ββΞΌ s * ββ(dirac y) t Tactic: refine' prod_eq fun s t hs ht => _ State Before: Ξ± : Type u_1
Ξ±' : Type ?u.4489185
Ξ² : Type u_2
Ξ²' : Type ?u.4489191
Ξ³ : Type ?u.4489194
E : Type ?u.4489197
instββ· : MeasurableSpace Ξ±
instββΆ : MeasurableSpace Ξ±'
instββ΅ : MeasurableSpace Ξ²
instββ΄ : MeasurableSpace Ξ²'
instβΒ³ : MeasurableSpace Ξ³
ΞΌ ΞΌ' : Measure Ξ±
Ξ½ Ξ½' : Measure Ξ²
Ο : Measure Ξ³
instβΒ² : NormedAddCommGroup E
instβΒΉ : SigmaFinite Ξ½
instβ : SigmaFinite ΞΌ
y : Ξ²
s : Set Ξ±
t : Set Ξ²
hs : MeasurableSet s
ht : MeasurableSet t
β’ ββ(map (fun x => (x, y)) ΞΌ) (s ΓΛ’ t) = ββΞΌ s * ββ(dirac y) t State After: no goals Tactic: simp_rw [map_apply measurable_prod_mk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if,
dirac_apply' _ ht, β indicator_mul_right _ fun _ => ΞΌ s, Pi.one_apply, mul_one] |
section \<open>Extending FOL by a modified version of HOL set theory\<close>
theory Set
imports "~~/src/FOL/FOL"
begin
declare [[eta_contract]]
typedecl 'a set
instance set :: ("term") "term" ..
subsection \<open>Set comprehension and membership\<close>
axiomatization Collect :: "['a \<Rightarrow> o] \<Rightarrow> 'a set"
and mem :: "['a, 'a set] \<Rightarrow> o" (infixl ":" 50)
where mem_Collect_iff: "(a : Collect(P)) \<longleftrightarrow> P(a)"
and set_extension: "A = B \<longleftrightarrow> (ALL x. x:A \<longleftrightarrow> x:B)"
syntax
"_Coll" :: "[idt, o] \<Rightarrow> 'a set" ("(1{_./ _})")
translations
"{x. P}" == "CONST Collect(\<lambda>x. P)"
lemma CollectI: "P(a) \<Longrightarrow> a : {x. P(x)}"
apply (rule mem_Collect_iff [THEN iffD2])
apply assumption
done
lemma CollectD: "a : {x. P(x)} \<Longrightarrow> P(a)"
apply (erule mem_Collect_iff [THEN iffD1])
done
lemmas CollectE = CollectD [elim_format]
lemma set_ext: "(\<And>x. x:A \<longleftrightarrow> x:B) \<Longrightarrow> A = B"
apply (rule set_extension [THEN iffD2])
apply simp
done
subsection \<open>Bounded quantifiers\<close>
definition Ball :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o"
where "Ball(A, P) == ALL x. x:A \<longrightarrow> P(x)"
definition Bex :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o"
where "Bex(A, P) == EX x. x:A \<and> P(x)"
syntax
"_Ball" :: "[idt, 'a set, o] \<Rightarrow> o" ("(ALL _:_./ _)" [0, 0, 0] 10)
"_Bex" :: "[idt, 'a set, o] \<Rightarrow> o" ("(EX _:_./ _)" [0, 0, 0] 10)
translations
"ALL x:A. P" == "CONST Ball(A, \<lambda>x. P)"
"EX x:A. P" == "CONST Bex(A, \<lambda>x. P)"
lemma ballI: "(\<And>x. x:A \<Longrightarrow> P(x)) \<Longrightarrow> ALL x:A. P(x)"
by (simp add: Ball_def)
lemma bspec: "\<lbrakk>ALL x:A. P(x); x:A\<rbrakk> \<Longrightarrow> P(x)"
by (simp add: Ball_def)
lemma ballE: "\<lbrakk>ALL x:A. P(x); P(x) \<Longrightarrow> Q; \<not> x:A \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Ball_def by blast
lemma bexI: "\<lbrakk>P(x); x:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)"
unfolding Bex_def by blast
lemma bexCI: "\<lbrakk>EX x:A. \<not>P(x) \<Longrightarrow> P(a); a:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)"
unfolding Bex_def by blast
lemma bexE: "\<lbrakk>EX x:A. P(x); \<And>x. \<lbrakk>x:A; P(x)\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Bex_def by blast
(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*)
lemma ball_rew: "(ALL x:A. True) \<longleftrightarrow> True"
by (blast intro: ballI)
subsubsection \<open>Congruence rules\<close>
lemma ball_cong:
"\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
(ALL x:A. P(x)) \<longleftrightarrow> (ALL x:A'. P'(x))"
by (blast intro: ballI elim: ballE)
lemma bex_cong:
"\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
(EX x:A. P(x)) \<longleftrightarrow> (EX x:A'. P'(x))"
by (blast intro: bexI elim: bexE)
subsection \<open>Further operations\<close>
definition subset :: "['a set, 'a set] \<Rightarrow> o" (infixl "<=" 50)
where "A <= B == ALL x:A. x:B"
definition mono :: "['a set \<Rightarrow> 'b set] \<Rightarrow> o"
where "mono(f) == (ALL A B. A <= B \<longrightarrow> f(A) <= f(B))"
definition singleton :: "'a \<Rightarrow> 'a set" ("{_}")
where "{a} == {x. x=a}"
definition empty :: "'a set" ("{}")
where "{} == {x. False}"
definition Un :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl "Un" 65)
where "A Un B == {x. x:A | x:B}"
definition Int :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl "Int" 70)
where "A Int B == {x. x:A \<and> x:B}"
definition Compl :: "('a set) \<Rightarrow> 'a set"
where "Compl(A) == {x. \<not>x:A}"
subsection \<open>Big Intersection / Union\<close>
definition INTER :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set"
where "INTER(A, B) == {y. ALL x:A. y: B(x)}"
definition UNION :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set"
where "UNION(A, B) == {y. EX x:A. y: B(x)}"
syntax
"_INTER" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" ("(INT _:_./ _)" [0, 0, 0] 10)
"_UNION" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" ("(UN _:_./ _)" [0, 0, 0] 10)
translations
"INT x:A. B" == "CONST INTER(A, \<lambda>x. B)"
"UN x:A. B" == "CONST UNION(A, \<lambda>x. B)"
definition Inter :: "(('a set)set) \<Rightarrow> 'a set"
where "Inter(S) == (INT x:S. x)"
definition Union :: "(('a set)set) \<Rightarrow> 'a set"
where "Union(S) == (UN x:S. x)"
subsection \<open>Rules for subsets\<close>
lemma subsetI: "(\<And>x. x:A \<Longrightarrow> x:B) \<Longrightarrow> A <= B"
unfolding subset_def by (blast intro: ballI)
(*Rule in Modus Ponens style*)
lemma subsetD: "\<lbrakk>A <= B; c:A\<rbrakk> \<Longrightarrow> c:B"
unfolding subset_def by (blast elim: ballE)
(*Classical elimination rule*)
lemma subsetCE: "\<lbrakk>A <= B; \<not>(c:A) \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast dest: subsetD)
lemma subset_refl: "A <= A"
by (blast intro: subsetI)
lemma subset_trans: "\<lbrakk>A <= B; B <= C\<rbrakk> \<Longrightarrow> A <= C"
by (blast intro: subsetI dest: subsetD)
subsection \<open>Rules for equality\<close>
(*Anti-symmetry of the subset relation*)
lemma subset_antisym: "\<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> A = B"
by (blast intro: set_ext dest: subsetD)
lemmas equalityI = subset_antisym
(* Equality rules from ZF set theory -- are they appropriate here? *)
lemma equalityD1: "A = B \<Longrightarrow> A<=B"
and equalityD2: "A = B \<Longrightarrow> B<=A"
by (simp_all add: subset_refl)
lemma equalityE: "\<lbrakk>A = B; \<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (simp add: subset_refl)
lemma equalityCE: "\<lbrakk>A = B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P; \<lbrakk>\<not> c:A; \<not> c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast elim: equalityE subsetCE)
lemma trivial_set: "{x. x:A} = A"
by (blast intro: equalityI subsetI CollectI dest: CollectD)
subsection \<open>Rules for binary union\<close>
lemma UnI1: "c:A \<Longrightarrow> c : A Un B"
and UnI2: "c:B \<Longrightarrow> c : A Un B"
unfolding Un_def by (blast intro: CollectI)+
(*Classical introduction rule: no commitment to A vs B*)
lemma UnCI: "(\<not>c:B \<Longrightarrow> c:A) \<Longrightarrow> c : A Un B"
by (blast intro: UnI1 UnI2)
lemma UnE: "\<lbrakk>c : A Un B; c:A \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
unfolding Un_def by (blast dest: CollectD)
subsection \<open>Rules for small intersection\<close>
lemma IntI: "\<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> c : A Int B"
unfolding Int_def by (blast intro: CollectI)
lemma IntD1: "c : A Int B \<Longrightarrow> c:A"
and IntD2: "c : A Int B \<Longrightarrow> c:B"
unfolding Int_def by (blast dest: CollectD)+
lemma IntE: "\<lbrakk>c : A Int B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast dest: IntD1 IntD2)
subsection \<open>Rules for set complement\<close>
lemma ComplI: "(c:A \<Longrightarrow> False) \<Longrightarrow> c : Compl(A)"
unfolding Compl_def by (blast intro: CollectI)
(*This form, with negated conclusion, works well with the Classical prover.
Negated assumptions behave like formulae on the right side of the notional
turnstile...*)
lemma ComplD: "c : Compl(A) \<Longrightarrow> \<not>c:A"
unfolding Compl_def by (blast dest: CollectD)
lemmas ComplE = ComplD [elim_format]
subsection \<open>Empty sets\<close>
lemma empty_eq: "{x. False} = {}"
by (simp add: empty_def)
lemma emptyD: "a : {} \<Longrightarrow> P"
unfolding empty_def by (blast dest: CollectD)
lemmas emptyE = emptyD [elim_format]
lemma not_emptyD:
assumes "\<not> A={}"
shows "EX x. x:A"
proof -
have "\<not> (EX x. x:A) \<Longrightarrow> A = {}"
by (rule equalityI) (blast intro!: subsetI elim!: emptyD)+
with assms show ?thesis by blast
qed
subsection \<open>Singleton sets\<close>
lemma singletonI: "a : {a}"
unfolding singleton_def by (blast intro: CollectI)
lemma singletonD: "b : {a} \<Longrightarrow> b=a"
unfolding singleton_def by (blast dest: CollectD)
lemmas singletonE = singletonD [elim_format]
subsection \<open>Unions of families\<close>
(*The order of the premises presupposes that A is rigid; b may be flexible*)
lemma UN_I: "\<lbrakk>a:A; b: B(a)\<rbrakk> \<Longrightarrow> b: (UN x:A. B(x))"
unfolding UNION_def by (blast intro: bexI CollectI)
lemma UN_E: "\<lbrakk>b : (UN x:A. B(x)); \<And>x. \<lbrakk>x:A; b: B(x)\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding UNION_def by (blast dest: CollectD elim: bexE)
lemma UN_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (UN x:A. C(x)) = (UN x:B. D(x))"
by (simp add: UNION_def cong: bex_cong)
subsection \<open>Intersections of families\<close>
lemma INT_I: "(\<And>x. x:A \<Longrightarrow> b: B(x)) \<Longrightarrow> b : (INT x:A. B(x))"
unfolding INTER_def by (blast intro: CollectI ballI)
lemma INT_D: "\<lbrakk>b : (INT x:A. B(x)); a:A\<rbrakk> \<Longrightarrow> b: B(a)"
unfolding INTER_def by (blast dest: CollectD bspec)
(*"Classical" elimination rule -- does not require proving X:C *)
lemma INT_E: "\<lbrakk>b : (INT x:A. B(x)); b: B(a) \<Longrightarrow> R; \<not> a:A \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding INTER_def by (blast dest: CollectD bspec)
lemma INT_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (INT x:A. C(x)) = (INT x:B. D(x))"
by (simp add: INTER_def cong: ball_cong)
subsection \<open>Rules for Unions\<close>
(*The order of the premises presupposes that C is rigid; A may be flexible*)
lemma UnionI: "\<lbrakk>X:C; A:X\<rbrakk> \<Longrightarrow> A : Union(C)"
unfolding Union_def by (blast intro: UN_I)
lemma UnionE: "\<lbrakk>A : Union(C); \<And>X. \<lbrakk> A:X; X:C\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding Union_def by (blast elim: UN_E)
subsection \<open>Rules for Inter\<close>
lemma InterI: "(\<And>X. X:C \<Longrightarrow> A:X) \<Longrightarrow> A : Inter(C)"
unfolding Inter_def by (blast intro: INT_I)
(*A "destruct" rule -- every X in C contains A as an element, but
A:X can hold when X:C does not! This rule is analogous to "spec". *)
lemma InterD: "\<lbrakk>A : Inter(C); X:C\<rbrakk> \<Longrightarrow> A:X"
unfolding Inter_def by (blast dest: INT_D)
(*"Classical" elimination rule -- does not require proving X:C *)
lemma InterE: "\<lbrakk>A : Inter(C); A:X \<Longrightarrow> R; \<not> X:C \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding Inter_def by (blast elim: INT_E)
section \<open>Derived rules involving subsets; Union and Intersection as lattice operations\<close>
subsection \<open>Big Union -- least upper bound of a set\<close>
lemma Union_upper: "B:A \<Longrightarrow> B <= Union(A)"
by (blast intro: subsetI UnionI)
lemma Union_least: "(\<And>X. X:A \<Longrightarrow> X<=C) \<Longrightarrow> Union(A) <= C"
by (blast intro: subsetI dest: subsetD elim: UnionE)
subsection \<open>Big Intersection -- greatest lower bound of a set\<close>
lemma Inter_lower: "B:A \<Longrightarrow> Inter(A) <= B"
by (blast intro: subsetI dest: InterD)
lemma Inter_greatest: "(\<And>X. X:A \<Longrightarrow> C<=X) \<Longrightarrow> C <= Inter(A)"
by (blast intro: subsetI InterI dest: subsetD)
subsection \<open>Finite Union -- the least upper bound of 2 sets\<close>
lemma Un_upper1: "A <= A Un B"
by (blast intro: subsetI UnI1)
lemma Un_upper2: "B <= A Un B"
by (blast intro: subsetI UnI2)
lemma Un_least: "\<lbrakk>A<=C; B<=C\<rbrakk> \<Longrightarrow> A Un B <= C"
by (blast intro: subsetI elim: UnE dest: subsetD)
subsection \<open>Finite Intersection -- the greatest lower bound of 2 sets\<close>
lemma Int_lower1: "A Int B <= A"
by (blast intro: subsetI elim: IntE)
lemma Int_lower2: "A Int B <= B"
by (blast intro: subsetI elim: IntE)
lemma Int_greatest: "\<lbrakk>C<=A; C<=B\<rbrakk> \<Longrightarrow> C <= A Int B"
by (blast intro: subsetI IntI dest: subsetD)
subsection \<open>Monotonicity\<close>
lemma monoI: "(\<And>A B. A <= B \<Longrightarrow> f(A) <= f(B)) \<Longrightarrow> mono(f)"
unfolding mono_def by blast
lemma monoD: "\<lbrakk>mono(f); A <= B\<rbrakk> \<Longrightarrow> f(A) <= f(B)"
unfolding mono_def by blast
lemma mono_Un: "mono(f) \<Longrightarrow> f(A) Un f(B) <= f(A Un B)"
by (blast intro: Un_least dest: monoD intro: Un_upper1 Un_upper2)
lemma mono_Int: "mono(f) \<Longrightarrow> f(A Int B) <= f(A) Int f(B)"
by (blast intro: Int_greatest dest: monoD intro: Int_lower1 Int_lower2)
subsection \<open>Automated reasoning setup\<close>
lemmas [intro!] = ballI subsetI InterI INT_I CollectI ComplI IntI UnCI singletonI
and [intro] = bexI UnionI UN_I
and [elim!] = bexE UnionE UN_E CollectE ComplE IntE UnE emptyE singletonE
and [elim] = ballE InterD InterE INT_D INT_E subsetD subsetCE
lemma mem_rews:
"(a : A Un B) \<longleftrightarrow> (a:A | a:B)"
"(a : A Int B) \<longleftrightarrow> (a:A \<and> a:B)"
"(a : Compl(B)) \<longleftrightarrow> (\<not>a:B)"
"(a : {b}) \<longleftrightarrow> (a=b)"
"(a : {}) \<longleftrightarrow> False"
"(a : {x. P(x)}) \<longleftrightarrow> P(a)"
by blast+
lemmas [simp] = trivial_set empty_eq mem_rews
and [cong] = ball_cong bex_cong INT_cong UN_cong
section \<open>Equalities involving union, intersection, inclusion, etc.\<close>
subsection \<open>Binary Intersection\<close>
lemma Int_absorb: "A Int A = A"
by (blast intro: equalityI)
lemma Int_commute: "A Int B = B Int A"
by (blast intro: equalityI)
lemma Int_assoc: "(A Int B) Int C = A Int (B Int C)"
by (blast intro: equalityI)
lemma Int_Un_distrib: "(A Un B) Int C = (A Int C) Un (B Int C)"
by (blast intro: equalityI)
lemma subset_Int_eq: "(A<=B) \<longleftrightarrow> (A Int B = A)"
by (blast intro: equalityI elim: equalityE)
subsection \<open>Binary Union\<close>
lemma Un_absorb: "A Un A = A"
by (blast intro: equalityI)
lemma Un_commute: "A Un B = B Un A"
by (blast intro: equalityI)
lemma Un_assoc: "(A Un B) Un C = A Un (B Un C)"
by (blast intro: equalityI)
lemma Un_Int_distrib: "(A Int B) Un C = (A Un C) Int (B Un C)"
by (blast intro: equalityI)
lemma Un_Int_crazy:
"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"
by (blast intro: equalityI)
lemma subset_Un_eq: "(A<=B) \<longleftrightarrow> (A Un B = B)"
by (blast intro: equalityI elim: equalityE)
subsection \<open>Simple properties of \<open>Compl\<close> -- complement of a set\<close>
lemma Compl_disjoint: "A Int Compl(A) = {x. False}"
by (blast intro: equalityI)
lemma Compl_partition: "A Un Compl(A) = {x. True}"
by (blast intro: equalityI)
lemma double_complement: "Compl(Compl(A)) = A"
by (blast intro: equalityI)
lemma Compl_Un: "Compl(A Un B) = Compl(A) Int Compl(B)"
by (blast intro: equalityI)
lemma Compl_Int: "Compl(A Int B) = Compl(A) Un Compl(B)"
by (blast intro: equalityI)
lemma Compl_UN: "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"
by (blast intro: equalityI)
lemma Compl_INT: "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"
by (blast intro: equalityI)
(*Halmos, Naive Set Theory, page 16.*)
lemma Un_Int_assoc_eq: "((A Int B) Un C = A Int (B Un C)) \<longleftrightarrow> (C<=A)"
by (blast intro: equalityI elim: equalityE)
subsection \<open>Big Union and Intersection\<close>
lemma Union_Un_distrib: "Union(A Un B) = Union(A) Un Union(B)"
by (blast intro: equalityI)
lemma Union_disjoint:
"(Union(C) Int A = {x. False}) \<longleftrightarrow> (ALL B:C. B Int A = {x. False})"
by (blast intro: equalityI elim: equalityE)
lemma Inter_Un_distrib: "Inter(A Un B) = Inter(A) Int Inter(B)"
by (blast intro: equalityI)
subsection \<open>Unions and Intersections of Families\<close>
lemma UN_eq: "(UN x:A. B(x)) = Union({Y. EX x:A. Y=B(x)})"
by (blast intro: equalityI)
(*Look: it has an EXISTENTIAL quantifier*)
lemma INT_eq: "(INT x:A. B(x)) = Inter({Y. EX x:A. Y=B(x)})"
by (blast intro: equalityI)
lemma Int_Union_image: "A Int Union(B) = (UN C:B. A Int C)"
by (blast intro: equalityI)
lemma Un_Inter_image: "A Un Inter(B) = (INT C:B. A Un C)"
by (blast intro: equalityI)
section \<open>Monotonicity of various operations\<close>
lemma Union_mono: "A<=B \<Longrightarrow> Union(A) <= Union(B)"
by blast
lemma Inter_anti_mono: "B <= A \<Longrightarrow> Inter(A) <= Inter(B)"
by blast
lemma UN_mono: "\<lbrakk>A <= B; \<And>x. x:A \<Longrightarrow> f(x)<=g(x)\<rbrakk> \<Longrightarrow> (UN x:A. f(x)) <= (UN x:B. g(x))"
by blast
lemma INT_anti_mono: "\<lbrakk>B <= A; \<And>x. x:A \<Longrightarrow> f(x) <= g(x)\<rbrakk> \<Longrightarrow> (INT x:A. f(x)) <= (INT x:A. g(x))"
by blast
lemma Un_mono: "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Un B <= C Un D"
by blast
lemma Int_mono: "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Int B <= C Int D"
by blast
lemma Compl_anti_mono: "A <= B \<Longrightarrow> Compl(B) <= Compl(A)"
by blast
end
|
module Compiler.Separate
import public Core.FC
import public Core.Name
import public Core.Name.Namespace
import public Core.CompileExpr
import public Compiler.VMCode
import public Libraries.Data.Graph
import public Libraries.Data.SortedMap
import public Libraries.Data.SortedSet
import public Libraries.Data.StringMap
import Core.Hash
import Core.TT
import Data.List
import Data.List1
import Data.Vect
import Data.Maybe
%default covering
-- Compilation unit IDs are intended to be opaque,
-- just to be able to express dependencies via keys in a map and such.
export
record CompilationUnitId where
constructor CUID
int : Int
export
Eq CompilationUnitId where
CUID x == CUID y = x == y
export
Ord CompilationUnitId where
compare (CUID x) (CUID y) = compare x y
export
Hashable CompilationUnitId where
hashWithSalt h (CUID int) = hashWithSalt h int
||| A compilation unit is a set of namespaces.
|||
||| The record is parameterised by the type of the definition,
||| which makes it reusable for various IRs provided by getCompileData.
public export
record CompilationUnit def where
constructor MkCompilationUnit
||| Unique identifier of a compilation unit within a CompilationUnitInfo record.
id : CompilationUnitId
||| Namespaces contained within the compilation unit.
namespaces : List1 Namespace
||| Other units that this unit depends on.
dependencies : SortedSet CompilationUnitId
||| The definitions belonging into this compilation unit.
definitions : List (Name, def)
export
Hashable def => Hashable (CompilationUnit def) where
hashWithSalt h cu =
h `hashWithSalt` SortedSet.toList cu.dependencies
`hashWithSalt` cu.definitions
private
getNS : Name -> Namespace
getNS (NS ns _) = ns
getNS _ = emptyNS
||| Group definitions by namespace.
private
splitByNS : List (Name, def) -> List (Namespace, List (Name, def))
splitByNS = SortedMap.toList . foldl addOne SortedMap.empty
where
addOne
: SortedMap Namespace (List (Name, def))
-> (Name, def)
-> SortedMap Namespace (List (Name, def))
addOne nss ndef@(n, _) =
SortedMap.mergeWith
(++)
(SortedMap.singleton (getNS n) [ndef])
nss
public export
interface HasNamespaces a where
||| Return the set of namespaces mentioned within
nsRefs : a -> SortedSet Namespace
-- For now, we have instances only for NamedDef and VMDef.
-- For other IR representations, we'll have to add more instances.
-- This is not hard, just a bit of tedious mechanical work.
mutual
export
HasNamespaces NamedCExp where
nsRefs (NmLocal fc n) = SortedSet.empty
nsRefs (NmRef fc n) = SortedSet.singleton $ getNS n
nsRefs (NmLam fc n rhs) = nsRefs rhs
nsRefs (NmLet fc n val rhs) = nsRefs val <+> nsRefs rhs
nsRefs (NmApp fc f args) = nsRefs f <+> concatMap nsRefs args
nsRefs (NmCon fc cn ci tag args) = concatMap nsRefs args
nsRefs (NmForce fc reason rhs) = nsRefs rhs
nsRefs (NmDelay fc reason rhs) = nsRefs rhs
nsRefs (NmErased fc) = SortedSet.empty
nsRefs (NmPrimVal ft x) = SortedSet.empty
nsRefs (NmOp fc op args) = concatMap nsRefs args
nsRefs (NmExtPrim fc n args) = concatMap nsRefs args
nsRefs (NmConCase fc scrut alts mbDflt) =
nsRefs scrut <+> concatMap nsRefs alts <+> concatMap nsRefs mbDflt
nsRefs (NmConstCase fc scrut alts mbDflt) =
nsRefs scrut <+> concatMap nsRefs alts <+> concatMap nsRefs mbDflt
nsRefs (NmCrash fc msg) = SortedSet.empty
export
HasNamespaces NamedConAlt where
nsRefs (MkNConAlt n ci tag args rhs) = nsRefs rhs
export
HasNamespaces NamedConstAlt where
nsRefs (MkNConstAlt c rhs) = nsRefs rhs
export
HasNamespaces NamedDef where
nsRefs (MkNmFun argNs rhs) = nsRefs rhs
nsRefs (MkNmCon tag arity nt) = SortedSet.empty
nsRefs (MkNmForeign ccs fargs rty) = SortedSet.empty
nsRefs (MkNmError rhs) = nsRefs rhs
export
HasNamespaces VMInst where
nsRefs (DECLARE x) = empty
nsRefs START = empty
nsRefs (ASSIGN x y) = empty
nsRefs (MKCON x tag args) = either (const empty) (singleton . getNS) tag
nsRefs (MKCLOSURE x n missing args) = singleton $ getNS n
nsRefs (MKCONSTANT x y) = empty
nsRefs (APPLY x f a) = empty
nsRefs (CALL x tailpos n args) = singleton $ getNS n
nsRefs (OP x y xs) = empty
nsRefs (EXTPRIM x n xs) = singleton $ getNS n
nsRefs (CASE x alts def) =
maybe empty (concatMap nsRefs) def <+>
concatMap ((concatMap nsRefs) . snd) alts <+>
concatMap ((either (const empty) (singleton . getNS)) . fst) alts
nsRefs (CONSTCASE x alts def) =
maybe empty (concatMap nsRefs) def <+>
concatMap ((concatMap nsRefs) . snd) alts
nsRefs (PROJECT x value pos) = empty
nsRefs (NULL x) = empty
nsRefs (ERROR x) = empty
export
HasNamespaces VMDef where
nsRefs (MkVMFun args is) = concatMap nsRefs is
nsRefs (MkVMForeign _ _ _) = empty
nsRefs (MkVMError is) = concatMap nsRefs is
-- a slight hack for convenient use with CompileData.namedDefs
export
HasNamespaces a => HasNamespaces (FC, a) where
nsRefs (_, x) = nsRefs x
-- another slight hack for convenient use with CompileData.namedDefs
export
Hashable def => Hashable (FC, def) where
-- ignore FC in hash, like everywhere else
hashWithSalt h (fc, x) = hashWithSalt h x
||| Output of the codegen separation algorithm.
||| Should contain everything you need in a separately compiling codegen.
public export
record CompilationUnitInfo def where
constructor MkCompilationUnitInfo
||| Compilation units computed from the given definitions,
||| ordered topologically, starting from units depending on no other unit.
compilationUnits : List (CompilationUnit def)
||| Mapping from ID to CompilationUnit.
byId : SortedMap CompilationUnitId (CompilationUnit def)
||| Maps each namespace to the compilation unit that contains it.
namespaceMap : SortedMap Namespace CompilationUnitId
||| Group the given definitions into compilation units for separate code generation.
export
getCompilationUnits : HasNamespaces def => List (Name, def) -> CompilationUnitInfo def
getCompilationUnits {def} defs =
let
-- Definitions grouped by namespace.
defsByNS : SortedMap Namespace (List (Name, def))
= SortedMap.fromList $ splitByNS defs
-- Mapping from a namespace to all namespaces mentioned within.
-- Represents graph edges pointing in that direction.
nsDeps : SortedMap Namespace (SortedSet Namespace)
= foldl (SortedMap.mergeWith SortedSet.union) SortedMap.empty
[ SortedMap.singleton (getNS n) (SortedSet.delete (getNS n) (nsRefs d))
| (n, d) <- defs
]
-- Strongly connected components of the NS dep graph,
-- ordered by output degree ascending.
--
-- Each SCC will become a compilation unit.
components : List (List1 Namespace)
= List.reverse $ tarjan nsDeps -- tarjan generates reverse toposort
-- Maps a namespace to the compilation unit that contains it.
nsMap : SortedMap Namespace CompilationUnitId
= SortedMap.fromList [(ns, cuid) | (cuid, nss) <- withCUID components, ns <- List1.forget nss]
-- List of all compilation units, ordered by number of dependencies, ascending.
units : List (CompilationUnit def)
= [mkUnit nsDeps nsMap defsByNS cuid nss | (cuid, nss) <- withCUID components]
in MkCompilationUnitInfo
{ compilationUnits = units
, byId = SortedMap.fromList [(unit.id, unit) | unit <- units]
, namespaceMap = nsMap
}
where
withCUID : List a -> List (CompilationUnitId, a)
withCUID xs = [(CUID $ cast i, x) | (i, x) <- zip [0..length xs] xs]
||| Wrap all information in a compilation unit record.
mkUnit :
SortedMap Namespace (SortedSet Namespace)
-> SortedMap Namespace CompilationUnitId
-> SortedMap Namespace (List (Name, def))
-> CompilationUnitId -> List1 Namespace -> CompilationUnit def
mkUnit nsDeps nsMap defsByNS cuid nss =
MkCompilationUnit
{ id = cuid
, namespaces = nss
, dependencies = SortedSet.delete cuid dependencies
, definitions = definitions
}
where
dependencies : SortedSet CompilationUnitId
dependencies = SortedSet.fromList $ do
ns <- List1.forget nss -- NS contained within
depsNS <- SortedSet.toList $ -- NS we depend on
fromMaybe SortedSet.empty $
SortedMap.lookup ns nsDeps
case SortedMap.lookup depsNS nsMap of
Nothing => []
Just depCUID => [depCUID]
definitions : List (Name, def)
definitions = concat [fromMaybe [] $ SortedMap.lookup ns defsByNS | ns <- nss]
|
{-# OPTIONS --copatterns --sized-types #-}
{- | The purpose of this module is to demonstrate how observational type theory
can be implemented for arbitrary types in Agda through the use of tests
and a corresponding bisimulation proof method.
To use the equivalence for a type A, the type has to be made _testable_ by
creating an instance of Testable for it, see the accompanying module
TestsInstances. Such an instance effectively encodes A as inductive or
coinductive type, by giving an observation map to a coproduct of product
of components of A. For now, we do not pose any restrictions on this encoding,
hence it can be used to induce trivial equivalences.
Using such an instance of Testable for A, we can define _tests_ and
_observational equivalence_ for A. Moreover, we give a bisimulation proof
method and prove its soundness.
-}
module Tests where
open import Size
open import Level
open import Data.Product
open import Data.Bool
open import Function
open import Relation.Binary.PropositionalEquality as P
open β‘-Reasoning
open import Relation.Binary
open import Isomorphisms
-- | Write dependent functions as Ξ -type to make duality between
-- inductive and coinducitve types clearer.
record Ξ {a b} (A : Set a) (B : A β Set b) : Set (a β b) where
field
app : (x : A) β B x
open Ξ public
-- | Generalised copairing for coproducts:
-- Turn an I-indexed tuple of maps fα΅’ : Bα΅’ β C into a map Ξ£ I B β C.
cotuple : {I C : Set} β {B : I β Set} β ((i : I) β B i β C) β (Ξ£ I B β C)
cotuple f x = f (projβ x) (projβ x)
-- | Distinguish inductive and coinductive types
data Kind : Set where
ind : Kind
coind : Kind
--- | Make a type observable. An inductive type shall be represented by
--- a coproduct, whereas a coinductive type is represented by a product.
ObsTy : (I : Set) (B : I β Set) β Kind β Set
ObsTy I B ind = Ξ£ I B
ObsTy I B coind = Ξ I B
-- | Make a type testable
record Testable (A : Set) : Setβ where
coinductive
field
index : Set
parts : index β Set
kind : Kind
obs : A β ObsTy index parts kind
partsTestable : (i : index) β Testable (parts i)
open Testable public
record IsoTestable (A : Set) : Setβ where
field
testable : Testable A
obsIso : IsIso (obs testable)
open IsoTestable public
SubTests : {l : Size} β {A : Set} β Testable A β Kind β Set
-- | Test formulae
data Test {i : Size} {A : Set} (T : Testable A) : Set where
β€ : Test T
β₯ : Test T
nonTriv : {j : Size< i} β SubTests {j} T (kind T) β Test {i} {A} T
SubTests {l} T ind = Ξ (index T) (Ξ» i β Test {l} (partsTestable T i))
SubTests {l} T coind = Ξ£ (index T) (Ξ» i β Test {l} (partsTestable T i))
-- | Satisfaction of subtests.
sat : {A : Set} {T : Testable A} {l : Size} β
(k : Kind) β SubTests {l} T k β ObsTy (index T) (parts T) k β Bool
-- | Test satisfaction
_β¨_ : {A : Set} {T : Testable A} β A β Test T β Bool
x β¨ β€ = true
x β¨ β₯ = false
_β¨_ {A} {T} x (nonTriv nt) = sat (kind T) nt (obs T x)
sat ind Οs o = cotuple (Ξ» i y β y β¨ app Οs i) o
sat coind (i , Ο) o = app o i β¨ Ο
-- | Observational equivalence: terms are equal if they satisfy the same tests.
record _ββ¨_β©_ {A : Set} (x : A) (T : Testable A) (y : A) : Setβ where
field
eqProof : (Ο : Test T) β (x β¨ Ο β‘ y β¨ Ο)
open _ββ¨_β©_ public
β‘ββ : {A : Set} β {T : Testable A} β
{a b : A} β a β‘ b β a ββ¨ T β© b
β‘ββ p = record { eqProof = Ξ» Ο β cong (Ξ» x β x β¨ Ο) p }
β-refl : {A : Set} β (T : Testable A) β
{a : A} β a ββ¨ T β© a
β-refl T = record { eqProof = Ξ» Ο β refl }
β-sym : {A : Set} β (T : Testable A) β
{a b : A} β a ββ¨ T β© b β b ββ¨ T β© a
β-sym T p = record { eqProof = sym β (eqProof p) }
β-trans : {A : Set} β (T : Testable A) β
{a b c : A} β a ββ¨ T β© b β b ββ¨ T β© c β a ββ¨ T β© c
β-trans T pβ pβ =
record { eqProof = Ξ» Ο β trans (eqProof pβ Ο) (eqProof pβ Ο) }
β-setoid : {A : Set} β (T : Testable A) β Setoid _ _
β-setoid {A} T = record
{ Carrier = A
;_β_ = Ξ» x y β x ββ¨ T β© y
; isEquivalence = record
{ refl = β-refl T
; sym = β-sym T
; trans = β-trans T }
}
-- Most likely impossible to prove within Agda.
-- Is it consistent with the system to postulate this for _IsoTestable_ ?
-- β-cong : {A B : Set} β {Tβ : Testable A} β {Tβ : Testable B} β {a b : A} β
-- (f : A β B) β a ββ¨ Tβ β© b β f a ββ¨ Tβ β© f b
-- If A is testable and there is a map B β A, then B is also testable.
comap-testable : {A B : Set} β (B β A) β Testable A β Testable B
comap-testable {A} {B} f T =
record
{ index = index T
; parts = parts T
; kind = kind T
; obs = (obs T) β f
; partsTestable = partsTestable T
}
-- | If A is testable and A β
B, then B is iso-testable as well.
iso-testable : {A B : Set} β Iso B A β IsoTestable A β IsoTestable B
iso-testable {A} {B} I T =
record
{ testable = comap-testable (Iso.iso I) (testable T)
; obsIso = iso-comp (Iso.indeedIso I) (obsIso T)
}
-- | Heterogeneous
record _~β¨_β₯_β©_ {A B : Set}
(x : A) (T : Testable A) (I : Iso B A) (y : B) : Setβ where
field
eqProofH : (Ο : Test T) β (x β¨ Ο β‘ (Iso.iso I y) β¨ Ο)
open _~β¨_β₯_β©_ public
-- | Helper to match on Kind in construction of β.
β-Proof : {A : Set} β
(k : Kind) β
(T : Testable A) β
(A β ObsTy (index T) (parts T) k) β
A β A β Set
-- | Bisimilarity induced from testable types.
record _ββ¨_β©_ {A : Set} (x : A) (T : Testable A) (y : A) : Set where
coinductive
field
proof : β-Proof (kind T) T (obs T) x y
open _ββ¨_β©_ public
-- | Helper to fiddle around with index in construction of IndProof.
ResolveIdx : {A : Set} β (T : Testable A) β
(i : index T) β (r s : ObsTy (index T) (parts T) ind) β
projβ r β‘ i β projβ s β‘ i β Set
ResolveIdx T i (.i , x') (.i , y') refl refl = x' ββ¨ partsTestable T i β© y'
-- | Proofs of bisimilarity on inductive types.
record IndProof {A : Set}
(T : Testable A)
(o : A β ObsTy (index T) (parts T) ind)
(x y : A) : Set where
coinductive
field
which : index T
eqIndexβ : projβ (o x) β‘ which
eqIndexβ : projβ (o y) β‘ which
eqTrans : ResolveIdx T which (o x) (o y) eqIndexβ eqIndexβ
open IndProof public
-- | Proofs of bisimilarity on coinductive types.
record CoindProof {A : Set}
(T : Testable A)
(o : A β ObsTy (index T) (parts T) coind)
(x y : A) : Set where
coinductive
field
eqStep : (i : index T) β app (o x) i ββ¨ partsTestable T i β© app (o y) i
open CoindProof public
β-Proof ind = IndProof
β-Proof coind = CoindProof
-- | Lemma for induction to prove soundness of bisimilarity
lem-βββ-testInduct : {j : Size} {A : Set} β (T : Testable A) β (x y : A) β
x ββ¨ T β© y β (Ο : Test {j} T) β x β¨ Ο β‘ y β¨ Ο
lem-βββ-testInduct _ _ _ _ β€ = refl
lem-βββ-testInduct _ _ _ _ β₯ = refl
lem-βββ-testInduct {j} {A} T x y xβy (nonTriv {l} nt)
= matchKind (kind T) nt (obs T) (proof xβy)
where
matchKind : (k : Kind) β
(nt : SubTests {l} T k) β
(o : A β ObsTy (index T) (parts T) k) β
β-Proof k T o x y β
sat k nt (o x) β‘ sat k nt (o y)
matchKind ind nt o p
= refine (which p) (o x) (o y) (eqIndexβ p) (eqIndexβ p) (eqTrans p)
where
--| Do pattern matching on IndProof
refine : (i : index T) β (r s : ObsTy (index T) (parts T) ind) β
(eqPβ : projβ r β‘ i) β (eqPβ : projβ s β‘ i) β
ResolveIdx T i r s eqPβ eqPβ β
(projβ r) β¨ app nt (projβ r) β‘ (projβ s) β¨ app nt (projβ s)
refine i (.i , x') (.i , y') refl refl p'
= lem-βββ-testInduct (partsTestable T i) x' y' p' (app nt i)
matchKind coind nt o p
= lem-βββ-testInduct (partsTestable T i) x' y' (eqStep p i) Ο
where
i : index T
i = projβ nt
Ο = projβ nt
x' = app (o x) i
y' = app (o y) i
-- | Bisimulation proofs are sound for observational equivalence.
βββ : {A : Set} β (T : Testable A) β (x y : A) β
x ββ¨ T β© y β x ββ¨ T β© y
βββ T x y xβy = record { eqProof = lem-βββ-testInduct T x y xβy }
{-
βββ : {A : Set} β (T : Testable A) β (x y : A) β
x ββ¨ T β© y β x ββ¨ T β© y
βββ {A} T x y xβy = record { proof = matchKind (kind T) (obs T) }
where
matchKind : (k : Kind) β
(o : A β ObsTy (index T) (parts T) k) β
β-Proof k T o x y
matchKind ind o = {!!}
matchKind coind o = {!!}
-}
|
SUBROUTINE QUIKVIS5(IDTARG,TARGNAMES,KTARGTYP,TARGPARM,IERR)
IMPLICIT REAL*8 (A-H,O-Z)
C
C THIS ROUTINE IS PART OF THE QUIKVIS PROGRAM. IT IS THE DRIVER FOR
C COMPUTING AND REPORTING TARGET AVAILABILITY. THESE ARE DONE BY LOWER
C LEVEL ROUTINES QUIKVIS5A AND QUIKVIS5B.
C
C
C VARIABLE DIM TYPE I/O DESCRIPTION
C -------- --- ---- --- -----------
C
C IDTARG MAXTARGS I*4 I DESCRIBED IN QUIKVIS(=MAIN) PROLOGUE.
C
C TARGNAMES MAXTARGS CH*16 I DESCRIBED IN QUIKVIS(=MAIN) PROLOGUE.
C
C KTARGTYP MAXTARGS I*4 I DESCRIBED IN QUIKVIS(=MAIN) PROLOGUE.
C
C TARGPARM NPARMS,MAXTARGS R*8 I DESCRIBED IN QUIKVIS(=MAIN) PROLOGUE.
C
C IERR 1 I*4 O ERROR RETURN FLAG
C =0, NO ERROR
C =OTHERWISE, ERROR.
C
C***********************************************************************
C
C BY C PETRUZZO/GFSC/742. 2/86.
C MODIFIED....
C
C***********************************************************************
C
INCLUDE 'QUIKVIS.INC'
C
CHARACTER*16 TARGNAMES(MAXTARGS)
INTEGER*4 IDTARG(MAXTARGS)
REAL*8 TARGPARM(NPARMS,MAXTARGS)
INTEGER*4 KTARGTYP(MAXTARGS)
C
IBUG = 0
LUBUG = 19
C
IF(IBUG.NE.0) WRITE(LUBUG,9001)
* (ITARG,TARGPARM(1,ITARG)*DEGRAD,TARGPARM(2,ITARG)*DEGRAD,
* ITARG=1,80)
9001 FORMAT(/,' QUIKVIS5. DEBUG. ENTRY VALUES. FIRST 80.'/,
* (' ITARG=',I3,' RA,DEC=',2G13.5))
C
C
C
C DO COMPUTATIONS AND OUTPUT THE RESULTS.
C
C*** RUN USING SPECIFIC TARGETS
C
IF(.NOT.DOSURVEY) THEN
CALL QUIKVIS5A(IDTARG,TARGNAMES,KTARGTYP,TARGPARM,IERR)
END IF
C
C*** SKY SURVEY RUN
C
IF(DOSURVEY) THEN
CALL QUIKVIS5B(IDTARG,TARGNAMES,KTARGTYP,TARGPARM,IERR)
END IF
C
RETURN
C
C***********************************************************************
C
C
C**** INITIALIZATION CALL. PUT GLOBAL PARAMETER VALUES INTO THIS
C ROUTINE'S LOCAL VARIABLES.
C
ENTRY QVINIT5
C
CALL QUIKVIS999(-1,R8DATA,I4DATA,L4DATA)
RETURN
C
C***********************************************************************
C
END
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
In the standard library we cannot assume the univalence axiom.
We say two types are equivalent if they are isomorphic.
Two equivalent types have the same cardinality.
-/
import data.sum data.nat
open function
structure equiv [class] (A B : Type) :=
(to_fun : A β B)
(inv_fun : B β A)
(left_inv : left_inverse inv_fun to_fun)
(right_inv : right_inverse inv_fun to_fun)
namespace equiv
definition perm [reducible] (A : Type) := equiv A A
infix ` β `:50 := equiv
definition fn {A B : Type} (e : equiv A B) : A β B :=
@equiv.to_fun A B e
infixr ` β `:100 := fn
definition inv {A B : Type} [e : equiv A B] : B β A :=
@equiv.inv_fun A B e
lemma eq_of_to_fun_eq {A B : Type} : β {eβ eβ : equiv A B}, fn eβ = fn eβ β eβ = eβ
| (mk fβ gβ lβ rβ) (mk fβ gβ lβ rβ) h :=
have fβ = fβ, from h,
have gβ = gβ, from funext (Ξ» x,
have fβ (gβ x) = fβ (gβ x), from eq.trans (rβ x) (eq.symm (rβ x)),
have fβ (gβ x) = fβ (gβ x), begin subst fβ, exact this end,
show gβ x = gβ x, from injective_of_left_inverse lβ this),
by congruence; repeat assumption
protected definition refl [refl] (A : Type) : A β A :=
mk (@id A) (@id A) (Ξ» x, rfl) (Ξ» x, rfl)
protected definition symm [symm] {A B : Type} : A β B β B β A
| (mk f g hβ hβ) := mk g f hβ hβ
protected definition trans [trans] {A B C : Type} : A β B β B β C β A β C
| (mk fβ gβ lβ rβ) (mk fβ gβ lβ rβ) :=
mk (fβ β fβ) (gβ β gβ)
(show β x, gβ (gβ (fβ (fβ x))) = x, by intros; rewrite [lβ, lβ]; reflexivity)
(show β x, fβ (fβ (gβ (gβ x))) = x, by intros; rewrite [rβ, rβ]; reflexivity)
abbreviation id {A : Type} := equiv.refl A
namespace ops
postfix β»ΒΉ := equiv.symm
postfix β»ΒΉ := equiv.inv
notation eβ β eβ := equiv.trans eβ eβ
end ops
open equiv.ops
lemma id_apply {A : Type} (x : A) : id β x = x :=
rfl
lemma comp_apply {A B C : Type} (g : B β C) (f : A β B) (x : A) : (g β f) β x = g β f β x :=
begin cases g, cases f, esimp end
lemma inverse_apply_apply {A B : Type} : β (e : A β B) (x : A), eβ»ΒΉ β e β x = x
| (mk fβ gβ lβ rβ) x := begin unfold [equiv.symm, fn], rewrite lβ end
lemma eq_iff_eq_of_injective {A B : Type} {f : A β B} (inj : injective f) (a b : A) : f a = f b β a = b :=
iff.intro
(suppose f a = f b, inj this)
(suppose a = b, by rewrite this)
lemma apply_eq_iff_eq {A B : Type} : β (f : A β B) (x y : A), f β x = f β y β x = y
| (mk fβ gβ lβ rβ) x y := eq_iff_eq_of_injective (injective_of_left_inverse lβ) x y
lemma apply_eq_iff_eq_inverse_apply {A B : Type} : β (f : A β B) (x : A) (y : B), f β x = y β x = fβ»ΒΉ β y
| (mk fβ gβ lβ rβ) x y :=
begin
esimp, unfold [equiv.symm, fn], apply iff.intro,
suppose fβ x = y, by subst y; rewrite lβ,
suppose x = gβ y, by subst x; rewrite rβ
end
definition false_equiv_empty : empty β false :=
mk (Ξ» e, empty.rec _ e) (Ξ» h, false.rec _ h) (Ξ» e, empty.rec _ e) (Ξ» h, false.rec _ h)
definition arrow_congr [congr] {Aβ Bβ Aβ Bβ : Type} : Aβ β Aβ β Bβ β Bβ β (Aβ β Bβ) β (Aβ β Bβ)
| (mk fβ gβ lβ rβ) (mk fβ gβ lβ rβ) :=
mk
(Ξ» (h : Aβ β Bβ) (a : Aβ), fβ (h (gβ a)))
(Ξ» (h : Aβ β Bβ) (a : Aβ), gβ (h (fβ a)))
(Ξ» h, funext (Ξ» a, by rewrite [lβ, lβ]; reflexivity))
(Ξ» h, funext (Ξ» a, by rewrite [rβ, rβ]; reflexivity))
section
open unit
definition arrow_unit_equiv_unit [simp] (A : Type) : (A β unit) β unit :=
mk (Ξ» f, star) (Ξ» u, (Ξ» f, star))
(Ξ» f, funext (Ξ» x, by cases (f x); reflexivity))
(Ξ» u, by cases u; reflexivity)
definition unit_arrow_equiv [simp] (A : Type) : (unit β A) β A :=
mk (Ξ» f, f star) (Ξ» a, (Ξ» u, a))
(Ξ» f, funext (Ξ» x, by cases x; reflexivity))
(Ξ» u, rfl)
definition empty_arrow_equiv_unit [simp] (A : Type) : (empty β A) β unit :=
mk (Ξ» f, star) (Ξ» u, Ξ» e, empty.rec _ e)
(Ξ» f, funext (Ξ» x, empty.rec _ x))
(Ξ» u, by cases u; reflexivity)
definition false_arrow_equiv_unit [simp] (A : Type) : (false β A) β unit :=
calc (false β A) β (empty β A) : arrow_congr false_equiv_empty !equiv.refl
... β unit : empty_arrow_equiv_unit
end
definition prod_congr [congr] {Aβ Bβ Aβ Bβ : Type} : Aβ β Aβ β Bβ β Bβ β (Aβ Γ Bβ) β (Aβ Γ Bβ)
| (mk fβ gβ lβ rβ) (mk fβ gβ lβ rβ) :=
mk
(Ξ» p, match p with (aβ, bβ) := (fβ aβ, fβ bβ) end)
(Ξ» p, match p with (aβ, bβ) := (gβ aβ, gβ bβ) end)
(Ξ» p, begin cases p, esimp, rewrite [lβ, lβ], reflexivity end)
(Ξ» p, begin cases p, esimp, rewrite [rβ, rβ], reflexivity end)
definition prod_comm [simp] (A B : Type) : (A Γ B) β (B Γ A) :=
mk (Ξ» p, match p with (a, b) := (b, a) end)
(Ξ» p, match p with (b, a) := (a, b) end)
(Ξ» p, begin cases p, esimp end)
(Ξ» p, begin cases p, esimp end)
definition prod_assoc [simp] (A B C : Type) : ((A Γ B) Γ C) β (A Γ (B Γ C)) :=
mk (Ξ» t, match t with ((a, b), c) := (a, (b, c)) end)
(Ξ» t, match t with (a, (b, c)) := ((a, b), c) end)
(Ξ» t, begin cases t with ab c, cases ab, esimp end)
(Ξ» t, begin cases t with a bc, cases bc, esimp end)
section
open unit prod.ops
definition prod_unit_right [simp] (A : Type) : (A Γ unit) β A :=
mk (Ξ» p, p.1)
(Ξ» a, (a, star))
(Ξ» p, begin cases p with a u, cases u, esimp end)
(Ξ» a, rfl)
definition prod_unit_left [simp] (A : Type) : (unit Γ A) β A :=
calc (unit Γ A) β (A Γ unit) : prod_comm
... β A : prod_unit_right
definition prod_empty_right [simp] (A : Type) : (A Γ empty) β empty :=
mk (Ξ» p, empty.rec _ p.2) (Ξ» e, empty.rec _ e) (Ξ» p, empty.rec _ p.2) (Ξ» e, empty.rec _ e)
definition prod_empty_left [simp] (A : Type) : (empty Γ A) β empty :=
calc (empty Γ A) β (A Γ empty) : prod_comm
... β empty : prod_empty_right
end
section
open sum
definition sum_congr [congr] {Aβ Bβ Aβ Bβ : Type} : Aβ β Aβ β Bβ β Bβ β (Aβ + Bβ) β (Aβ + Bβ)
| (mk fβ gβ lβ rβ) (mk fβ gβ lβ rβ) :=
mk
(Ξ» s, match s with inl aβ := inl (fβ aβ) | inr bβ := inr (fβ bβ) end)
(Ξ» s, match s with inl aβ := inl (gβ aβ) | inr bβ := inr (gβ bβ) end)
(Ξ» s, begin cases s, {esimp, rewrite lβ, reflexivity}, {esimp, rewrite lβ, reflexivity} end)
(Ξ» s, begin cases s, {esimp, rewrite rβ, reflexivity}, {esimp, rewrite rβ, reflexivity} end)
open bool unit
definition bool_equiv_unit_sum_unit : bool β (unit + unit) :=
mk (Ξ» b, match b with tt := inl star | ff := inr star end)
(Ξ» s, match s with inl star := tt | inr star := ff end)
(Ξ» b, begin cases b, esimp, esimp end)
(Ξ» s, begin cases s with u u, {cases u, esimp}, {cases u, esimp} end)
definition sum_comm [simp] (A B : Type) : (A + B) β (B + A) :=
mk (Ξ» s, match s with inl a := inr a | inr b := inl b end)
(Ξ» s, match s with inl b := inr b | inr a := inl a end)
(Ξ» s, begin cases s, esimp, esimp end)
(Ξ» s, begin cases s, esimp, esimp end)
definition sum_assoc [simp] (A B C : Type) : ((A + B) + C) β (A + (B + C)) :=
mk (Ξ» s, match s with inl (inl a) := inl a | inl (inr b) := inr (inl b) | inr c := inr (inr c) end)
(Ξ» s, match s with inl a := inl (inl a) | inr (inl b) := inl (inr b) | inr (inr c) := inr c end)
(Ξ» s, begin cases s with ab c, cases ab, repeat esimp end)
(Ξ» s, begin cases s with a bc, esimp, cases bc, repeat esimp end)
definition sum_empty_right [simp] (A : Type) : (A + empty) β A :=
mk (Ξ» s, match s with inl a := a | inr e := empty.rec _ e end)
(Ξ» a, inl a)
(Ξ» s, begin cases s with a e, esimp, exact empty.rec _ e end)
(Ξ» a, rfl)
definition sum_empty_left [simp] (A : Type) : (empty + A) β A :=
calc (empty + A) β (A + empty) : sum_comm
... β A : sum_empty_right
end
section
open prod.ops
definition arrow_prod_equiv_prod_arrow (A B C : Type) : (C β A Γ B) β ((C β A) Γ (C β B)) :=
mk (Ξ» f, (Ξ» c, (f c).1, Ξ» c, (f c).2))
(Ξ» p, Ξ» c, (p.1 c, p.2 c))
(Ξ» f, funext (Ξ» c, begin esimp, cases f c, esimp end))
(Ξ» p, begin cases p, esimp end)
definition arrow_arrow_equiv_prod_arrow (A B C : Type) : (A β B β C) β (A Γ B β C) :=
mk (Ξ» f, Ξ» p, f p.1 p.2)
(Ξ» f, Ξ» a b, f (a, b))
(Ξ» f, rfl)
(Ξ» f, funext (Ξ» p, begin cases p, esimp end))
open sum
definition sum_arrow_equiv_prod_arrow (A B C : Type) : ((A + B) β C) β ((A β C) Γ (B β C)) :=
mk (Ξ» f, (Ξ» a, f (inl a), Ξ» b, f (inr b)))
(Ξ» p, (Ξ» s, match s with inl a := p.1 a | inr b := p.2 b end))
(Ξ» f, funext (Ξ» s, begin cases s, esimp, esimp end))
(Ξ» p, begin cases p, esimp end)
definition sum_prod_distrib (A B C : Type) : ((A + B) Γ C) β ((A Γ C) + (B Γ C)) :=
mk (Ξ» p, match p with (inl a, c) := inl (a, c) | (inr b, c) := inr (b, c) end)
(Ξ» s, match s with inl (a, c) := (inl a, c) | inr (b, c) := (inr b, c) end)
(Ξ» p, begin cases p with ab c, cases ab, repeat esimp end)
(Ξ» s, begin cases s with ac bc, cases ac, esimp, cases bc, esimp end)
definition prod_sum_distrib (A B C : Type) : (A Γ (B + C)) β ((A Γ B) + (A Γ C)) :=
calc (A Γ (B + C)) β ((B + C) Γ A) : prod_comm
... β ((B Γ A) + (C Γ A)) : sum_prod_distrib
... β ((A Γ B) + (A Γ C)) : sum_congr !prod_comm !prod_comm
definition bool_prod_equiv_sum (A : Type) : (bool Γ A) β (A + A) :=
calc (bool Γ A) β ((unit + unit) Γ A) : prod_congr bool_equiv_unit_sum_unit !equiv.refl
... β (A Γ (unit + unit)) : prod_comm
... β ((A Γ unit) + (A Γ unit)) : prod_sum_distrib
... β (A + A) : sum_congr !prod_unit_right !prod_unit_right
end
section
open sum nat unit prod.ops
definition nat_equiv_nat_sum_unit : nat β (nat + unit) :=
mk (Ξ» n, match n with zero := inr star | succ a := inl a end)
(Ξ» s, match s with inl n := succ n | inr star := zero end)
(Ξ» n, begin cases n, repeat esimp end)
(Ξ» s, begin cases s with a u, esimp, {cases u, esimp} end)
definition nat_sum_unit_equiv_nat [simp] : (nat + unit) β nat :=
equiv.symm nat_equiv_nat_sum_unit
definition nat_prod_nat_equiv_nat [simp] : (nat Γ nat) β nat :=
mk (Ξ» p, mkpair p.1 p.2)
(Ξ» n, unpair n)
(Ξ» p, begin cases p, apply unpair_mkpair end)
(Ξ» n, mkpair_unpair n)
definition nat_sum_bool_equiv_nat [simp] : (nat + bool) β nat :=
calc (nat + bool) β (nat + (unit + unit)) : sum_congr !equiv.refl bool_equiv_unit_sum_unit
... β ((nat + unit) + unit) : sum_assoc
... β (nat + unit) : sum_congr nat_sum_unit_equiv_nat !equiv.refl
... β nat : nat_sum_unit_equiv_nat
open decidable
definition nat_sum_nat_equiv_nat [simp] : (nat + nat) β nat :=
mk (Ξ» s, match s with inl n := 2*n | inr n := 2*n+1 end)
(Ξ» n, if even n then inl (n / 2) else inr ((n - 1) / 2))
(Ξ» s, begin
have two_gt_0 : 2 > zero, from dec_trivial,
cases s,
{esimp, rewrite [if_pos (even_two_mul _), nat.mul_div_cancel_left _ two_gt_0]},
{esimp, rewrite [if_neg (not_even_two_mul_plus_one _), nat.add_sub_cancel,
nat.mul_div_cancel_left _ two_gt_0]}
end)
(Ξ» n, by_cases
(Ξ» h : even n,
by rewrite [if_pos h]; esimp; rewrite [nat.mul_div_cancel' (dvd_of_even h)])
(Ξ» h : Β¬ even n,
begin
rewrite [if_neg h], esimp,
cases n,
{exact absurd even_zero h},
{rewrite [-(add_one a), nat.add_sub_cancel,
nat.mul_div_cancel' (dvd_of_even (even_of_odd_succ (odd_of_not_even h)))]}
end))
definition prod_equiv_of_equiv_nat {A : Type} : A β nat β (A Γ A) β A :=
take e, calc
(A Γ A) β (nat Γ nat) : prod_congr e e
... β nat : nat_prod_nat_equiv_nat
... β A : equiv.symm e
end
section
open decidable
definition decidable_eq_of_equiv {A B : Type} [h : decidable_eq A] : A β B β decidable_eq B
| (mk f g l r) :=
take bβ bβ, match h (g bβ) (g bβ) with
| inl he := inl (have aux : f (g bβ) = f (g bβ), from congr_arg f he,
begin rewrite *r at aux, exact aux end)
| inr hn := inr (Ξ» bβeqbβ, by subst bβeqbβ; exact absurd rfl hn)
end
end
definition inhabited_of_equiv {A B : Type} [h : inhabited A] : A β B β inhabited B
| (mk f g l r) := inhabited.mk (f (inhabited.value h))
section
open subtype
definition subtype_equiv_of_subtype {A B : Type} {p : A β Prop} : A β B β {a : A | p a} β {b : B | p bβ»ΒΉ}
| (mk f g l r) :=
mk (Ξ» s, match s with tag v h := tag (f v) (eq.rec_on (eq.symm (l v)) h) end)
(Ξ» s, match s with tag v h := tag (g v) (eq.rec_on (eq.symm (r v)) h) end)
(Ξ» s, begin cases s, esimp, congruence, rewrite l, reflexivity end)
(Ξ» s, begin cases s, esimp, congruence, rewrite r, reflexivity end)
end
section swap
variable {A : Type}
variable [h : decidable_eq A]
include h
open decidable
definition swap_core (a b r : A) : A :=
if r = a then b
else if r = b then a
else r
lemma swap_core_swap_core (r a b : A) : swap_core a b (swap_core a b r) = r :=
by_cases
(suppose r = a, by_cases
(suppose r = b, begin unfold swap_core, rewrite [if_pos `r = a`, if_pos (eq.refl b), -`r = a`, -`r = b`, if_pos (eq.refl r)] end)
(suppose Β¬ r = b,
have b β a, from assume h, begin rewrite h at this, contradiction end,
begin unfold swap_core, rewrite [*if_pos `r = a`, if_pos (eq.refl b), if_neg `b β a`, `r = a`] end))
(suppose Β¬ r = a, by_cases
(suppose r = b, begin unfold swap_core, rewrite [if_neg `Β¬ r = a`, *if_pos `r = b`, if_pos (eq.refl a), this] end)
(suppose Β¬ r = b, begin unfold swap_core, rewrite [*if_neg `Β¬ r = a`, *if_neg `Β¬ r = b`, if_neg `Β¬ r = a`] end))
lemma swap_core_self (r a : A) : swap_core a a r = r :=
by_cases
(suppose r = a, begin unfold swap_core, rewrite [*if_pos this, this] end)
(suppose r β a, begin unfold swap_core, rewrite [*if_neg this] end)
lemma swap_core_comm (r a b : A) : swap_core a b r = swap_core b a r :=
by_cases
(suppose r = a, by_cases
(suppose r = b, begin unfold swap_core, rewrite [if_pos `r = a`, if_pos `r = b`, -`r = a`, -`r = b`] end)
(suppose Β¬ r = b, begin unfold swap_core, rewrite [*if_pos `r = a`, if_neg `Β¬ r = b`] end))
(suppose Β¬ r = a, by_cases
(suppose r = b, begin unfold swap_core, rewrite [if_neg `Β¬ r = a`, *if_pos `r = b`] end)
(suppose Β¬ r = b, begin unfold swap_core, rewrite [*if_neg `Β¬ r = a`, *if_neg `Β¬ r = b`] end))
definition swap (a b : A) : perm A :=
mk (swap_core a b)
(swap_core a b)
(Ξ» x, abstract by rewrite swap_core_swap_core end)
(Ξ» x, abstract by rewrite swap_core_swap_core end)
lemma swap_self (a : A) : swap a a = id :=
eq_of_to_fun_eq (funext (Ξ» x, begin unfold [swap, fn], rewrite swap_core_self end))
lemma swap_comm (a b : A) : swap a b = swap b a :=
eq_of_to_fun_eq (funext (Ξ» x, begin unfold [swap, fn], rewrite swap_core_comm end))
lemma swap_apply_def (a b : A) (x : A) : swap a b β x = if x = a then b else if x = b then a else x :=
rfl
lemma swap_apply_left (a b : A) : swap a b β a = b :=
if_pos rfl
lemma swap_apply_right (a b : A) : swap a b β b = a :=
by_cases
(suppose b = a, by rewrite [swap_apply_def, this, *if_pos rfl])
(suppose b β a, by rewrite [swap_apply_def, if_pos rfl, if_neg this])
lemma swap_apply_of_ne_of_ne {a b : A} {x : A} : x β a β x β b β swap a b β x = x :=
assume hβ hβ, by rewrite [swap_apply_def, if_neg hβ, if_neg hβ]
lemma swap_swap (a b : A) : swap a b β swap a b = id :=
eq_of_to_fun_eq (funext (Ξ» x, begin unfold [swap, fn, equiv.trans, equiv.refl], rewrite swap_core_swap_core end))
lemma swap_comp_apply (a b : A) (Ο : perm A) (x : A) : (swap a b β Ο) β x = if Ο β x = a then b else if Ο β x = b then a else Ο β x :=
begin cases Ο, reflexivity end
end swap
end equiv
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
universes u v
namespace Mathlib
namespace smt
def array (Ξ± : Type u) (Ξ² : Type v) :=
Ξ± β Ξ²
def select {Ξ± : Type u} {Ξ² : Type v} (a : array Ξ± Ξ²) (i : Ξ±) : Ξ² :=
a i
theorem arrayext {Ξ± : Type u} {Ξ² : Type v} (aβ : array Ξ± Ξ²) (aβ : array Ξ± Ξ²) : (β (i : Ξ±), select aβ i = select aβ i) β aβ = aβ :=
funext
def store {Ξ± : Type u} {Ξ² : Type v} [DecidableEq Ξ±] (a : array Ξ± Ξ²) (i : Ξ±) (v : Ξ²) : array Ξ± Ξ² :=
fun (j : Ξ±) => ite (j = i) v (select a j)
@[simp] theorem select_store {Ξ± : Type u} {Ξ² : Type v} [DecidableEq Ξ±] (a : array Ξ± Ξ²) (i : Ξ±) (v : Ξ²) : select (store a i v) i = v := sorry
@[simp] theorem select_store_ne {Ξ± : Type u} {Ξ² : Type v} [DecidableEq Ξ±] (a : array Ξ± Ξ²) (i : Ξ±) (j : Ξ±) (v : Ξ²) : j β i β select (store a i v) j = select a j := sorry
|
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_cdf.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_permutation.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_blas.h>
#define DECORRCRITS 1
#define SET 1
int pcl_ran_mvgaussian(gsl_rng *rseed, gsl_vector *mean, gsl_matrix *Sigma, gsl_vector *output)
{
int i;
gsl_linalg_cholesky_decomp(Sigma);
for (i = 0; i < output->size; i++){
gsl_vector_set (output, i , gsl_ran_gaussian(rseed,1));
}
gsl_blas_dtrmv (CblasLower, CblasNoTrans, CblasNonUnit, Sigma, output);
gsl_blas_daxpy (1.0,mean, output);
return 0;
}
unsigned long int random_seed()
{
unsigned int seed;
FILE *devrandom;
if ((devrandom = fopen("/dev/random","r")) == NULL) {
fprintf(stderr,"Cannot open /dev/random, setting seed to 0\n");
seed = 0;
} else {
fread(&seed,sizeof(seed),1,devrandom);
fclose(devrandom);
}
return(seed);
}
double ran_inv_gamma(const gsl_rng * r, double a, double b){
return(1/gsl_ran_gamma(r, a, 1/b));
}
double ran_trunc_norm_upper(const gsl_rng * r, double mu, double sigma, double a){
double u = gsl_rng_uniform_pos(r);
double v = gsl_cdf_ugaussian_P((-a+mu)/sigma);
return(-sigma * gsl_cdf_ugaussian_Pinv(u*v)+mu);
}
double ran_trunc_norm_lower(const gsl_rng * r, double mu, double sigma, double a){
double u = gsl_rng_uniform_pos(r);
double v = gsl_cdf_ugaussian_P((a-mu)/sigma);
return(sigma * gsl_cdf_ugaussian_Pinv(u*v)+mu);
}
double ran_trunc_norm_both(const gsl_rng * r, double mu, double sigma, double a, double b){
double low = gsl_cdf_ugaussian_P((a-mu)/sigma);
double high = gsl_cdf_ugaussian_P((b-mu)/sigma);
double u = gsl_ran_flat(r,low,high);
return(sigma * gsl_cdf_ugaussian_Pinv(u)+mu);
}
int main(int argc, char *argv[]){
unsigned long int seed;
seed = random_seed();
int ITERS,iter,i,j,k,K,I,J,nRespCat;
char *fnEstimates,*fnChains,*fnInput;
int yij,sij;
int errs=0;
char *name;
double a,b,e,f,sigDecorr,mu_crits,sd_crits;
double sig2mu,mumu;
////This is all data and prior entry code.
if(argc<9){
fprintf(stderr,"Arguments needed: #iterations nRespCat sig2_mu a b e f sigDecorr [analysis.name] [input.file]\n");
fprintf(stderr,"\n****Not enough arguments.\n\n");
return(1);
}
if(argc>9){
name=argv[9];
}else{
name="analysis";
}
ITERS=atoi(argv[1]);
nRespCat=atoi(argv[2]);
sig2mu=atof(argv[3]);
a=atof(argv[4]);
b=atof(argv[5]);
e=atof(argv[6]);
f=atof(argv[7]);
sigDecorr=atof(argv[8]);
//confirm and check command line arguments.
printf("Name : %s\n",name);
printf("nRespCat = %i\n",nRespCat);
printf("sig2mu = %.2f\n",sig2mu);
printf("a = %.2f\n",a);
printf("b = %.2f\n",b);
printf("e = %.2f\n",e);
printf("f = %.2f\n",f);
printf("sigDecorr= %.2f\n",sigDecorr);
if(nRespCat<=0){printf("\n*****nRespCat must be greater than 0.\n");errs++;}
if(sig2mu<=0){printf("\n*****sig2mu must be greater than 0.\n");errs++;}
if(a<=0){printf("\n*****a must be greater than 0.\n");errs++;}
if(b<=0){printf("\n*****b must be greater than 0.\n");errs++;}
if(e<=0){printf("\n*****e must be greater than 0.\n");errs++;}
if(f<=0){printf("\n*****f must be greater than 0.\n");errs++;}
if(sigDecorr<=0){printf("\n*****sigDecorr must be greater than 0.\n");errs++;}
if(errs>0){printf("Exiting...\n\n");exit(1);}
asprintf(&fnEstimates,"%s.est",name);
asprintf(&fnChains,"%s.chn",name);
if(argc>10){
fnInput=argv[10];
}else{
asprintf(&fnInput,"%s.dat",name);
}
K=nRespCat-1;
//open files.
FILE *CHAINS;
if ((CHAINS = fopen(fnChains,"w")) == NULL) {
fprintf(stderr,"\n****Cannot open chain output file!\n\n");
return(1);
}
FILE *ESTIMATES;
if ((ESTIMATES = fopen(fnEstimates,"w")) == NULL) {
fprintf(stderr,"\n****Cannot open estimate output file!\n\n");
fclose(CHAINS);
return(1);
}
FILE *INPUT;
if ((INPUT = fopen(fnInput,"r")) == NULL) {
fprintf(stderr,"\n****Cannot open input file %s!\n\n",fnInput);
fclose(CHAINS);
fclose(ESTIMATES);
return(1);
}
//Now that we have opened the files, get the data.
i=0;
int newline=0,spaces=0;
int got;
while(!newline&&!feof(INPUT)){
i++;
got=fgetc(INPUT);
//printf("Character %i: %i %c\n",i,got,got);
if(got==32) spaces++;
if(got==10) newline=1;
}
rewind(INPUT);
if(!newline){
printf("Could not determine number of conditions!\n");
exit(1);
}
J=(spaces+1)/2;
i=0;
int totsig=0;
int totnoi=0;
printf("Reading data...\n");
while(!feof(INPUT)){
for(j=0;j<J;j++){
if(j==(J-1)){
fscanf(INPUT,"%i %i\n",&yij,&sij);
}else{
fscanf(INPUT,"%i %i ",&yij,&sij);
}
if((yij>=nRespCat)||(sij!=0&&sij!=1)){
printf("\n****Invalid data for participant %i, item %i: (yij=%i,sij=%i)\n\n",i+1,j+1,yij,sij);
fclose(CHAINS);
fclose(ESTIMATES);
fclose(INPUT);
return(1);
}
if(yij!=-1){
totsig+=sij;
totnoi+=1-sij;
}
}
i++;
}
rewind(INPUT);
I=i;
//printf("Read %d subjects and %d items, with %d total signal trials.\n",I,J,totsig);
//initialize matrices
printf("Reserving memory for matrices...\n");
int Ys[totsig],Yn[totnoi],Subs[totsig],Subn[totnoi];
gsl_vector *Ws,*Wn,*Ts,*Tn,*WsMean,*WnMean,*WsDev,*TsMean,*TnMean;
gsl_matrix *Xs,*Xn,*Ss,*Sn,*TpSs,*TpSn,*XstXs,*XntXn,*Vs,*Vn;
Xs = gsl_matrix_calloc(totsig, I+J+1);
Xn = gsl_matrix_calloc(totnoi, I+J+1);
XstXs = gsl_matrix_calloc(I+J+1, I+J+1);
XntXn = gsl_matrix_calloc(I+J+1, I+J+1);
Vs = gsl_matrix_calloc(I+J+1, I+J+1);
Vn = gsl_matrix_calloc(I+J+1, I+J+1);
gsl_matrix *Vn0 = gsl_matrix_calloc(I+J+1, I+J+1);
gsl_matrix *Vs0 = gsl_matrix_calloc(I+J+1, I+J+1);
Ws = gsl_vector_calloc(totsig);
Wn = gsl_vector_calloc(totnoi);
Ts = gsl_vector_calloc(I+J+1);
Tn = gsl_vector_calloc(I+J+1);
WsMean = gsl_vector_calloc(totsig);
WnMean = gsl_vector_calloc(totnoi);
WsDev = gsl_vector_calloc(totsig);
TsMean= gsl_vector_calloc(I+J+1);
TnMean= gsl_vector_calloc(I+J+1);
gsl_vector *TsMean0= gsl_vector_calloc(I+J+1);
gsl_vector *TnMean0= gsl_vector_calloc(I+J+1);
gsl_vector *isAlph = gsl_vector_calloc(I+J+1);
gsl_vector *isBeta = gsl_vector_calloc(I+J+1);
TpSs = gsl_matrix_calloc(I+J+1,I+J+1);
TpSn = gsl_matrix_calloc(I+J+1,I+J+1);
gsl_matrix_set_identity(TpSs);
gsl_matrix_set_identity(TpSn);
Ss = gsl_matrix_calloc(totsig, totsig);
gsl_matrix_set_identity(Ss);
Sn = gsl_matrix_calloc(totnoi, totnoi);
gsl_matrix_set_identity(Sn);
gsl_permutation *gslPerm = gsl_permutation_alloc(I+J+1);
gsl_permutation_init(gslPerm);
gsl_matrix_set(TpSs,0,0,1.0/sig2mu);
gsl_matrix_set(TpSn,0,0,1.0/sig2mu);
//end initialize matrices
printf("Creating design matrix...\n");
int y[I][J],s[I][J],sigindex=0,noiindex=0;
i=0;
while(!feof(INPUT)){
for(j=0;j<J;j++){
if(j==(J-1)){
fscanf(INPUT,"%i %i\n",&yij,&sij);
}else{
fscanf(INPUT,"%i %i ",&yij,&sij);
}
y[i][j]=yij;
s[i][j]=sij;
if(yij!=-1){
if(sij){
Ys[sigindex] = y[i][j];
Subs[sigindex] = i;
gsl_matrix_set(Xs,sigindex,0,1);
gsl_matrix_set(Xs,sigindex,i+1,1);
gsl_matrix_set(Xs,sigindex++,I+1+j,1);
}else{
Yn[noiindex] = y[i][j];
Subn[noiindex] = i;
gsl_matrix_set(Xn,noiindex,0,1);
gsl_matrix_set(Xn,noiindex,i+1,1);
gsl_matrix_set(Xn,noiindex++,I+1+j,1);
}
}
}
i++;
}
gsl_blas_dgemm(CblasTrans,CblasNoTrans,1,Xn,Xn,1,XntXn);
gsl_blas_dgemm(CblasTrans,CblasNoTrans,1,Xs,Xs,1,XstXs);
//gsl_vector_set(Tn,0,-.6);
//gsl_vector_set(Ts,0,.6);
//gsl_matrix_fprintf(stdout,Xs,"%f");
//exit(0);
printf("Read %i subjects values for %i items.\n\n",I,J);
fclose(INPUT);
////End data entry.
//Create starting values
int keq0=0;
double crits[I][K],critCand[I][K],w[I][J];
double maxw[I][K+1];
double minw[I][K+1];
double sig2=1;
double sumWnSqr=0,sumWsSqr=0;
int nSignal[I],totSignal=0;
int nRespS[I][K+1];
int nRespN[I][K+1];
double sumwN[I];
double sumwS[I];
double zDecorr[I][K];
double sig2N=1;
double sig2beta0=.2,sig2beta1=.2,sig2alph0=.2,sig2alph1=.2;
double bDecorr[I];
int accDecorr[I],badCrits[I];
double decorrRate=0,sumAlp20=0,sumAlp21=0,sumBet20=0,sumBet21=0;
for(i=0;i<I;i++){
accDecorr[i]=0;
nSignal[i]=0;
nRespN[i][K]=0;
nRespS[i][K]=0;
crits[i][0]=0;
crits[i][K-1]=SET;
for(k=0;k<K;k++){
nRespS[i][k]=0;
nRespN[i][k]=0;
if(k!=0 && k<(K-1)){
crits[i][k]=k*SET/((float)(K-1));
//printf("i: %i k: %i crit: %f\n",i,k,crits[i][k]);
}
}
for(j=0;j<J;j++){
nSignal[i]+=s[i][j];
totSignal+=s[i][j];
if(y[i][j]!=-1){
nRespN[i][y[i][j]]+=1-s[i][j];
nRespS[i][y[i][j]]+=s[i][j];
}
}
}
//for(k=0;k<(K+1);k++)
//{
// printf("nRespN[0][%d]=%d\n",k,nRespN[0][k]);
// printf("nRespS[0][%d]=%d\n",k,nRespS[0][k]);
//}
//end starting values
//Initialize GSL
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
//gsl_rng_set(r,seed);
gsl_rng_set(r,0);
gsl_blas_dgemv(CblasNoTrans,1,Xs,Ts,0,WsMean);
gsl_blas_dgemv(CblasNoTrans,1,Xn,Tn,0,WnMean);
printf("Total signal trials: %i\nTotal noise trials: %i\n\n",totsig,totnoi);
//begin MCMC loop.
i=0;
printf("Starting %i MCMC iterations...\n",ITERS);
for(iter=0;iter<ITERS;iter++){
for(i=0;i<I;i++){
sumwS[i]=0;
sumwN[i]=0;
for(k=0;k<(K+1);k++){
minw[i][k]=GSL_POSINF;
maxw[i][k]=GSL_NEGINF;
}
}
//sig2=1;
//sample latent variables
for(i=0;i<totsig;i++){
//printf("%d %d %f %f\n",Ys[i],Subs[i],crits[Subs[i]][0],crits[Subs[i]][1]);
if( Ys[i]==0 ){
gsl_vector_set(Ws,i,ran_trunc_norm_lower(r, gsl_vector_get(WsMean,i), sqrt(sig2), crits[Subs[i]][0]));
if(gsl_vector_get(Ws,i)>maxw[Subs[i]][0]) { maxw[Subs[i]][0]=gsl_vector_get(Ws,i);}
}else if( Ys[i]==K ){
gsl_vector_set(Ws,i,ran_trunc_norm_upper(r, gsl_vector_get(WsMean,i), sqrt(sig2), crits[Subs[i]][K-1]));
if(gsl_vector_get(Ws,i)<minw[Subs[i]][K]) { minw[Subs[i]][K]=gsl_vector_get(Ws,i);}
}else{
gsl_vector_set(Ws,i,ran_trunc_norm_both(r, gsl_vector_get(WsMean,i), sqrt(sig2), crits[Subs[i]][Ys[i]-1],crits[Subs[i]][Ys[i]]));
if(gsl_vector_get(Ws,i)>maxw[Subs[i]][Ys[i]]) { maxw[Subs[i]][Ys[i]]=gsl_vector_get(Ws,i);}
if(gsl_vector_get(Ws,i)<minw[Subs[i]][Ys[i]]) { minw[Subs[i]][Ys[i]]=gsl_vector_get(Ws,i);}
}
}
for(i=0;i<totnoi;i++){
//if(i==511) printf("%d %d %f %f\n",Yn[i],Subn[i],crits[Subn[i]][Yn[i]-1],crits[Subs[i]][Yn[i]]);
if( Yn[i]==0 ){
gsl_vector_set(Wn,i,ran_trunc_norm_lower(r, gsl_vector_get(WnMean,i), sqrt(sig2N), crits[Subn[i]][0]));
if(gsl_vector_get(Wn,i)>maxw[Subn[i]][0]) { maxw[Subn[i]][0]=gsl_vector_get(Wn,i);}
}else if( Yn[i]==K ){
gsl_vector_set(Wn,i,ran_trunc_norm_upper(r, gsl_vector_get(WnMean,i), sqrt(sig2N), crits[Subn[i]][K-1]));
if(gsl_vector_get(Wn,i)<minw[Subn[i]][K]) { minw[Subn[i]][K]=gsl_vector_get(Wn,i);}
}else{
gsl_vector_set(Wn,i,ran_trunc_norm_both(r, gsl_vector_get(WnMean,i), sqrt(sig2N), crits[Subn[i]][Yn[i]-1],crits[Subn[i]][Yn[i]]));
if(gsl_vector_get(Wn,i)>maxw[Subn[i]][Yn[i]]) { maxw[Subn[i]][Yn[i]]=gsl_vector_get(Wn,i);}
if(gsl_vector_get(Wn,i)<minw[Subn[i]][Yn[i]]) { minw[Subn[i]][Yn[i]]=gsl_vector_get(Wn,i);}
}
}
//printf("***\n");
//gsl_vector_fprintf(stderr,Wn,"%f");
//gsl_vector_fprintf(stderr,WnMean,"%f");
//for(i=0;i<totnoi;i++) printf("%i : %f\n",i,gsl_vector_get(WnMean,i));
gsl_blas_daxpy(-1,Ws,WsMean);
gsl_blas_daxpy(-1,Wn,WnMean);
gsl_blas_ddot(WsMean,WsMean,&sumWsSqr);
gsl_blas_ddot(WnMean,WnMean,&sumWnSqr);
//gsl_vector_fprintf(stderr,WnMean,"%f");
//for(i=0;i<totnoi;i++) printf("%i : %f\n",i,gsl_vector_get(Wn,i));
//printf("\nSS: %f,%f,%f,%d\n--------\n",sumWnSqr,a,b,totnoi);
//sample sigma^2
sig2=ran_inv_gamma(r, a+.5*totsig,b+(.5*sumWsSqr));
//sig2=1;
//printf("IG: %f %f\n",a0+.5*totSignal,(1.0)*b0+(.5*sumWSqr));
//sig2=1.3;
fwrite(&sig2,sizeof(double),1,CHAINS);
sig2N=ran_inv_gamma(r, a+.5*totnoi,b+(.5*sumWnSqr));
fwrite(&sig2N,sizeof(double),1,CHAINS);
// set up variances for full conditional on parameter vector
for(i=0;i<(I+J+1);i++){
if(i>I){
gsl_matrix_set(TpSs,i,i,1.0/sig2beta1);
gsl_matrix_set(TpSn,i,i,1.0/sig2beta0);
}else if(i>0){
gsl_matrix_set(TpSs,i,i,1.0/sig2alph1);
gsl_matrix_set(TpSn,i,i,1.0/sig2alph0);
}
//printf("i: %d | %f\n",i,gsl_matrix_get(TpSs,i,i));
}
gsl_matrix_memcpy(Vs,TpSs);
gsl_matrix_memcpy(Vn,TpSn);
gsl_matrix_scale(Vs,sig2);
gsl_matrix_scale(Vn,sig2N);
gsl_matrix_add(Vn,XntXn);
gsl_matrix_add(Vs,XstXs);
gsl_matrix_scale(Vs,1.0/sig2);
gsl_matrix_scale(Vn,1.0/sig2N);
//for(i=0;i<(I+J+1);i++){
// for(j=0;j<(I+J+1);j++){
// printf("%f ",gsl_matrix_get(Vs,i,j));
// }
// printf("\n");
//}
gsl_linalg_LU_decomp(Vn,gslPerm,&i);
gsl_linalg_LU_invert(Vn,gslPerm,Vn0);
gsl_linalg_LU_decomp(Vs,gslPerm,&i);
gsl_linalg_LU_invert(Vs,gslPerm,Vs0);
//for(i=0;i<(I+J+1);i++){
// for(j=0;j<(I+J+1);j++){
// printf("%f ",gsl_matrix_get(Vs0,i,j));
// }
// printf("\n");
//}
//sample parameter vectors
gsl_blas_dgemv(CblasTrans,1.0/sig2,Xs,Ws,0,TsMean);
gsl_blas_dgemv(CblasNoTrans,1,Vs0,TsMean,0,TsMean0);
gsl_blas_dgemv(CblasTrans,1.0/sig2N,Xn,Wn,0,TnMean);
gsl_blas_dgemv(CblasNoTrans,1,Vn0,TnMean,0,TnMean0);
pcl_ran_mvgaussian(r, TsMean0, Vs0, Ts);
pcl_ran_mvgaussian(r, TnMean0, Vn0, Tn);
gsl_vector_fwrite(CHAINS,Ts);
gsl_vector_fwrite(CHAINS,Tn);
//Sample variances of parameters
sumAlp20=0;
sumAlp21=0;
sumBet20=0;
sumBet21=0;
for(i=0;i<(I+J+1);i++){
if(i>I){
sumBet20+=pow(gsl_vector_get(Tn,i),2);
sumBet21+=pow(gsl_vector_get(Ts,i),2);
}else if(i>0){
sumAlp20+=pow(gsl_vector_get(Tn,i),2);
sumAlp21+=pow(gsl_vector_get(Ts,i),2);
}
}
//printf("%f\n",sumBet20);
//sample sigma^2
sig2alph0=ran_inv_gamma(r, e+.5*I,f+(.5*sumAlp20));
sig2alph1=ran_inv_gamma(r, e+.5*I,f+(.5*sumAlp21));
sig2beta0=ran_inv_gamma(r, e+.5*J,f+(.5*sumBet20));
sig2beta1=ran_inv_gamma(r, e+.5*J,f+(.5*sumBet21));
//printf("IG: %f %f\n",a0+.5*totSignal,(1.0)*b0+(.5*sumWSqr));
//sig2=1.3;
fwrite(&sig2alph0,sizeof(double),1,CHAINS);
fwrite(&sig2alph1,sizeof(double),1,CHAINS);
fwrite(&sig2beta0,sizeof(double),1,CHAINS);
fwrite(&sig2beta1,sizeof(double),1,CHAINS);
gsl_blas_dgemv(CblasNoTrans,1,Xs,Ts,0,WsMean);
gsl_blas_dgemv(CblasNoTrans,1,Xn,Tn,0,WnMean);
//gsl_vector_fprintf(stderr,WsMean,"%f");
//printf("\n\n");
//gsl_vector_fprintf(stderr,WnMean,"%f");
//printf("\n\n");
//gsl_vector_fprintf(stderr,WsMean,"%g");
//printf("\ns2=%f\n%f %f,%f %f\n",sig2,sumwN[i]/(1.0*(J-nSignal[i])),1.0/sqrt(1.0*(J-nSignal[i])),sumwS[i]/(1.0*nSignal[i]),sqrt(sig2/(1.0*(nSignal[i]))));
//sample criteria
for(i=0;i<I;i++){
//zDecorr[i]=gsl_ran_gaussian(r,sigDecorr);
bDecorr[i]=1;
badCrits[i]=0;
crits[i][0]=0; crits[i][K-1]=SET;
for(k=0;k<K;k++){
zDecorr[i][k]=gsl_ran_gaussian(r,sigDecorr);
//printf("i=%d k=%i max=%f min=%f\n",i,k,maxw[i][k],minw[i][k+1]);
//if((k!=0) && (k!=(K-1))) crits[i][k]=(gsl_rng_uniform_pos(r)*(minw[i][k+1]-maxw[i][k])+maxw[i][k]);
//crits[i][k]=ran_trunc_norm_both(r,mu_crits,sd_crits,maxw[i][k],minw[i][k+1])*(k!=keq0);
//printf("newcrit: %f\n",crits[i][k]);
//for decorrelating
critCand[i][k]=crits[i][k]+zDecorr[i][k]*(k!=0 && k!=(K-1));
//bDecorr[i]=bDecorr[i]*exp(-.5*pow((critCand[i][k]-mu_crits)/sd_crits,2)+.5*pow((crits[i][k]-mu_crits)/sd_crits,2));
//printf("candcrit: %f\n",critCand[i][k]);
if(k>0 && critCand[i][k]<critCand[i][k-1] ) badCrits[i]=1;
//printf("CRITS: %d %d d:%f s:%f : %f %f\n",iter,k,ds[i],sig2,crits[i][k],critCand[i][k]);
}
}
if(DECORRCRITS){
for(i=0;i<totsig;i++){
if(!badCrits[Subs[i]])
if(Ys[i]==0){
bDecorr[Subs[i]]=bDecorr[Subs[i]]*(
gsl_cdf_ugaussian_P((critCand[Subs[i]][0]-gsl_vector_get(WsMean,i))/sqrt(sig2)) /
gsl_cdf_ugaussian_P((crits[Subs[i]][0] -gsl_vector_get(WsMean,i))/sqrt(sig2))
);
}else if(Ys[i]==K){
bDecorr[Subs[i]]=bDecorr[Subs[i]]*(
(1-gsl_cdf_ugaussian_P((critCand[Subs[i]][K-1]-gsl_vector_get(WsMean,i))/sqrt(sig2))) /
(1-gsl_cdf_ugaussian_P((crits[Subs[i]][K-1] -gsl_vector_get(WsMean,i))/sqrt(sig2)))
);
}else{
bDecorr[Subs[i]]=bDecorr[Subs[i]]*(
(gsl_cdf_ugaussian_P((critCand[Subs[i]][Ys[i]]-gsl_vector_get(WsMean,i))/sqrt(sig2)) - gsl_cdf_ugaussian_P((critCand[Subs[i]][Ys[i]-1]-gsl_vector_get(WsMean,i))/sqrt(sig2))) /
(gsl_cdf_ugaussian_P((crits[Subs[i]][Ys[i]] -gsl_vector_get(WsMean,i))/sqrt(sig2)) - gsl_cdf_ugaussian_P((crits[Subs[i]][Ys[i]-1] -gsl_vector_get(WsMean,i))/sqrt(sig2)))
);
}
}
for(i=0;i<totnoi;i++){
if(!badCrits[Subn[i]])
if(Yn[i]==0){
bDecorr[Subn[i]]=bDecorr[Subn[i]]*(
gsl_cdf_ugaussian_P((critCand[Subn[i]][0]-gsl_vector_get(WnMean,i))/sqrt(sig2N)) /
gsl_cdf_ugaussian_P((crits[Subn[i]][0] -gsl_vector_get(WnMean,i))/sqrt(sig2N))
);
}else if(Yn[i]==K){
bDecorr[Subn[i]]=bDecorr[Subn[i]]*(
(1-gsl_cdf_ugaussian_P((critCand[Subn[i]][K-1]-gsl_vector_get(WnMean,i))/sqrt(sig2N))) /
(1-gsl_cdf_ugaussian_P((crits[Subn[i]][K-1] -gsl_vector_get(WnMean,i))/sqrt(sig2N)))
);
}else{
bDecorr[Subn[i]]=bDecorr[Subn[i]]*(
(gsl_cdf_ugaussian_P((critCand[Subn[i]][Yn[i]]-gsl_vector_get(WnMean,i))/sqrt(sig2N)) - gsl_cdf_ugaussian_P((critCand[Subn[i]][Yn[i]-1]-gsl_vector_get(WnMean,i))/sqrt(sig2N))) /
(gsl_cdf_ugaussian_P((crits[Subn[i]][Yn[i]] -gsl_vector_get(WnMean,i))/sqrt(sig2N)) - gsl_cdf_ugaussian_P((crits[Subn[i]][Yn[i]-1] -gsl_vector_get(WnMean,i))/sqrt(sig2N)))
);
}
}
//for(i=0;i<I;i++)
//for(k=0;k<K;k++)
//printf("m: %d, i: %d, k: %d, z: %f crit: %f cand: %f, BAD: %d\n",iter,i,k,zDecorr[i][k],crits[i][k],critCand[i][k],badCrits[i]);
for(i=0;i<I;i++){
if(!badCrits[i]){
accDecorr[i]=gsl_ran_bernoulli(r,(bDecorr[i]>1)?1:bDecorr[i]);
//printf("BDECORR: m%d i%d bdecorr %f\n",iter,i,bDecorr[i]);
if(accDecorr[i]){
decorrRate+=1/(1.0*I*ITERS);
for(k=0;k<K;k++){
//printf("m: %d, i: %d, k: %d, z: %f crit: %f cand: %f\n",iter,i,k,zDecorr[i][k],crits[i][k],critCand[i][k]);
crits[i][k]=critCand[i][k];
}
}
}
}
}
fwrite(crits,sizeof(double),I*K,CHAINS);
if(!((iter+1)%25)) { printf("Iteration %i\n",iter+1); }
}//End MCMC loop
printf("\nDone. Decorrelating step acceptance rate: %f\n\n",decorrRate);
//close files
fclose(ESTIMATES);
fclose(CHAINS);
return 0;
}
|
For any point $a$ in the unit disk, there exists a biholomorphism $f$ of the unit disk onto itself such that $f(a) = 0$. |
# Copyright (c) 2018-2021, Carnegie Mellon University
# See LICENSE for details
Class(InterpolateDFT, TaggedNonTerminal, rec(
abbrevs := [ (dsfunc,usfunc) -> [dsfunc, usfunc] ],
dims := self >> [self.params[1].domain(), self.params[2].domain()],
terminate := self >> let(n:=self.params[2].domain(), N := self.params[1].range(),
Downsample(self.params[1]).terminate() *
DFT(N, 1).terminate() *
Upsample(self.params[2]).terminate() *
Scale(1/n, DFT(n, -1).terminate())),
isReal := False,
normalizedArithCost := self >> let(n:=self.params[2].domain(), N := self.params[1].range(),
n + IntDouble(5 * n * d_log(n) / d_log(2)) + IntDouble(5 * N * d_log(N) / d_log(2))),
TType := T_Complex(T_Real(64))
));
NewRulesFor(InterpolateDFT, rec(
InterpolateDFT_base := rec(
applicable := (self, nt) >> not nt.hasTags(),
children := nt -> let(n:=nt.params[2].domain(), N := nt.params[1].range(),
[[ Downsample(nt.params[1]), DFT(N, 1), Upsample(nt.params[2]), DFT(n, -1) ]]),
apply := (nt, C, cnt) -> C[1] * C[2] * C[3] * Scale(1/nt.params[2].domain(), C[4])
),
InterpolateDFT_PrunedDFT := rec(
applicable := (self, nt) >> not nt.hasTags() and ObjId(nt.params[2]) = fZeroPadMiddle,
children := nt -> let(n:=nt.params[2].domain(), N := nt.params[1].range(), us := N/n, blk := n/2,
[[ PrunedDFT(N, 1, blk, [0, 2*us-1]), DFT(n, -1) ]]),
apply := (nt, C, cnt) -> let(n:=nt.params[2].domain(), N := nt.params[1].range(), us := N/n, blk := n/2,
Gath(nt.params[1]) * C[1] * Scale(1/n, C[2])
)
)
));
|
{-# OPTIONS --type-in-type #-}
Ty : Set
Ty =
(Ty : Set)
(nat top bot : Ty)
(arr prod sum : Ty β Ty β Ty)
β Ty
nat : Ty; nat = Ξ» _ nat _ _ _ _ _ β nat
top : Ty; top = Ξ» _ _ top _ _ _ _ β top
bot : Ty; bot = Ξ» _ _ _ bot _ _ _ β bot
arr : Ty β Ty β Ty; arr
= Ξ» A B Ty nat top bot arr prod sum β
arr (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
prod : Ty β Ty β Ty; prod
= Ξ» A B Ty nat top bot arr prod sum β
prod (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
sum : Ty β Ty β Ty; sum
= Ξ» A B Ty nat top bot arr prod sum β
sum (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
Con : Set; Con
= (Con : Set)
(nil : Con)
(snoc : Con β Ty β Con)
β Con
nil : Con; nil
= Ξ» Con nil snoc β nil
snoc : Con β Ty β Con; snoc
= Ξ» Ξ A Con nil snoc β snoc (Ξ Con nil snoc) A
Var : Con β Ty β Set; Var
= Ξ» Ξ A β
(Var : Con β Ty β Set)
(vz : β Ξ A β Var (snoc Ξ A) A)
(vs : β Ξ B A β Var Ξ A β Var (snoc Ξ B) A)
β Var Ξ A
vz : β{Ξ A} β Var (snoc Ξ A) A; vz
= Ξ» Var vz vs β vz _ _
vs : β{Ξ B A} β Var Ξ A β Var (snoc Ξ B) A; vs
= Ξ» x Var vz vs β vs _ _ _ (x Var vz vs)
Tm : Con β Ty β Set; Tm
= Ξ» Ξ A β
(Tm : Con β Ty β Set)
(var : β Ξ A β Var Ξ A β Tm Ξ A)
(lam : β Ξ A B β Tm (snoc Ξ A) B β Tm Ξ (arr A B))
(app : β Ξ A B β Tm Ξ (arr A B) β Tm Ξ A β Tm Ξ B)
(tt : β Ξ β Tm Ξ top)
(pair : β Ξ A B β Tm Ξ A β Tm Ξ B β Tm Ξ (prod A B))
(fst : β Ξ A B β Tm Ξ (prod A B) β Tm Ξ A)
(snd : β Ξ A B β Tm Ξ (prod A B) β Tm Ξ B)
(left : β Ξ A B β Tm Ξ A β Tm Ξ (sum A B))
(right : β Ξ A B β Tm Ξ B β Tm Ξ (sum A B))
(case : β Ξ A B C β Tm Ξ (sum A B) β Tm Ξ (arr A C) β Tm Ξ (arr B C) β Tm Ξ C)
(zero : β Ξ β Tm Ξ nat)
(suc : β Ξ β Tm Ξ nat β Tm Ξ nat)
(rec : β Ξ A β Tm Ξ nat β Tm Ξ (arr nat (arr A A)) β Tm Ξ A β Tm Ξ A)
β Tm Ξ A
var : β{Ξ A} β Var Ξ A β Tm Ξ A; var
= Ξ» x Tm var lam app tt pair fst snd left right case zero suc rec β
var _ _ x
lam : β{Ξ A B} β Tm (snoc Ξ A) B β Tm Ξ (arr A B); lam
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
lam _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
app : β{Ξ A B} β Tm Ξ (arr A B) β Tm Ξ A β Tm Ξ B; app
= Ξ» t u Tm var lam app tt pair fst snd left right case zero suc rec β
app _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
tt : β{Ξ} β Tm Ξ top; tt
= Ξ» Tm var lam app tt pair fst snd left right case zero suc rec β tt _
pair : β{Ξ A B} β Tm Ξ A β Tm Ξ B β Tm Ξ (prod A B); pair
= Ξ» t u Tm var lam app tt pair fst snd left right case zero suc rec β
pair _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
fst : β{Ξ A B} β Tm Ξ (prod A B) β Tm Ξ A; fst
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
fst _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
snd : β{Ξ A B} β Tm Ξ (prod A B) β Tm Ξ B; snd
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
snd _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
left : β{Ξ A B} β Tm Ξ A β Tm Ξ (sum A B); left
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
left _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
right : β{Ξ A B} β Tm Ξ B β Tm Ξ (sum A B); right
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
right _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
case : β{Ξ A B C} β Tm Ξ (sum A B) β Tm Ξ (arr A C) β Tm Ξ (arr B C) β Tm Ξ C; case
= Ξ» t u v Tm var lam app tt pair fst snd left right case zero suc rec β
case _ _ _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
zero : β{Ξ} β Tm Ξ nat; zero
= Ξ» Tm var lam app tt pair fst snd left right case zero suc rec β zero _
suc : β{Ξ} β Tm Ξ nat β Tm Ξ nat; suc
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
suc _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
rec : β{Ξ A} β Tm Ξ nat β Tm Ξ (arr nat (arr A A)) β Tm Ξ A β Tm Ξ A; rec
= Ξ» t u v Tm var lam app tt pair fst snd left right case zero suc rec β
rec _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
v0 : β{Ξ A} β Tm (snoc Ξ A) A; v0
= var vz
v1 : β{Ξ A B} β Tm (snoc (snoc Ξ A) B) A; v1
= var (vs vz)
v2 : β{Ξ A B C} β Tm (snoc (snoc (snoc Ξ A) B) C) A; v2
= var (vs (vs vz))
v3 : β{Ξ A B C D} β Tm (snoc (snoc (snoc (snoc Ξ A) B) C) D) A; v3
= var (vs (vs (vs vz)))
tbool : Ty; tbool
= sum top top
true : β{Ξ} β Tm Ξ tbool; true
= left tt
tfalse : β{Ξ} β Tm Ξ tbool; tfalse
= right tt
ifthenelse : β{Ξ A} β Tm Ξ (arr tbool (arr A (arr A A))); ifthenelse
= lam (lam (lam (case v2 (lam v2) (lam v1))))
times4 : β{Ξ A} β Tm Ξ (arr (arr A A) (arr A A)); times4
= lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))
add : β{Ξ} β Tm Ξ (arr nat (arr nat nat)); add
= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0))
mul : β{Ξ} β Tm Ξ (arr nat (arr nat nat)); mul
= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero))
fact : β{Ξ} β Tm Ξ (arr nat nat); fact
= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero))
{-# OPTIONS --type-in-type #-}
Ty1 : Set
Ty1 =
(Ty1 : Set)
(nat top bot : Ty1)
(arr prod sum : Ty1 β Ty1 β Ty1)
β Ty1
nat1 : Ty1; nat1 = Ξ» _ nat1 _ _ _ _ _ β nat1
top1 : Ty1; top1 = Ξ» _ _ top1 _ _ _ _ β top1
bot1 : Ty1; bot1 = Ξ» _ _ _ bot1 _ _ _ β bot1
arr1 : Ty1 β Ty1 β Ty1; arr1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod sum β
arr1 (A Ty1 nat1 top1 bot1 arr1 prod sum) (B Ty1 nat1 top1 bot1 arr1 prod sum)
prod1 : Ty1 β Ty1 β Ty1; prod1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod1 sum β
prod1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum) (B Ty1 nat1 top1 bot1 arr1 prod1 sum)
sum1 : Ty1 β Ty1 β Ty1; sum1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod1 sum1 β
sum1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum1) (B Ty1 nat1 top1 bot1 arr1 prod1 sum1)
Con1 : Set; Con1
= (Con1 : Set)
(nil : Con1)
(snoc : Con1 β Ty1 β Con1)
β Con1
nil1 : Con1; nil1
= Ξ» Con1 nil1 snoc β nil1
snoc1 : Con1 β Ty1 β Con1; snoc1
= Ξ» Ξ A Con1 nil1 snoc1 β snoc1 (Ξ Con1 nil1 snoc1) A
Var1 : Con1 β Ty1 β Set; Var1
= Ξ» Ξ A β
(Var1 : Con1 β Ty1 β Set)
(vz : β Ξ A β Var1 (snoc1 Ξ A) A)
(vs : β Ξ B A β Var1 Ξ A β Var1 (snoc1 Ξ B) A)
β Var1 Ξ A
vz1 : β{Ξ A} β Var1 (snoc1 Ξ A) A; vz1
= Ξ» Var1 vz1 vs β vz1 _ _
vs1 : β{Ξ B A} β Var1 Ξ A β Var1 (snoc1 Ξ B) A; vs1
= Ξ» x Var1 vz1 vs1 β vs1 _ _ _ (x Var1 vz1 vs1)
Tm1 : Con1 β Ty1 β Set; Tm1
= Ξ» Ξ A β
(Tm1 : Con1 β Ty1 β Set)
(var : β Ξ A β Var1 Ξ A β Tm1 Ξ A)
(lam : β Ξ A B β Tm1 (snoc1 Ξ A) B β Tm1 Ξ (arr1 A B))
(app : β Ξ A B β Tm1 Ξ (arr1 A B) β Tm1 Ξ A β Tm1 Ξ B)
(tt : β Ξ β Tm1 Ξ top1)
(pair : β Ξ A B β Tm1 Ξ A β Tm1 Ξ B β Tm1 Ξ (prod1 A B))
(fst : β Ξ A B β Tm1 Ξ (prod1 A B) β Tm1 Ξ A)
(snd : β Ξ A B β Tm1 Ξ (prod1 A B) β Tm1 Ξ B)
(left : β Ξ A B β Tm1 Ξ A β Tm1 Ξ (sum1 A B))
(right : β Ξ A B β Tm1 Ξ B β Tm1 Ξ (sum1 A B))
(case : β Ξ A B C β Tm1 Ξ (sum1 A B) β Tm1 Ξ (arr1 A C) β Tm1 Ξ (arr1 B C) β Tm1 Ξ C)
(zero : β Ξ β Tm1 Ξ nat1)
(suc : β Ξ β Tm1 Ξ nat1 β Tm1 Ξ nat1)
(rec : β Ξ A β Tm1 Ξ nat1 β Tm1 Ξ (arr1 nat1 (arr1 A A)) β Tm1 Ξ A β Tm1 Ξ A)
β Tm1 Ξ A
var1 : β{Ξ A} β Var1 Ξ A β Tm1 Ξ A; var1
= Ξ» x Tm1 var1 lam app tt pair fst snd left right case zero suc rec β
var1 _ _ x
lam1 : β{Ξ A B} β Tm1 (snoc1 Ξ A) B β Tm1 Ξ (arr1 A B); lam1
= Ξ» t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec β
lam1 _ _ _ (t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec)
app1 : β{Ξ A B} β Tm1 Ξ (arr1 A B) β Tm1 Ξ A β Tm1 Ξ B; app1
= Ξ» t u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec β
app1 _ _ _ (t Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
tt1 : β{Ξ} β Tm1 Ξ top1; tt1
= Ξ» Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec β tt1 _
pair1 : β{Ξ A B} β Tm1 Ξ A β Tm1 Ξ B β Tm1 Ξ (prod1 A B); pair1
= Ξ» t u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec β
pair1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
fst1 : β{Ξ A B} β Tm1 Ξ (prod1 A B) β Tm1 Ξ A; fst1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec β
fst1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec)
snd1 : β{Ξ A B} β Tm1 Ξ (prod1 A B) β Tm1 Ξ B; snd1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec β
snd1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec)
left1 : β{Ξ A B} β Tm1 Ξ A β Tm1 Ξ (sum1 A B); left1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec β
left1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec)
right1 : β{Ξ A B} β Tm1 Ξ B β Tm1 Ξ (sum1 A B); right1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec β
right1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec)
case1 : β{Ξ A B C} β Tm1 Ξ (sum1 A B) β Tm1 Ξ (arr1 A C) β Tm1 Ξ (arr1 B C) β Tm1 Ξ C; case1
= Ξ» t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec β
case1 _ _ _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
zero1 : β{Ξ} β Tm1 Ξ nat1; zero1
= Ξ» Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec β zero1 _
suc1 : β{Ξ} β Tm1 Ξ nat1 β Tm1 Ξ nat1; suc1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec β
suc1 _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec)
rec1 : β{Ξ A} β Tm1 Ξ nat1 β Tm1 Ξ (arr1 nat1 (arr1 A A)) β Tm1 Ξ A β Tm1 Ξ A; rec1
= Ξ» t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1 β
rec1 _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
v01 : β{Ξ A} β Tm1 (snoc1 Ξ A) A; v01
= var1 vz1
v11 : β{Ξ A B} β Tm1 (snoc1 (snoc1 Ξ A) B) A; v11
= var1 (vs1 vz1)
v21 : β{Ξ A B C} β Tm1 (snoc1 (snoc1 (snoc1 Ξ A) B) C) A; v21
= var1 (vs1 (vs1 vz1))
v31 : β{Ξ A B C D} β Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Ξ A) B) C) D) A; v31
= var1 (vs1 (vs1 (vs1 vz1)))
tbool1 : Ty1; tbool1
= sum1 top1 top1
true1 : β{Ξ} β Tm1 Ξ tbool1; true1
= left1 tt1
tfalse1 : β{Ξ} β Tm1 Ξ tbool1; tfalse1
= right1 tt1
ifthenelse1 : β{Ξ A} β Tm1 Ξ (arr1 tbool1 (arr1 A (arr1 A A))); ifthenelse1
= lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11))))
times41 : β{Ξ A} β Tm1 Ξ (arr1 (arr1 A A) (arr1 A A)); times41
= lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01)))))
add1 : β{Ξ} β Tm1 Ξ (arr1 nat1 (arr1 nat1 nat1)); add1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))
(lam1 v01))
mul1 : β{Ξ} β Tm1 Ξ (arr1 nat1 (arr1 nat1 nat1)); mul1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))
(lam1 zero1))
fact1 : β{Ξ} β Tm1 Ξ (arr1 nat1 nat1); fact1
= lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))
(suc1 zero1))
{-# OPTIONS --type-in-type #-}
Ty2 : Set
Ty2 =
(Ty2 : Set)
(nat top bot : Ty2)
(arr prod sum : Ty2 β Ty2 β Ty2)
β Ty2
nat2 : Ty2; nat2 = Ξ» _ nat2 _ _ _ _ _ β nat2
top2 : Ty2; top2 = Ξ» _ _ top2 _ _ _ _ β top2
bot2 : Ty2; bot2 = Ξ» _ _ _ bot2 _ _ _ β bot2
arr2 : Ty2 β Ty2 β Ty2; arr2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod sum β
arr2 (A Ty2 nat2 top2 bot2 arr2 prod sum) (B Ty2 nat2 top2 bot2 arr2 prod sum)
prod2 : Ty2 β Ty2 β Ty2; prod2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod2 sum β
prod2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum) (B Ty2 nat2 top2 bot2 arr2 prod2 sum)
sum2 : Ty2 β Ty2 β Ty2; sum2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod2 sum2 β
sum2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum2) (B Ty2 nat2 top2 bot2 arr2 prod2 sum2)
Con2 : Set; Con2
= (Con2 : Set)
(nil : Con2)
(snoc : Con2 β Ty2 β Con2)
β Con2
nil2 : Con2; nil2
= Ξ» Con2 nil2 snoc β nil2
snoc2 : Con2 β Ty2 β Con2; snoc2
= Ξ» Ξ A Con2 nil2 snoc2 β snoc2 (Ξ Con2 nil2 snoc2) A
Var2 : Con2 β Ty2 β Set; Var2
= Ξ» Ξ A β
(Var2 : Con2 β Ty2 β Set)
(vz : β Ξ A β Var2 (snoc2 Ξ A) A)
(vs : β Ξ B A β Var2 Ξ A β Var2 (snoc2 Ξ B) A)
β Var2 Ξ A
vz2 : β{Ξ A} β Var2 (snoc2 Ξ A) A; vz2
= Ξ» Var2 vz2 vs β vz2 _ _
vs2 : β{Ξ B A} β Var2 Ξ A β Var2 (snoc2 Ξ B) A; vs2
= Ξ» x Var2 vz2 vs2 β vs2 _ _ _ (x Var2 vz2 vs2)
Tm2 : Con2 β Ty2 β Set; Tm2
= Ξ» Ξ A β
(Tm2 : Con2 β Ty2 β Set)
(var : β Ξ A β Var2 Ξ A β Tm2 Ξ A)
(lam : β Ξ A B β Tm2 (snoc2 Ξ A) B β Tm2 Ξ (arr2 A B))
(app : β Ξ A B β Tm2 Ξ (arr2 A B) β Tm2 Ξ A β Tm2 Ξ B)
(tt : β Ξ β Tm2 Ξ top2)
(pair : β Ξ A B β Tm2 Ξ A β Tm2 Ξ B β Tm2 Ξ (prod2 A B))
(fst : β Ξ A B β Tm2 Ξ (prod2 A B) β Tm2 Ξ A)
(snd : β Ξ A B β Tm2 Ξ (prod2 A B) β Tm2 Ξ B)
(left : β Ξ A B β Tm2 Ξ A β Tm2 Ξ (sum2 A B))
(right : β Ξ A B β Tm2 Ξ B β Tm2 Ξ (sum2 A B))
(case : β Ξ A B C β Tm2 Ξ (sum2 A B) β Tm2 Ξ (arr2 A C) β Tm2 Ξ (arr2 B C) β Tm2 Ξ C)
(zero : β Ξ β Tm2 Ξ nat2)
(suc : β Ξ β Tm2 Ξ nat2 β Tm2 Ξ nat2)
(rec : β Ξ A β Tm2 Ξ nat2 β Tm2 Ξ (arr2 nat2 (arr2 A A)) β Tm2 Ξ A β Tm2 Ξ A)
β Tm2 Ξ A
var2 : β{Ξ A} β Var2 Ξ A β Tm2 Ξ A; var2
= Ξ» x Tm2 var2 lam app tt pair fst snd left right case zero suc rec β
var2 _ _ x
lam2 : β{Ξ A B} β Tm2 (snoc2 Ξ A) B β Tm2 Ξ (arr2 A B); lam2
= Ξ» t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec β
lam2 _ _ _ (t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec)
app2 : β{Ξ A B} β Tm2 Ξ (arr2 A B) β Tm2 Ξ A β Tm2 Ξ B; app2
= Ξ» t u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec β
app2 _ _ _ (t Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
tt2 : β{Ξ} β Tm2 Ξ top2; tt2
= Ξ» Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec β tt2 _
pair2 : β{Ξ A B} β Tm2 Ξ A β Tm2 Ξ B β Tm2 Ξ (prod2 A B); pair2
= Ξ» t u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec β
pair2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
fst2 : β{Ξ A B} β Tm2 Ξ (prod2 A B) β Tm2 Ξ A; fst2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec β
fst2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec)
snd2 : β{Ξ A B} β Tm2 Ξ (prod2 A B) β Tm2 Ξ B; snd2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec β
snd2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec)
left2 : β{Ξ A B} β Tm2 Ξ A β Tm2 Ξ (sum2 A B); left2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec β
left2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec)
right2 : β{Ξ A B} β Tm2 Ξ B β Tm2 Ξ (sum2 A B); right2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec β
right2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec)
case2 : β{Ξ A B C} β Tm2 Ξ (sum2 A B) β Tm2 Ξ (arr2 A C) β Tm2 Ξ (arr2 B C) β Tm2 Ξ C; case2
= Ξ» t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec β
case2 _ _ _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
zero2 : β{Ξ} β Tm2 Ξ nat2; zero2
= Ξ» Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec β zero2 _
suc2 : β{Ξ} β Tm2 Ξ nat2 β Tm2 Ξ nat2; suc2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec β
suc2 _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec)
rec2 : β{Ξ A} β Tm2 Ξ nat2 β Tm2 Ξ (arr2 nat2 (arr2 A A)) β Tm2 Ξ A β Tm2 Ξ A; rec2
= Ξ» t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2 β
rec2 _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
v02 : β{Ξ A} β Tm2 (snoc2 Ξ A) A; v02
= var2 vz2
v12 : β{Ξ A B} β Tm2 (snoc2 (snoc2 Ξ A) B) A; v12
= var2 (vs2 vz2)
v22 : β{Ξ A B C} β Tm2 (snoc2 (snoc2 (snoc2 Ξ A) B) C) A; v22
= var2 (vs2 (vs2 vz2))
v32 : β{Ξ A B C D} β Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Ξ A) B) C) D) A; v32
= var2 (vs2 (vs2 (vs2 vz2)))
tbool2 : Ty2; tbool2
= sum2 top2 top2
true2 : β{Ξ} β Tm2 Ξ tbool2; true2
= left2 tt2
tfalse2 : β{Ξ} β Tm2 Ξ tbool2; tfalse2
= right2 tt2
ifthenelse2 : β{Ξ A} β Tm2 Ξ (arr2 tbool2 (arr2 A (arr2 A A))); ifthenelse2
= lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12))))
times42 : β{Ξ A} β Tm2 Ξ (arr2 (arr2 A A) (arr2 A A)); times42
= lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02)))))
add2 : β{Ξ} β Tm2 Ξ (arr2 nat2 (arr2 nat2 nat2)); add2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))
(lam2 v02))
mul2 : β{Ξ} β Tm2 Ξ (arr2 nat2 (arr2 nat2 nat2)); mul2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))
(lam2 zero2))
fact2 : β{Ξ} β Tm2 Ξ (arr2 nat2 nat2); fact2
= lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))
(suc2 zero2))
{-# OPTIONS --type-in-type #-}
Ty3 : Set
Ty3 =
(Ty3 : Set)
(nat top bot : Ty3)
(arr prod sum : Ty3 β Ty3 β Ty3)
β Ty3
nat3 : Ty3; nat3 = Ξ» _ nat3 _ _ _ _ _ β nat3
top3 : Ty3; top3 = Ξ» _ _ top3 _ _ _ _ β top3
bot3 : Ty3; bot3 = Ξ» _ _ _ bot3 _ _ _ β bot3
arr3 : Ty3 β Ty3 β Ty3; arr3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod sum β
arr3 (A Ty3 nat3 top3 bot3 arr3 prod sum) (B Ty3 nat3 top3 bot3 arr3 prod sum)
prod3 : Ty3 β Ty3 β Ty3; prod3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod3 sum β
prod3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum) (B Ty3 nat3 top3 bot3 arr3 prod3 sum)
sum3 : Ty3 β Ty3 β Ty3; sum3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod3 sum3 β
sum3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum3) (B Ty3 nat3 top3 bot3 arr3 prod3 sum3)
Con3 : Set; Con3
= (Con3 : Set)
(nil : Con3)
(snoc : Con3 β Ty3 β Con3)
β Con3
nil3 : Con3; nil3
= Ξ» Con3 nil3 snoc β nil3
snoc3 : Con3 β Ty3 β Con3; snoc3
= Ξ» Ξ A Con3 nil3 snoc3 β snoc3 (Ξ Con3 nil3 snoc3) A
Var3 : Con3 β Ty3 β Set; Var3
= Ξ» Ξ A β
(Var3 : Con3 β Ty3 β Set)
(vz : β Ξ A β Var3 (snoc3 Ξ A) A)
(vs : β Ξ B A β Var3 Ξ A β Var3 (snoc3 Ξ B) A)
β Var3 Ξ A
vz3 : β{Ξ A} β Var3 (snoc3 Ξ A) A; vz3
= Ξ» Var3 vz3 vs β vz3 _ _
vs3 : β{Ξ B A} β Var3 Ξ A β Var3 (snoc3 Ξ B) A; vs3
= Ξ» x Var3 vz3 vs3 β vs3 _ _ _ (x Var3 vz3 vs3)
Tm3 : Con3 β Ty3 β Set; Tm3
= Ξ» Ξ A β
(Tm3 : Con3 β Ty3 β Set)
(var : β Ξ A β Var3 Ξ A β Tm3 Ξ A)
(lam : β Ξ A B β Tm3 (snoc3 Ξ A) B β Tm3 Ξ (arr3 A B))
(app : β Ξ A B β Tm3 Ξ (arr3 A B) β Tm3 Ξ A β Tm3 Ξ B)
(tt : β Ξ β Tm3 Ξ top3)
(pair : β Ξ A B β Tm3 Ξ A β Tm3 Ξ B β Tm3 Ξ (prod3 A B))
(fst : β Ξ A B β Tm3 Ξ (prod3 A B) β Tm3 Ξ A)
(snd : β Ξ A B β Tm3 Ξ (prod3 A B) β Tm3 Ξ B)
(left : β Ξ A B β Tm3 Ξ A β Tm3 Ξ (sum3 A B))
(right : β Ξ A B β Tm3 Ξ B β Tm3 Ξ (sum3 A B))
(case : β Ξ A B C β Tm3 Ξ (sum3 A B) β Tm3 Ξ (arr3 A C) β Tm3 Ξ (arr3 B C) β Tm3 Ξ C)
(zero : β Ξ β Tm3 Ξ nat3)
(suc : β Ξ β Tm3 Ξ nat3 β Tm3 Ξ nat3)
(rec : β Ξ A β Tm3 Ξ nat3 β Tm3 Ξ (arr3 nat3 (arr3 A A)) β Tm3 Ξ A β Tm3 Ξ A)
β Tm3 Ξ A
var3 : β{Ξ A} β Var3 Ξ A β Tm3 Ξ A; var3
= Ξ» x Tm3 var3 lam app tt pair fst snd left right case zero suc rec β
var3 _ _ x
lam3 : β{Ξ A B} β Tm3 (snoc3 Ξ A) B β Tm3 Ξ (arr3 A B); lam3
= Ξ» t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec β
lam3 _ _ _ (t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec)
app3 : β{Ξ A B} β Tm3 Ξ (arr3 A B) β Tm3 Ξ A β Tm3 Ξ B; app3
= Ξ» t u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec β
app3 _ _ _ (t Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
tt3 : β{Ξ} β Tm3 Ξ top3; tt3
= Ξ» Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec β tt3 _
pair3 : β{Ξ A B} β Tm3 Ξ A β Tm3 Ξ B β Tm3 Ξ (prod3 A B); pair3
= Ξ» t u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec β
pair3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
fst3 : β{Ξ A B} β Tm3 Ξ (prod3 A B) β Tm3 Ξ A; fst3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec β
fst3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec)
snd3 : β{Ξ A B} β Tm3 Ξ (prod3 A B) β Tm3 Ξ B; snd3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec β
snd3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec)
left3 : β{Ξ A B} β Tm3 Ξ A β Tm3 Ξ (sum3 A B); left3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec β
left3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec)
right3 : β{Ξ A B} β Tm3 Ξ B β Tm3 Ξ (sum3 A B); right3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec β
right3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec)
case3 : β{Ξ A B C} β Tm3 Ξ (sum3 A B) β Tm3 Ξ (arr3 A C) β Tm3 Ξ (arr3 B C) β Tm3 Ξ C; case3
= Ξ» t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec β
case3 _ _ _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
zero3 : β{Ξ} β Tm3 Ξ nat3; zero3
= Ξ» Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec β zero3 _
suc3 : β{Ξ} β Tm3 Ξ nat3 β Tm3 Ξ nat3; suc3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec β
suc3 _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec)
rec3 : β{Ξ A} β Tm3 Ξ nat3 β Tm3 Ξ (arr3 nat3 (arr3 A A)) β Tm3 Ξ A β Tm3 Ξ A; rec3
= Ξ» t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3 β
rec3 _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
v03 : β{Ξ A} β Tm3 (snoc3 Ξ A) A; v03
= var3 vz3
v13 : β{Ξ A B} β Tm3 (snoc3 (snoc3 Ξ A) B) A; v13
= var3 (vs3 vz3)
v23 : β{Ξ A B C} β Tm3 (snoc3 (snoc3 (snoc3 Ξ A) B) C) A; v23
= var3 (vs3 (vs3 vz3))
v33 : β{Ξ A B C D} β Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Ξ A) B) C) D) A; v33
= var3 (vs3 (vs3 (vs3 vz3)))
tbool3 : Ty3; tbool3
= sum3 top3 top3
true3 : β{Ξ} β Tm3 Ξ tbool3; true3
= left3 tt3
tfalse3 : β{Ξ} β Tm3 Ξ tbool3; tfalse3
= right3 tt3
ifthenelse3 : β{Ξ A} β Tm3 Ξ (arr3 tbool3 (arr3 A (arr3 A A))); ifthenelse3
= lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13))))
times43 : β{Ξ A} β Tm3 Ξ (arr3 (arr3 A A) (arr3 A A)); times43
= lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03)))))
add3 : β{Ξ} β Tm3 Ξ (arr3 nat3 (arr3 nat3 nat3)); add3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))
(lam3 v03))
mul3 : β{Ξ} β Tm3 Ξ (arr3 nat3 (arr3 nat3 nat3)); mul3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))
(lam3 zero3))
fact3 : β{Ξ} β Tm3 Ξ (arr3 nat3 nat3); fact3
= lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))
(suc3 zero3))
{-# OPTIONS --type-in-type #-}
Ty4 : Set
Ty4 =
(Ty4 : Set)
(nat top bot : Ty4)
(arr prod sum : Ty4 β Ty4 β Ty4)
β Ty4
nat4 : Ty4; nat4 = Ξ» _ nat4 _ _ _ _ _ β nat4
top4 : Ty4; top4 = Ξ» _ _ top4 _ _ _ _ β top4
bot4 : Ty4; bot4 = Ξ» _ _ _ bot4 _ _ _ β bot4
arr4 : Ty4 β Ty4 β Ty4; arr4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod sum β
arr4 (A Ty4 nat4 top4 bot4 arr4 prod sum) (B Ty4 nat4 top4 bot4 arr4 prod sum)
prod4 : Ty4 β Ty4 β Ty4; prod4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod4 sum β
prod4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum) (B Ty4 nat4 top4 bot4 arr4 prod4 sum)
sum4 : Ty4 β Ty4 β Ty4; sum4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod4 sum4 β
sum4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum4) (B Ty4 nat4 top4 bot4 arr4 prod4 sum4)
Con4 : Set; Con4
= (Con4 : Set)
(nil : Con4)
(snoc : Con4 β Ty4 β Con4)
β Con4
nil4 : Con4; nil4
= Ξ» Con4 nil4 snoc β nil4
snoc4 : Con4 β Ty4 β Con4; snoc4
= Ξ» Ξ A Con4 nil4 snoc4 β snoc4 (Ξ Con4 nil4 snoc4) A
Var4 : Con4 β Ty4 β Set; Var4
= Ξ» Ξ A β
(Var4 : Con4 β Ty4 β Set)
(vz : β Ξ A β Var4 (snoc4 Ξ A) A)
(vs : β Ξ B A β Var4 Ξ A β Var4 (snoc4 Ξ B) A)
β Var4 Ξ A
vz4 : β{Ξ A} β Var4 (snoc4 Ξ A) A; vz4
= Ξ» Var4 vz4 vs β vz4 _ _
vs4 : β{Ξ B A} β Var4 Ξ A β Var4 (snoc4 Ξ B) A; vs4
= Ξ» x Var4 vz4 vs4 β vs4 _ _ _ (x Var4 vz4 vs4)
Tm4 : Con4 β Ty4 β Set; Tm4
= Ξ» Ξ A β
(Tm4 : Con4 β Ty4 β Set)
(var : β Ξ A β Var4 Ξ A β Tm4 Ξ A)
(lam : β Ξ A B β Tm4 (snoc4 Ξ A) B β Tm4 Ξ (arr4 A B))
(app : β Ξ A B β Tm4 Ξ (arr4 A B) β Tm4 Ξ A β Tm4 Ξ B)
(tt : β Ξ β Tm4 Ξ top4)
(pair : β Ξ A B β Tm4 Ξ A β Tm4 Ξ B β Tm4 Ξ (prod4 A B))
(fst : β Ξ A B β Tm4 Ξ (prod4 A B) β Tm4 Ξ A)
(snd : β Ξ A B β Tm4 Ξ (prod4 A B) β Tm4 Ξ B)
(left : β Ξ A B β Tm4 Ξ A β Tm4 Ξ (sum4 A B))
(right : β Ξ A B β Tm4 Ξ B β Tm4 Ξ (sum4 A B))
(case : β Ξ A B C β Tm4 Ξ (sum4 A B) β Tm4 Ξ (arr4 A C) β Tm4 Ξ (arr4 B C) β Tm4 Ξ C)
(zero : β Ξ β Tm4 Ξ nat4)
(suc : β Ξ β Tm4 Ξ nat4 β Tm4 Ξ nat4)
(rec : β Ξ A β Tm4 Ξ nat4 β Tm4 Ξ (arr4 nat4 (arr4 A A)) β Tm4 Ξ A β Tm4 Ξ A)
β Tm4 Ξ A
var4 : β{Ξ A} β Var4 Ξ A β Tm4 Ξ A; var4
= Ξ» x Tm4 var4 lam app tt pair fst snd left right case zero suc rec β
var4 _ _ x
lam4 : β{Ξ A B} β Tm4 (snoc4 Ξ A) B β Tm4 Ξ (arr4 A B); lam4
= Ξ» t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec β
lam4 _ _ _ (t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec)
app4 : β{Ξ A B} β Tm4 Ξ (arr4 A B) β Tm4 Ξ A β Tm4 Ξ B; app4
= Ξ» t u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec β
app4 _ _ _ (t Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
tt4 : β{Ξ} β Tm4 Ξ top4; tt4
= Ξ» Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec β tt4 _
pair4 : β{Ξ A B} β Tm4 Ξ A β Tm4 Ξ B β Tm4 Ξ (prod4 A B); pair4
= Ξ» t u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec β
pair4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
fst4 : β{Ξ A B} β Tm4 Ξ (prod4 A B) β Tm4 Ξ A; fst4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec β
fst4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec)
snd4 : β{Ξ A B} β Tm4 Ξ (prod4 A B) β Tm4 Ξ B; snd4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec β
snd4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec)
left4 : β{Ξ A B} β Tm4 Ξ A β Tm4 Ξ (sum4 A B); left4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec β
left4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec)
right4 : β{Ξ A B} β Tm4 Ξ B β Tm4 Ξ (sum4 A B); right4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec β
right4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec)
case4 : β{Ξ A B C} β Tm4 Ξ (sum4 A B) β Tm4 Ξ (arr4 A C) β Tm4 Ξ (arr4 B C) β Tm4 Ξ C; case4
= Ξ» t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec β
case4 _ _ _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
zero4 : β{Ξ} β Tm4 Ξ nat4; zero4
= Ξ» Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec β zero4 _
suc4 : β{Ξ} β Tm4 Ξ nat4 β Tm4 Ξ nat4; suc4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec β
suc4 _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec)
rec4 : β{Ξ A} β Tm4 Ξ nat4 β Tm4 Ξ (arr4 nat4 (arr4 A A)) β Tm4 Ξ A β Tm4 Ξ A; rec4
= Ξ» t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4 β
rec4 _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
v04 : β{Ξ A} β Tm4 (snoc4 Ξ A) A; v04
= var4 vz4
v14 : β{Ξ A B} β Tm4 (snoc4 (snoc4 Ξ A) B) A; v14
= var4 (vs4 vz4)
v24 : β{Ξ A B C} β Tm4 (snoc4 (snoc4 (snoc4 Ξ A) B) C) A; v24
= var4 (vs4 (vs4 vz4))
v34 : β{Ξ A B C D} β Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Ξ A) B) C) D) A; v34
= var4 (vs4 (vs4 (vs4 vz4)))
tbool4 : Ty4; tbool4
= sum4 top4 top4
true4 : β{Ξ} β Tm4 Ξ tbool4; true4
= left4 tt4
tfalse4 : β{Ξ} β Tm4 Ξ tbool4; tfalse4
= right4 tt4
ifthenelse4 : β{Ξ A} β Tm4 Ξ (arr4 tbool4 (arr4 A (arr4 A A))); ifthenelse4
= lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14))))
times44 : β{Ξ A} β Tm4 Ξ (arr4 (arr4 A A) (arr4 A A)); times44
= lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04)))))
add4 : β{Ξ} β Tm4 Ξ (arr4 nat4 (arr4 nat4 nat4)); add4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))
(lam4 v04))
mul4 : β{Ξ} β Tm4 Ξ (arr4 nat4 (arr4 nat4 nat4)); mul4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))
(lam4 zero4))
fact4 : β{Ξ} β Tm4 Ξ (arr4 nat4 nat4); fact4
= lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))
(suc4 zero4))
{-# OPTIONS --type-in-type #-}
Ty5 : Set
Ty5 =
(Ty5 : Set)
(nat top bot : Ty5)
(arr prod sum : Ty5 β Ty5 β Ty5)
β Ty5
nat5 : Ty5; nat5 = Ξ» _ nat5 _ _ _ _ _ β nat5
top5 : Ty5; top5 = Ξ» _ _ top5 _ _ _ _ β top5
bot5 : Ty5; bot5 = Ξ» _ _ _ bot5 _ _ _ β bot5
arr5 : Ty5 β Ty5 β Ty5; arr5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod sum β
arr5 (A Ty5 nat5 top5 bot5 arr5 prod sum) (B Ty5 nat5 top5 bot5 arr5 prod sum)
prod5 : Ty5 β Ty5 β Ty5; prod5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod5 sum β
prod5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum) (B Ty5 nat5 top5 bot5 arr5 prod5 sum)
sum5 : Ty5 β Ty5 β Ty5; sum5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod5 sum5 β
sum5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum5) (B Ty5 nat5 top5 bot5 arr5 prod5 sum5)
Con5 : Set; Con5
= (Con5 : Set)
(nil : Con5)
(snoc : Con5 β Ty5 β Con5)
β Con5
nil5 : Con5; nil5
= Ξ» Con5 nil5 snoc β nil5
snoc5 : Con5 β Ty5 β Con5; snoc5
= Ξ» Ξ A Con5 nil5 snoc5 β snoc5 (Ξ Con5 nil5 snoc5) A
Var5 : Con5 β Ty5 β Set; Var5
= Ξ» Ξ A β
(Var5 : Con5 β Ty5 β Set)
(vz : β Ξ A β Var5 (snoc5 Ξ A) A)
(vs : β Ξ B A β Var5 Ξ A β Var5 (snoc5 Ξ B) A)
β Var5 Ξ A
vz5 : β{Ξ A} β Var5 (snoc5 Ξ A) A; vz5
= Ξ» Var5 vz5 vs β vz5 _ _
vs5 : β{Ξ B A} β Var5 Ξ A β Var5 (snoc5 Ξ B) A; vs5
= Ξ» x Var5 vz5 vs5 β vs5 _ _ _ (x Var5 vz5 vs5)
Tm5 : Con5 β Ty5 β Set; Tm5
= Ξ» Ξ A β
(Tm5 : Con5 β Ty5 β Set)
(var : β Ξ A β Var5 Ξ A β Tm5 Ξ A)
(lam : β Ξ A B β Tm5 (snoc5 Ξ A) B β Tm5 Ξ (arr5 A B))
(app : β Ξ A B β Tm5 Ξ (arr5 A B) β Tm5 Ξ A β Tm5 Ξ B)
(tt : β Ξ β Tm5 Ξ top5)
(pair : β Ξ A B β Tm5 Ξ A β Tm5 Ξ B β Tm5 Ξ (prod5 A B))
(fst : β Ξ A B β Tm5 Ξ (prod5 A B) β Tm5 Ξ A)
(snd : β Ξ A B β Tm5 Ξ (prod5 A B) β Tm5 Ξ B)
(left : β Ξ A B β Tm5 Ξ A β Tm5 Ξ (sum5 A B))
(right : β Ξ A B β Tm5 Ξ B β Tm5 Ξ (sum5 A B))
(case : β Ξ A B C β Tm5 Ξ (sum5 A B) β Tm5 Ξ (arr5 A C) β Tm5 Ξ (arr5 B C) β Tm5 Ξ C)
(zero : β Ξ β Tm5 Ξ nat5)
(suc : β Ξ β Tm5 Ξ nat5 β Tm5 Ξ nat5)
(rec : β Ξ A β Tm5 Ξ nat5 β Tm5 Ξ (arr5 nat5 (arr5 A A)) β Tm5 Ξ A β Tm5 Ξ A)
β Tm5 Ξ A
var5 : β{Ξ A} β Var5 Ξ A β Tm5 Ξ A; var5
= Ξ» x Tm5 var5 lam app tt pair fst snd left right case zero suc rec β
var5 _ _ x
lam5 : β{Ξ A B} β Tm5 (snoc5 Ξ A) B β Tm5 Ξ (arr5 A B); lam5
= Ξ» t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec β
lam5 _ _ _ (t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec)
app5 : β{Ξ A B} β Tm5 Ξ (arr5 A B) β Tm5 Ξ A β Tm5 Ξ B; app5
= Ξ» t u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec β
app5 _ _ _ (t Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
tt5 : β{Ξ} β Tm5 Ξ top5; tt5
= Ξ» Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec β tt5 _
pair5 : β{Ξ A B} β Tm5 Ξ A β Tm5 Ξ B β Tm5 Ξ (prod5 A B); pair5
= Ξ» t u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec β
pair5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
fst5 : β{Ξ A B} β Tm5 Ξ (prod5 A B) β Tm5 Ξ A; fst5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec β
fst5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec)
snd5 : β{Ξ A B} β Tm5 Ξ (prod5 A B) β Tm5 Ξ B; snd5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec β
snd5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec)
left5 : β{Ξ A B} β Tm5 Ξ A β Tm5 Ξ (sum5 A B); left5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec β
left5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec)
right5 : β{Ξ A B} β Tm5 Ξ B β Tm5 Ξ (sum5 A B); right5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec β
right5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec)
case5 : β{Ξ A B C} β Tm5 Ξ (sum5 A B) β Tm5 Ξ (arr5 A C) β Tm5 Ξ (arr5 B C) β Tm5 Ξ C; case5
= Ξ» t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec β
case5 _ _ _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
zero5 : β{Ξ} β Tm5 Ξ nat5; zero5
= Ξ» Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec β zero5 _
suc5 : β{Ξ} β Tm5 Ξ nat5 β Tm5 Ξ nat5; suc5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec β
suc5 _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec)
rec5 : β{Ξ A} β Tm5 Ξ nat5 β Tm5 Ξ (arr5 nat5 (arr5 A A)) β Tm5 Ξ A β Tm5 Ξ A; rec5
= Ξ» t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5 β
rec5 _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
v05 : β{Ξ A} β Tm5 (snoc5 Ξ A) A; v05
= var5 vz5
v15 : β{Ξ A B} β Tm5 (snoc5 (snoc5 Ξ A) B) A; v15
= var5 (vs5 vz5)
v25 : β{Ξ A B C} β Tm5 (snoc5 (snoc5 (snoc5 Ξ A) B) C) A; v25
= var5 (vs5 (vs5 vz5))
v35 : β{Ξ A B C D} β Tm5 (snoc5 (snoc5 (snoc5 (snoc5 Ξ A) B) C) D) A; v35
= var5 (vs5 (vs5 (vs5 vz5)))
tbool5 : Ty5; tbool5
= sum5 top5 top5
true5 : β{Ξ} β Tm5 Ξ tbool5; true5
= left5 tt5
tfalse5 : β{Ξ} β Tm5 Ξ tbool5; tfalse5
= right5 tt5
ifthenelse5 : β{Ξ A} β Tm5 Ξ (arr5 tbool5 (arr5 A (arr5 A A))); ifthenelse5
= lam5 (lam5 (lam5 (case5 v25 (lam5 v25) (lam5 v15))))
times45 : β{Ξ A} β Tm5 Ξ (arr5 (arr5 A A) (arr5 A A)); times45
= lam5 (lam5 (app5 v15 (app5 v15 (app5 v15 (app5 v15 v05)))))
add5 : β{Ξ} β Tm5 Ξ (arr5 nat5 (arr5 nat5 nat5)); add5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (suc5 (app5 v15 v05)))))
(lam5 v05))
mul5 : β{Ξ} β Tm5 Ξ (arr5 nat5 (arr5 nat5 nat5)); mul5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (app5 (app5 add5 (app5 v15 v05)) v05))))
(lam5 zero5))
fact5 : β{Ξ} β Tm5 Ξ (arr5 nat5 nat5); fact5
= lam5 (rec5 v05 (lam5 (lam5 (app5 (app5 mul5 (suc5 v15)) v05)))
(suc5 zero5))
{-# OPTIONS --type-in-type #-}
Ty6 : Set
Ty6 =
(Ty6 : Set)
(nat top bot : Ty6)
(arr prod sum : Ty6 β Ty6 β Ty6)
β Ty6
nat6 : Ty6; nat6 = Ξ» _ nat6 _ _ _ _ _ β nat6
top6 : Ty6; top6 = Ξ» _ _ top6 _ _ _ _ β top6
bot6 : Ty6; bot6 = Ξ» _ _ _ bot6 _ _ _ β bot6
arr6 : Ty6 β Ty6 β Ty6; arr6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod sum β
arr6 (A Ty6 nat6 top6 bot6 arr6 prod sum) (B Ty6 nat6 top6 bot6 arr6 prod sum)
prod6 : Ty6 β Ty6 β Ty6; prod6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod6 sum β
prod6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum) (B Ty6 nat6 top6 bot6 arr6 prod6 sum)
sum6 : Ty6 β Ty6 β Ty6; sum6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod6 sum6 β
sum6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum6) (B Ty6 nat6 top6 bot6 arr6 prod6 sum6)
Con6 : Set; Con6
= (Con6 : Set)
(nil : Con6)
(snoc : Con6 β Ty6 β Con6)
β Con6
nil6 : Con6; nil6
= Ξ» Con6 nil6 snoc β nil6
snoc6 : Con6 β Ty6 β Con6; snoc6
= Ξ» Ξ A Con6 nil6 snoc6 β snoc6 (Ξ Con6 nil6 snoc6) A
Var6 : Con6 β Ty6 β Set; Var6
= Ξ» Ξ A β
(Var6 : Con6 β Ty6 β Set)
(vz : β Ξ A β Var6 (snoc6 Ξ A) A)
(vs : β Ξ B A β Var6 Ξ A β Var6 (snoc6 Ξ B) A)
β Var6 Ξ A
vz6 : β{Ξ A} β Var6 (snoc6 Ξ A) A; vz6
= Ξ» Var6 vz6 vs β vz6 _ _
vs6 : β{Ξ B A} β Var6 Ξ A β Var6 (snoc6 Ξ B) A; vs6
= Ξ» x Var6 vz6 vs6 β vs6 _ _ _ (x Var6 vz6 vs6)
Tm6 : Con6 β Ty6 β Set; Tm6
= Ξ» Ξ A β
(Tm6 : Con6 β Ty6 β Set)
(var : β Ξ A β Var6 Ξ A β Tm6 Ξ A)
(lam : β Ξ A B β Tm6 (snoc6 Ξ A) B β Tm6 Ξ (arr6 A B))
(app : β Ξ A B β Tm6 Ξ (arr6 A B) β Tm6 Ξ A β Tm6 Ξ B)
(tt : β Ξ β Tm6 Ξ top6)
(pair : β Ξ A B β Tm6 Ξ A β Tm6 Ξ B β Tm6 Ξ (prod6 A B))
(fst : β Ξ A B β Tm6 Ξ (prod6 A B) β Tm6 Ξ A)
(snd : β Ξ A B β Tm6 Ξ (prod6 A B) β Tm6 Ξ B)
(left : β Ξ A B β Tm6 Ξ A β Tm6 Ξ (sum6 A B))
(right : β Ξ A B β Tm6 Ξ B β Tm6 Ξ (sum6 A B))
(case : β Ξ A B C β Tm6 Ξ (sum6 A B) β Tm6 Ξ (arr6 A C) β Tm6 Ξ (arr6 B C) β Tm6 Ξ C)
(zero : β Ξ β Tm6 Ξ nat6)
(suc : β Ξ β Tm6 Ξ nat6 β Tm6 Ξ nat6)
(rec : β Ξ A β Tm6 Ξ nat6 β Tm6 Ξ (arr6 nat6 (arr6 A A)) β Tm6 Ξ A β Tm6 Ξ A)
β Tm6 Ξ A
var6 : β{Ξ A} β Var6 Ξ A β Tm6 Ξ A; var6
= Ξ» x Tm6 var6 lam app tt pair fst snd left right case zero suc rec β
var6 _ _ x
lam6 : β{Ξ A B} β Tm6 (snoc6 Ξ A) B β Tm6 Ξ (arr6 A B); lam6
= Ξ» t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec β
lam6 _ _ _ (t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec)
app6 : β{Ξ A B} β Tm6 Ξ (arr6 A B) β Tm6 Ξ A β Tm6 Ξ B; app6
= Ξ» t u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec β
app6 _ _ _ (t Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
tt6 : β{Ξ} β Tm6 Ξ top6; tt6
= Ξ» Tm6 var6 lam6 app6 tt6 pair fst snd left right case zero suc rec β tt6 _
pair6 : β{Ξ A B} β Tm6 Ξ A β Tm6 Ξ B β Tm6 Ξ (prod6 A B); pair6
= Ξ» t u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec β
pair6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
fst6 : β{Ξ A B} β Tm6 Ξ (prod6 A B) β Tm6 Ξ A; fst6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec β
fst6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec)
snd6 : β{Ξ A B} β Tm6 Ξ (prod6 A B) β Tm6 Ξ B; snd6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec β
snd6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec)
left6 : β{Ξ A B} β Tm6 Ξ A β Tm6 Ξ (sum6 A B); left6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec β
left6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec)
right6 : β{Ξ A B} β Tm6 Ξ B β Tm6 Ξ (sum6 A B); right6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec β
right6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec)
case6 : β{Ξ A B C} β Tm6 Ξ (sum6 A B) β Tm6 Ξ (arr6 A C) β Tm6 Ξ (arr6 B C) β Tm6 Ξ C; case6
= Ξ» t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec β
case6 _ _ _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
zero6 : β{Ξ} β Tm6 Ξ nat6; zero6
= Ξ» Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc rec β zero6 _
suc6 : β{Ξ} β Tm6 Ξ nat6 β Tm6 Ξ nat6; suc6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec β
suc6 _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec)
rec6 : β{Ξ A} β Tm6 Ξ nat6 β Tm6 Ξ (arr6 nat6 (arr6 A A)) β Tm6 Ξ A β Tm6 Ξ A; rec6
= Ξ» t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6 β
rec6 _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
v06 : β{Ξ A} β Tm6 (snoc6 Ξ A) A; v06
= var6 vz6
v16 : β{Ξ A B} β Tm6 (snoc6 (snoc6 Ξ A) B) A; v16
= var6 (vs6 vz6)
v26 : β{Ξ A B C} β Tm6 (snoc6 (snoc6 (snoc6 Ξ A) B) C) A; v26
= var6 (vs6 (vs6 vz6))
v36 : β{Ξ A B C D} β Tm6 (snoc6 (snoc6 (snoc6 (snoc6 Ξ A) B) C) D) A; v36
= var6 (vs6 (vs6 (vs6 vz6)))
tbool6 : Ty6; tbool6
= sum6 top6 top6
true6 : β{Ξ} β Tm6 Ξ tbool6; true6
= left6 tt6
tfalse6 : β{Ξ} β Tm6 Ξ tbool6; tfalse6
= right6 tt6
ifthenelse6 : β{Ξ A} β Tm6 Ξ (arr6 tbool6 (arr6 A (arr6 A A))); ifthenelse6
= lam6 (lam6 (lam6 (case6 v26 (lam6 v26) (lam6 v16))))
times46 : β{Ξ A} β Tm6 Ξ (arr6 (arr6 A A) (arr6 A A)); times46
= lam6 (lam6 (app6 v16 (app6 v16 (app6 v16 (app6 v16 v06)))))
add6 : β{Ξ} β Tm6 Ξ (arr6 nat6 (arr6 nat6 nat6)); add6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (suc6 (app6 v16 v06)))))
(lam6 v06))
mul6 : β{Ξ} β Tm6 Ξ (arr6 nat6 (arr6 nat6 nat6)); mul6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (app6 (app6 add6 (app6 v16 v06)) v06))))
(lam6 zero6))
fact6 : β{Ξ} β Tm6 Ξ (arr6 nat6 nat6); fact6
= lam6 (rec6 v06 (lam6 (lam6 (app6 (app6 mul6 (suc6 v16)) v06)))
(suc6 zero6))
{-# OPTIONS --type-in-type #-}
Ty7 : Set
Ty7 =
(Ty7 : Set)
(nat top bot : Ty7)
(arr prod sum : Ty7 β Ty7 β Ty7)
β Ty7
nat7 : Ty7; nat7 = Ξ» _ nat7 _ _ _ _ _ β nat7
top7 : Ty7; top7 = Ξ» _ _ top7 _ _ _ _ β top7
bot7 : Ty7; bot7 = Ξ» _ _ _ bot7 _ _ _ β bot7
arr7 : Ty7 β Ty7 β Ty7; arr7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod sum β
arr7 (A Ty7 nat7 top7 bot7 arr7 prod sum) (B Ty7 nat7 top7 bot7 arr7 prod sum)
prod7 : Ty7 β Ty7 β Ty7; prod7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod7 sum β
prod7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum) (B Ty7 nat7 top7 bot7 arr7 prod7 sum)
sum7 : Ty7 β Ty7 β Ty7; sum7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod7 sum7 β
sum7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum7) (B Ty7 nat7 top7 bot7 arr7 prod7 sum7)
Con7 : Set; Con7
= (Con7 : Set)
(nil : Con7)
(snoc : Con7 β Ty7 β Con7)
β Con7
nil7 : Con7; nil7
= Ξ» Con7 nil7 snoc β nil7
snoc7 : Con7 β Ty7 β Con7; snoc7
= Ξ» Ξ A Con7 nil7 snoc7 β snoc7 (Ξ Con7 nil7 snoc7) A
Var7 : Con7 β Ty7 β Set; Var7
= Ξ» Ξ A β
(Var7 : Con7 β Ty7 β Set)
(vz : β Ξ A β Var7 (snoc7 Ξ A) A)
(vs : β Ξ B A β Var7 Ξ A β Var7 (snoc7 Ξ B) A)
β Var7 Ξ A
vz7 : β{Ξ A} β Var7 (snoc7 Ξ A) A; vz7
= Ξ» Var7 vz7 vs β vz7 _ _
vs7 : β{Ξ B A} β Var7 Ξ A β Var7 (snoc7 Ξ B) A; vs7
= Ξ» x Var7 vz7 vs7 β vs7 _ _ _ (x Var7 vz7 vs7)
Tm7 : Con7 β Ty7 β Set; Tm7
= Ξ» Ξ A β
(Tm7 : Con7 β Ty7 β Set)
(var : β Ξ A β Var7 Ξ A β Tm7 Ξ A)
(lam : β Ξ A B β Tm7 (snoc7 Ξ A) B β Tm7 Ξ (arr7 A B))
(app : β Ξ A B β Tm7 Ξ (arr7 A B) β Tm7 Ξ A β Tm7 Ξ B)
(tt : β Ξ β Tm7 Ξ top7)
(pair : β Ξ A B β Tm7 Ξ A β Tm7 Ξ B β Tm7 Ξ (prod7 A B))
(fst : β Ξ A B β Tm7 Ξ (prod7 A B) β Tm7 Ξ A)
(snd : β Ξ A B β Tm7 Ξ (prod7 A B) β Tm7 Ξ B)
(left : β Ξ A B β Tm7 Ξ A β Tm7 Ξ (sum7 A B))
(right : β Ξ A B β Tm7 Ξ B β Tm7 Ξ (sum7 A B))
(case : β Ξ A B C β Tm7 Ξ (sum7 A B) β Tm7 Ξ (arr7 A C) β Tm7 Ξ (arr7 B C) β Tm7 Ξ C)
(zero : β Ξ β Tm7 Ξ nat7)
(suc : β Ξ β Tm7 Ξ nat7 β Tm7 Ξ nat7)
(rec : β Ξ A β Tm7 Ξ nat7 β Tm7 Ξ (arr7 nat7 (arr7 A A)) β Tm7 Ξ A β Tm7 Ξ A)
β Tm7 Ξ A
var7 : β{Ξ A} β Var7 Ξ A β Tm7 Ξ A; var7
= Ξ» x Tm7 var7 lam app tt pair fst snd left right case zero suc rec β
var7 _ _ x
lam7 : β{Ξ A B} β Tm7 (snoc7 Ξ A) B β Tm7 Ξ (arr7 A B); lam7
= Ξ» t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec β
lam7 _ _ _ (t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec)
app7 : β{Ξ A B} β Tm7 Ξ (arr7 A B) β Tm7 Ξ A β Tm7 Ξ B; app7
= Ξ» t u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec β
app7 _ _ _ (t Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
tt7 : β{Ξ} β Tm7 Ξ top7; tt7
= Ξ» Tm7 var7 lam7 app7 tt7 pair fst snd left right case zero suc rec β tt7 _
pair7 : β{Ξ A B} β Tm7 Ξ A β Tm7 Ξ B β Tm7 Ξ (prod7 A B); pair7
= Ξ» t u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec β
pair7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
fst7 : β{Ξ A B} β Tm7 Ξ (prod7 A B) β Tm7 Ξ A; fst7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec β
fst7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec)
snd7 : β{Ξ A B} β Tm7 Ξ (prod7 A B) β Tm7 Ξ B; snd7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec β
snd7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec)
left7 : β{Ξ A B} β Tm7 Ξ A β Tm7 Ξ (sum7 A B); left7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec β
left7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec)
right7 : β{Ξ A B} β Tm7 Ξ B β Tm7 Ξ (sum7 A B); right7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec β
right7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec)
case7 : β{Ξ A B C} β Tm7 Ξ (sum7 A B) β Tm7 Ξ (arr7 A C) β Tm7 Ξ (arr7 B C) β Tm7 Ξ C; case7
= Ξ» t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec β
case7 _ _ _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
zero7 : β{Ξ} β Tm7 Ξ nat7; zero7
= Ξ» Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc rec β zero7 _
suc7 : β{Ξ} β Tm7 Ξ nat7 β Tm7 Ξ nat7; suc7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec β
suc7 _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec)
rec7 : β{Ξ A} β Tm7 Ξ nat7 β Tm7 Ξ (arr7 nat7 (arr7 A A)) β Tm7 Ξ A β Tm7 Ξ A; rec7
= Ξ» t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7 β
rec7 _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
v07 : β{Ξ A} β Tm7 (snoc7 Ξ A) A; v07
= var7 vz7
v17 : β{Ξ A B} β Tm7 (snoc7 (snoc7 Ξ A) B) A; v17
= var7 (vs7 vz7)
v27 : β{Ξ A B C} β Tm7 (snoc7 (snoc7 (snoc7 Ξ A) B) C) A; v27
= var7 (vs7 (vs7 vz7))
v37 : β{Ξ A B C D} β Tm7 (snoc7 (snoc7 (snoc7 (snoc7 Ξ A) B) C) D) A; v37
= var7 (vs7 (vs7 (vs7 vz7)))
tbool7 : Ty7; tbool7
= sum7 top7 top7
true7 : β{Ξ} β Tm7 Ξ tbool7; true7
= left7 tt7
tfalse7 : β{Ξ} β Tm7 Ξ tbool7; tfalse7
= right7 tt7
ifthenelse7 : β{Ξ A} β Tm7 Ξ (arr7 tbool7 (arr7 A (arr7 A A))); ifthenelse7
= lam7 (lam7 (lam7 (case7 v27 (lam7 v27) (lam7 v17))))
times47 : β{Ξ A} β Tm7 Ξ (arr7 (arr7 A A) (arr7 A A)); times47
= lam7 (lam7 (app7 v17 (app7 v17 (app7 v17 (app7 v17 v07)))))
add7 : β{Ξ} β Tm7 Ξ (arr7 nat7 (arr7 nat7 nat7)); add7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (suc7 (app7 v17 v07)))))
(lam7 v07))
mul7 : β{Ξ} β Tm7 Ξ (arr7 nat7 (arr7 nat7 nat7)); mul7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (app7 (app7 add7 (app7 v17 v07)) v07))))
(lam7 zero7))
fact7 : β{Ξ} β Tm7 Ξ (arr7 nat7 nat7); fact7
= lam7 (rec7 v07 (lam7 (lam7 (app7 (app7 mul7 (suc7 v17)) v07)))
(suc7 zero7))
{-# OPTIONS --type-in-type #-}
Ty8 : Set
Ty8 =
(Ty8 : Set)
(nat top bot : Ty8)
(arr prod sum : Ty8 β Ty8 β Ty8)
β Ty8
nat8 : Ty8; nat8 = Ξ» _ nat8 _ _ _ _ _ β nat8
top8 : Ty8; top8 = Ξ» _ _ top8 _ _ _ _ β top8
bot8 : Ty8; bot8 = Ξ» _ _ _ bot8 _ _ _ β bot8
arr8 : Ty8 β Ty8 β Ty8; arr8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod sum β
arr8 (A Ty8 nat8 top8 bot8 arr8 prod sum) (B Ty8 nat8 top8 bot8 arr8 prod sum)
prod8 : Ty8 β Ty8 β Ty8; prod8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod8 sum β
prod8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum) (B Ty8 nat8 top8 bot8 arr8 prod8 sum)
sum8 : Ty8 β Ty8 β Ty8; sum8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod8 sum8 β
sum8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum8) (B Ty8 nat8 top8 bot8 arr8 prod8 sum8)
Con8 : Set; Con8
= (Con8 : Set)
(nil : Con8)
(snoc : Con8 β Ty8 β Con8)
β Con8
nil8 : Con8; nil8
= Ξ» Con8 nil8 snoc β nil8
snoc8 : Con8 β Ty8 β Con8; snoc8
= Ξ» Ξ A Con8 nil8 snoc8 β snoc8 (Ξ Con8 nil8 snoc8) A
Var8 : Con8 β Ty8 β Set; Var8
= Ξ» Ξ A β
(Var8 : Con8 β Ty8 β Set)
(vz : β Ξ A β Var8 (snoc8 Ξ A) A)
(vs : β Ξ B A β Var8 Ξ A β Var8 (snoc8 Ξ B) A)
β Var8 Ξ A
vz8 : β{Ξ A} β Var8 (snoc8 Ξ A) A; vz8
= Ξ» Var8 vz8 vs β vz8 _ _
vs8 : β{Ξ B A} β Var8 Ξ A β Var8 (snoc8 Ξ B) A; vs8
= Ξ» x Var8 vz8 vs8 β vs8 _ _ _ (x Var8 vz8 vs8)
Tm8 : Con8 β Ty8 β Set; Tm8
= Ξ» Ξ A β
(Tm8 : Con8 β Ty8 β Set)
(var : β Ξ A β Var8 Ξ A β Tm8 Ξ A)
(lam : β Ξ A B β Tm8 (snoc8 Ξ A) B β Tm8 Ξ (arr8 A B))
(app : β Ξ A B β Tm8 Ξ (arr8 A B) β Tm8 Ξ A β Tm8 Ξ B)
(tt : β Ξ β Tm8 Ξ top8)
(pair : β Ξ A B β Tm8 Ξ A β Tm8 Ξ B β Tm8 Ξ (prod8 A B))
(fst : β Ξ A B β Tm8 Ξ (prod8 A B) β Tm8 Ξ A)
(snd : β Ξ A B β Tm8 Ξ (prod8 A B) β Tm8 Ξ B)
(left : β Ξ A B β Tm8 Ξ A β Tm8 Ξ (sum8 A B))
(right : β Ξ A B β Tm8 Ξ B β Tm8 Ξ (sum8 A B))
(case : β Ξ A B C β Tm8 Ξ (sum8 A B) β Tm8 Ξ (arr8 A C) β Tm8 Ξ (arr8 B C) β Tm8 Ξ C)
(zero : β Ξ β Tm8 Ξ nat8)
(suc : β Ξ β Tm8 Ξ nat8 β Tm8 Ξ nat8)
(rec : β Ξ A β Tm8 Ξ nat8 β Tm8 Ξ (arr8 nat8 (arr8 A A)) β Tm8 Ξ A β Tm8 Ξ A)
β Tm8 Ξ A
var8 : β{Ξ A} β Var8 Ξ A β Tm8 Ξ A; var8
= Ξ» x Tm8 var8 lam app tt pair fst snd left right case zero suc rec β
var8 _ _ x
lam8 : β{Ξ A B} β Tm8 (snoc8 Ξ A) B β Tm8 Ξ (arr8 A B); lam8
= Ξ» t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec β
lam8 _ _ _ (t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec)
app8 : β{Ξ A B} β Tm8 Ξ (arr8 A B) β Tm8 Ξ A β Tm8 Ξ B; app8
= Ξ» t u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec β
app8 _ _ _ (t Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
tt8 : β{Ξ} β Tm8 Ξ top8; tt8
= Ξ» Tm8 var8 lam8 app8 tt8 pair fst snd left right case zero suc rec β tt8 _
pair8 : β{Ξ A B} β Tm8 Ξ A β Tm8 Ξ B β Tm8 Ξ (prod8 A B); pair8
= Ξ» t u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec β
pair8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
fst8 : β{Ξ A B} β Tm8 Ξ (prod8 A B) β Tm8 Ξ A; fst8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec β
fst8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec)
snd8 : β{Ξ A B} β Tm8 Ξ (prod8 A B) β Tm8 Ξ B; snd8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec β
snd8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec)
left8 : β{Ξ A B} β Tm8 Ξ A β Tm8 Ξ (sum8 A B); left8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec β
left8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec)
right8 : β{Ξ A B} β Tm8 Ξ B β Tm8 Ξ (sum8 A B); right8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec β
right8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec)
case8 : β{Ξ A B C} β Tm8 Ξ (sum8 A B) β Tm8 Ξ (arr8 A C) β Tm8 Ξ (arr8 B C) β Tm8 Ξ C; case8
= Ξ» t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec β
case8 _ _ _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
zero8 : β{Ξ} β Tm8 Ξ nat8; zero8
= Ξ» Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc rec β zero8 _
suc8 : β{Ξ} β Tm8 Ξ nat8 β Tm8 Ξ nat8; suc8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec β
suc8 _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec)
rec8 : β{Ξ A} β Tm8 Ξ nat8 β Tm8 Ξ (arr8 nat8 (arr8 A A)) β Tm8 Ξ A β Tm8 Ξ A; rec8
= Ξ» t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8 β
rec8 _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
v08 : β{Ξ A} β Tm8 (snoc8 Ξ A) A; v08
= var8 vz8
v18 : β{Ξ A B} β Tm8 (snoc8 (snoc8 Ξ A) B) A; v18
= var8 (vs8 vz8)
v28 : β{Ξ A B C} β Tm8 (snoc8 (snoc8 (snoc8 Ξ A) B) C) A; v28
= var8 (vs8 (vs8 vz8))
v38 : β{Ξ A B C D} β Tm8 (snoc8 (snoc8 (snoc8 (snoc8 Ξ A) B) C) D) A; v38
= var8 (vs8 (vs8 (vs8 vz8)))
tbool8 : Ty8; tbool8
= sum8 top8 top8
true8 : β{Ξ} β Tm8 Ξ tbool8; true8
= left8 tt8
tfalse8 : β{Ξ} β Tm8 Ξ tbool8; tfalse8
= right8 tt8
ifthenelse8 : β{Ξ A} β Tm8 Ξ (arr8 tbool8 (arr8 A (arr8 A A))); ifthenelse8
= lam8 (lam8 (lam8 (case8 v28 (lam8 v28) (lam8 v18))))
times48 : β{Ξ A} β Tm8 Ξ (arr8 (arr8 A A) (arr8 A A)); times48
= lam8 (lam8 (app8 v18 (app8 v18 (app8 v18 (app8 v18 v08)))))
add8 : β{Ξ} β Tm8 Ξ (arr8 nat8 (arr8 nat8 nat8)); add8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (suc8 (app8 v18 v08)))))
(lam8 v08))
mul8 : β{Ξ} β Tm8 Ξ (arr8 nat8 (arr8 nat8 nat8)); mul8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (app8 (app8 add8 (app8 v18 v08)) v08))))
(lam8 zero8))
fact8 : β{Ξ} β Tm8 Ξ (arr8 nat8 nat8); fact8
= lam8 (rec8 v08 (lam8 (lam8 (app8 (app8 mul8 (suc8 v18)) v08)))
(suc8 zero8))
{-# OPTIONS --type-in-type #-}
Ty9 : Set
Ty9 =
(Ty9 : Set)
(nat top bot : Ty9)
(arr prod sum : Ty9 β Ty9 β Ty9)
β Ty9
nat9 : Ty9; nat9 = Ξ» _ nat9 _ _ _ _ _ β nat9
top9 : Ty9; top9 = Ξ» _ _ top9 _ _ _ _ β top9
bot9 : Ty9; bot9 = Ξ» _ _ _ bot9 _ _ _ β bot9
arr9 : Ty9 β Ty9 β Ty9; arr9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod sum β
arr9 (A Ty9 nat9 top9 bot9 arr9 prod sum) (B Ty9 nat9 top9 bot9 arr9 prod sum)
prod9 : Ty9 β Ty9 β Ty9; prod9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod9 sum β
prod9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum) (B Ty9 nat9 top9 bot9 arr9 prod9 sum)
sum9 : Ty9 β Ty9 β Ty9; sum9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod9 sum9 β
sum9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum9) (B Ty9 nat9 top9 bot9 arr9 prod9 sum9)
Con9 : Set; Con9
= (Con9 : Set)
(nil : Con9)
(snoc : Con9 β Ty9 β Con9)
β Con9
nil9 : Con9; nil9
= Ξ» Con9 nil9 snoc β nil9
snoc9 : Con9 β Ty9 β Con9; snoc9
= Ξ» Ξ A Con9 nil9 snoc9 β snoc9 (Ξ Con9 nil9 snoc9) A
Var9 : Con9 β Ty9 β Set; Var9
= Ξ» Ξ A β
(Var9 : Con9 β Ty9 β Set)
(vz : β Ξ A β Var9 (snoc9 Ξ A) A)
(vs : β Ξ B A β Var9 Ξ A β Var9 (snoc9 Ξ B) A)
β Var9 Ξ A
vz9 : β{Ξ A} β Var9 (snoc9 Ξ A) A; vz9
= Ξ» Var9 vz9 vs β vz9 _ _
vs9 : β{Ξ B A} β Var9 Ξ A β Var9 (snoc9 Ξ B) A; vs9
= Ξ» x Var9 vz9 vs9 β vs9 _ _ _ (x Var9 vz9 vs9)
Tm9 : Con9 β Ty9 β Set; Tm9
= Ξ» Ξ A β
(Tm9 : Con9 β Ty9 β Set)
(var : β Ξ A β Var9 Ξ A β Tm9 Ξ A)
(lam : β Ξ A B β Tm9 (snoc9 Ξ A) B β Tm9 Ξ (arr9 A B))
(app : β Ξ A B β Tm9 Ξ (arr9 A B) β Tm9 Ξ A β Tm9 Ξ B)
(tt : β Ξ β Tm9 Ξ top9)
(pair : β Ξ A B β Tm9 Ξ A β Tm9 Ξ B β Tm9 Ξ (prod9 A B))
(fst : β Ξ A B β Tm9 Ξ (prod9 A B) β Tm9 Ξ A)
(snd : β Ξ A B β Tm9 Ξ (prod9 A B) β Tm9 Ξ B)
(left : β Ξ A B β Tm9 Ξ A β Tm9 Ξ (sum9 A B))
(right : β Ξ A B β Tm9 Ξ B β Tm9 Ξ (sum9 A B))
(case : β Ξ A B C β Tm9 Ξ (sum9 A B) β Tm9 Ξ (arr9 A C) β Tm9 Ξ (arr9 B C) β Tm9 Ξ C)
(zero : β Ξ β Tm9 Ξ nat9)
(suc : β Ξ β Tm9 Ξ nat9 β Tm9 Ξ nat9)
(rec : β Ξ A β Tm9 Ξ nat9 β Tm9 Ξ (arr9 nat9 (arr9 A A)) β Tm9 Ξ A β Tm9 Ξ A)
β Tm9 Ξ A
var9 : β{Ξ A} β Var9 Ξ A β Tm9 Ξ A; var9
= Ξ» x Tm9 var9 lam app tt pair fst snd left right case zero suc rec β
var9 _ _ x
lam9 : β{Ξ A B} β Tm9 (snoc9 Ξ A) B β Tm9 Ξ (arr9 A B); lam9
= Ξ» t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec β
lam9 _ _ _ (t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec)
app9 : β{Ξ A B} β Tm9 Ξ (arr9 A B) β Tm9 Ξ A β Tm9 Ξ B; app9
= Ξ» t u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec β
app9 _ _ _ (t Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
tt9 : β{Ξ} β Tm9 Ξ top9; tt9
= Ξ» Tm9 var9 lam9 app9 tt9 pair fst snd left right case zero suc rec β tt9 _
pair9 : β{Ξ A B} β Tm9 Ξ A β Tm9 Ξ B β Tm9 Ξ (prod9 A B); pair9
= Ξ» t u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec β
pair9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
fst9 : β{Ξ A B} β Tm9 Ξ (prod9 A B) β Tm9 Ξ A; fst9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec β
fst9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec)
snd9 : β{Ξ A B} β Tm9 Ξ (prod9 A B) β Tm9 Ξ B; snd9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec β
snd9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec)
left9 : β{Ξ A B} β Tm9 Ξ A β Tm9 Ξ (sum9 A B); left9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec β
left9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec)
right9 : β{Ξ A B} β Tm9 Ξ B β Tm9 Ξ (sum9 A B); right9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec β
right9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec)
case9 : β{Ξ A B C} β Tm9 Ξ (sum9 A B) β Tm9 Ξ (arr9 A C) β Tm9 Ξ (arr9 B C) β Tm9 Ξ C; case9
= Ξ» t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec β
case9 _ _ _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
zero9 : β{Ξ} β Tm9 Ξ nat9; zero9
= Ξ» Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc rec β zero9 _
suc9 : β{Ξ} β Tm9 Ξ nat9 β Tm9 Ξ nat9; suc9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec β
suc9 _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec)
rec9 : β{Ξ A} β Tm9 Ξ nat9 β Tm9 Ξ (arr9 nat9 (arr9 A A)) β Tm9 Ξ A β Tm9 Ξ A; rec9
= Ξ» t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9 β
rec9 _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
v09 : β{Ξ A} β Tm9 (snoc9 Ξ A) A; v09
= var9 vz9
v19 : β{Ξ A B} β Tm9 (snoc9 (snoc9 Ξ A) B) A; v19
= var9 (vs9 vz9)
v29 : β{Ξ A B C} β Tm9 (snoc9 (snoc9 (snoc9 Ξ A) B) C) A; v29
= var9 (vs9 (vs9 vz9))
v39 : β{Ξ A B C D} β Tm9 (snoc9 (snoc9 (snoc9 (snoc9 Ξ A) B) C) D) A; v39
= var9 (vs9 (vs9 (vs9 vz9)))
tbool9 : Ty9; tbool9
= sum9 top9 top9
true9 : β{Ξ} β Tm9 Ξ tbool9; true9
= left9 tt9
tfalse9 : β{Ξ} β Tm9 Ξ tbool9; tfalse9
= right9 tt9
ifthenelse9 : β{Ξ A} β Tm9 Ξ (arr9 tbool9 (arr9 A (arr9 A A))); ifthenelse9
= lam9 (lam9 (lam9 (case9 v29 (lam9 v29) (lam9 v19))))
times49 : β{Ξ A} β Tm9 Ξ (arr9 (arr9 A A) (arr9 A A)); times49
= lam9 (lam9 (app9 v19 (app9 v19 (app9 v19 (app9 v19 v09)))))
add9 : β{Ξ} β Tm9 Ξ (arr9 nat9 (arr9 nat9 nat9)); add9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (suc9 (app9 v19 v09)))))
(lam9 v09))
mul9 : β{Ξ} β Tm9 Ξ (arr9 nat9 (arr9 nat9 nat9)); mul9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (app9 (app9 add9 (app9 v19 v09)) v09))))
(lam9 zero9))
fact9 : β{Ξ} β Tm9 Ξ (arr9 nat9 nat9); fact9
= lam9 (rec9 v09 (lam9 (lam9 (app9 (app9 mul9 (suc9 v19)) v09)))
(suc9 zero9))
{-# OPTIONS --type-in-type #-}
Ty10 : Set
Ty10 =
(Ty10 : Set)
(nat top bot : Ty10)
(arr prod sum : Ty10 β Ty10 β Ty10)
β Ty10
nat10 : Ty10; nat10 = Ξ» _ nat10 _ _ _ _ _ β nat10
top10 : Ty10; top10 = Ξ» _ _ top10 _ _ _ _ β top10
bot10 : Ty10; bot10 = Ξ» _ _ _ bot10 _ _ _ β bot10
arr10 : Ty10 β Ty10 β Ty10; arr10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod sum β
arr10 (A Ty10 nat10 top10 bot10 arr10 prod sum) (B Ty10 nat10 top10 bot10 arr10 prod sum)
prod10 : Ty10 β Ty10 β Ty10; prod10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod10 sum β
prod10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum) (B Ty10 nat10 top10 bot10 arr10 prod10 sum)
sum10 : Ty10 β Ty10 β Ty10; sum10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod10 sum10 β
sum10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum10) (B Ty10 nat10 top10 bot10 arr10 prod10 sum10)
Con10 : Set; Con10
= (Con10 : Set)
(nil : Con10)
(snoc : Con10 β Ty10 β Con10)
β Con10
nil10 : Con10; nil10
= Ξ» Con10 nil10 snoc β nil10
snoc10 : Con10 β Ty10 β Con10; snoc10
= Ξ» Ξ A Con10 nil10 snoc10 β snoc10 (Ξ Con10 nil10 snoc10) A
Var10 : Con10 β Ty10 β Set; Var10
= Ξ» Ξ A β
(Var10 : Con10 β Ty10 β Set)
(vz : β Ξ A β Var10 (snoc10 Ξ A) A)
(vs : β Ξ B A β Var10 Ξ A β Var10 (snoc10 Ξ B) A)
β Var10 Ξ A
vz10 : β{Ξ A} β Var10 (snoc10 Ξ A) A; vz10
= Ξ» Var10 vz10 vs β vz10 _ _
vs10 : β{Ξ B A} β Var10 Ξ A β Var10 (snoc10 Ξ B) A; vs10
= Ξ» x Var10 vz10 vs10 β vs10 _ _ _ (x Var10 vz10 vs10)
Tm10 : Con10 β Ty10 β Set; Tm10
= Ξ» Ξ A β
(Tm10 : Con10 β Ty10 β Set)
(var : β Ξ A β Var10 Ξ A β Tm10 Ξ A)
(lam : β Ξ A B β Tm10 (snoc10 Ξ A) B β Tm10 Ξ (arr10 A B))
(app : β Ξ A B β Tm10 Ξ (arr10 A B) β Tm10 Ξ A β Tm10 Ξ B)
(tt : β Ξ β Tm10 Ξ top10)
(pair : β Ξ A B β Tm10 Ξ A β Tm10 Ξ B β Tm10 Ξ (prod10 A B))
(fst : β Ξ A B β Tm10 Ξ (prod10 A B) β Tm10 Ξ A)
(snd : β Ξ A B β Tm10 Ξ (prod10 A B) β Tm10 Ξ B)
(left : β Ξ A B β Tm10 Ξ A β Tm10 Ξ (sum10 A B))
(right : β Ξ A B β Tm10 Ξ B β Tm10 Ξ (sum10 A B))
(case : β Ξ A B C β Tm10 Ξ (sum10 A B) β Tm10 Ξ (arr10 A C) β Tm10 Ξ (arr10 B C) β Tm10 Ξ C)
(zero : β Ξ β Tm10 Ξ nat10)
(suc : β Ξ β Tm10 Ξ nat10 β Tm10 Ξ nat10)
(rec : β Ξ A β Tm10 Ξ nat10 β Tm10 Ξ (arr10 nat10 (arr10 A A)) β Tm10 Ξ A β Tm10 Ξ A)
β Tm10 Ξ A
var10 : β{Ξ A} β Var10 Ξ A β Tm10 Ξ A; var10
= Ξ» x Tm10 var10 lam app tt pair fst snd left right case zero suc rec β
var10 _ _ x
lam10 : β{Ξ A B} β Tm10 (snoc10 Ξ A) B β Tm10 Ξ (arr10 A B); lam10
= Ξ» t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec β
lam10 _ _ _ (t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec)
app10 : β{Ξ A B} β Tm10 Ξ (arr10 A B) β Tm10 Ξ A β Tm10 Ξ B; app10
= Ξ» t u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec β
app10 _ _ _ (t Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
tt10 : β{Ξ} β Tm10 Ξ top10; tt10
= Ξ» Tm10 var10 lam10 app10 tt10 pair fst snd left right case zero suc rec β tt10 _
pair10 : β{Ξ A B} β Tm10 Ξ A β Tm10 Ξ B β Tm10 Ξ (prod10 A B); pair10
= Ξ» t u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec β
pair10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
fst10 : β{Ξ A B} β Tm10 Ξ (prod10 A B) β Tm10 Ξ A; fst10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec β
fst10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec)
snd10 : β{Ξ A B} β Tm10 Ξ (prod10 A B) β Tm10 Ξ B; snd10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec β
snd10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec)
left10 : β{Ξ A B} β Tm10 Ξ A β Tm10 Ξ (sum10 A B); left10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec β
left10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec)
right10 : β{Ξ A B} β Tm10 Ξ B β Tm10 Ξ (sum10 A B); right10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec β
right10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec)
case10 : β{Ξ A B C} β Tm10 Ξ (sum10 A B) β Tm10 Ξ (arr10 A C) β Tm10 Ξ (arr10 B C) β Tm10 Ξ C; case10
= Ξ» t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec β
case10 _ _ _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
zero10 : β{Ξ} β Tm10 Ξ nat10; zero10
= Ξ» Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc rec β zero10 _
suc10 : β{Ξ} β Tm10 Ξ nat10 β Tm10 Ξ nat10; suc10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec β
suc10 _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec)
rec10 : β{Ξ A} β Tm10 Ξ nat10 β Tm10 Ξ (arr10 nat10 (arr10 A A)) β Tm10 Ξ A β Tm10 Ξ A; rec10
= Ξ» t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10 β
rec10 _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
v010 : β{Ξ A} β Tm10 (snoc10 Ξ A) A; v010
= var10 vz10
v110 : β{Ξ A B} β Tm10 (snoc10 (snoc10 Ξ A) B) A; v110
= var10 (vs10 vz10)
v210 : β{Ξ A B C} β Tm10 (snoc10 (snoc10 (snoc10 Ξ A) B) C) A; v210
= var10 (vs10 (vs10 vz10))
v310 : β{Ξ A B C D} β Tm10 (snoc10 (snoc10 (snoc10 (snoc10 Ξ A) B) C) D) A; v310
= var10 (vs10 (vs10 (vs10 vz10)))
tbool10 : Ty10; tbool10
= sum10 top10 top10
true10 : β{Ξ} β Tm10 Ξ tbool10; true10
= left10 tt10
tfalse10 : β{Ξ} β Tm10 Ξ tbool10; tfalse10
= right10 tt10
ifthenelse10 : β{Ξ A} β Tm10 Ξ (arr10 tbool10 (arr10 A (arr10 A A))); ifthenelse10
= lam10 (lam10 (lam10 (case10 v210 (lam10 v210) (lam10 v110))))
times410 : β{Ξ A} β Tm10 Ξ (arr10 (arr10 A A) (arr10 A A)); times410
= lam10 (lam10 (app10 v110 (app10 v110 (app10 v110 (app10 v110 v010)))))
add10 : β{Ξ} β Tm10 Ξ (arr10 nat10 (arr10 nat10 nat10)); add10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (suc10 (app10 v110 v010)))))
(lam10 v010))
mul10 : β{Ξ} β Tm10 Ξ (arr10 nat10 (arr10 nat10 nat10)); mul10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (app10 (app10 add10 (app10 v110 v010)) v010))))
(lam10 zero10))
fact10 : β{Ξ} β Tm10 Ξ (arr10 nat10 nat10); fact10
= lam10 (rec10 v010 (lam10 (lam10 (app10 (app10 mul10 (suc10 v110)) v010)))
(suc10 zero10))
{-# OPTIONS --type-in-type #-}
Ty11 : Set
Ty11 =
(Ty11 : Set)
(nat top bot : Ty11)
(arr prod sum : Ty11 β Ty11 β Ty11)
β Ty11
nat11 : Ty11; nat11 = Ξ» _ nat11 _ _ _ _ _ β nat11
top11 : Ty11; top11 = Ξ» _ _ top11 _ _ _ _ β top11
bot11 : Ty11; bot11 = Ξ» _ _ _ bot11 _ _ _ β bot11
arr11 : Ty11 β Ty11 β Ty11; arr11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod sum β
arr11 (A Ty11 nat11 top11 bot11 arr11 prod sum) (B Ty11 nat11 top11 bot11 arr11 prod sum)
prod11 : Ty11 β Ty11 β Ty11; prod11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod11 sum β
prod11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum) (B Ty11 nat11 top11 bot11 arr11 prod11 sum)
sum11 : Ty11 β Ty11 β Ty11; sum11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod11 sum11 β
sum11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum11) (B Ty11 nat11 top11 bot11 arr11 prod11 sum11)
Con11 : Set; Con11
= (Con11 : Set)
(nil : Con11)
(snoc : Con11 β Ty11 β Con11)
β Con11
nil11 : Con11; nil11
= Ξ» Con11 nil11 snoc β nil11
snoc11 : Con11 β Ty11 β Con11; snoc11
= Ξ» Ξ A Con11 nil11 snoc11 β snoc11 (Ξ Con11 nil11 snoc11) A
Var11 : Con11 β Ty11 β Set; Var11
= Ξ» Ξ A β
(Var11 : Con11 β Ty11 β Set)
(vz : β Ξ A β Var11 (snoc11 Ξ A) A)
(vs : β Ξ B A β Var11 Ξ A β Var11 (snoc11 Ξ B) A)
β Var11 Ξ A
vz11 : β{Ξ A} β Var11 (snoc11 Ξ A) A; vz11
= Ξ» Var11 vz11 vs β vz11 _ _
vs11 : β{Ξ B A} β Var11 Ξ A β Var11 (snoc11 Ξ B) A; vs11
= Ξ» x Var11 vz11 vs11 β vs11 _ _ _ (x Var11 vz11 vs11)
Tm11 : Con11 β Ty11 β Set; Tm11
= Ξ» Ξ A β
(Tm11 : Con11 β Ty11 β Set)
(var : β Ξ A β Var11 Ξ A β Tm11 Ξ A)
(lam : β Ξ A B β Tm11 (snoc11 Ξ A) B β Tm11 Ξ (arr11 A B))
(app : β Ξ A B β Tm11 Ξ (arr11 A B) β Tm11 Ξ A β Tm11 Ξ B)
(tt : β Ξ β Tm11 Ξ top11)
(pair : β Ξ A B β Tm11 Ξ A β Tm11 Ξ B β Tm11 Ξ (prod11 A B))
(fst : β Ξ A B β Tm11 Ξ (prod11 A B) β Tm11 Ξ A)
(snd : β Ξ A B β Tm11 Ξ (prod11 A B) β Tm11 Ξ B)
(left : β Ξ A B β Tm11 Ξ A β Tm11 Ξ (sum11 A B))
(right : β Ξ A B β Tm11 Ξ B β Tm11 Ξ (sum11 A B))
(case : β Ξ A B C β Tm11 Ξ (sum11 A B) β Tm11 Ξ (arr11 A C) β Tm11 Ξ (arr11 B C) β Tm11 Ξ C)
(zero : β Ξ β Tm11 Ξ nat11)
(suc : β Ξ β Tm11 Ξ nat11 β Tm11 Ξ nat11)
(rec : β Ξ A β Tm11 Ξ nat11 β Tm11 Ξ (arr11 nat11 (arr11 A A)) β Tm11 Ξ A β Tm11 Ξ A)
β Tm11 Ξ A
var11 : β{Ξ A} β Var11 Ξ A β Tm11 Ξ A; var11
= Ξ» x Tm11 var11 lam app tt pair fst snd left right case zero suc rec β
var11 _ _ x
lam11 : β{Ξ A B} β Tm11 (snoc11 Ξ A) B β Tm11 Ξ (arr11 A B); lam11
= Ξ» t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec β
lam11 _ _ _ (t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec)
app11 : β{Ξ A B} β Tm11 Ξ (arr11 A B) β Tm11 Ξ A β Tm11 Ξ B; app11
= Ξ» t u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec β
app11 _ _ _ (t Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
tt11 : β{Ξ} β Tm11 Ξ top11; tt11
= Ξ» Tm11 var11 lam11 app11 tt11 pair fst snd left right case zero suc rec β tt11 _
pair11 : β{Ξ A B} β Tm11 Ξ A β Tm11 Ξ B β Tm11 Ξ (prod11 A B); pair11
= Ξ» t u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec β
pair11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
fst11 : β{Ξ A B} β Tm11 Ξ (prod11 A B) β Tm11 Ξ A; fst11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec β
fst11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec)
snd11 : β{Ξ A B} β Tm11 Ξ (prod11 A B) β Tm11 Ξ B; snd11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec β
snd11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec)
left11 : β{Ξ A B} β Tm11 Ξ A β Tm11 Ξ (sum11 A B); left11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec β
left11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec)
right11 : β{Ξ A B} β Tm11 Ξ B β Tm11 Ξ (sum11 A B); right11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec β
right11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec)
case11 : β{Ξ A B C} β Tm11 Ξ (sum11 A B) β Tm11 Ξ (arr11 A C) β Tm11 Ξ (arr11 B C) β Tm11 Ξ C; case11
= Ξ» t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec β
case11 _ _ _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
zero11 : β{Ξ} β Tm11 Ξ nat11; zero11
= Ξ» Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc rec β zero11 _
suc11 : β{Ξ} β Tm11 Ξ nat11 β Tm11 Ξ nat11; suc11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec β
suc11 _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec)
rec11 : β{Ξ A} β Tm11 Ξ nat11 β Tm11 Ξ (arr11 nat11 (arr11 A A)) β Tm11 Ξ A β Tm11 Ξ A; rec11
= Ξ» t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11 β
rec11 _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
v011 : β{Ξ A} β Tm11 (snoc11 Ξ A) A; v011
= var11 vz11
v111 : β{Ξ A B} β Tm11 (snoc11 (snoc11 Ξ A) B) A; v111
= var11 (vs11 vz11)
v211 : β{Ξ A B C} β Tm11 (snoc11 (snoc11 (snoc11 Ξ A) B) C) A; v211
= var11 (vs11 (vs11 vz11))
v311 : β{Ξ A B C D} β Tm11 (snoc11 (snoc11 (snoc11 (snoc11 Ξ A) B) C) D) A; v311
= var11 (vs11 (vs11 (vs11 vz11)))
tbool11 : Ty11; tbool11
= sum11 top11 top11
true11 : β{Ξ} β Tm11 Ξ tbool11; true11
= left11 tt11
tfalse11 : β{Ξ} β Tm11 Ξ tbool11; tfalse11
= right11 tt11
ifthenelse11 : β{Ξ A} β Tm11 Ξ (arr11 tbool11 (arr11 A (arr11 A A))); ifthenelse11
= lam11 (lam11 (lam11 (case11 v211 (lam11 v211) (lam11 v111))))
times411 : β{Ξ A} β Tm11 Ξ (arr11 (arr11 A A) (arr11 A A)); times411
= lam11 (lam11 (app11 v111 (app11 v111 (app11 v111 (app11 v111 v011)))))
add11 : β{Ξ} β Tm11 Ξ (arr11 nat11 (arr11 nat11 nat11)); add11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (suc11 (app11 v111 v011)))))
(lam11 v011))
mul11 : β{Ξ} β Tm11 Ξ (arr11 nat11 (arr11 nat11 nat11)); mul11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (app11 (app11 add11 (app11 v111 v011)) v011))))
(lam11 zero11))
fact11 : β{Ξ} β Tm11 Ξ (arr11 nat11 nat11); fact11
= lam11 (rec11 v011 (lam11 (lam11 (app11 (app11 mul11 (suc11 v111)) v011)))
(suc11 zero11))
{-# OPTIONS --type-in-type #-}
Ty12 : Set
Ty12 =
(Ty12 : Set)
(nat top bot : Ty12)
(arr prod sum : Ty12 β Ty12 β Ty12)
β Ty12
nat12 : Ty12; nat12 = Ξ» _ nat12 _ _ _ _ _ β nat12
top12 : Ty12; top12 = Ξ» _ _ top12 _ _ _ _ β top12
bot12 : Ty12; bot12 = Ξ» _ _ _ bot12 _ _ _ β bot12
arr12 : Ty12 β Ty12 β Ty12; arr12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod sum β
arr12 (A Ty12 nat12 top12 bot12 arr12 prod sum) (B Ty12 nat12 top12 bot12 arr12 prod sum)
prod12 : Ty12 β Ty12 β Ty12; prod12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod12 sum β
prod12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum) (B Ty12 nat12 top12 bot12 arr12 prod12 sum)
sum12 : Ty12 β Ty12 β Ty12; sum12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod12 sum12 β
sum12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum12) (B Ty12 nat12 top12 bot12 arr12 prod12 sum12)
Con12 : Set; Con12
= (Con12 : Set)
(nil : Con12)
(snoc : Con12 β Ty12 β Con12)
β Con12
nil12 : Con12; nil12
= Ξ» Con12 nil12 snoc β nil12
snoc12 : Con12 β Ty12 β Con12; snoc12
= Ξ» Ξ A Con12 nil12 snoc12 β snoc12 (Ξ Con12 nil12 snoc12) A
Var12 : Con12 β Ty12 β Set; Var12
= Ξ» Ξ A β
(Var12 : Con12 β Ty12 β Set)
(vz : β Ξ A β Var12 (snoc12 Ξ A) A)
(vs : β Ξ B A β Var12 Ξ A β Var12 (snoc12 Ξ B) A)
β Var12 Ξ A
vz12 : β{Ξ A} β Var12 (snoc12 Ξ A) A; vz12
= Ξ» Var12 vz12 vs β vz12 _ _
vs12 : β{Ξ B A} β Var12 Ξ A β Var12 (snoc12 Ξ B) A; vs12
= Ξ» x Var12 vz12 vs12 β vs12 _ _ _ (x Var12 vz12 vs12)
Tm12 : Con12 β Ty12 β Set; Tm12
= Ξ» Ξ A β
(Tm12 : Con12 β Ty12 β Set)
(var : β Ξ A β Var12 Ξ A β Tm12 Ξ A)
(lam : β Ξ A B β Tm12 (snoc12 Ξ A) B β Tm12 Ξ (arr12 A B))
(app : β Ξ A B β Tm12 Ξ (arr12 A B) β Tm12 Ξ A β Tm12 Ξ B)
(tt : β Ξ β Tm12 Ξ top12)
(pair : β Ξ A B β Tm12 Ξ A β Tm12 Ξ B β Tm12 Ξ (prod12 A B))
(fst : β Ξ A B β Tm12 Ξ (prod12 A B) β Tm12 Ξ A)
(snd : β Ξ A B β Tm12 Ξ (prod12 A B) β Tm12 Ξ B)
(left : β Ξ A B β Tm12 Ξ A β Tm12 Ξ (sum12 A B))
(right : β Ξ A B β Tm12 Ξ B β Tm12 Ξ (sum12 A B))
(case : β Ξ A B C β Tm12 Ξ (sum12 A B) β Tm12 Ξ (arr12 A C) β Tm12 Ξ (arr12 B C) β Tm12 Ξ C)
(zero : β Ξ β Tm12 Ξ nat12)
(suc : β Ξ β Tm12 Ξ nat12 β Tm12 Ξ nat12)
(rec : β Ξ A β Tm12 Ξ nat12 β Tm12 Ξ (arr12 nat12 (arr12 A A)) β Tm12 Ξ A β Tm12 Ξ A)
β Tm12 Ξ A
var12 : β{Ξ A} β Var12 Ξ A β Tm12 Ξ A; var12
= Ξ» x Tm12 var12 lam app tt pair fst snd left right case zero suc rec β
var12 _ _ x
lam12 : β{Ξ A B} β Tm12 (snoc12 Ξ A) B β Tm12 Ξ (arr12 A B); lam12
= Ξ» t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec β
lam12 _ _ _ (t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec)
app12 : β{Ξ A B} β Tm12 Ξ (arr12 A B) β Tm12 Ξ A β Tm12 Ξ B; app12
= Ξ» t u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec β
app12 _ _ _ (t Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
tt12 : β{Ξ} β Tm12 Ξ top12; tt12
= Ξ» Tm12 var12 lam12 app12 tt12 pair fst snd left right case zero suc rec β tt12 _
pair12 : β{Ξ A B} β Tm12 Ξ A β Tm12 Ξ B β Tm12 Ξ (prod12 A B); pair12
= Ξ» t u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec β
pair12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
fst12 : β{Ξ A B} β Tm12 Ξ (prod12 A B) β Tm12 Ξ A; fst12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec β
fst12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec)
snd12 : β{Ξ A B} β Tm12 Ξ (prod12 A B) β Tm12 Ξ B; snd12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec β
snd12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec)
left12 : β{Ξ A B} β Tm12 Ξ A β Tm12 Ξ (sum12 A B); left12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec β
left12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec)
right12 : β{Ξ A B} β Tm12 Ξ B β Tm12 Ξ (sum12 A B); right12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec β
right12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec)
case12 : β{Ξ A B C} β Tm12 Ξ (sum12 A B) β Tm12 Ξ (arr12 A C) β Tm12 Ξ (arr12 B C) β Tm12 Ξ C; case12
= Ξ» t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec β
case12 _ _ _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
zero12 : β{Ξ} β Tm12 Ξ nat12; zero12
= Ξ» Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc rec β zero12 _
suc12 : β{Ξ} β Tm12 Ξ nat12 β Tm12 Ξ nat12; suc12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec β
suc12 _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec)
rec12 : β{Ξ A} β Tm12 Ξ nat12 β Tm12 Ξ (arr12 nat12 (arr12 A A)) β Tm12 Ξ A β Tm12 Ξ A; rec12
= Ξ» t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12 β
rec12 _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
v012 : β{Ξ A} β Tm12 (snoc12 Ξ A) A; v012
= var12 vz12
v112 : β{Ξ A B} β Tm12 (snoc12 (snoc12 Ξ A) B) A; v112
= var12 (vs12 vz12)
v212 : β{Ξ A B C} β Tm12 (snoc12 (snoc12 (snoc12 Ξ A) B) C) A; v212
= var12 (vs12 (vs12 vz12))
v312 : β{Ξ A B C D} β Tm12 (snoc12 (snoc12 (snoc12 (snoc12 Ξ A) B) C) D) A; v312
= var12 (vs12 (vs12 (vs12 vz12)))
tbool12 : Ty12; tbool12
= sum12 top12 top12
true12 : β{Ξ} β Tm12 Ξ tbool12; true12
= left12 tt12
tfalse12 : β{Ξ} β Tm12 Ξ tbool12; tfalse12
= right12 tt12
ifthenelse12 : β{Ξ A} β Tm12 Ξ (arr12 tbool12 (arr12 A (arr12 A A))); ifthenelse12
= lam12 (lam12 (lam12 (case12 v212 (lam12 v212) (lam12 v112))))
times412 : β{Ξ A} β Tm12 Ξ (arr12 (arr12 A A) (arr12 A A)); times412
= lam12 (lam12 (app12 v112 (app12 v112 (app12 v112 (app12 v112 v012)))))
add12 : β{Ξ} β Tm12 Ξ (arr12 nat12 (arr12 nat12 nat12)); add12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (suc12 (app12 v112 v012)))))
(lam12 v012))
mul12 : β{Ξ} β Tm12 Ξ (arr12 nat12 (arr12 nat12 nat12)); mul12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (app12 (app12 add12 (app12 v112 v012)) v012))))
(lam12 zero12))
fact12 : β{Ξ} β Tm12 Ξ (arr12 nat12 nat12); fact12
= lam12 (rec12 v012 (lam12 (lam12 (app12 (app12 mul12 (suc12 v112)) v012)))
(suc12 zero12))
{-# OPTIONS --type-in-type #-}
Ty13 : Set
Ty13 =
(Ty13 : Set)
(nat top bot : Ty13)
(arr prod sum : Ty13 β Ty13 β Ty13)
β Ty13
nat13 : Ty13; nat13 = Ξ» _ nat13 _ _ _ _ _ β nat13
top13 : Ty13; top13 = Ξ» _ _ top13 _ _ _ _ β top13
bot13 : Ty13; bot13 = Ξ» _ _ _ bot13 _ _ _ β bot13
arr13 : Ty13 β Ty13 β Ty13; arr13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod sum β
arr13 (A Ty13 nat13 top13 bot13 arr13 prod sum) (B Ty13 nat13 top13 bot13 arr13 prod sum)
prod13 : Ty13 β Ty13 β Ty13; prod13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod13 sum β
prod13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum) (B Ty13 nat13 top13 bot13 arr13 prod13 sum)
sum13 : Ty13 β Ty13 β Ty13; sum13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod13 sum13 β
sum13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum13) (B Ty13 nat13 top13 bot13 arr13 prod13 sum13)
Con13 : Set; Con13
= (Con13 : Set)
(nil : Con13)
(snoc : Con13 β Ty13 β Con13)
β Con13
nil13 : Con13; nil13
= Ξ» Con13 nil13 snoc β nil13
snoc13 : Con13 β Ty13 β Con13; snoc13
= Ξ» Ξ A Con13 nil13 snoc13 β snoc13 (Ξ Con13 nil13 snoc13) A
Var13 : Con13 β Ty13 β Set; Var13
= Ξ» Ξ A β
(Var13 : Con13 β Ty13 β Set)
(vz : β Ξ A β Var13 (snoc13 Ξ A) A)
(vs : β Ξ B A β Var13 Ξ A β Var13 (snoc13 Ξ B) A)
β Var13 Ξ A
vz13 : β{Ξ A} β Var13 (snoc13 Ξ A) A; vz13
= Ξ» Var13 vz13 vs β vz13 _ _
vs13 : β{Ξ B A} β Var13 Ξ A β Var13 (snoc13 Ξ B) A; vs13
= Ξ» x Var13 vz13 vs13 β vs13 _ _ _ (x Var13 vz13 vs13)
Tm13 : Con13 β Ty13 β Set; Tm13
= Ξ» Ξ A β
(Tm13 : Con13 β Ty13 β Set)
(var : β Ξ A β Var13 Ξ A β Tm13 Ξ A)
(lam : β Ξ A B β Tm13 (snoc13 Ξ A) B β Tm13 Ξ (arr13 A B))
(app : β Ξ A B β Tm13 Ξ (arr13 A B) β Tm13 Ξ A β Tm13 Ξ B)
(tt : β Ξ β Tm13 Ξ top13)
(pair : β Ξ A B β Tm13 Ξ A β Tm13 Ξ B β Tm13 Ξ (prod13 A B))
(fst : β Ξ A B β Tm13 Ξ (prod13 A B) β Tm13 Ξ A)
(snd : β Ξ A B β Tm13 Ξ (prod13 A B) β Tm13 Ξ B)
(left : β Ξ A B β Tm13 Ξ A β Tm13 Ξ (sum13 A B))
(right : β Ξ A B β Tm13 Ξ B β Tm13 Ξ (sum13 A B))
(case : β Ξ A B C β Tm13 Ξ (sum13 A B) β Tm13 Ξ (arr13 A C) β Tm13 Ξ (arr13 B C) β Tm13 Ξ C)
(zero : β Ξ β Tm13 Ξ nat13)
(suc : β Ξ β Tm13 Ξ nat13 β Tm13 Ξ nat13)
(rec : β Ξ A β Tm13 Ξ nat13 β Tm13 Ξ (arr13 nat13 (arr13 A A)) β Tm13 Ξ A β Tm13 Ξ A)
β Tm13 Ξ A
var13 : β{Ξ A} β Var13 Ξ A β Tm13 Ξ A; var13
= Ξ» x Tm13 var13 lam app tt pair fst snd left right case zero suc rec β
var13 _ _ x
lam13 : β{Ξ A B} β Tm13 (snoc13 Ξ A) B β Tm13 Ξ (arr13 A B); lam13
= Ξ» t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec β
lam13 _ _ _ (t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec)
app13 : β{Ξ A B} β Tm13 Ξ (arr13 A B) β Tm13 Ξ A β Tm13 Ξ B; app13
= Ξ» t u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec β
app13 _ _ _ (t Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
tt13 : β{Ξ} β Tm13 Ξ top13; tt13
= Ξ» Tm13 var13 lam13 app13 tt13 pair fst snd left right case zero suc rec β tt13 _
pair13 : β{Ξ A B} β Tm13 Ξ A β Tm13 Ξ B β Tm13 Ξ (prod13 A B); pair13
= Ξ» t u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec β
pair13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
fst13 : β{Ξ A B} β Tm13 Ξ (prod13 A B) β Tm13 Ξ A; fst13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec β
fst13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec)
snd13 : β{Ξ A B} β Tm13 Ξ (prod13 A B) β Tm13 Ξ B; snd13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec β
snd13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec)
left13 : β{Ξ A B} β Tm13 Ξ A β Tm13 Ξ (sum13 A B); left13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec β
left13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec)
right13 : β{Ξ A B} β Tm13 Ξ B β Tm13 Ξ (sum13 A B); right13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec β
right13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec)
case13 : β{Ξ A B C} β Tm13 Ξ (sum13 A B) β Tm13 Ξ (arr13 A C) β Tm13 Ξ (arr13 B C) β Tm13 Ξ C; case13
= Ξ» t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec β
case13 _ _ _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
zero13 : β{Ξ} β Tm13 Ξ nat13; zero13
= Ξ» Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc rec β zero13 _
suc13 : β{Ξ} β Tm13 Ξ nat13 β Tm13 Ξ nat13; suc13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec β
suc13 _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec)
rec13 : β{Ξ A} β Tm13 Ξ nat13 β Tm13 Ξ (arr13 nat13 (arr13 A A)) β Tm13 Ξ A β Tm13 Ξ A; rec13
= Ξ» t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13 β
rec13 _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
v013 : β{Ξ A} β Tm13 (snoc13 Ξ A) A; v013
= var13 vz13
v113 : β{Ξ A B} β Tm13 (snoc13 (snoc13 Ξ A) B) A; v113
= var13 (vs13 vz13)
v213 : β{Ξ A B C} β Tm13 (snoc13 (snoc13 (snoc13 Ξ A) B) C) A; v213
= var13 (vs13 (vs13 vz13))
v313 : β{Ξ A B C D} β Tm13 (snoc13 (snoc13 (snoc13 (snoc13 Ξ A) B) C) D) A; v313
= var13 (vs13 (vs13 (vs13 vz13)))
tbool13 : Ty13; tbool13
= sum13 top13 top13
true13 : β{Ξ} β Tm13 Ξ tbool13; true13
= left13 tt13
tfalse13 : β{Ξ} β Tm13 Ξ tbool13; tfalse13
= right13 tt13
ifthenelse13 : β{Ξ A} β Tm13 Ξ (arr13 tbool13 (arr13 A (arr13 A A))); ifthenelse13
= lam13 (lam13 (lam13 (case13 v213 (lam13 v213) (lam13 v113))))
times413 : β{Ξ A} β Tm13 Ξ (arr13 (arr13 A A) (arr13 A A)); times413
= lam13 (lam13 (app13 v113 (app13 v113 (app13 v113 (app13 v113 v013)))))
add13 : β{Ξ} β Tm13 Ξ (arr13 nat13 (arr13 nat13 nat13)); add13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (suc13 (app13 v113 v013)))))
(lam13 v013))
mul13 : β{Ξ} β Tm13 Ξ (arr13 nat13 (arr13 nat13 nat13)); mul13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (app13 (app13 add13 (app13 v113 v013)) v013))))
(lam13 zero13))
fact13 : β{Ξ} β Tm13 Ξ (arr13 nat13 nat13); fact13
= lam13 (rec13 v013 (lam13 (lam13 (app13 (app13 mul13 (suc13 v113)) v013)))
(suc13 zero13))
{-# OPTIONS --type-in-type #-}
Ty14 : Set
Ty14 =
(Ty14 : Set)
(nat top bot : Ty14)
(arr prod sum : Ty14 β Ty14 β Ty14)
β Ty14
nat14 : Ty14; nat14 = Ξ» _ nat14 _ _ _ _ _ β nat14
top14 : Ty14; top14 = Ξ» _ _ top14 _ _ _ _ β top14
bot14 : Ty14; bot14 = Ξ» _ _ _ bot14 _ _ _ β bot14
arr14 : Ty14 β Ty14 β Ty14; arr14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod sum β
arr14 (A Ty14 nat14 top14 bot14 arr14 prod sum) (B Ty14 nat14 top14 bot14 arr14 prod sum)
prod14 : Ty14 β Ty14 β Ty14; prod14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod14 sum β
prod14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum) (B Ty14 nat14 top14 bot14 arr14 prod14 sum)
sum14 : Ty14 β Ty14 β Ty14; sum14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod14 sum14 β
sum14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum14) (B Ty14 nat14 top14 bot14 arr14 prod14 sum14)
Con14 : Set; Con14
= (Con14 : Set)
(nil : Con14)
(snoc : Con14 β Ty14 β Con14)
β Con14
nil14 : Con14; nil14
= Ξ» Con14 nil14 snoc β nil14
snoc14 : Con14 β Ty14 β Con14; snoc14
= Ξ» Ξ A Con14 nil14 snoc14 β snoc14 (Ξ Con14 nil14 snoc14) A
Var14 : Con14 β Ty14 β Set; Var14
= Ξ» Ξ A β
(Var14 : Con14 β Ty14 β Set)
(vz : β Ξ A β Var14 (snoc14 Ξ A) A)
(vs : β Ξ B A β Var14 Ξ A β Var14 (snoc14 Ξ B) A)
β Var14 Ξ A
vz14 : β{Ξ A} β Var14 (snoc14 Ξ A) A; vz14
= Ξ» Var14 vz14 vs β vz14 _ _
vs14 : β{Ξ B A} β Var14 Ξ A β Var14 (snoc14 Ξ B) A; vs14
= Ξ» x Var14 vz14 vs14 β vs14 _ _ _ (x Var14 vz14 vs14)
Tm14 : Con14 β Ty14 β Set; Tm14
= Ξ» Ξ A β
(Tm14 : Con14 β Ty14 β Set)
(var : β Ξ A β Var14 Ξ A β Tm14 Ξ A)
(lam : β Ξ A B β Tm14 (snoc14 Ξ A) B β Tm14 Ξ (arr14 A B))
(app : β Ξ A B β Tm14 Ξ (arr14 A B) β Tm14 Ξ A β Tm14 Ξ B)
(tt : β Ξ β Tm14 Ξ top14)
(pair : β Ξ A B β Tm14 Ξ A β Tm14 Ξ B β Tm14 Ξ (prod14 A B))
(fst : β Ξ A B β Tm14 Ξ (prod14 A B) β Tm14 Ξ A)
(snd : β Ξ A B β Tm14 Ξ (prod14 A B) β Tm14 Ξ B)
(left : β Ξ A B β Tm14 Ξ A β Tm14 Ξ (sum14 A B))
(right : β Ξ A B β Tm14 Ξ B β Tm14 Ξ (sum14 A B))
(case : β Ξ A B C β Tm14 Ξ (sum14 A B) β Tm14 Ξ (arr14 A C) β Tm14 Ξ (arr14 B C) β Tm14 Ξ C)
(zero : β Ξ β Tm14 Ξ nat14)
(suc : β Ξ β Tm14 Ξ nat14 β Tm14 Ξ nat14)
(rec : β Ξ A β Tm14 Ξ nat14 β Tm14 Ξ (arr14 nat14 (arr14 A A)) β Tm14 Ξ A β Tm14 Ξ A)
β Tm14 Ξ A
var14 : β{Ξ A} β Var14 Ξ A β Tm14 Ξ A; var14
= Ξ» x Tm14 var14 lam app tt pair fst snd left right case zero suc rec β
var14 _ _ x
lam14 : β{Ξ A B} β Tm14 (snoc14 Ξ A) B β Tm14 Ξ (arr14 A B); lam14
= Ξ» t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec β
lam14 _ _ _ (t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec)
app14 : β{Ξ A B} β Tm14 Ξ (arr14 A B) β Tm14 Ξ A β Tm14 Ξ B; app14
= Ξ» t u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec β
app14 _ _ _ (t Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
tt14 : β{Ξ} β Tm14 Ξ top14; tt14
= Ξ» Tm14 var14 lam14 app14 tt14 pair fst snd left right case zero suc rec β tt14 _
pair14 : β{Ξ A B} β Tm14 Ξ A β Tm14 Ξ B β Tm14 Ξ (prod14 A B); pair14
= Ξ» t u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec β
pair14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
fst14 : β{Ξ A B} β Tm14 Ξ (prod14 A B) β Tm14 Ξ A; fst14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec β
fst14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec)
snd14 : β{Ξ A B} β Tm14 Ξ (prod14 A B) β Tm14 Ξ B; snd14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec β
snd14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec)
left14 : β{Ξ A B} β Tm14 Ξ A β Tm14 Ξ (sum14 A B); left14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec β
left14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec)
right14 : β{Ξ A B} β Tm14 Ξ B β Tm14 Ξ (sum14 A B); right14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec β
right14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec)
case14 : β{Ξ A B C} β Tm14 Ξ (sum14 A B) β Tm14 Ξ (arr14 A C) β Tm14 Ξ (arr14 B C) β Tm14 Ξ C; case14
= Ξ» t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec β
case14 _ _ _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
zero14 : β{Ξ} β Tm14 Ξ nat14; zero14
= Ξ» Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc rec β zero14 _
suc14 : β{Ξ} β Tm14 Ξ nat14 β Tm14 Ξ nat14; suc14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec β
suc14 _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec)
rec14 : β{Ξ A} β Tm14 Ξ nat14 β Tm14 Ξ (arr14 nat14 (arr14 A A)) β Tm14 Ξ A β Tm14 Ξ A; rec14
= Ξ» t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14 β
rec14 _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
v014 : β{Ξ A} β Tm14 (snoc14 Ξ A) A; v014
= var14 vz14
v114 : β{Ξ A B} β Tm14 (snoc14 (snoc14 Ξ A) B) A; v114
= var14 (vs14 vz14)
v214 : β{Ξ A B C} β Tm14 (snoc14 (snoc14 (snoc14 Ξ A) B) C) A; v214
= var14 (vs14 (vs14 vz14))
v314 : β{Ξ A B C D} β Tm14 (snoc14 (snoc14 (snoc14 (snoc14 Ξ A) B) C) D) A; v314
= var14 (vs14 (vs14 (vs14 vz14)))
tbool14 : Ty14; tbool14
= sum14 top14 top14
true14 : β{Ξ} β Tm14 Ξ tbool14; true14
= left14 tt14
tfalse14 : β{Ξ} β Tm14 Ξ tbool14; tfalse14
= right14 tt14
ifthenelse14 : β{Ξ A} β Tm14 Ξ (arr14 tbool14 (arr14 A (arr14 A A))); ifthenelse14
= lam14 (lam14 (lam14 (case14 v214 (lam14 v214) (lam14 v114))))
times414 : β{Ξ A} β Tm14 Ξ (arr14 (arr14 A A) (arr14 A A)); times414
= lam14 (lam14 (app14 v114 (app14 v114 (app14 v114 (app14 v114 v014)))))
add14 : β{Ξ} β Tm14 Ξ (arr14 nat14 (arr14 nat14 nat14)); add14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (suc14 (app14 v114 v014)))))
(lam14 v014))
mul14 : β{Ξ} β Tm14 Ξ (arr14 nat14 (arr14 nat14 nat14)); mul14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (app14 (app14 add14 (app14 v114 v014)) v014))))
(lam14 zero14))
fact14 : β{Ξ} β Tm14 Ξ (arr14 nat14 nat14); fact14
= lam14 (rec14 v014 (lam14 (lam14 (app14 (app14 mul14 (suc14 v114)) v014)))
(suc14 zero14))
{-# OPTIONS --type-in-type #-}
Ty15 : Set
Ty15 =
(Ty15 : Set)
(nat top bot : Ty15)
(arr prod sum : Ty15 β Ty15 β Ty15)
β Ty15
nat15 : Ty15; nat15 = Ξ» _ nat15 _ _ _ _ _ β nat15
top15 : Ty15; top15 = Ξ» _ _ top15 _ _ _ _ β top15
bot15 : Ty15; bot15 = Ξ» _ _ _ bot15 _ _ _ β bot15
arr15 : Ty15 β Ty15 β Ty15; arr15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod sum β
arr15 (A Ty15 nat15 top15 bot15 arr15 prod sum) (B Ty15 nat15 top15 bot15 arr15 prod sum)
prod15 : Ty15 β Ty15 β Ty15; prod15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod15 sum β
prod15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum) (B Ty15 nat15 top15 bot15 arr15 prod15 sum)
sum15 : Ty15 β Ty15 β Ty15; sum15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod15 sum15 β
sum15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum15) (B Ty15 nat15 top15 bot15 arr15 prod15 sum15)
Con15 : Set; Con15
= (Con15 : Set)
(nil : Con15)
(snoc : Con15 β Ty15 β Con15)
β Con15
nil15 : Con15; nil15
= Ξ» Con15 nil15 snoc β nil15
snoc15 : Con15 β Ty15 β Con15; snoc15
= Ξ» Ξ A Con15 nil15 snoc15 β snoc15 (Ξ Con15 nil15 snoc15) A
Var15 : Con15 β Ty15 β Set; Var15
= Ξ» Ξ A β
(Var15 : Con15 β Ty15 β Set)
(vz : β Ξ A β Var15 (snoc15 Ξ A) A)
(vs : β Ξ B A β Var15 Ξ A β Var15 (snoc15 Ξ B) A)
β Var15 Ξ A
vz15 : β{Ξ A} β Var15 (snoc15 Ξ A) A; vz15
= Ξ» Var15 vz15 vs β vz15 _ _
vs15 : β{Ξ B A} β Var15 Ξ A β Var15 (snoc15 Ξ B) A; vs15
= Ξ» x Var15 vz15 vs15 β vs15 _ _ _ (x Var15 vz15 vs15)
Tm15 : Con15 β Ty15 β Set; Tm15
= Ξ» Ξ A β
(Tm15 : Con15 β Ty15 β Set)
(var : β Ξ A β Var15 Ξ A β Tm15 Ξ A)
(lam : β Ξ A B β Tm15 (snoc15 Ξ A) B β Tm15 Ξ (arr15 A B))
(app : β Ξ A B β Tm15 Ξ (arr15 A B) β Tm15 Ξ A β Tm15 Ξ B)
(tt : β Ξ β Tm15 Ξ top15)
(pair : β Ξ A B β Tm15 Ξ A β Tm15 Ξ B β Tm15 Ξ (prod15 A B))
(fst : β Ξ A B β Tm15 Ξ (prod15 A B) β Tm15 Ξ A)
(snd : β Ξ A B β Tm15 Ξ (prod15 A B) β Tm15 Ξ B)
(left : β Ξ A B β Tm15 Ξ A β Tm15 Ξ (sum15 A B))
(right : β Ξ A B β Tm15 Ξ B β Tm15 Ξ (sum15 A B))
(case : β Ξ A B C β Tm15 Ξ (sum15 A B) β Tm15 Ξ (arr15 A C) β Tm15 Ξ (arr15 B C) β Tm15 Ξ C)
(zero : β Ξ β Tm15 Ξ nat15)
(suc : β Ξ β Tm15 Ξ nat15 β Tm15 Ξ nat15)
(rec : β Ξ A β Tm15 Ξ nat15 β Tm15 Ξ (arr15 nat15 (arr15 A A)) β Tm15 Ξ A β Tm15 Ξ A)
β Tm15 Ξ A
var15 : β{Ξ A} β Var15 Ξ A β Tm15 Ξ A; var15
= Ξ» x Tm15 var15 lam app tt pair fst snd left right case zero suc rec β
var15 _ _ x
lam15 : β{Ξ A B} β Tm15 (snoc15 Ξ A) B β Tm15 Ξ (arr15 A B); lam15
= Ξ» t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec β
lam15 _ _ _ (t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec)
app15 : β{Ξ A B} β Tm15 Ξ (arr15 A B) β Tm15 Ξ A β Tm15 Ξ B; app15
= Ξ» t u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec β
app15 _ _ _ (t Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
tt15 : β{Ξ} β Tm15 Ξ top15; tt15
= Ξ» Tm15 var15 lam15 app15 tt15 pair fst snd left right case zero suc rec β tt15 _
pair15 : β{Ξ A B} β Tm15 Ξ A β Tm15 Ξ B β Tm15 Ξ (prod15 A B); pair15
= Ξ» t u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec β
pair15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
fst15 : β{Ξ A B} β Tm15 Ξ (prod15 A B) β Tm15 Ξ A; fst15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec β
fst15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec)
snd15 : β{Ξ A B} β Tm15 Ξ (prod15 A B) β Tm15 Ξ B; snd15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec β
snd15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec)
left15 : β{Ξ A B} β Tm15 Ξ A β Tm15 Ξ (sum15 A B); left15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec β
left15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec)
right15 : β{Ξ A B} β Tm15 Ξ B β Tm15 Ξ (sum15 A B); right15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec β
right15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec)
case15 : β{Ξ A B C} β Tm15 Ξ (sum15 A B) β Tm15 Ξ (arr15 A C) β Tm15 Ξ (arr15 B C) β Tm15 Ξ C; case15
= Ξ» t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec β
case15 _ _ _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
zero15 : β{Ξ} β Tm15 Ξ nat15; zero15
= Ξ» Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc rec β zero15 _
suc15 : β{Ξ} β Tm15 Ξ nat15 β Tm15 Ξ nat15; suc15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec β
suc15 _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec)
rec15 : β{Ξ A} β Tm15 Ξ nat15 β Tm15 Ξ (arr15 nat15 (arr15 A A)) β Tm15 Ξ A β Tm15 Ξ A; rec15
= Ξ» t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15 β
rec15 _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
v015 : β{Ξ A} β Tm15 (snoc15 Ξ A) A; v015
= var15 vz15
v115 : β{Ξ A B} β Tm15 (snoc15 (snoc15 Ξ A) B) A; v115
= var15 (vs15 vz15)
v215 : β{Ξ A B C} β Tm15 (snoc15 (snoc15 (snoc15 Ξ A) B) C) A; v215
= var15 (vs15 (vs15 vz15))
v315 : β{Ξ A B C D} β Tm15 (snoc15 (snoc15 (snoc15 (snoc15 Ξ A) B) C) D) A; v315
= var15 (vs15 (vs15 (vs15 vz15)))
tbool15 : Ty15; tbool15
= sum15 top15 top15
true15 : β{Ξ} β Tm15 Ξ tbool15; true15
= left15 tt15
tfalse15 : β{Ξ} β Tm15 Ξ tbool15; tfalse15
= right15 tt15
ifthenelse15 : β{Ξ A} β Tm15 Ξ (arr15 tbool15 (arr15 A (arr15 A A))); ifthenelse15
= lam15 (lam15 (lam15 (case15 v215 (lam15 v215) (lam15 v115))))
times415 : β{Ξ A} β Tm15 Ξ (arr15 (arr15 A A) (arr15 A A)); times415
= lam15 (lam15 (app15 v115 (app15 v115 (app15 v115 (app15 v115 v015)))))
add15 : β{Ξ} β Tm15 Ξ (arr15 nat15 (arr15 nat15 nat15)); add15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (suc15 (app15 v115 v015)))))
(lam15 v015))
mul15 : β{Ξ} β Tm15 Ξ (arr15 nat15 (arr15 nat15 nat15)); mul15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (app15 (app15 add15 (app15 v115 v015)) v015))))
(lam15 zero15))
fact15 : β{Ξ} β Tm15 Ξ (arr15 nat15 nat15); fact15
= lam15 (rec15 v015 (lam15 (lam15 (app15 (app15 mul15 (suc15 v115)) v015)))
(suc15 zero15))
{-# OPTIONS --type-in-type #-}
Ty16 : Set
Ty16 =
(Ty16 : Set)
(nat top bot : Ty16)
(arr prod sum : Ty16 β Ty16 β Ty16)
β Ty16
nat16 : Ty16; nat16 = Ξ» _ nat16 _ _ _ _ _ β nat16
top16 : Ty16; top16 = Ξ» _ _ top16 _ _ _ _ β top16
bot16 : Ty16; bot16 = Ξ» _ _ _ bot16 _ _ _ β bot16
arr16 : Ty16 β Ty16 β Ty16; arr16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod sum β
arr16 (A Ty16 nat16 top16 bot16 arr16 prod sum) (B Ty16 nat16 top16 bot16 arr16 prod sum)
prod16 : Ty16 β Ty16 β Ty16; prod16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod16 sum β
prod16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum) (B Ty16 nat16 top16 bot16 arr16 prod16 sum)
sum16 : Ty16 β Ty16 β Ty16; sum16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod16 sum16 β
sum16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum16) (B Ty16 nat16 top16 bot16 arr16 prod16 sum16)
Con16 : Set; Con16
= (Con16 : Set)
(nil : Con16)
(snoc : Con16 β Ty16 β Con16)
β Con16
nil16 : Con16; nil16
= Ξ» Con16 nil16 snoc β nil16
snoc16 : Con16 β Ty16 β Con16; snoc16
= Ξ» Ξ A Con16 nil16 snoc16 β snoc16 (Ξ Con16 nil16 snoc16) A
Var16 : Con16 β Ty16 β Set; Var16
= Ξ» Ξ A β
(Var16 : Con16 β Ty16 β Set)
(vz : β Ξ A β Var16 (snoc16 Ξ A) A)
(vs : β Ξ B A β Var16 Ξ A β Var16 (snoc16 Ξ B) A)
β Var16 Ξ A
vz16 : β{Ξ A} β Var16 (snoc16 Ξ A) A; vz16
= Ξ» Var16 vz16 vs β vz16 _ _
vs16 : β{Ξ B A} β Var16 Ξ A β Var16 (snoc16 Ξ B) A; vs16
= Ξ» x Var16 vz16 vs16 β vs16 _ _ _ (x Var16 vz16 vs16)
Tm16 : Con16 β Ty16 β Set; Tm16
= Ξ» Ξ A β
(Tm16 : Con16 β Ty16 β Set)
(var : β Ξ A β Var16 Ξ A β Tm16 Ξ A)
(lam : β Ξ A B β Tm16 (snoc16 Ξ A) B β Tm16 Ξ (arr16 A B))
(app : β Ξ A B β Tm16 Ξ (arr16 A B) β Tm16 Ξ A β Tm16 Ξ B)
(tt : β Ξ β Tm16 Ξ top16)
(pair : β Ξ A B β Tm16 Ξ A β Tm16 Ξ B β Tm16 Ξ (prod16 A B))
(fst : β Ξ A B β Tm16 Ξ (prod16 A B) β Tm16 Ξ A)
(snd : β Ξ A B β Tm16 Ξ (prod16 A B) β Tm16 Ξ B)
(left : β Ξ A B β Tm16 Ξ A β Tm16 Ξ (sum16 A B))
(right : β Ξ A B β Tm16 Ξ B β Tm16 Ξ (sum16 A B))
(case : β Ξ A B C β Tm16 Ξ (sum16 A B) β Tm16 Ξ (arr16 A C) β Tm16 Ξ (arr16 B C) β Tm16 Ξ C)
(zero : β Ξ β Tm16 Ξ nat16)
(suc : β Ξ β Tm16 Ξ nat16 β Tm16 Ξ nat16)
(rec : β Ξ A β Tm16 Ξ nat16 β Tm16 Ξ (arr16 nat16 (arr16 A A)) β Tm16 Ξ A β Tm16 Ξ A)
β Tm16 Ξ A
var16 : β{Ξ A} β Var16 Ξ A β Tm16 Ξ A; var16
= Ξ» x Tm16 var16 lam app tt pair fst snd left right case zero suc rec β
var16 _ _ x
lam16 : β{Ξ A B} β Tm16 (snoc16 Ξ A) B β Tm16 Ξ (arr16 A B); lam16
= Ξ» t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec β
lam16 _ _ _ (t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec)
app16 : β{Ξ A B} β Tm16 Ξ (arr16 A B) β Tm16 Ξ A β Tm16 Ξ B; app16
= Ξ» t u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec β
app16 _ _ _ (t Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
tt16 : β{Ξ} β Tm16 Ξ top16; tt16
= Ξ» Tm16 var16 lam16 app16 tt16 pair fst snd left right case zero suc rec β tt16 _
pair16 : β{Ξ A B} β Tm16 Ξ A β Tm16 Ξ B β Tm16 Ξ (prod16 A B); pair16
= Ξ» t u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec β
pair16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
fst16 : β{Ξ A B} β Tm16 Ξ (prod16 A B) β Tm16 Ξ A; fst16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec β
fst16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec)
snd16 : β{Ξ A B} β Tm16 Ξ (prod16 A B) β Tm16 Ξ B; snd16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec β
snd16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec)
left16 : β{Ξ A B} β Tm16 Ξ A β Tm16 Ξ (sum16 A B); left16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec β
left16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec)
right16 : β{Ξ A B} β Tm16 Ξ B β Tm16 Ξ (sum16 A B); right16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec β
right16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec)
case16 : β{Ξ A B C} β Tm16 Ξ (sum16 A B) β Tm16 Ξ (arr16 A C) β Tm16 Ξ (arr16 B C) β Tm16 Ξ C; case16
= Ξ» t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec β
case16 _ _ _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
zero16 : β{Ξ} β Tm16 Ξ nat16; zero16
= Ξ» Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc rec β zero16 _
suc16 : β{Ξ} β Tm16 Ξ nat16 β Tm16 Ξ nat16; suc16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec β
suc16 _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec)
rec16 : β{Ξ A} β Tm16 Ξ nat16 β Tm16 Ξ (arr16 nat16 (arr16 A A)) β Tm16 Ξ A β Tm16 Ξ A; rec16
= Ξ» t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16 β
rec16 _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
v016 : β{Ξ A} β Tm16 (snoc16 Ξ A) A; v016
= var16 vz16
v116 : β{Ξ A B} β Tm16 (snoc16 (snoc16 Ξ A) B) A; v116
= var16 (vs16 vz16)
v216 : β{Ξ A B C} β Tm16 (snoc16 (snoc16 (snoc16 Ξ A) B) C) A; v216
= var16 (vs16 (vs16 vz16))
v316 : β{Ξ A B C D} β Tm16 (snoc16 (snoc16 (snoc16 (snoc16 Ξ A) B) C) D) A; v316
= var16 (vs16 (vs16 (vs16 vz16)))
tbool16 : Ty16; tbool16
= sum16 top16 top16
true16 : β{Ξ} β Tm16 Ξ tbool16; true16
= left16 tt16
tfalse16 : β{Ξ} β Tm16 Ξ tbool16; tfalse16
= right16 tt16
ifthenelse16 : β{Ξ A} β Tm16 Ξ (arr16 tbool16 (arr16 A (arr16 A A))); ifthenelse16
= lam16 (lam16 (lam16 (case16 v216 (lam16 v216) (lam16 v116))))
times416 : β{Ξ A} β Tm16 Ξ (arr16 (arr16 A A) (arr16 A A)); times416
= lam16 (lam16 (app16 v116 (app16 v116 (app16 v116 (app16 v116 v016)))))
add16 : β{Ξ} β Tm16 Ξ (arr16 nat16 (arr16 nat16 nat16)); add16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (suc16 (app16 v116 v016)))))
(lam16 v016))
mul16 : β{Ξ} β Tm16 Ξ (arr16 nat16 (arr16 nat16 nat16)); mul16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (app16 (app16 add16 (app16 v116 v016)) v016))))
(lam16 zero16))
fact16 : β{Ξ} β Tm16 Ξ (arr16 nat16 nat16); fact16
= lam16 (rec16 v016 (lam16 (lam16 (app16 (app16 mul16 (suc16 v116)) v016)))
(suc16 zero16))
{-# OPTIONS --type-in-type #-}
Ty17 : Set
Ty17 =
(Ty17 : Set)
(nat top bot : Ty17)
(arr prod sum : Ty17 β Ty17 β Ty17)
β Ty17
nat17 : Ty17; nat17 = Ξ» _ nat17 _ _ _ _ _ β nat17
top17 : Ty17; top17 = Ξ» _ _ top17 _ _ _ _ β top17
bot17 : Ty17; bot17 = Ξ» _ _ _ bot17 _ _ _ β bot17
arr17 : Ty17 β Ty17 β Ty17; arr17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod sum β
arr17 (A Ty17 nat17 top17 bot17 arr17 prod sum) (B Ty17 nat17 top17 bot17 arr17 prod sum)
prod17 : Ty17 β Ty17 β Ty17; prod17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod17 sum β
prod17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum) (B Ty17 nat17 top17 bot17 arr17 prod17 sum)
sum17 : Ty17 β Ty17 β Ty17; sum17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod17 sum17 β
sum17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum17) (B Ty17 nat17 top17 bot17 arr17 prod17 sum17)
Con17 : Set; Con17
= (Con17 : Set)
(nil : Con17)
(snoc : Con17 β Ty17 β Con17)
β Con17
nil17 : Con17; nil17
= Ξ» Con17 nil17 snoc β nil17
snoc17 : Con17 β Ty17 β Con17; snoc17
= Ξ» Ξ A Con17 nil17 snoc17 β snoc17 (Ξ Con17 nil17 snoc17) A
Var17 : Con17 β Ty17 β Set; Var17
= Ξ» Ξ A β
(Var17 : Con17 β Ty17 β Set)
(vz : β Ξ A β Var17 (snoc17 Ξ A) A)
(vs : β Ξ B A β Var17 Ξ A β Var17 (snoc17 Ξ B) A)
β Var17 Ξ A
vz17 : β{Ξ A} β Var17 (snoc17 Ξ A) A; vz17
= Ξ» Var17 vz17 vs β vz17 _ _
vs17 : β{Ξ B A} β Var17 Ξ A β Var17 (snoc17 Ξ B) A; vs17
= Ξ» x Var17 vz17 vs17 β vs17 _ _ _ (x Var17 vz17 vs17)
Tm17 : Con17 β Ty17 β Set; Tm17
= Ξ» Ξ A β
(Tm17 : Con17 β Ty17 β Set)
(var : β Ξ A β Var17 Ξ A β Tm17 Ξ A)
(lam : β Ξ A B β Tm17 (snoc17 Ξ A) B β Tm17 Ξ (arr17 A B))
(app : β Ξ A B β Tm17 Ξ (arr17 A B) β Tm17 Ξ A β Tm17 Ξ B)
(tt : β Ξ β Tm17 Ξ top17)
(pair : β Ξ A B β Tm17 Ξ A β Tm17 Ξ B β Tm17 Ξ (prod17 A B))
(fst : β Ξ A B β Tm17 Ξ (prod17 A B) β Tm17 Ξ A)
(snd : β Ξ A B β Tm17 Ξ (prod17 A B) β Tm17 Ξ B)
(left : β Ξ A B β Tm17 Ξ A β Tm17 Ξ (sum17 A B))
(right : β Ξ A B β Tm17 Ξ B β Tm17 Ξ (sum17 A B))
(case : β Ξ A B C β Tm17 Ξ (sum17 A B) β Tm17 Ξ (arr17 A C) β Tm17 Ξ (arr17 B C) β Tm17 Ξ C)
(zero : β Ξ β Tm17 Ξ nat17)
(suc : β Ξ β Tm17 Ξ nat17 β Tm17 Ξ nat17)
(rec : β Ξ A β Tm17 Ξ nat17 β Tm17 Ξ (arr17 nat17 (arr17 A A)) β Tm17 Ξ A β Tm17 Ξ A)
β Tm17 Ξ A
var17 : β{Ξ A} β Var17 Ξ A β Tm17 Ξ A; var17
= Ξ» x Tm17 var17 lam app tt pair fst snd left right case zero suc rec β
var17 _ _ x
lam17 : β{Ξ A B} β Tm17 (snoc17 Ξ A) B β Tm17 Ξ (arr17 A B); lam17
= Ξ» t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec β
lam17 _ _ _ (t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec)
app17 : β{Ξ A B} β Tm17 Ξ (arr17 A B) β Tm17 Ξ A β Tm17 Ξ B; app17
= Ξ» t u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec β
app17 _ _ _ (t Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
tt17 : β{Ξ} β Tm17 Ξ top17; tt17
= Ξ» Tm17 var17 lam17 app17 tt17 pair fst snd left right case zero suc rec β tt17 _
pair17 : β{Ξ A B} β Tm17 Ξ A β Tm17 Ξ B β Tm17 Ξ (prod17 A B); pair17
= Ξ» t u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec β
pair17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
fst17 : β{Ξ A B} β Tm17 Ξ (prod17 A B) β Tm17 Ξ A; fst17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec β
fst17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec)
snd17 : β{Ξ A B} β Tm17 Ξ (prod17 A B) β Tm17 Ξ B; snd17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec β
snd17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec)
left17 : β{Ξ A B} β Tm17 Ξ A β Tm17 Ξ (sum17 A B); left17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec β
left17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec)
right17 : β{Ξ A B} β Tm17 Ξ B β Tm17 Ξ (sum17 A B); right17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec β
right17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec)
case17 : β{Ξ A B C} β Tm17 Ξ (sum17 A B) β Tm17 Ξ (arr17 A C) β Tm17 Ξ (arr17 B C) β Tm17 Ξ C; case17
= Ξ» t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec β
case17 _ _ _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
zero17 : β{Ξ} β Tm17 Ξ nat17; zero17
= Ξ» Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc rec β zero17 _
suc17 : β{Ξ} β Tm17 Ξ nat17 β Tm17 Ξ nat17; suc17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec β
suc17 _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec)
rec17 : β{Ξ A} β Tm17 Ξ nat17 β Tm17 Ξ (arr17 nat17 (arr17 A A)) β Tm17 Ξ A β Tm17 Ξ A; rec17
= Ξ» t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17 β
rec17 _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
v017 : β{Ξ A} β Tm17 (snoc17 Ξ A) A; v017
= var17 vz17
v117 : β{Ξ A B} β Tm17 (snoc17 (snoc17 Ξ A) B) A; v117
= var17 (vs17 vz17)
v217 : β{Ξ A B C} β Tm17 (snoc17 (snoc17 (snoc17 Ξ A) B) C) A; v217
= var17 (vs17 (vs17 vz17))
v317 : β{Ξ A B C D} β Tm17 (snoc17 (snoc17 (snoc17 (snoc17 Ξ A) B) C) D) A; v317
= var17 (vs17 (vs17 (vs17 vz17)))
tbool17 : Ty17; tbool17
= sum17 top17 top17
true17 : β{Ξ} β Tm17 Ξ tbool17; true17
= left17 tt17
tfalse17 : β{Ξ} β Tm17 Ξ tbool17; tfalse17
= right17 tt17
ifthenelse17 : β{Ξ A} β Tm17 Ξ (arr17 tbool17 (arr17 A (arr17 A A))); ifthenelse17
= lam17 (lam17 (lam17 (case17 v217 (lam17 v217) (lam17 v117))))
times417 : β{Ξ A} β Tm17 Ξ (arr17 (arr17 A A) (arr17 A A)); times417
= lam17 (lam17 (app17 v117 (app17 v117 (app17 v117 (app17 v117 v017)))))
add17 : β{Ξ} β Tm17 Ξ (arr17 nat17 (arr17 nat17 nat17)); add17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (suc17 (app17 v117 v017)))))
(lam17 v017))
mul17 : β{Ξ} β Tm17 Ξ (arr17 nat17 (arr17 nat17 nat17)); mul17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (app17 (app17 add17 (app17 v117 v017)) v017))))
(lam17 zero17))
fact17 : β{Ξ} β Tm17 Ξ (arr17 nat17 nat17); fact17
= lam17 (rec17 v017 (lam17 (lam17 (app17 (app17 mul17 (suc17 v117)) v017)))
(suc17 zero17))
{-# OPTIONS --type-in-type #-}
Ty18 : Set
Ty18 =
(Ty18 : Set)
(nat top bot : Ty18)
(arr prod sum : Ty18 β Ty18 β Ty18)
β Ty18
nat18 : Ty18; nat18 = Ξ» _ nat18 _ _ _ _ _ β nat18
top18 : Ty18; top18 = Ξ» _ _ top18 _ _ _ _ β top18
bot18 : Ty18; bot18 = Ξ» _ _ _ bot18 _ _ _ β bot18
arr18 : Ty18 β Ty18 β Ty18; arr18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod sum β
arr18 (A Ty18 nat18 top18 bot18 arr18 prod sum) (B Ty18 nat18 top18 bot18 arr18 prod sum)
prod18 : Ty18 β Ty18 β Ty18; prod18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod18 sum β
prod18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum) (B Ty18 nat18 top18 bot18 arr18 prod18 sum)
sum18 : Ty18 β Ty18 β Ty18; sum18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod18 sum18 β
sum18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum18) (B Ty18 nat18 top18 bot18 arr18 prod18 sum18)
Con18 : Set; Con18
= (Con18 : Set)
(nil : Con18)
(snoc : Con18 β Ty18 β Con18)
β Con18
nil18 : Con18; nil18
= Ξ» Con18 nil18 snoc β nil18
snoc18 : Con18 β Ty18 β Con18; snoc18
= Ξ» Ξ A Con18 nil18 snoc18 β snoc18 (Ξ Con18 nil18 snoc18) A
Var18 : Con18 β Ty18 β Set; Var18
= Ξ» Ξ A β
(Var18 : Con18 β Ty18 β Set)
(vz : β Ξ A β Var18 (snoc18 Ξ A) A)
(vs : β Ξ B A β Var18 Ξ A β Var18 (snoc18 Ξ B) A)
β Var18 Ξ A
vz18 : β{Ξ A} β Var18 (snoc18 Ξ A) A; vz18
= Ξ» Var18 vz18 vs β vz18 _ _
vs18 : β{Ξ B A} β Var18 Ξ A β Var18 (snoc18 Ξ B) A; vs18
= Ξ» x Var18 vz18 vs18 β vs18 _ _ _ (x Var18 vz18 vs18)
Tm18 : Con18 β Ty18 β Set; Tm18
= Ξ» Ξ A β
(Tm18 : Con18 β Ty18 β Set)
(var : β Ξ A β Var18 Ξ A β Tm18 Ξ A)
(lam : β Ξ A B β Tm18 (snoc18 Ξ A) B β Tm18 Ξ (arr18 A B))
(app : β Ξ A B β Tm18 Ξ (arr18 A B) β Tm18 Ξ A β Tm18 Ξ B)
(tt : β Ξ β Tm18 Ξ top18)
(pair : β Ξ A B β Tm18 Ξ A β Tm18 Ξ B β Tm18 Ξ (prod18 A B))
(fst : β Ξ A B β Tm18 Ξ (prod18 A B) β Tm18 Ξ A)
(snd : β Ξ A B β Tm18 Ξ (prod18 A B) β Tm18 Ξ B)
(left : β Ξ A B β Tm18 Ξ A β Tm18 Ξ (sum18 A B))
(right : β Ξ A B β Tm18 Ξ B β Tm18 Ξ (sum18 A B))
(case : β Ξ A B C β Tm18 Ξ (sum18 A B) β Tm18 Ξ (arr18 A C) β Tm18 Ξ (arr18 B C) β Tm18 Ξ C)
(zero : β Ξ β Tm18 Ξ nat18)
(suc : β Ξ β Tm18 Ξ nat18 β Tm18 Ξ nat18)
(rec : β Ξ A β Tm18 Ξ nat18 β Tm18 Ξ (arr18 nat18 (arr18 A A)) β Tm18 Ξ A β Tm18 Ξ A)
β Tm18 Ξ A
var18 : β{Ξ A} β Var18 Ξ A β Tm18 Ξ A; var18
= Ξ» x Tm18 var18 lam app tt pair fst snd left right case zero suc rec β
var18 _ _ x
lam18 : β{Ξ A B} β Tm18 (snoc18 Ξ A) B β Tm18 Ξ (arr18 A B); lam18
= Ξ» t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec β
lam18 _ _ _ (t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec)
app18 : β{Ξ A B} β Tm18 Ξ (arr18 A B) β Tm18 Ξ A β Tm18 Ξ B; app18
= Ξ» t u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec β
app18 _ _ _ (t Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
tt18 : β{Ξ} β Tm18 Ξ top18; tt18
= Ξ» Tm18 var18 lam18 app18 tt18 pair fst snd left right case zero suc rec β tt18 _
pair18 : β{Ξ A B} β Tm18 Ξ A β Tm18 Ξ B β Tm18 Ξ (prod18 A B); pair18
= Ξ» t u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec β
pair18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
fst18 : β{Ξ A B} β Tm18 Ξ (prod18 A B) β Tm18 Ξ A; fst18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec β
fst18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec)
snd18 : β{Ξ A B} β Tm18 Ξ (prod18 A B) β Tm18 Ξ B; snd18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec β
snd18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec)
left18 : β{Ξ A B} β Tm18 Ξ A β Tm18 Ξ (sum18 A B); left18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec β
left18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec)
right18 : β{Ξ A B} β Tm18 Ξ B β Tm18 Ξ (sum18 A B); right18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec β
right18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec)
case18 : β{Ξ A B C} β Tm18 Ξ (sum18 A B) β Tm18 Ξ (arr18 A C) β Tm18 Ξ (arr18 B C) β Tm18 Ξ C; case18
= Ξ» t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec β
case18 _ _ _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
zero18 : β{Ξ} β Tm18 Ξ nat18; zero18
= Ξ» Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc rec β zero18 _
suc18 : β{Ξ} β Tm18 Ξ nat18 β Tm18 Ξ nat18; suc18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec β
suc18 _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec)
rec18 : β{Ξ A} β Tm18 Ξ nat18 β Tm18 Ξ (arr18 nat18 (arr18 A A)) β Tm18 Ξ A β Tm18 Ξ A; rec18
= Ξ» t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18 β
rec18 _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
v018 : β{Ξ A} β Tm18 (snoc18 Ξ A) A; v018
= var18 vz18
v118 : β{Ξ A B} β Tm18 (snoc18 (snoc18 Ξ A) B) A; v118
= var18 (vs18 vz18)
v218 : β{Ξ A B C} β Tm18 (snoc18 (snoc18 (snoc18 Ξ A) B) C) A; v218
= var18 (vs18 (vs18 vz18))
v318 : β{Ξ A B C D} β Tm18 (snoc18 (snoc18 (snoc18 (snoc18 Ξ A) B) C) D) A; v318
= var18 (vs18 (vs18 (vs18 vz18)))
tbool18 : Ty18; tbool18
= sum18 top18 top18
true18 : β{Ξ} β Tm18 Ξ tbool18; true18
= left18 tt18
tfalse18 : β{Ξ} β Tm18 Ξ tbool18; tfalse18
= right18 tt18
ifthenelse18 : β{Ξ A} β Tm18 Ξ (arr18 tbool18 (arr18 A (arr18 A A))); ifthenelse18
= lam18 (lam18 (lam18 (case18 v218 (lam18 v218) (lam18 v118))))
times418 : β{Ξ A} β Tm18 Ξ (arr18 (arr18 A A) (arr18 A A)); times418
= lam18 (lam18 (app18 v118 (app18 v118 (app18 v118 (app18 v118 v018)))))
add18 : β{Ξ} β Tm18 Ξ (arr18 nat18 (arr18 nat18 nat18)); add18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (suc18 (app18 v118 v018)))))
(lam18 v018))
mul18 : β{Ξ} β Tm18 Ξ (arr18 nat18 (arr18 nat18 nat18)); mul18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (app18 (app18 add18 (app18 v118 v018)) v018))))
(lam18 zero18))
fact18 : β{Ξ} β Tm18 Ξ (arr18 nat18 nat18); fact18
= lam18 (rec18 v018 (lam18 (lam18 (app18 (app18 mul18 (suc18 v118)) v018)))
(suc18 zero18))
{-# OPTIONS --type-in-type #-}
Ty19 : Set
Ty19 =
(Ty19 : Set)
(nat top bot : Ty19)
(arr prod sum : Ty19 β Ty19 β Ty19)
β Ty19
nat19 : Ty19; nat19 = Ξ» _ nat19 _ _ _ _ _ β nat19
top19 : Ty19; top19 = Ξ» _ _ top19 _ _ _ _ β top19
bot19 : Ty19; bot19 = Ξ» _ _ _ bot19 _ _ _ β bot19
arr19 : Ty19 β Ty19 β Ty19; arr19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod sum β
arr19 (A Ty19 nat19 top19 bot19 arr19 prod sum) (B Ty19 nat19 top19 bot19 arr19 prod sum)
prod19 : Ty19 β Ty19 β Ty19; prod19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod19 sum β
prod19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum) (B Ty19 nat19 top19 bot19 arr19 prod19 sum)
sum19 : Ty19 β Ty19 β Ty19; sum19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod19 sum19 β
sum19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum19) (B Ty19 nat19 top19 bot19 arr19 prod19 sum19)
Con19 : Set; Con19
= (Con19 : Set)
(nil : Con19)
(snoc : Con19 β Ty19 β Con19)
β Con19
nil19 : Con19; nil19
= Ξ» Con19 nil19 snoc β nil19
snoc19 : Con19 β Ty19 β Con19; snoc19
= Ξ» Ξ A Con19 nil19 snoc19 β snoc19 (Ξ Con19 nil19 snoc19) A
Var19 : Con19 β Ty19 β Set; Var19
= Ξ» Ξ A β
(Var19 : Con19 β Ty19 β Set)
(vz : β Ξ A β Var19 (snoc19 Ξ A) A)
(vs : β Ξ B A β Var19 Ξ A β Var19 (snoc19 Ξ B) A)
β Var19 Ξ A
vz19 : β{Ξ A} β Var19 (snoc19 Ξ A) A; vz19
= Ξ» Var19 vz19 vs β vz19 _ _
vs19 : β{Ξ B A} β Var19 Ξ A β Var19 (snoc19 Ξ B) A; vs19
= Ξ» x Var19 vz19 vs19 β vs19 _ _ _ (x Var19 vz19 vs19)
Tm19 : Con19 β Ty19 β Set; Tm19
= Ξ» Ξ A β
(Tm19 : Con19 β Ty19 β Set)
(var : β Ξ A β Var19 Ξ A β Tm19 Ξ A)
(lam : β Ξ A B β Tm19 (snoc19 Ξ A) B β Tm19 Ξ (arr19 A B))
(app : β Ξ A B β Tm19 Ξ (arr19 A B) β Tm19 Ξ A β Tm19 Ξ B)
(tt : β Ξ β Tm19 Ξ top19)
(pair : β Ξ A B β Tm19 Ξ A β Tm19 Ξ B β Tm19 Ξ (prod19 A B))
(fst : β Ξ A B β Tm19 Ξ (prod19 A B) β Tm19 Ξ A)
(snd : β Ξ A B β Tm19 Ξ (prod19 A B) β Tm19 Ξ B)
(left : β Ξ A B β Tm19 Ξ A β Tm19 Ξ (sum19 A B))
(right : β Ξ A B β Tm19 Ξ B β Tm19 Ξ (sum19 A B))
(case : β Ξ A B C β Tm19 Ξ (sum19 A B) β Tm19 Ξ (arr19 A C) β Tm19 Ξ (arr19 B C) β Tm19 Ξ C)
(zero : β Ξ β Tm19 Ξ nat19)
(suc : β Ξ β Tm19 Ξ nat19 β Tm19 Ξ nat19)
(rec : β Ξ A β Tm19 Ξ nat19 β Tm19 Ξ (arr19 nat19 (arr19 A A)) β Tm19 Ξ A β Tm19 Ξ A)
β Tm19 Ξ A
var19 : β{Ξ A} β Var19 Ξ A β Tm19 Ξ A; var19
= Ξ» x Tm19 var19 lam app tt pair fst snd left right case zero suc rec β
var19 _ _ x
lam19 : β{Ξ A B} β Tm19 (snoc19 Ξ A) B β Tm19 Ξ (arr19 A B); lam19
= Ξ» t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec β
lam19 _ _ _ (t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec)
app19 : β{Ξ A B} β Tm19 Ξ (arr19 A B) β Tm19 Ξ A β Tm19 Ξ B; app19
= Ξ» t u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec β
app19 _ _ _ (t Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
tt19 : β{Ξ} β Tm19 Ξ top19; tt19
= Ξ» Tm19 var19 lam19 app19 tt19 pair fst snd left right case zero suc rec β tt19 _
pair19 : β{Ξ A B} β Tm19 Ξ A β Tm19 Ξ B β Tm19 Ξ (prod19 A B); pair19
= Ξ» t u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec β
pair19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
fst19 : β{Ξ A B} β Tm19 Ξ (prod19 A B) β Tm19 Ξ A; fst19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec β
fst19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec)
snd19 : β{Ξ A B} β Tm19 Ξ (prod19 A B) β Tm19 Ξ B; snd19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec β
snd19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec)
left19 : β{Ξ A B} β Tm19 Ξ A β Tm19 Ξ (sum19 A B); left19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec β
left19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec)
right19 : β{Ξ A B} β Tm19 Ξ B β Tm19 Ξ (sum19 A B); right19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec β
right19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec)
case19 : β{Ξ A B C} β Tm19 Ξ (sum19 A B) β Tm19 Ξ (arr19 A C) β Tm19 Ξ (arr19 B C) β Tm19 Ξ C; case19
= Ξ» t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec β
case19 _ _ _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
zero19 : β{Ξ} β Tm19 Ξ nat19; zero19
= Ξ» Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc rec β zero19 _
suc19 : β{Ξ} β Tm19 Ξ nat19 β Tm19 Ξ nat19; suc19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec β
suc19 _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec)
rec19 : β{Ξ A} β Tm19 Ξ nat19 β Tm19 Ξ (arr19 nat19 (arr19 A A)) β Tm19 Ξ A β Tm19 Ξ A; rec19
= Ξ» t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19 β
rec19 _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
v019 : β{Ξ A} β Tm19 (snoc19 Ξ A) A; v019
= var19 vz19
v119 : β{Ξ A B} β Tm19 (snoc19 (snoc19 Ξ A) B) A; v119
= var19 (vs19 vz19)
v219 : β{Ξ A B C} β Tm19 (snoc19 (snoc19 (snoc19 Ξ A) B) C) A; v219
= var19 (vs19 (vs19 vz19))
v319 : β{Ξ A B C D} β Tm19 (snoc19 (snoc19 (snoc19 (snoc19 Ξ A) B) C) D) A; v319
= var19 (vs19 (vs19 (vs19 vz19)))
tbool19 : Ty19; tbool19
= sum19 top19 top19
true19 : β{Ξ} β Tm19 Ξ tbool19; true19
= left19 tt19
tfalse19 : β{Ξ} β Tm19 Ξ tbool19; tfalse19
= right19 tt19
ifthenelse19 : β{Ξ A} β Tm19 Ξ (arr19 tbool19 (arr19 A (arr19 A A))); ifthenelse19
= lam19 (lam19 (lam19 (case19 v219 (lam19 v219) (lam19 v119))))
times419 : β{Ξ A} β Tm19 Ξ (arr19 (arr19 A A) (arr19 A A)); times419
= lam19 (lam19 (app19 v119 (app19 v119 (app19 v119 (app19 v119 v019)))))
add19 : β{Ξ} β Tm19 Ξ (arr19 nat19 (arr19 nat19 nat19)); add19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (suc19 (app19 v119 v019)))))
(lam19 v019))
mul19 : β{Ξ} β Tm19 Ξ (arr19 nat19 (arr19 nat19 nat19)); mul19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (app19 (app19 add19 (app19 v119 v019)) v019))))
(lam19 zero19))
fact19 : β{Ξ} β Tm19 Ξ (arr19 nat19 nat19); fact19
= lam19 (rec19 v019 (lam19 (lam19 (app19 (app19 mul19 (suc19 v119)) v019)))
(suc19 zero19))
{-# OPTIONS --type-in-type #-}
Ty20 : Set
Ty20 =
(Ty20 : Set)
(nat top bot : Ty20)
(arr prod sum : Ty20 β Ty20 β Ty20)
β Ty20
nat20 : Ty20; nat20 = Ξ» _ nat20 _ _ _ _ _ β nat20
top20 : Ty20; top20 = Ξ» _ _ top20 _ _ _ _ β top20
bot20 : Ty20; bot20 = Ξ» _ _ _ bot20 _ _ _ β bot20
arr20 : Ty20 β Ty20 β Ty20; arr20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod sum β
arr20 (A Ty20 nat20 top20 bot20 arr20 prod sum) (B Ty20 nat20 top20 bot20 arr20 prod sum)
prod20 : Ty20 β Ty20 β Ty20; prod20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod20 sum β
prod20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum) (B Ty20 nat20 top20 bot20 arr20 prod20 sum)
sum20 : Ty20 β Ty20 β Ty20; sum20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod20 sum20 β
sum20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum20) (B Ty20 nat20 top20 bot20 arr20 prod20 sum20)
Con20 : Set; Con20
= (Con20 : Set)
(nil : Con20)
(snoc : Con20 β Ty20 β Con20)
β Con20
nil20 : Con20; nil20
= Ξ» Con20 nil20 snoc β nil20
snoc20 : Con20 β Ty20 β Con20; snoc20
= Ξ» Ξ A Con20 nil20 snoc20 β snoc20 (Ξ Con20 nil20 snoc20) A
Var20 : Con20 β Ty20 β Set; Var20
= Ξ» Ξ A β
(Var20 : Con20 β Ty20 β Set)
(vz : β Ξ A β Var20 (snoc20 Ξ A) A)
(vs : β Ξ B A β Var20 Ξ A β Var20 (snoc20 Ξ B) A)
β Var20 Ξ A
vz20 : β{Ξ A} β Var20 (snoc20 Ξ A) A; vz20
= Ξ» Var20 vz20 vs β vz20 _ _
vs20 : β{Ξ B A} β Var20 Ξ A β Var20 (snoc20 Ξ B) A; vs20
= Ξ» x Var20 vz20 vs20 β vs20 _ _ _ (x Var20 vz20 vs20)
Tm20 : Con20 β Ty20 β Set; Tm20
= Ξ» Ξ A β
(Tm20 : Con20 β Ty20 β Set)
(var : β Ξ A β Var20 Ξ A β Tm20 Ξ A)
(lam : β Ξ A B β Tm20 (snoc20 Ξ A) B β Tm20 Ξ (arr20 A B))
(app : β Ξ A B β Tm20 Ξ (arr20 A B) β Tm20 Ξ A β Tm20 Ξ B)
(tt : β Ξ β Tm20 Ξ top20)
(pair : β Ξ A B β Tm20 Ξ A β Tm20 Ξ B β Tm20 Ξ (prod20 A B))
(fst : β Ξ A B β Tm20 Ξ (prod20 A B) β Tm20 Ξ A)
(snd : β Ξ A B β Tm20 Ξ (prod20 A B) β Tm20 Ξ B)
(left : β Ξ A B β Tm20 Ξ A β Tm20 Ξ (sum20 A B))
(right : β Ξ A B β Tm20 Ξ B β Tm20 Ξ (sum20 A B))
(case : β Ξ A B C β Tm20 Ξ (sum20 A B) β Tm20 Ξ (arr20 A C) β Tm20 Ξ (arr20 B C) β Tm20 Ξ C)
(zero : β Ξ β Tm20 Ξ nat20)
(suc : β Ξ β Tm20 Ξ nat20 β Tm20 Ξ nat20)
(rec : β Ξ A β Tm20 Ξ nat20 β Tm20 Ξ (arr20 nat20 (arr20 A A)) β Tm20 Ξ A β Tm20 Ξ A)
β Tm20 Ξ A
var20 : β{Ξ A} β Var20 Ξ A β Tm20 Ξ A; var20
= Ξ» x Tm20 var20 lam app tt pair fst snd left right case zero suc rec β
var20 _ _ x
lam20 : β{Ξ A B} β Tm20 (snoc20 Ξ A) B β Tm20 Ξ (arr20 A B); lam20
= Ξ» t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec β
lam20 _ _ _ (t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec)
app20 : β{Ξ A B} β Tm20 Ξ (arr20 A B) β Tm20 Ξ A β Tm20 Ξ B; app20
= Ξ» t u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec β
app20 _ _ _ (t Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
tt20 : β{Ξ} β Tm20 Ξ top20; tt20
= Ξ» Tm20 var20 lam20 app20 tt20 pair fst snd left right case zero suc rec β tt20 _
pair20 : β{Ξ A B} β Tm20 Ξ A β Tm20 Ξ B β Tm20 Ξ (prod20 A B); pair20
= Ξ» t u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec β
pair20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
fst20 : β{Ξ A B} β Tm20 Ξ (prod20 A B) β Tm20 Ξ A; fst20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec β
fst20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec)
snd20 : β{Ξ A B} β Tm20 Ξ (prod20 A B) β Tm20 Ξ B; snd20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec β
snd20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec)
left20 : β{Ξ A B} β Tm20 Ξ A β Tm20 Ξ (sum20 A B); left20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec β
left20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec)
right20 : β{Ξ A B} β Tm20 Ξ B β Tm20 Ξ (sum20 A B); right20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec β
right20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec)
case20 : β{Ξ A B C} β Tm20 Ξ (sum20 A B) β Tm20 Ξ (arr20 A C) β Tm20 Ξ (arr20 B C) β Tm20 Ξ C; case20
= Ξ» t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec β
case20 _ _ _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
zero20 : β{Ξ} β Tm20 Ξ nat20; zero20
= Ξ» Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc rec β zero20 _
suc20 : β{Ξ} β Tm20 Ξ nat20 β Tm20 Ξ nat20; suc20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec β
suc20 _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec)
rec20 : β{Ξ A} β Tm20 Ξ nat20 β Tm20 Ξ (arr20 nat20 (arr20 A A)) β Tm20 Ξ A β Tm20 Ξ A; rec20
= Ξ» t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20 β
rec20 _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
v020 : β{Ξ A} β Tm20 (snoc20 Ξ A) A; v020
= var20 vz20
v120 : β{Ξ A B} β Tm20 (snoc20 (snoc20 Ξ A) B) A; v120
= var20 (vs20 vz20)
v220 : β{Ξ A B C} β Tm20 (snoc20 (snoc20 (snoc20 Ξ A) B) C) A; v220
= var20 (vs20 (vs20 vz20))
v320 : β{Ξ A B C D} β Tm20 (snoc20 (snoc20 (snoc20 (snoc20 Ξ A) B) C) D) A; v320
= var20 (vs20 (vs20 (vs20 vz20)))
tbool20 : Ty20; tbool20
= sum20 top20 top20
true20 : β{Ξ} β Tm20 Ξ tbool20; true20
= left20 tt20
tfalse20 : β{Ξ} β Tm20 Ξ tbool20; tfalse20
= right20 tt20
ifthenelse20 : β{Ξ A} β Tm20 Ξ (arr20 tbool20 (arr20 A (arr20 A A))); ifthenelse20
= lam20 (lam20 (lam20 (case20 v220 (lam20 v220) (lam20 v120))))
times420 : β{Ξ A} β Tm20 Ξ (arr20 (arr20 A A) (arr20 A A)); times420
= lam20 (lam20 (app20 v120 (app20 v120 (app20 v120 (app20 v120 v020)))))
add20 : β{Ξ} β Tm20 Ξ (arr20 nat20 (arr20 nat20 nat20)); add20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (suc20 (app20 v120 v020)))))
(lam20 v020))
mul20 : β{Ξ} β Tm20 Ξ (arr20 nat20 (arr20 nat20 nat20)); mul20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (app20 (app20 add20 (app20 v120 v020)) v020))))
(lam20 zero20))
fact20 : β{Ξ} β Tm20 Ξ (arr20 nat20 nat20); fact20
= lam20 (rec20 v020 (lam20 (lam20 (app20 (app20 mul20 (suc20 v120)) v020)))
(suc20 zero20))
{-# OPTIONS --type-in-type #-}
Ty21 : Set
Ty21 =
(Ty21 : Set)
(nat top bot : Ty21)
(arr prod sum : Ty21 β Ty21 β Ty21)
β Ty21
nat21 : Ty21; nat21 = Ξ» _ nat21 _ _ _ _ _ β nat21
top21 : Ty21; top21 = Ξ» _ _ top21 _ _ _ _ β top21
bot21 : Ty21; bot21 = Ξ» _ _ _ bot21 _ _ _ β bot21
arr21 : Ty21 β Ty21 β Ty21; arr21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod sum β
arr21 (A Ty21 nat21 top21 bot21 arr21 prod sum) (B Ty21 nat21 top21 bot21 arr21 prod sum)
prod21 : Ty21 β Ty21 β Ty21; prod21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod21 sum β
prod21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum) (B Ty21 nat21 top21 bot21 arr21 prod21 sum)
sum21 : Ty21 β Ty21 β Ty21; sum21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod21 sum21 β
sum21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum21) (B Ty21 nat21 top21 bot21 arr21 prod21 sum21)
Con21 : Set; Con21
= (Con21 : Set)
(nil : Con21)
(snoc : Con21 β Ty21 β Con21)
β Con21
nil21 : Con21; nil21
= Ξ» Con21 nil21 snoc β nil21
snoc21 : Con21 β Ty21 β Con21; snoc21
= Ξ» Ξ A Con21 nil21 snoc21 β snoc21 (Ξ Con21 nil21 snoc21) A
Var21 : Con21 β Ty21 β Set; Var21
= Ξ» Ξ A β
(Var21 : Con21 β Ty21 β Set)
(vz : β Ξ A β Var21 (snoc21 Ξ A) A)
(vs : β Ξ B A β Var21 Ξ A β Var21 (snoc21 Ξ B) A)
β Var21 Ξ A
vz21 : β{Ξ A} β Var21 (snoc21 Ξ A) A; vz21
= Ξ» Var21 vz21 vs β vz21 _ _
vs21 : β{Ξ B A} β Var21 Ξ A β Var21 (snoc21 Ξ B) A; vs21
= Ξ» x Var21 vz21 vs21 β vs21 _ _ _ (x Var21 vz21 vs21)
Tm21 : Con21 β Ty21 β Set; Tm21
= Ξ» Ξ A β
(Tm21 : Con21 β Ty21 β Set)
(var : β Ξ A β Var21 Ξ A β Tm21 Ξ A)
(lam : β Ξ A B β Tm21 (snoc21 Ξ A) B β Tm21 Ξ (arr21 A B))
(app : β Ξ A B β Tm21 Ξ (arr21 A B) β Tm21 Ξ A β Tm21 Ξ B)
(tt : β Ξ β Tm21 Ξ top21)
(pair : β Ξ A B β Tm21 Ξ A β Tm21 Ξ B β Tm21 Ξ (prod21 A B))
(fst : β Ξ A B β Tm21 Ξ (prod21 A B) β Tm21 Ξ A)
(snd : β Ξ A B β Tm21 Ξ (prod21 A B) β Tm21 Ξ B)
(left : β Ξ A B β Tm21 Ξ A β Tm21 Ξ (sum21 A B))
(right : β Ξ A B β Tm21 Ξ B β Tm21 Ξ (sum21 A B))
(case : β Ξ A B C β Tm21 Ξ (sum21 A B) β Tm21 Ξ (arr21 A C) β Tm21 Ξ (arr21 B C) β Tm21 Ξ C)
(zero : β Ξ β Tm21 Ξ nat21)
(suc : β Ξ β Tm21 Ξ nat21 β Tm21 Ξ nat21)
(rec : β Ξ A β Tm21 Ξ nat21 β Tm21 Ξ (arr21 nat21 (arr21 A A)) β Tm21 Ξ A β Tm21 Ξ A)
β Tm21 Ξ A
var21 : β{Ξ A} β Var21 Ξ A β Tm21 Ξ A; var21
= Ξ» x Tm21 var21 lam app tt pair fst snd left right case zero suc rec β
var21 _ _ x
lam21 : β{Ξ A B} β Tm21 (snoc21 Ξ A) B β Tm21 Ξ (arr21 A B); lam21
= Ξ» t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec β
lam21 _ _ _ (t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec)
app21 : β{Ξ A B} β Tm21 Ξ (arr21 A B) β Tm21 Ξ A β Tm21 Ξ B; app21
= Ξ» t u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec β
app21 _ _ _ (t Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
tt21 : β{Ξ} β Tm21 Ξ top21; tt21
= Ξ» Tm21 var21 lam21 app21 tt21 pair fst snd left right case zero suc rec β tt21 _
pair21 : β{Ξ A B} β Tm21 Ξ A β Tm21 Ξ B β Tm21 Ξ (prod21 A B); pair21
= Ξ» t u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec β
pair21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
fst21 : β{Ξ A B} β Tm21 Ξ (prod21 A B) β Tm21 Ξ A; fst21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec β
fst21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec)
snd21 : β{Ξ A B} β Tm21 Ξ (prod21 A B) β Tm21 Ξ B; snd21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec β
snd21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec)
left21 : β{Ξ A B} β Tm21 Ξ A β Tm21 Ξ (sum21 A B); left21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec β
left21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec)
right21 : β{Ξ A B} β Tm21 Ξ B β Tm21 Ξ (sum21 A B); right21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec β
right21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec)
case21 : β{Ξ A B C} β Tm21 Ξ (sum21 A B) β Tm21 Ξ (arr21 A C) β Tm21 Ξ (arr21 B C) β Tm21 Ξ C; case21
= Ξ» t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec β
case21 _ _ _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
zero21 : β{Ξ} β Tm21 Ξ nat21; zero21
= Ξ» Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc rec β zero21 _
suc21 : β{Ξ} β Tm21 Ξ nat21 β Tm21 Ξ nat21; suc21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec β
suc21 _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec)
rec21 : β{Ξ A} β Tm21 Ξ nat21 β Tm21 Ξ (arr21 nat21 (arr21 A A)) β Tm21 Ξ A β Tm21 Ξ A; rec21
= Ξ» t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21 β
rec21 _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
v021 : β{Ξ A} β Tm21 (snoc21 Ξ A) A; v021
= var21 vz21
v121 : β{Ξ A B} β Tm21 (snoc21 (snoc21 Ξ A) B) A; v121
= var21 (vs21 vz21)
v221 : β{Ξ A B C} β Tm21 (snoc21 (snoc21 (snoc21 Ξ A) B) C) A; v221
= var21 (vs21 (vs21 vz21))
v321 : β{Ξ A B C D} β Tm21 (snoc21 (snoc21 (snoc21 (snoc21 Ξ A) B) C) D) A; v321
= var21 (vs21 (vs21 (vs21 vz21)))
tbool21 : Ty21; tbool21
= sum21 top21 top21
true21 : β{Ξ} β Tm21 Ξ tbool21; true21
= left21 tt21
tfalse21 : β{Ξ} β Tm21 Ξ tbool21; tfalse21
= right21 tt21
ifthenelse21 : β{Ξ A} β Tm21 Ξ (arr21 tbool21 (arr21 A (arr21 A A))); ifthenelse21
= lam21 (lam21 (lam21 (case21 v221 (lam21 v221) (lam21 v121))))
times421 : β{Ξ A} β Tm21 Ξ (arr21 (arr21 A A) (arr21 A A)); times421
= lam21 (lam21 (app21 v121 (app21 v121 (app21 v121 (app21 v121 v021)))))
add21 : β{Ξ} β Tm21 Ξ (arr21 nat21 (arr21 nat21 nat21)); add21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (suc21 (app21 v121 v021)))))
(lam21 v021))
mul21 : β{Ξ} β Tm21 Ξ (arr21 nat21 (arr21 nat21 nat21)); mul21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (app21 (app21 add21 (app21 v121 v021)) v021))))
(lam21 zero21))
fact21 : β{Ξ} β Tm21 Ξ (arr21 nat21 nat21); fact21
= lam21 (rec21 v021 (lam21 (lam21 (app21 (app21 mul21 (suc21 v121)) v021)))
(suc21 zero21))
{-# OPTIONS --type-in-type #-}
Ty22 : Set
Ty22 =
(Ty22 : Set)
(nat top bot : Ty22)
(arr prod sum : Ty22 β Ty22 β Ty22)
β Ty22
nat22 : Ty22; nat22 = Ξ» _ nat22 _ _ _ _ _ β nat22
top22 : Ty22; top22 = Ξ» _ _ top22 _ _ _ _ β top22
bot22 : Ty22; bot22 = Ξ» _ _ _ bot22 _ _ _ β bot22
arr22 : Ty22 β Ty22 β Ty22; arr22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod sum β
arr22 (A Ty22 nat22 top22 bot22 arr22 prod sum) (B Ty22 nat22 top22 bot22 arr22 prod sum)
prod22 : Ty22 β Ty22 β Ty22; prod22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod22 sum β
prod22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum) (B Ty22 nat22 top22 bot22 arr22 prod22 sum)
sum22 : Ty22 β Ty22 β Ty22; sum22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod22 sum22 β
sum22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum22) (B Ty22 nat22 top22 bot22 arr22 prod22 sum22)
Con22 : Set; Con22
= (Con22 : Set)
(nil : Con22)
(snoc : Con22 β Ty22 β Con22)
β Con22
nil22 : Con22; nil22
= Ξ» Con22 nil22 snoc β nil22
snoc22 : Con22 β Ty22 β Con22; snoc22
= Ξ» Ξ A Con22 nil22 snoc22 β snoc22 (Ξ Con22 nil22 snoc22) A
Var22 : Con22 β Ty22 β Set; Var22
= Ξ» Ξ A β
(Var22 : Con22 β Ty22 β Set)
(vz : β Ξ A β Var22 (snoc22 Ξ A) A)
(vs : β Ξ B A β Var22 Ξ A β Var22 (snoc22 Ξ B) A)
β Var22 Ξ A
vz22 : β{Ξ A} β Var22 (snoc22 Ξ A) A; vz22
= Ξ» Var22 vz22 vs β vz22 _ _
vs22 : β{Ξ B A} β Var22 Ξ A β Var22 (snoc22 Ξ B) A; vs22
= Ξ» x Var22 vz22 vs22 β vs22 _ _ _ (x Var22 vz22 vs22)
Tm22 : Con22 β Ty22 β Set; Tm22
= Ξ» Ξ A β
(Tm22 : Con22 β Ty22 β Set)
(var : β Ξ A β Var22 Ξ A β Tm22 Ξ A)
(lam : β Ξ A B β Tm22 (snoc22 Ξ A) B β Tm22 Ξ (arr22 A B))
(app : β Ξ A B β Tm22 Ξ (arr22 A B) β Tm22 Ξ A β Tm22 Ξ B)
(tt : β Ξ β Tm22 Ξ top22)
(pair : β Ξ A B β Tm22 Ξ A β Tm22 Ξ B β Tm22 Ξ (prod22 A B))
(fst : β Ξ A B β Tm22 Ξ (prod22 A B) β Tm22 Ξ A)
(snd : β Ξ A B β Tm22 Ξ (prod22 A B) β Tm22 Ξ B)
(left : β Ξ A B β Tm22 Ξ A β Tm22 Ξ (sum22 A B))
(right : β Ξ A B β Tm22 Ξ B β Tm22 Ξ (sum22 A B))
(case : β Ξ A B C β Tm22 Ξ (sum22 A B) β Tm22 Ξ (arr22 A C) β Tm22 Ξ (arr22 B C) β Tm22 Ξ C)
(zero : β Ξ β Tm22 Ξ nat22)
(suc : β Ξ β Tm22 Ξ nat22 β Tm22 Ξ nat22)
(rec : β Ξ A β Tm22 Ξ nat22 β Tm22 Ξ (arr22 nat22 (arr22 A A)) β Tm22 Ξ A β Tm22 Ξ A)
β Tm22 Ξ A
var22 : β{Ξ A} β Var22 Ξ A β Tm22 Ξ A; var22
= Ξ» x Tm22 var22 lam app tt pair fst snd left right case zero suc rec β
var22 _ _ x
lam22 : β{Ξ A B} β Tm22 (snoc22 Ξ A) B β Tm22 Ξ (arr22 A B); lam22
= Ξ» t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec β
lam22 _ _ _ (t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec)
app22 : β{Ξ A B} β Tm22 Ξ (arr22 A B) β Tm22 Ξ A β Tm22 Ξ B; app22
= Ξ» t u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec β
app22 _ _ _ (t Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
tt22 : β{Ξ} β Tm22 Ξ top22; tt22
= Ξ» Tm22 var22 lam22 app22 tt22 pair fst snd left right case zero suc rec β tt22 _
pair22 : β{Ξ A B} β Tm22 Ξ A β Tm22 Ξ B β Tm22 Ξ (prod22 A B); pair22
= Ξ» t u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec β
pair22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
fst22 : β{Ξ A B} β Tm22 Ξ (prod22 A B) β Tm22 Ξ A; fst22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec β
fst22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec)
snd22 : β{Ξ A B} β Tm22 Ξ (prod22 A B) β Tm22 Ξ B; snd22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec β
snd22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec)
left22 : β{Ξ A B} β Tm22 Ξ A β Tm22 Ξ (sum22 A B); left22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec β
left22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec)
right22 : β{Ξ A B} β Tm22 Ξ B β Tm22 Ξ (sum22 A B); right22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec β
right22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec)
case22 : β{Ξ A B C} β Tm22 Ξ (sum22 A B) β Tm22 Ξ (arr22 A C) β Tm22 Ξ (arr22 B C) β Tm22 Ξ C; case22
= Ξ» t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec β
case22 _ _ _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
zero22 : β{Ξ} β Tm22 Ξ nat22; zero22
= Ξ» Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc rec β zero22 _
suc22 : β{Ξ} β Tm22 Ξ nat22 β Tm22 Ξ nat22; suc22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec β
suc22 _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec)
rec22 : β{Ξ A} β Tm22 Ξ nat22 β Tm22 Ξ (arr22 nat22 (arr22 A A)) β Tm22 Ξ A β Tm22 Ξ A; rec22
= Ξ» t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22 β
rec22 _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
v022 : β{Ξ A} β Tm22 (snoc22 Ξ A) A; v022
= var22 vz22
v122 : β{Ξ A B} β Tm22 (snoc22 (snoc22 Ξ A) B) A; v122
= var22 (vs22 vz22)
v222 : β{Ξ A B C} β Tm22 (snoc22 (snoc22 (snoc22 Ξ A) B) C) A; v222
= var22 (vs22 (vs22 vz22))
v322 : β{Ξ A B C D} β Tm22 (snoc22 (snoc22 (snoc22 (snoc22 Ξ A) B) C) D) A; v322
= var22 (vs22 (vs22 (vs22 vz22)))
tbool22 : Ty22; tbool22
= sum22 top22 top22
true22 : β{Ξ} β Tm22 Ξ tbool22; true22
= left22 tt22
tfalse22 : β{Ξ} β Tm22 Ξ tbool22; tfalse22
= right22 tt22
ifthenelse22 : β{Ξ A} β Tm22 Ξ (arr22 tbool22 (arr22 A (arr22 A A))); ifthenelse22
= lam22 (lam22 (lam22 (case22 v222 (lam22 v222) (lam22 v122))))
times422 : β{Ξ A} β Tm22 Ξ (arr22 (arr22 A A) (arr22 A A)); times422
= lam22 (lam22 (app22 v122 (app22 v122 (app22 v122 (app22 v122 v022)))))
add22 : β{Ξ} β Tm22 Ξ (arr22 nat22 (arr22 nat22 nat22)); add22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (suc22 (app22 v122 v022)))))
(lam22 v022))
mul22 : β{Ξ} β Tm22 Ξ (arr22 nat22 (arr22 nat22 nat22)); mul22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (app22 (app22 add22 (app22 v122 v022)) v022))))
(lam22 zero22))
fact22 : β{Ξ} β Tm22 Ξ (arr22 nat22 nat22); fact22
= lam22 (rec22 v022 (lam22 (lam22 (app22 (app22 mul22 (suc22 v122)) v022)))
(suc22 zero22))
{-# OPTIONS --type-in-type #-}
Ty23 : Set
Ty23 =
(Ty23 : Set)
(nat top bot : Ty23)
(arr prod sum : Ty23 β Ty23 β Ty23)
β Ty23
nat23 : Ty23; nat23 = Ξ» _ nat23 _ _ _ _ _ β nat23
top23 : Ty23; top23 = Ξ» _ _ top23 _ _ _ _ β top23
bot23 : Ty23; bot23 = Ξ» _ _ _ bot23 _ _ _ β bot23
arr23 : Ty23 β Ty23 β Ty23; arr23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod sum β
arr23 (A Ty23 nat23 top23 bot23 arr23 prod sum) (B Ty23 nat23 top23 bot23 arr23 prod sum)
prod23 : Ty23 β Ty23 β Ty23; prod23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod23 sum β
prod23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum) (B Ty23 nat23 top23 bot23 arr23 prod23 sum)
sum23 : Ty23 β Ty23 β Ty23; sum23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod23 sum23 β
sum23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum23) (B Ty23 nat23 top23 bot23 arr23 prod23 sum23)
Con23 : Set; Con23
= (Con23 : Set)
(nil : Con23)
(snoc : Con23 β Ty23 β Con23)
β Con23
nil23 : Con23; nil23
= Ξ» Con23 nil23 snoc β nil23
snoc23 : Con23 β Ty23 β Con23; snoc23
= Ξ» Ξ A Con23 nil23 snoc23 β snoc23 (Ξ Con23 nil23 snoc23) A
Var23 : Con23 β Ty23 β Set; Var23
= Ξ» Ξ A β
(Var23 : Con23 β Ty23 β Set)
(vz : β Ξ A β Var23 (snoc23 Ξ A) A)
(vs : β Ξ B A β Var23 Ξ A β Var23 (snoc23 Ξ B) A)
β Var23 Ξ A
vz23 : β{Ξ A} β Var23 (snoc23 Ξ A) A; vz23
= Ξ» Var23 vz23 vs β vz23 _ _
vs23 : β{Ξ B A} β Var23 Ξ A β Var23 (snoc23 Ξ B) A; vs23
= Ξ» x Var23 vz23 vs23 β vs23 _ _ _ (x Var23 vz23 vs23)
Tm23 : Con23 β Ty23 β Set; Tm23
= Ξ» Ξ A β
(Tm23 : Con23 β Ty23 β Set)
(var : β Ξ A β Var23 Ξ A β Tm23 Ξ A)
(lam : β Ξ A B β Tm23 (snoc23 Ξ A) B β Tm23 Ξ (arr23 A B))
(app : β Ξ A B β Tm23 Ξ (arr23 A B) β Tm23 Ξ A β Tm23 Ξ B)
(tt : β Ξ β Tm23 Ξ top23)
(pair : β Ξ A B β Tm23 Ξ A β Tm23 Ξ B β Tm23 Ξ (prod23 A B))
(fst : β Ξ A B β Tm23 Ξ (prod23 A B) β Tm23 Ξ A)
(snd : β Ξ A B β Tm23 Ξ (prod23 A B) β Tm23 Ξ B)
(left : β Ξ A B β Tm23 Ξ A β Tm23 Ξ (sum23 A B))
(right : β Ξ A B β Tm23 Ξ B β Tm23 Ξ (sum23 A B))
(case : β Ξ A B C β Tm23 Ξ (sum23 A B) β Tm23 Ξ (arr23 A C) β Tm23 Ξ (arr23 B C) β Tm23 Ξ C)
(zero : β Ξ β Tm23 Ξ nat23)
(suc : β Ξ β Tm23 Ξ nat23 β Tm23 Ξ nat23)
(rec : β Ξ A β Tm23 Ξ nat23 β Tm23 Ξ (arr23 nat23 (arr23 A A)) β Tm23 Ξ A β Tm23 Ξ A)
β Tm23 Ξ A
var23 : β{Ξ A} β Var23 Ξ A β Tm23 Ξ A; var23
= Ξ» x Tm23 var23 lam app tt pair fst snd left right case zero suc rec β
var23 _ _ x
lam23 : β{Ξ A B} β Tm23 (snoc23 Ξ A) B β Tm23 Ξ (arr23 A B); lam23
= Ξ» t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec β
lam23 _ _ _ (t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec)
app23 : β{Ξ A B} β Tm23 Ξ (arr23 A B) β Tm23 Ξ A β Tm23 Ξ B; app23
= Ξ» t u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec β
app23 _ _ _ (t Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
tt23 : β{Ξ} β Tm23 Ξ top23; tt23
= Ξ» Tm23 var23 lam23 app23 tt23 pair fst snd left right case zero suc rec β tt23 _
pair23 : β{Ξ A B} β Tm23 Ξ A β Tm23 Ξ B β Tm23 Ξ (prod23 A B); pair23
= Ξ» t u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec β
pair23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
fst23 : β{Ξ A B} β Tm23 Ξ (prod23 A B) β Tm23 Ξ A; fst23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec β
fst23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec)
snd23 : β{Ξ A B} β Tm23 Ξ (prod23 A B) β Tm23 Ξ B; snd23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec β
snd23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec)
left23 : β{Ξ A B} β Tm23 Ξ A β Tm23 Ξ (sum23 A B); left23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec β
left23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec)
right23 : β{Ξ A B} β Tm23 Ξ B β Tm23 Ξ (sum23 A B); right23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec β
right23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec)
case23 : β{Ξ A B C} β Tm23 Ξ (sum23 A B) β Tm23 Ξ (arr23 A C) β Tm23 Ξ (arr23 B C) β Tm23 Ξ C; case23
= Ξ» t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec β
case23 _ _ _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
zero23 : β{Ξ} β Tm23 Ξ nat23; zero23
= Ξ» Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc rec β zero23 _
suc23 : β{Ξ} β Tm23 Ξ nat23 β Tm23 Ξ nat23; suc23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec β
suc23 _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec)
rec23 : β{Ξ A} β Tm23 Ξ nat23 β Tm23 Ξ (arr23 nat23 (arr23 A A)) β Tm23 Ξ A β Tm23 Ξ A; rec23
= Ξ» t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23 β
rec23 _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
v023 : β{Ξ A} β Tm23 (snoc23 Ξ A) A; v023
= var23 vz23
v123 : β{Ξ A B} β Tm23 (snoc23 (snoc23 Ξ A) B) A; v123
= var23 (vs23 vz23)
v223 : β{Ξ A B C} β Tm23 (snoc23 (snoc23 (snoc23 Ξ A) B) C) A; v223
= var23 (vs23 (vs23 vz23))
v323 : β{Ξ A B C D} β Tm23 (snoc23 (snoc23 (snoc23 (snoc23 Ξ A) B) C) D) A; v323
= var23 (vs23 (vs23 (vs23 vz23)))
tbool23 : Ty23; tbool23
= sum23 top23 top23
true23 : β{Ξ} β Tm23 Ξ tbool23; true23
= left23 tt23
tfalse23 : β{Ξ} β Tm23 Ξ tbool23; tfalse23
= right23 tt23
ifthenelse23 : β{Ξ A} β Tm23 Ξ (arr23 tbool23 (arr23 A (arr23 A A))); ifthenelse23
= lam23 (lam23 (lam23 (case23 v223 (lam23 v223) (lam23 v123))))
times423 : β{Ξ A} β Tm23 Ξ (arr23 (arr23 A A) (arr23 A A)); times423
= lam23 (lam23 (app23 v123 (app23 v123 (app23 v123 (app23 v123 v023)))))
add23 : β{Ξ} β Tm23 Ξ (arr23 nat23 (arr23 nat23 nat23)); add23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (suc23 (app23 v123 v023)))))
(lam23 v023))
mul23 : β{Ξ} β Tm23 Ξ (arr23 nat23 (arr23 nat23 nat23)); mul23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (app23 (app23 add23 (app23 v123 v023)) v023))))
(lam23 zero23))
fact23 : β{Ξ} β Tm23 Ξ (arr23 nat23 nat23); fact23
= lam23 (rec23 v023 (lam23 (lam23 (app23 (app23 mul23 (suc23 v123)) v023)))
(suc23 zero23))
{-# OPTIONS --type-in-type #-}
Ty24 : Set
Ty24 =
(Ty24 : Set)
(nat top bot : Ty24)
(arr prod sum : Ty24 β Ty24 β Ty24)
β Ty24
nat24 : Ty24; nat24 = Ξ» _ nat24 _ _ _ _ _ β nat24
top24 : Ty24; top24 = Ξ» _ _ top24 _ _ _ _ β top24
bot24 : Ty24; bot24 = Ξ» _ _ _ bot24 _ _ _ β bot24
arr24 : Ty24 β Ty24 β Ty24; arr24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod sum β
arr24 (A Ty24 nat24 top24 bot24 arr24 prod sum) (B Ty24 nat24 top24 bot24 arr24 prod sum)
prod24 : Ty24 β Ty24 β Ty24; prod24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod24 sum β
prod24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum) (B Ty24 nat24 top24 bot24 arr24 prod24 sum)
sum24 : Ty24 β Ty24 β Ty24; sum24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod24 sum24 β
sum24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum24) (B Ty24 nat24 top24 bot24 arr24 prod24 sum24)
Con24 : Set; Con24
= (Con24 : Set)
(nil : Con24)
(snoc : Con24 β Ty24 β Con24)
β Con24
nil24 : Con24; nil24
= Ξ» Con24 nil24 snoc β nil24
snoc24 : Con24 β Ty24 β Con24; snoc24
= Ξ» Ξ A Con24 nil24 snoc24 β snoc24 (Ξ Con24 nil24 snoc24) A
Var24 : Con24 β Ty24 β Set; Var24
= Ξ» Ξ A β
(Var24 : Con24 β Ty24 β Set)
(vz : β Ξ A β Var24 (snoc24 Ξ A) A)
(vs : β Ξ B A β Var24 Ξ A β Var24 (snoc24 Ξ B) A)
β Var24 Ξ A
vz24 : β{Ξ A} β Var24 (snoc24 Ξ A) A; vz24
= Ξ» Var24 vz24 vs β vz24 _ _
vs24 : β{Ξ B A} β Var24 Ξ A β Var24 (snoc24 Ξ B) A; vs24
= Ξ» x Var24 vz24 vs24 β vs24 _ _ _ (x Var24 vz24 vs24)
Tm24 : Con24 β Ty24 β Set; Tm24
= Ξ» Ξ A β
(Tm24 : Con24 β Ty24 β Set)
(var : β Ξ A β Var24 Ξ A β Tm24 Ξ A)
(lam : β Ξ A B β Tm24 (snoc24 Ξ A) B β Tm24 Ξ (arr24 A B))
(app : β Ξ A B β Tm24 Ξ (arr24 A B) β Tm24 Ξ A β Tm24 Ξ B)
(tt : β Ξ β Tm24 Ξ top24)
(pair : β Ξ A B β Tm24 Ξ A β Tm24 Ξ B β Tm24 Ξ (prod24 A B))
(fst : β Ξ A B β Tm24 Ξ (prod24 A B) β Tm24 Ξ A)
(snd : β Ξ A B β Tm24 Ξ (prod24 A B) β Tm24 Ξ B)
(left : β Ξ A B β Tm24 Ξ A β Tm24 Ξ (sum24 A B))
(right : β Ξ A B β Tm24 Ξ B β Tm24 Ξ (sum24 A B))
(case : β Ξ A B C β Tm24 Ξ (sum24 A B) β Tm24 Ξ (arr24 A C) β Tm24 Ξ (arr24 B C) β Tm24 Ξ C)
(zero : β Ξ β Tm24 Ξ nat24)
(suc : β Ξ β Tm24 Ξ nat24 β Tm24 Ξ nat24)
(rec : β Ξ A β Tm24 Ξ nat24 β Tm24 Ξ (arr24 nat24 (arr24 A A)) β Tm24 Ξ A β Tm24 Ξ A)
β Tm24 Ξ A
var24 : β{Ξ A} β Var24 Ξ A β Tm24 Ξ A; var24
= Ξ» x Tm24 var24 lam app tt pair fst snd left right case zero suc rec β
var24 _ _ x
lam24 : β{Ξ A B} β Tm24 (snoc24 Ξ A) B β Tm24 Ξ (arr24 A B); lam24
= Ξ» t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec β
lam24 _ _ _ (t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec)
app24 : β{Ξ A B} β Tm24 Ξ (arr24 A B) β Tm24 Ξ A β Tm24 Ξ B; app24
= Ξ» t u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec β
app24 _ _ _ (t Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
tt24 : β{Ξ} β Tm24 Ξ top24; tt24
= Ξ» Tm24 var24 lam24 app24 tt24 pair fst snd left right case zero suc rec β tt24 _
pair24 : β{Ξ A B} β Tm24 Ξ A β Tm24 Ξ B β Tm24 Ξ (prod24 A B); pair24
= Ξ» t u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec β
pair24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
fst24 : β{Ξ A B} β Tm24 Ξ (prod24 A B) β Tm24 Ξ A; fst24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec β
fst24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec)
snd24 : β{Ξ A B} β Tm24 Ξ (prod24 A B) β Tm24 Ξ B; snd24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec β
snd24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec)
left24 : β{Ξ A B} β Tm24 Ξ A β Tm24 Ξ (sum24 A B); left24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec β
left24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec)
right24 : β{Ξ A B} β Tm24 Ξ B β Tm24 Ξ (sum24 A B); right24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec β
right24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec)
case24 : β{Ξ A B C} β Tm24 Ξ (sum24 A B) β Tm24 Ξ (arr24 A C) β Tm24 Ξ (arr24 B C) β Tm24 Ξ C; case24
= Ξ» t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec β
case24 _ _ _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
zero24 : β{Ξ} β Tm24 Ξ nat24; zero24
= Ξ» Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc rec β zero24 _
suc24 : β{Ξ} β Tm24 Ξ nat24 β Tm24 Ξ nat24; suc24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec β
suc24 _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec)
rec24 : β{Ξ A} β Tm24 Ξ nat24 β Tm24 Ξ (arr24 nat24 (arr24 A A)) β Tm24 Ξ A β Tm24 Ξ A; rec24
= Ξ» t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24 β
rec24 _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
v024 : β{Ξ A} β Tm24 (snoc24 Ξ A) A; v024
= var24 vz24
v124 : β{Ξ A B} β Tm24 (snoc24 (snoc24 Ξ A) B) A; v124
= var24 (vs24 vz24)
v224 : β{Ξ A B C} β Tm24 (snoc24 (snoc24 (snoc24 Ξ A) B) C) A; v224
= var24 (vs24 (vs24 vz24))
v324 : β{Ξ A B C D} β Tm24 (snoc24 (snoc24 (snoc24 (snoc24 Ξ A) B) C) D) A; v324
= var24 (vs24 (vs24 (vs24 vz24)))
tbool24 : Ty24; tbool24
= sum24 top24 top24
true24 : β{Ξ} β Tm24 Ξ tbool24; true24
= left24 tt24
tfalse24 : β{Ξ} β Tm24 Ξ tbool24; tfalse24
= right24 tt24
ifthenelse24 : β{Ξ A} β Tm24 Ξ (arr24 tbool24 (arr24 A (arr24 A A))); ifthenelse24
= lam24 (lam24 (lam24 (case24 v224 (lam24 v224) (lam24 v124))))
times424 : β{Ξ A} β Tm24 Ξ (arr24 (arr24 A A) (arr24 A A)); times424
= lam24 (lam24 (app24 v124 (app24 v124 (app24 v124 (app24 v124 v024)))))
add24 : β{Ξ} β Tm24 Ξ (arr24 nat24 (arr24 nat24 nat24)); add24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (suc24 (app24 v124 v024)))))
(lam24 v024))
mul24 : β{Ξ} β Tm24 Ξ (arr24 nat24 (arr24 nat24 nat24)); mul24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (app24 (app24 add24 (app24 v124 v024)) v024))))
(lam24 zero24))
fact24 : β{Ξ} β Tm24 Ξ (arr24 nat24 nat24); fact24
= lam24 (rec24 v024 (lam24 (lam24 (app24 (app24 mul24 (suc24 v124)) v024)))
(suc24 zero24))
{-# OPTIONS --type-in-type #-}
Ty25 : Set
Ty25 =
(Ty25 : Set)
(nat top bot : Ty25)
(arr prod sum : Ty25 β Ty25 β Ty25)
β Ty25
nat25 : Ty25; nat25 = Ξ» _ nat25 _ _ _ _ _ β nat25
top25 : Ty25; top25 = Ξ» _ _ top25 _ _ _ _ β top25
bot25 : Ty25; bot25 = Ξ» _ _ _ bot25 _ _ _ β bot25
arr25 : Ty25 β Ty25 β Ty25; arr25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod sum β
arr25 (A Ty25 nat25 top25 bot25 arr25 prod sum) (B Ty25 nat25 top25 bot25 arr25 prod sum)
prod25 : Ty25 β Ty25 β Ty25; prod25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod25 sum β
prod25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum) (B Ty25 nat25 top25 bot25 arr25 prod25 sum)
sum25 : Ty25 β Ty25 β Ty25; sum25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod25 sum25 β
sum25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum25) (B Ty25 nat25 top25 bot25 arr25 prod25 sum25)
Con25 : Set; Con25
= (Con25 : Set)
(nil : Con25)
(snoc : Con25 β Ty25 β Con25)
β Con25
nil25 : Con25; nil25
= Ξ» Con25 nil25 snoc β nil25
snoc25 : Con25 β Ty25 β Con25; snoc25
= Ξ» Ξ A Con25 nil25 snoc25 β snoc25 (Ξ Con25 nil25 snoc25) A
Var25 : Con25 β Ty25 β Set; Var25
= Ξ» Ξ A β
(Var25 : Con25 β Ty25 β Set)
(vz : β Ξ A β Var25 (snoc25 Ξ A) A)
(vs : β Ξ B A β Var25 Ξ A β Var25 (snoc25 Ξ B) A)
β Var25 Ξ A
vz25 : β{Ξ A} β Var25 (snoc25 Ξ A) A; vz25
= Ξ» Var25 vz25 vs β vz25 _ _
vs25 : β{Ξ B A} β Var25 Ξ A β Var25 (snoc25 Ξ B) A; vs25
= Ξ» x Var25 vz25 vs25 β vs25 _ _ _ (x Var25 vz25 vs25)
Tm25 : Con25 β Ty25 β Set; Tm25
= Ξ» Ξ A β
(Tm25 : Con25 β Ty25 β Set)
(var : β Ξ A β Var25 Ξ A β Tm25 Ξ A)
(lam : β Ξ A B β Tm25 (snoc25 Ξ A) B β Tm25 Ξ (arr25 A B))
(app : β Ξ A B β Tm25 Ξ (arr25 A B) β Tm25 Ξ A β Tm25 Ξ B)
(tt : β Ξ β Tm25 Ξ top25)
(pair : β Ξ A B β Tm25 Ξ A β Tm25 Ξ B β Tm25 Ξ (prod25 A B))
(fst : β Ξ A B β Tm25 Ξ (prod25 A B) β Tm25 Ξ A)
(snd : β Ξ A B β Tm25 Ξ (prod25 A B) β Tm25 Ξ B)
(left : β Ξ A B β Tm25 Ξ A β Tm25 Ξ (sum25 A B))
(right : β Ξ A B β Tm25 Ξ B β Tm25 Ξ (sum25 A B))
(case : β Ξ A B C β Tm25 Ξ (sum25 A B) β Tm25 Ξ (arr25 A C) β Tm25 Ξ (arr25 B C) β Tm25 Ξ C)
(zero : β Ξ β Tm25 Ξ nat25)
(suc : β Ξ β Tm25 Ξ nat25 β Tm25 Ξ nat25)
(rec : β Ξ A β Tm25 Ξ nat25 β Tm25 Ξ (arr25 nat25 (arr25 A A)) β Tm25 Ξ A β Tm25 Ξ A)
β Tm25 Ξ A
var25 : β{Ξ A} β Var25 Ξ A β Tm25 Ξ A; var25
= Ξ» x Tm25 var25 lam app tt pair fst snd left right case zero suc rec β
var25 _ _ x
lam25 : β{Ξ A B} β Tm25 (snoc25 Ξ A) B β Tm25 Ξ (arr25 A B); lam25
= Ξ» t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec β
lam25 _ _ _ (t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec)
app25 : β{Ξ A B} β Tm25 Ξ (arr25 A B) β Tm25 Ξ A β Tm25 Ξ B; app25
= Ξ» t u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec β
app25 _ _ _ (t Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
tt25 : β{Ξ} β Tm25 Ξ top25; tt25
= Ξ» Tm25 var25 lam25 app25 tt25 pair fst snd left right case zero suc rec β tt25 _
pair25 : β{Ξ A B} β Tm25 Ξ A β Tm25 Ξ B β Tm25 Ξ (prod25 A B); pair25
= Ξ» t u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec β
pair25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
fst25 : β{Ξ A B} β Tm25 Ξ (prod25 A B) β Tm25 Ξ A; fst25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec β
fst25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec)
snd25 : β{Ξ A B} β Tm25 Ξ (prod25 A B) β Tm25 Ξ B; snd25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec β
snd25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec)
left25 : β{Ξ A B} β Tm25 Ξ A β Tm25 Ξ (sum25 A B); left25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec β
left25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec)
right25 : β{Ξ A B} β Tm25 Ξ B β Tm25 Ξ (sum25 A B); right25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec β
right25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec)
case25 : β{Ξ A B C} β Tm25 Ξ (sum25 A B) β Tm25 Ξ (arr25 A C) β Tm25 Ξ (arr25 B C) β Tm25 Ξ C; case25
= Ξ» t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec β
case25 _ _ _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
zero25 : β{Ξ} β Tm25 Ξ nat25; zero25
= Ξ» Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc rec β zero25 _
suc25 : β{Ξ} β Tm25 Ξ nat25 β Tm25 Ξ nat25; suc25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec β
suc25 _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec)
rec25 : β{Ξ A} β Tm25 Ξ nat25 β Tm25 Ξ (arr25 nat25 (arr25 A A)) β Tm25 Ξ A β Tm25 Ξ A; rec25
= Ξ» t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25 β
rec25 _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
v025 : β{Ξ A} β Tm25 (snoc25 Ξ A) A; v025
= var25 vz25
v125 : β{Ξ A B} β Tm25 (snoc25 (snoc25 Ξ A) B) A; v125
= var25 (vs25 vz25)
v225 : β{Ξ A B C} β Tm25 (snoc25 (snoc25 (snoc25 Ξ A) B) C) A; v225
= var25 (vs25 (vs25 vz25))
v325 : β{Ξ A B C D} β Tm25 (snoc25 (snoc25 (snoc25 (snoc25 Ξ A) B) C) D) A; v325
= var25 (vs25 (vs25 (vs25 vz25)))
tbool25 : Ty25; tbool25
= sum25 top25 top25
true25 : β{Ξ} β Tm25 Ξ tbool25; true25
= left25 tt25
tfalse25 : β{Ξ} β Tm25 Ξ tbool25; tfalse25
= right25 tt25
ifthenelse25 : β{Ξ A} β Tm25 Ξ (arr25 tbool25 (arr25 A (arr25 A A))); ifthenelse25
= lam25 (lam25 (lam25 (case25 v225 (lam25 v225) (lam25 v125))))
times425 : β{Ξ A} β Tm25 Ξ (arr25 (arr25 A A) (arr25 A A)); times425
= lam25 (lam25 (app25 v125 (app25 v125 (app25 v125 (app25 v125 v025)))))
add25 : β{Ξ} β Tm25 Ξ (arr25 nat25 (arr25 nat25 nat25)); add25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (suc25 (app25 v125 v025)))))
(lam25 v025))
mul25 : β{Ξ} β Tm25 Ξ (arr25 nat25 (arr25 nat25 nat25)); mul25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (app25 (app25 add25 (app25 v125 v025)) v025))))
(lam25 zero25))
fact25 : β{Ξ} β Tm25 Ξ (arr25 nat25 nat25); fact25
= lam25 (rec25 v025 (lam25 (lam25 (app25 (app25 mul25 (suc25 v125)) v025)))
(suc25 zero25))
{-# OPTIONS --type-in-type #-}
Ty26 : Set
Ty26 =
(Ty26 : Set)
(nat top bot : Ty26)
(arr prod sum : Ty26 β Ty26 β Ty26)
β Ty26
nat26 : Ty26; nat26 = Ξ» _ nat26 _ _ _ _ _ β nat26
top26 : Ty26; top26 = Ξ» _ _ top26 _ _ _ _ β top26
bot26 : Ty26; bot26 = Ξ» _ _ _ bot26 _ _ _ β bot26
arr26 : Ty26 β Ty26 β Ty26; arr26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod sum β
arr26 (A Ty26 nat26 top26 bot26 arr26 prod sum) (B Ty26 nat26 top26 bot26 arr26 prod sum)
prod26 : Ty26 β Ty26 β Ty26; prod26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod26 sum β
prod26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum) (B Ty26 nat26 top26 bot26 arr26 prod26 sum)
sum26 : Ty26 β Ty26 β Ty26; sum26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod26 sum26 β
sum26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum26) (B Ty26 nat26 top26 bot26 arr26 prod26 sum26)
Con26 : Set; Con26
= (Con26 : Set)
(nil : Con26)
(snoc : Con26 β Ty26 β Con26)
β Con26
nil26 : Con26; nil26
= Ξ» Con26 nil26 snoc β nil26
snoc26 : Con26 β Ty26 β Con26; snoc26
= Ξ» Ξ A Con26 nil26 snoc26 β snoc26 (Ξ Con26 nil26 snoc26) A
Var26 : Con26 β Ty26 β Set; Var26
= Ξ» Ξ A β
(Var26 : Con26 β Ty26 β Set)
(vz : β Ξ A β Var26 (snoc26 Ξ A) A)
(vs : β Ξ B A β Var26 Ξ A β Var26 (snoc26 Ξ B) A)
β Var26 Ξ A
vz26 : β{Ξ A} β Var26 (snoc26 Ξ A) A; vz26
= Ξ» Var26 vz26 vs β vz26 _ _
vs26 : β{Ξ B A} β Var26 Ξ A β Var26 (snoc26 Ξ B) A; vs26
= Ξ» x Var26 vz26 vs26 β vs26 _ _ _ (x Var26 vz26 vs26)
Tm26 : Con26 β Ty26 β Set; Tm26
= Ξ» Ξ A β
(Tm26 : Con26 β Ty26 β Set)
(var : β Ξ A β Var26 Ξ A β Tm26 Ξ A)
(lam : β Ξ A B β Tm26 (snoc26 Ξ A) B β Tm26 Ξ (arr26 A B))
(app : β Ξ A B β Tm26 Ξ (arr26 A B) β Tm26 Ξ A β Tm26 Ξ B)
(tt : β Ξ β Tm26 Ξ top26)
(pair : β Ξ A B β Tm26 Ξ A β Tm26 Ξ B β Tm26 Ξ (prod26 A B))
(fst : β Ξ A B β Tm26 Ξ (prod26 A B) β Tm26 Ξ A)
(snd : β Ξ A B β Tm26 Ξ (prod26 A B) β Tm26 Ξ B)
(left : β Ξ A B β Tm26 Ξ A β Tm26 Ξ (sum26 A B))
(right : β Ξ A B β Tm26 Ξ B β Tm26 Ξ (sum26 A B))
(case : β Ξ A B C β Tm26 Ξ (sum26 A B) β Tm26 Ξ (arr26 A C) β Tm26 Ξ (arr26 B C) β Tm26 Ξ C)
(zero : β Ξ β Tm26 Ξ nat26)
(suc : β Ξ β Tm26 Ξ nat26 β Tm26 Ξ nat26)
(rec : β Ξ A β Tm26 Ξ nat26 β Tm26 Ξ (arr26 nat26 (arr26 A A)) β Tm26 Ξ A β Tm26 Ξ A)
β Tm26 Ξ A
var26 : β{Ξ A} β Var26 Ξ A β Tm26 Ξ A; var26
= Ξ» x Tm26 var26 lam app tt pair fst snd left right case zero suc rec β
var26 _ _ x
lam26 : β{Ξ A B} β Tm26 (snoc26 Ξ A) B β Tm26 Ξ (arr26 A B); lam26
= Ξ» t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec β
lam26 _ _ _ (t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec)
app26 : β{Ξ A B} β Tm26 Ξ (arr26 A B) β Tm26 Ξ A β Tm26 Ξ B; app26
= Ξ» t u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec β
app26 _ _ _ (t Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
tt26 : β{Ξ} β Tm26 Ξ top26; tt26
= Ξ» Tm26 var26 lam26 app26 tt26 pair fst snd left right case zero suc rec β tt26 _
pair26 : β{Ξ A B} β Tm26 Ξ A β Tm26 Ξ B β Tm26 Ξ (prod26 A B); pair26
= Ξ» t u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec β
pair26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
fst26 : β{Ξ A B} β Tm26 Ξ (prod26 A B) β Tm26 Ξ A; fst26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec β
fst26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec)
snd26 : β{Ξ A B} β Tm26 Ξ (prod26 A B) β Tm26 Ξ B; snd26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec β
snd26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec)
left26 : β{Ξ A B} β Tm26 Ξ A β Tm26 Ξ (sum26 A B); left26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec β
left26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec)
right26 : β{Ξ A B} β Tm26 Ξ B β Tm26 Ξ (sum26 A B); right26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec β
right26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec)
case26 : β{Ξ A B C} β Tm26 Ξ (sum26 A B) β Tm26 Ξ (arr26 A C) β Tm26 Ξ (arr26 B C) β Tm26 Ξ C; case26
= Ξ» t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec β
case26 _ _ _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
zero26 : β{Ξ} β Tm26 Ξ nat26; zero26
= Ξ» Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc rec β zero26 _
suc26 : β{Ξ} β Tm26 Ξ nat26 β Tm26 Ξ nat26; suc26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec β
suc26 _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec)
rec26 : β{Ξ A} β Tm26 Ξ nat26 β Tm26 Ξ (arr26 nat26 (arr26 A A)) β Tm26 Ξ A β Tm26 Ξ A; rec26
= Ξ» t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26 β
rec26 _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
v026 : β{Ξ A} β Tm26 (snoc26 Ξ A) A; v026
= var26 vz26
v126 : β{Ξ A B} β Tm26 (snoc26 (snoc26 Ξ A) B) A; v126
= var26 (vs26 vz26)
v226 : β{Ξ A B C} β Tm26 (snoc26 (snoc26 (snoc26 Ξ A) B) C) A; v226
= var26 (vs26 (vs26 vz26))
v326 : β{Ξ A B C D} β Tm26 (snoc26 (snoc26 (snoc26 (snoc26 Ξ A) B) C) D) A; v326
= var26 (vs26 (vs26 (vs26 vz26)))
tbool26 : Ty26; tbool26
= sum26 top26 top26
true26 : β{Ξ} β Tm26 Ξ tbool26; true26
= left26 tt26
tfalse26 : β{Ξ} β Tm26 Ξ tbool26; tfalse26
= right26 tt26
ifthenelse26 : β{Ξ A} β Tm26 Ξ (arr26 tbool26 (arr26 A (arr26 A A))); ifthenelse26
= lam26 (lam26 (lam26 (case26 v226 (lam26 v226) (lam26 v126))))
times426 : β{Ξ A} β Tm26 Ξ (arr26 (arr26 A A) (arr26 A A)); times426
= lam26 (lam26 (app26 v126 (app26 v126 (app26 v126 (app26 v126 v026)))))
add26 : β{Ξ} β Tm26 Ξ (arr26 nat26 (arr26 nat26 nat26)); add26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (suc26 (app26 v126 v026)))))
(lam26 v026))
mul26 : β{Ξ} β Tm26 Ξ (arr26 nat26 (arr26 nat26 nat26)); mul26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (app26 (app26 add26 (app26 v126 v026)) v026))))
(lam26 zero26))
fact26 : β{Ξ} β Tm26 Ξ (arr26 nat26 nat26); fact26
= lam26 (rec26 v026 (lam26 (lam26 (app26 (app26 mul26 (suc26 v126)) v026)))
(suc26 zero26))
{-# OPTIONS --type-in-type #-}
Ty27 : Set
Ty27 =
(Ty27 : Set)
(nat top bot : Ty27)
(arr prod sum : Ty27 β Ty27 β Ty27)
β Ty27
nat27 : Ty27; nat27 = Ξ» _ nat27 _ _ _ _ _ β nat27
top27 : Ty27; top27 = Ξ» _ _ top27 _ _ _ _ β top27
bot27 : Ty27; bot27 = Ξ» _ _ _ bot27 _ _ _ β bot27
arr27 : Ty27 β Ty27 β Ty27; arr27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod sum β
arr27 (A Ty27 nat27 top27 bot27 arr27 prod sum) (B Ty27 nat27 top27 bot27 arr27 prod sum)
prod27 : Ty27 β Ty27 β Ty27; prod27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod27 sum β
prod27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum) (B Ty27 nat27 top27 bot27 arr27 prod27 sum)
sum27 : Ty27 β Ty27 β Ty27; sum27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod27 sum27 β
sum27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum27) (B Ty27 nat27 top27 bot27 arr27 prod27 sum27)
Con27 : Set; Con27
= (Con27 : Set)
(nil : Con27)
(snoc : Con27 β Ty27 β Con27)
β Con27
nil27 : Con27; nil27
= Ξ» Con27 nil27 snoc β nil27
snoc27 : Con27 β Ty27 β Con27; snoc27
= Ξ» Ξ A Con27 nil27 snoc27 β snoc27 (Ξ Con27 nil27 snoc27) A
Var27 : Con27 β Ty27 β Set; Var27
= Ξ» Ξ A β
(Var27 : Con27 β Ty27 β Set)
(vz : β Ξ A β Var27 (snoc27 Ξ A) A)
(vs : β Ξ B A β Var27 Ξ A β Var27 (snoc27 Ξ B) A)
β Var27 Ξ A
vz27 : β{Ξ A} β Var27 (snoc27 Ξ A) A; vz27
= Ξ» Var27 vz27 vs β vz27 _ _
vs27 : β{Ξ B A} β Var27 Ξ A β Var27 (snoc27 Ξ B) A; vs27
= Ξ» x Var27 vz27 vs27 β vs27 _ _ _ (x Var27 vz27 vs27)
Tm27 : Con27 β Ty27 β Set; Tm27
= Ξ» Ξ A β
(Tm27 : Con27 β Ty27 β Set)
(var : β Ξ A β Var27 Ξ A β Tm27 Ξ A)
(lam : β Ξ A B β Tm27 (snoc27 Ξ A) B β Tm27 Ξ (arr27 A B))
(app : β Ξ A B β Tm27 Ξ (arr27 A B) β Tm27 Ξ A β Tm27 Ξ B)
(tt : β Ξ β Tm27 Ξ top27)
(pair : β Ξ A B β Tm27 Ξ A β Tm27 Ξ B β Tm27 Ξ (prod27 A B))
(fst : β Ξ A B β Tm27 Ξ (prod27 A B) β Tm27 Ξ A)
(snd : β Ξ A B β Tm27 Ξ (prod27 A B) β Tm27 Ξ B)
(left : β Ξ A B β Tm27 Ξ A β Tm27 Ξ (sum27 A B))
(right : β Ξ A B β Tm27 Ξ B β Tm27 Ξ (sum27 A B))
(case : β Ξ A B C β Tm27 Ξ (sum27 A B) β Tm27 Ξ (arr27 A C) β Tm27 Ξ (arr27 B C) β Tm27 Ξ C)
(zero : β Ξ β Tm27 Ξ nat27)
(suc : β Ξ β Tm27 Ξ nat27 β Tm27 Ξ nat27)
(rec : β Ξ A β Tm27 Ξ nat27 β Tm27 Ξ (arr27 nat27 (arr27 A A)) β Tm27 Ξ A β Tm27 Ξ A)
β Tm27 Ξ A
var27 : β{Ξ A} β Var27 Ξ A β Tm27 Ξ A; var27
= Ξ» x Tm27 var27 lam app tt pair fst snd left right case zero suc rec β
var27 _ _ x
lam27 : β{Ξ A B} β Tm27 (snoc27 Ξ A) B β Tm27 Ξ (arr27 A B); lam27
= Ξ» t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec β
lam27 _ _ _ (t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec)
app27 : β{Ξ A B} β Tm27 Ξ (arr27 A B) β Tm27 Ξ A β Tm27 Ξ B; app27
= Ξ» t u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec β
app27 _ _ _ (t Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
tt27 : β{Ξ} β Tm27 Ξ top27; tt27
= Ξ» Tm27 var27 lam27 app27 tt27 pair fst snd left right case zero suc rec β tt27 _
pair27 : β{Ξ A B} β Tm27 Ξ A β Tm27 Ξ B β Tm27 Ξ (prod27 A B); pair27
= Ξ» t u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec β
pair27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
fst27 : β{Ξ A B} β Tm27 Ξ (prod27 A B) β Tm27 Ξ A; fst27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec β
fst27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec)
snd27 : β{Ξ A B} β Tm27 Ξ (prod27 A B) β Tm27 Ξ B; snd27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec β
snd27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec)
left27 : β{Ξ A B} β Tm27 Ξ A β Tm27 Ξ (sum27 A B); left27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec β
left27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec)
right27 : β{Ξ A B} β Tm27 Ξ B β Tm27 Ξ (sum27 A B); right27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec β
right27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec)
case27 : β{Ξ A B C} β Tm27 Ξ (sum27 A B) β Tm27 Ξ (arr27 A C) β Tm27 Ξ (arr27 B C) β Tm27 Ξ C; case27
= Ξ» t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec β
case27 _ _ _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
zero27 : β{Ξ} β Tm27 Ξ nat27; zero27
= Ξ» Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc rec β zero27 _
suc27 : β{Ξ} β Tm27 Ξ nat27 β Tm27 Ξ nat27; suc27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec β
suc27 _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec)
rec27 : β{Ξ A} β Tm27 Ξ nat27 β Tm27 Ξ (arr27 nat27 (arr27 A A)) β Tm27 Ξ A β Tm27 Ξ A; rec27
= Ξ» t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27 β
rec27 _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
v027 : β{Ξ A} β Tm27 (snoc27 Ξ A) A; v027
= var27 vz27
v127 : β{Ξ A B} β Tm27 (snoc27 (snoc27 Ξ A) B) A; v127
= var27 (vs27 vz27)
v227 : β{Ξ A B C} β Tm27 (snoc27 (snoc27 (snoc27 Ξ A) B) C) A; v227
= var27 (vs27 (vs27 vz27))
v327 : β{Ξ A B C D} β Tm27 (snoc27 (snoc27 (snoc27 (snoc27 Ξ A) B) C) D) A; v327
= var27 (vs27 (vs27 (vs27 vz27)))
tbool27 : Ty27; tbool27
= sum27 top27 top27
true27 : β{Ξ} β Tm27 Ξ tbool27; true27
= left27 tt27
tfalse27 : β{Ξ} β Tm27 Ξ tbool27; tfalse27
= right27 tt27
ifthenelse27 : β{Ξ A} β Tm27 Ξ (arr27 tbool27 (arr27 A (arr27 A A))); ifthenelse27
= lam27 (lam27 (lam27 (case27 v227 (lam27 v227) (lam27 v127))))
times427 : β{Ξ A} β Tm27 Ξ (arr27 (arr27 A A) (arr27 A A)); times427
= lam27 (lam27 (app27 v127 (app27 v127 (app27 v127 (app27 v127 v027)))))
add27 : β{Ξ} β Tm27 Ξ (arr27 nat27 (arr27 nat27 nat27)); add27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (suc27 (app27 v127 v027)))))
(lam27 v027))
mul27 : β{Ξ} β Tm27 Ξ (arr27 nat27 (arr27 nat27 nat27)); mul27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (app27 (app27 add27 (app27 v127 v027)) v027))))
(lam27 zero27))
fact27 : β{Ξ} β Tm27 Ξ (arr27 nat27 nat27); fact27
= lam27 (rec27 v027 (lam27 (lam27 (app27 (app27 mul27 (suc27 v127)) v027)))
(suc27 zero27))
{-# OPTIONS --type-in-type #-}
Ty28 : Set
Ty28 =
(Ty28 : Set)
(nat top bot : Ty28)
(arr prod sum : Ty28 β Ty28 β Ty28)
β Ty28
nat28 : Ty28; nat28 = Ξ» _ nat28 _ _ _ _ _ β nat28
top28 : Ty28; top28 = Ξ» _ _ top28 _ _ _ _ β top28
bot28 : Ty28; bot28 = Ξ» _ _ _ bot28 _ _ _ β bot28
arr28 : Ty28 β Ty28 β Ty28; arr28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod sum β
arr28 (A Ty28 nat28 top28 bot28 arr28 prod sum) (B Ty28 nat28 top28 bot28 arr28 prod sum)
prod28 : Ty28 β Ty28 β Ty28; prod28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod28 sum β
prod28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum) (B Ty28 nat28 top28 bot28 arr28 prod28 sum)
sum28 : Ty28 β Ty28 β Ty28; sum28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod28 sum28 β
sum28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum28) (B Ty28 nat28 top28 bot28 arr28 prod28 sum28)
Con28 : Set; Con28
= (Con28 : Set)
(nil : Con28)
(snoc : Con28 β Ty28 β Con28)
β Con28
nil28 : Con28; nil28
= Ξ» Con28 nil28 snoc β nil28
snoc28 : Con28 β Ty28 β Con28; snoc28
= Ξ» Ξ A Con28 nil28 snoc28 β snoc28 (Ξ Con28 nil28 snoc28) A
Var28 : Con28 β Ty28 β Set; Var28
= Ξ» Ξ A β
(Var28 : Con28 β Ty28 β Set)
(vz : β Ξ A β Var28 (snoc28 Ξ A) A)
(vs : β Ξ B A β Var28 Ξ A β Var28 (snoc28 Ξ B) A)
β Var28 Ξ A
vz28 : β{Ξ A} β Var28 (snoc28 Ξ A) A; vz28
= Ξ» Var28 vz28 vs β vz28 _ _
vs28 : β{Ξ B A} β Var28 Ξ A β Var28 (snoc28 Ξ B) A; vs28
= Ξ» x Var28 vz28 vs28 β vs28 _ _ _ (x Var28 vz28 vs28)
Tm28 : Con28 β Ty28 β Set; Tm28
= Ξ» Ξ A β
(Tm28 : Con28 β Ty28 β Set)
(var : β Ξ A β Var28 Ξ A β Tm28 Ξ A)
(lam : β Ξ A B β Tm28 (snoc28 Ξ A) B β Tm28 Ξ (arr28 A B))
(app : β Ξ A B β Tm28 Ξ (arr28 A B) β Tm28 Ξ A β Tm28 Ξ B)
(tt : β Ξ β Tm28 Ξ top28)
(pair : β Ξ A B β Tm28 Ξ A β Tm28 Ξ B β Tm28 Ξ (prod28 A B))
(fst : β Ξ A B β Tm28 Ξ (prod28 A B) β Tm28 Ξ A)
(snd : β Ξ A B β Tm28 Ξ (prod28 A B) β Tm28 Ξ B)
(left : β Ξ A B β Tm28 Ξ A β Tm28 Ξ (sum28 A B))
(right : β Ξ A B β Tm28 Ξ B β Tm28 Ξ (sum28 A B))
(case : β Ξ A B C β Tm28 Ξ (sum28 A B) β Tm28 Ξ (arr28 A C) β Tm28 Ξ (arr28 B C) β Tm28 Ξ C)
(zero : β Ξ β Tm28 Ξ nat28)
(suc : β Ξ β Tm28 Ξ nat28 β Tm28 Ξ nat28)
(rec : β Ξ A β Tm28 Ξ nat28 β Tm28 Ξ (arr28 nat28 (arr28 A A)) β Tm28 Ξ A β Tm28 Ξ A)
β Tm28 Ξ A
var28 : β{Ξ A} β Var28 Ξ A β Tm28 Ξ A; var28
= Ξ» x Tm28 var28 lam app tt pair fst snd left right case zero suc rec β
var28 _ _ x
lam28 : β{Ξ A B} β Tm28 (snoc28 Ξ A) B β Tm28 Ξ (arr28 A B); lam28
= Ξ» t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec β
lam28 _ _ _ (t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec)
app28 : β{Ξ A B} β Tm28 Ξ (arr28 A B) β Tm28 Ξ A β Tm28 Ξ B; app28
= Ξ» t u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec β
app28 _ _ _ (t Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
tt28 : β{Ξ} β Tm28 Ξ top28; tt28
= Ξ» Tm28 var28 lam28 app28 tt28 pair fst snd left right case zero suc rec β tt28 _
pair28 : β{Ξ A B} β Tm28 Ξ A β Tm28 Ξ B β Tm28 Ξ (prod28 A B); pair28
= Ξ» t u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec β
pair28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
fst28 : β{Ξ A B} β Tm28 Ξ (prod28 A B) β Tm28 Ξ A; fst28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec β
fst28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec)
snd28 : β{Ξ A B} β Tm28 Ξ (prod28 A B) β Tm28 Ξ B; snd28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec β
snd28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec)
left28 : β{Ξ A B} β Tm28 Ξ A β Tm28 Ξ (sum28 A B); left28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec β
left28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec)
right28 : β{Ξ A B} β Tm28 Ξ B β Tm28 Ξ (sum28 A B); right28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec β
right28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec)
case28 : β{Ξ A B C} β Tm28 Ξ (sum28 A B) β Tm28 Ξ (arr28 A C) β Tm28 Ξ (arr28 B C) β Tm28 Ξ C; case28
= Ξ» t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec β
case28 _ _ _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
zero28 : β{Ξ} β Tm28 Ξ nat28; zero28
= Ξ» Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc rec β zero28 _
suc28 : β{Ξ} β Tm28 Ξ nat28 β Tm28 Ξ nat28; suc28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec β
suc28 _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec)
rec28 : β{Ξ A} β Tm28 Ξ nat28 β Tm28 Ξ (arr28 nat28 (arr28 A A)) β Tm28 Ξ A β Tm28 Ξ A; rec28
= Ξ» t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28 β
rec28 _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
v028 : β{Ξ A} β Tm28 (snoc28 Ξ A) A; v028
= var28 vz28
v128 : β{Ξ A B} β Tm28 (snoc28 (snoc28 Ξ A) B) A; v128
= var28 (vs28 vz28)
v228 : β{Ξ A B C} β Tm28 (snoc28 (snoc28 (snoc28 Ξ A) B) C) A; v228
= var28 (vs28 (vs28 vz28))
v328 : β{Ξ A B C D} β Tm28 (snoc28 (snoc28 (snoc28 (snoc28 Ξ A) B) C) D) A; v328
= var28 (vs28 (vs28 (vs28 vz28)))
tbool28 : Ty28; tbool28
= sum28 top28 top28
true28 : β{Ξ} β Tm28 Ξ tbool28; true28
= left28 tt28
tfalse28 : β{Ξ} β Tm28 Ξ tbool28; tfalse28
= right28 tt28
ifthenelse28 : β{Ξ A} β Tm28 Ξ (arr28 tbool28 (arr28 A (arr28 A A))); ifthenelse28
= lam28 (lam28 (lam28 (case28 v228 (lam28 v228) (lam28 v128))))
times428 : β{Ξ A} β Tm28 Ξ (arr28 (arr28 A A) (arr28 A A)); times428
= lam28 (lam28 (app28 v128 (app28 v128 (app28 v128 (app28 v128 v028)))))
add28 : β{Ξ} β Tm28 Ξ (arr28 nat28 (arr28 nat28 nat28)); add28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (suc28 (app28 v128 v028)))))
(lam28 v028))
mul28 : β{Ξ} β Tm28 Ξ (arr28 nat28 (arr28 nat28 nat28)); mul28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (app28 (app28 add28 (app28 v128 v028)) v028))))
(lam28 zero28))
fact28 : β{Ξ} β Tm28 Ξ (arr28 nat28 nat28); fact28
= lam28 (rec28 v028 (lam28 (lam28 (app28 (app28 mul28 (suc28 v128)) v028)))
(suc28 zero28))
{-# OPTIONS --type-in-type #-}
Ty29 : Set
Ty29 =
(Ty29 : Set)
(nat top bot : Ty29)
(arr prod sum : Ty29 β Ty29 β Ty29)
β Ty29
nat29 : Ty29; nat29 = Ξ» _ nat29 _ _ _ _ _ β nat29
top29 : Ty29; top29 = Ξ» _ _ top29 _ _ _ _ β top29
bot29 : Ty29; bot29 = Ξ» _ _ _ bot29 _ _ _ β bot29
arr29 : Ty29 β Ty29 β Ty29; arr29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod sum β
arr29 (A Ty29 nat29 top29 bot29 arr29 prod sum) (B Ty29 nat29 top29 bot29 arr29 prod sum)
prod29 : Ty29 β Ty29 β Ty29; prod29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod29 sum β
prod29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum) (B Ty29 nat29 top29 bot29 arr29 prod29 sum)
sum29 : Ty29 β Ty29 β Ty29; sum29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod29 sum29 β
sum29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum29) (B Ty29 nat29 top29 bot29 arr29 prod29 sum29)
Con29 : Set; Con29
= (Con29 : Set)
(nil : Con29)
(snoc : Con29 β Ty29 β Con29)
β Con29
nil29 : Con29; nil29
= Ξ» Con29 nil29 snoc β nil29
snoc29 : Con29 β Ty29 β Con29; snoc29
= Ξ» Ξ A Con29 nil29 snoc29 β snoc29 (Ξ Con29 nil29 snoc29) A
Var29 : Con29 β Ty29 β Set; Var29
= Ξ» Ξ A β
(Var29 : Con29 β Ty29 β Set)
(vz : β Ξ A β Var29 (snoc29 Ξ A) A)
(vs : β Ξ B A β Var29 Ξ A β Var29 (snoc29 Ξ B) A)
β Var29 Ξ A
vz29 : β{Ξ A} β Var29 (snoc29 Ξ A) A; vz29
= Ξ» Var29 vz29 vs β vz29 _ _
vs29 : β{Ξ B A} β Var29 Ξ A β Var29 (snoc29 Ξ B) A; vs29
= Ξ» x Var29 vz29 vs29 β vs29 _ _ _ (x Var29 vz29 vs29)
Tm29 : Con29 β Ty29 β Set; Tm29
= Ξ» Ξ A β
(Tm29 : Con29 β Ty29 β Set)
(var : β Ξ A β Var29 Ξ A β Tm29 Ξ A)
(lam : β Ξ A B β Tm29 (snoc29 Ξ A) B β Tm29 Ξ (arr29 A B))
(app : β Ξ A B β Tm29 Ξ (arr29 A B) β Tm29 Ξ A β Tm29 Ξ B)
(tt : β Ξ β Tm29 Ξ top29)
(pair : β Ξ A B β Tm29 Ξ A β Tm29 Ξ B β Tm29 Ξ (prod29 A B))
(fst : β Ξ A B β Tm29 Ξ (prod29 A B) β Tm29 Ξ A)
(snd : β Ξ A B β Tm29 Ξ (prod29 A B) β Tm29 Ξ B)
(left : β Ξ A B β Tm29 Ξ A β Tm29 Ξ (sum29 A B))
(right : β Ξ A B β Tm29 Ξ B β Tm29 Ξ (sum29 A B))
(case : β Ξ A B C β Tm29 Ξ (sum29 A B) β Tm29 Ξ (arr29 A C) β Tm29 Ξ (arr29 B C) β Tm29 Ξ C)
(zero : β Ξ β Tm29 Ξ nat29)
(suc : β Ξ β Tm29 Ξ nat29 β Tm29 Ξ nat29)
(rec : β Ξ A β Tm29 Ξ nat29 β Tm29 Ξ (arr29 nat29 (arr29 A A)) β Tm29 Ξ A β Tm29 Ξ A)
β Tm29 Ξ A
var29 : β{Ξ A} β Var29 Ξ A β Tm29 Ξ A; var29
= Ξ» x Tm29 var29 lam app tt pair fst snd left right case zero suc rec β
var29 _ _ x
lam29 : β{Ξ A B} β Tm29 (snoc29 Ξ A) B β Tm29 Ξ (arr29 A B); lam29
= Ξ» t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec β
lam29 _ _ _ (t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec)
app29 : β{Ξ A B} β Tm29 Ξ (arr29 A B) β Tm29 Ξ A β Tm29 Ξ B; app29
= Ξ» t u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec β
app29 _ _ _ (t Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
tt29 : β{Ξ} β Tm29 Ξ top29; tt29
= Ξ» Tm29 var29 lam29 app29 tt29 pair fst snd left right case zero suc rec β tt29 _
pair29 : β{Ξ A B} β Tm29 Ξ A β Tm29 Ξ B β Tm29 Ξ (prod29 A B); pair29
= Ξ» t u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec β
pair29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
fst29 : β{Ξ A B} β Tm29 Ξ (prod29 A B) β Tm29 Ξ A; fst29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec β
fst29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec)
snd29 : β{Ξ A B} β Tm29 Ξ (prod29 A B) β Tm29 Ξ B; snd29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec β
snd29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec)
left29 : β{Ξ A B} β Tm29 Ξ A β Tm29 Ξ (sum29 A B); left29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec β
left29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec)
right29 : β{Ξ A B} β Tm29 Ξ B β Tm29 Ξ (sum29 A B); right29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec β
right29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec)
case29 : β{Ξ A B C} β Tm29 Ξ (sum29 A B) β Tm29 Ξ (arr29 A C) β Tm29 Ξ (arr29 B C) β Tm29 Ξ C; case29
= Ξ» t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec β
case29 _ _ _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
zero29 : β{Ξ} β Tm29 Ξ nat29; zero29
= Ξ» Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc rec β zero29 _
suc29 : β{Ξ} β Tm29 Ξ nat29 β Tm29 Ξ nat29; suc29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec β
suc29 _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec)
rec29 : β{Ξ A} β Tm29 Ξ nat29 β Tm29 Ξ (arr29 nat29 (arr29 A A)) β Tm29 Ξ A β Tm29 Ξ A; rec29
= Ξ» t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29 β
rec29 _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
v029 : β{Ξ A} β Tm29 (snoc29 Ξ A) A; v029
= var29 vz29
v129 : β{Ξ A B} β Tm29 (snoc29 (snoc29 Ξ A) B) A; v129
= var29 (vs29 vz29)
v229 : β{Ξ A B C} β Tm29 (snoc29 (snoc29 (snoc29 Ξ A) B) C) A; v229
= var29 (vs29 (vs29 vz29))
v329 : β{Ξ A B C D} β Tm29 (snoc29 (snoc29 (snoc29 (snoc29 Ξ A) B) C) D) A; v329
= var29 (vs29 (vs29 (vs29 vz29)))
tbool29 : Ty29; tbool29
= sum29 top29 top29
true29 : β{Ξ} β Tm29 Ξ tbool29; true29
= left29 tt29
tfalse29 : β{Ξ} β Tm29 Ξ tbool29; tfalse29
= right29 tt29
ifthenelse29 : β{Ξ A} β Tm29 Ξ (arr29 tbool29 (arr29 A (arr29 A A))); ifthenelse29
= lam29 (lam29 (lam29 (case29 v229 (lam29 v229) (lam29 v129))))
times429 : β{Ξ A} β Tm29 Ξ (arr29 (arr29 A A) (arr29 A A)); times429
= lam29 (lam29 (app29 v129 (app29 v129 (app29 v129 (app29 v129 v029)))))
add29 : β{Ξ} β Tm29 Ξ (arr29 nat29 (arr29 nat29 nat29)); add29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (suc29 (app29 v129 v029)))))
(lam29 v029))
mul29 : β{Ξ} β Tm29 Ξ (arr29 nat29 (arr29 nat29 nat29)); mul29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (app29 (app29 add29 (app29 v129 v029)) v029))))
(lam29 zero29))
fact29 : β{Ξ} β Tm29 Ξ (arr29 nat29 nat29); fact29
= lam29 (rec29 v029 (lam29 (lam29 (app29 (app29 mul29 (suc29 v129)) v029)))
(suc29 zero29))
{-# OPTIONS --type-in-type #-}
Ty30 : Set
Ty30 =
(Ty30 : Set)
(nat top bot : Ty30)
(arr prod sum : Ty30 β Ty30 β Ty30)
β Ty30
nat30 : Ty30; nat30 = Ξ» _ nat30 _ _ _ _ _ β nat30
top30 : Ty30; top30 = Ξ» _ _ top30 _ _ _ _ β top30
bot30 : Ty30; bot30 = Ξ» _ _ _ bot30 _ _ _ β bot30
arr30 : Ty30 β Ty30 β Ty30; arr30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod sum β
arr30 (A Ty30 nat30 top30 bot30 arr30 prod sum) (B Ty30 nat30 top30 bot30 arr30 prod sum)
prod30 : Ty30 β Ty30 β Ty30; prod30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod30 sum β
prod30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum) (B Ty30 nat30 top30 bot30 arr30 prod30 sum)
sum30 : Ty30 β Ty30 β Ty30; sum30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod30 sum30 β
sum30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum30) (B Ty30 nat30 top30 bot30 arr30 prod30 sum30)
Con30 : Set; Con30
= (Con30 : Set)
(nil : Con30)
(snoc : Con30 β Ty30 β Con30)
β Con30
nil30 : Con30; nil30
= Ξ» Con30 nil30 snoc β nil30
snoc30 : Con30 β Ty30 β Con30; snoc30
= Ξ» Ξ A Con30 nil30 snoc30 β snoc30 (Ξ Con30 nil30 snoc30) A
Var30 : Con30 β Ty30 β Set; Var30
= Ξ» Ξ A β
(Var30 : Con30 β Ty30 β Set)
(vz : β Ξ A β Var30 (snoc30 Ξ A) A)
(vs : β Ξ B A β Var30 Ξ A β Var30 (snoc30 Ξ B) A)
β Var30 Ξ A
vz30 : β{Ξ A} β Var30 (snoc30 Ξ A) A; vz30
= Ξ» Var30 vz30 vs β vz30 _ _
vs30 : β{Ξ B A} β Var30 Ξ A β Var30 (snoc30 Ξ B) A; vs30
= Ξ» x Var30 vz30 vs30 β vs30 _ _ _ (x Var30 vz30 vs30)
Tm30 : Con30 β Ty30 β Set; Tm30
= Ξ» Ξ A β
(Tm30 : Con30 β Ty30 β Set)
(var : β Ξ A β Var30 Ξ A β Tm30 Ξ A)
(lam : β Ξ A B β Tm30 (snoc30 Ξ A) B β Tm30 Ξ (arr30 A B))
(app : β Ξ A B β Tm30 Ξ (arr30 A B) β Tm30 Ξ A β Tm30 Ξ B)
(tt : β Ξ β Tm30 Ξ top30)
(pair : β Ξ A B β Tm30 Ξ A β Tm30 Ξ B β Tm30 Ξ (prod30 A B))
(fst : β Ξ A B β Tm30 Ξ (prod30 A B) β Tm30 Ξ A)
(snd : β Ξ A B β Tm30 Ξ (prod30 A B) β Tm30 Ξ B)
(left : β Ξ A B β Tm30 Ξ A β Tm30 Ξ (sum30 A B))
(right : β Ξ A B β Tm30 Ξ B β Tm30 Ξ (sum30 A B))
(case : β Ξ A B C β Tm30 Ξ (sum30 A B) β Tm30 Ξ (arr30 A C) β Tm30 Ξ (arr30 B C) β Tm30 Ξ C)
(zero : β Ξ β Tm30 Ξ nat30)
(suc : β Ξ β Tm30 Ξ nat30 β Tm30 Ξ nat30)
(rec : β Ξ A β Tm30 Ξ nat30 β Tm30 Ξ (arr30 nat30 (arr30 A A)) β Tm30 Ξ A β Tm30 Ξ A)
β Tm30 Ξ A
var30 : β{Ξ A} β Var30 Ξ A β Tm30 Ξ A; var30
= Ξ» x Tm30 var30 lam app tt pair fst snd left right case zero suc rec β
var30 _ _ x
lam30 : β{Ξ A B} β Tm30 (snoc30 Ξ A) B β Tm30 Ξ (arr30 A B); lam30
= Ξ» t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec β
lam30 _ _ _ (t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec)
app30 : β{Ξ A B} β Tm30 Ξ (arr30 A B) β Tm30 Ξ A β Tm30 Ξ B; app30
= Ξ» t u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec β
app30 _ _ _ (t Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
tt30 : β{Ξ} β Tm30 Ξ top30; tt30
= Ξ» Tm30 var30 lam30 app30 tt30 pair fst snd left right case zero suc rec β tt30 _
pair30 : β{Ξ A B} β Tm30 Ξ A β Tm30 Ξ B β Tm30 Ξ (prod30 A B); pair30
= Ξ» t u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec β
pair30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
fst30 : β{Ξ A B} β Tm30 Ξ (prod30 A B) β Tm30 Ξ A; fst30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec β
fst30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec)
snd30 : β{Ξ A B} β Tm30 Ξ (prod30 A B) β Tm30 Ξ B; snd30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec β
snd30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec)
left30 : β{Ξ A B} β Tm30 Ξ A β Tm30 Ξ (sum30 A B); left30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec β
left30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec)
right30 : β{Ξ A B} β Tm30 Ξ B β Tm30 Ξ (sum30 A B); right30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec β
right30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec)
case30 : β{Ξ A B C} β Tm30 Ξ (sum30 A B) β Tm30 Ξ (arr30 A C) β Tm30 Ξ (arr30 B C) β Tm30 Ξ C; case30
= Ξ» t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec β
case30 _ _ _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
zero30 : β{Ξ} β Tm30 Ξ nat30; zero30
= Ξ» Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc rec β zero30 _
suc30 : β{Ξ} β Tm30 Ξ nat30 β Tm30 Ξ nat30; suc30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec β
suc30 _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec)
rec30 : β{Ξ A} β Tm30 Ξ nat30 β Tm30 Ξ (arr30 nat30 (arr30 A A)) β Tm30 Ξ A β Tm30 Ξ A; rec30
= Ξ» t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30 β
rec30 _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
v030 : β{Ξ A} β Tm30 (snoc30 Ξ A) A; v030
= var30 vz30
v130 : β{Ξ A B} β Tm30 (snoc30 (snoc30 Ξ A) B) A; v130
= var30 (vs30 vz30)
v230 : β{Ξ A B C} β Tm30 (snoc30 (snoc30 (snoc30 Ξ A) B) C) A; v230
= var30 (vs30 (vs30 vz30))
v330 : β{Ξ A B C D} β Tm30 (snoc30 (snoc30 (snoc30 (snoc30 Ξ A) B) C) D) A; v330
= var30 (vs30 (vs30 (vs30 vz30)))
tbool30 : Ty30; tbool30
= sum30 top30 top30
true30 : β{Ξ} β Tm30 Ξ tbool30; true30
= left30 tt30
tfalse30 : β{Ξ} β Tm30 Ξ tbool30; tfalse30
= right30 tt30
ifthenelse30 : β{Ξ A} β Tm30 Ξ (arr30 tbool30 (arr30 A (arr30 A A))); ifthenelse30
= lam30 (lam30 (lam30 (case30 v230 (lam30 v230) (lam30 v130))))
times430 : β{Ξ A} β Tm30 Ξ (arr30 (arr30 A A) (arr30 A A)); times430
= lam30 (lam30 (app30 v130 (app30 v130 (app30 v130 (app30 v130 v030)))))
add30 : β{Ξ} β Tm30 Ξ (arr30 nat30 (arr30 nat30 nat30)); add30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (suc30 (app30 v130 v030)))))
(lam30 v030))
mul30 : β{Ξ} β Tm30 Ξ (arr30 nat30 (arr30 nat30 nat30)); mul30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (app30 (app30 add30 (app30 v130 v030)) v030))))
(lam30 zero30))
fact30 : β{Ξ} β Tm30 Ξ (arr30 nat30 nat30); fact30
= lam30 (rec30 v030 (lam30 (lam30 (app30 (app30 mul30 (suc30 v130)) v030)))
(suc30 zero30))
{-# OPTIONS --type-in-type #-}
Ty31 : Set
Ty31 =
(Ty31 : Set)
(nat top bot : Ty31)
(arr prod sum : Ty31 β Ty31 β Ty31)
β Ty31
nat31 : Ty31; nat31 = Ξ» _ nat31 _ _ _ _ _ β nat31
top31 : Ty31; top31 = Ξ» _ _ top31 _ _ _ _ β top31
bot31 : Ty31; bot31 = Ξ» _ _ _ bot31 _ _ _ β bot31
arr31 : Ty31 β Ty31 β Ty31; arr31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod sum β
arr31 (A Ty31 nat31 top31 bot31 arr31 prod sum) (B Ty31 nat31 top31 bot31 arr31 prod sum)
prod31 : Ty31 β Ty31 β Ty31; prod31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod31 sum β
prod31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum) (B Ty31 nat31 top31 bot31 arr31 prod31 sum)
sum31 : Ty31 β Ty31 β Ty31; sum31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod31 sum31 β
sum31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum31) (B Ty31 nat31 top31 bot31 arr31 prod31 sum31)
Con31 : Set; Con31
= (Con31 : Set)
(nil : Con31)
(snoc : Con31 β Ty31 β Con31)
β Con31
nil31 : Con31; nil31
= Ξ» Con31 nil31 snoc β nil31
snoc31 : Con31 β Ty31 β Con31; snoc31
= Ξ» Ξ A Con31 nil31 snoc31 β snoc31 (Ξ Con31 nil31 snoc31) A
Var31 : Con31 β Ty31 β Set; Var31
= Ξ» Ξ A β
(Var31 : Con31 β Ty31 β Set)
(vz : β Ξ A β Var31 (snoc31 Ξ A) A)
(vs : β Ξ B A β Var31 Ξ A β Var31 (snoc31 Ξ B) A)
β Var31 Ξ A
vz31 : β{Ξ A} β Var31 (snoc31 Ξ A) A; vz31
= Ξ» Var31 vz31 vs β vz31 _ _
vs31 : β{Ξ B A} β Var31 Ξ A β Var31 (snoc31 Ξ B) A; vs31
= Ξ» x Var31 vz31 vs31 β vs31 _ _ _ (x Var31 vz31 vs31)
Tm31 : Con31 β Ty31 β Set; Tm31
= Ξ» Ξ A β
(Tm31 : Con31 β Ty31 β Set)
(var : β Ξ A β Var31 Ξ A β Tm31 Ξ A)
(lam : β Ξ A B β Tm31 (snoc31 Ξ A) B β Tm31 Ξ (arr31 A B))
(app : β Ξ A B β Tm31 Ξ (arr31 A B) β Tm31 Ξ A β Tm31 Ξ B)
(tt : β Ξ β Tm31 Ξ top31)
(pair : β Ξ A B β Tm31 Ξ A β Tm31 Ξ B β Tm31 Ξ (prod31 A B))
(fst : β Ξ A B β Tm31 Ξ (prod31 A B) β Tm31 Ξ A)
(snd : β Ξ A B β Tm31 Ξ (prod31 A B) β Tm31 Ξ B)
(left : β Ξ A B β Tm31 Ξ A β Tm31 Ξ (sum31 A B))
(right : β Ξ A B β Tm31 Ξ B β Tm31 Ξ (sum31 A B))
(case : β Ξ A B C β Tm31 Ξ (sum31 A B) β Tm31 Ξ (arr31 A C) β Tm31 Ξ (arr31 B C) β Tm31 Ξ C)
(zero : β Ξ β Tm31 Ξ nat31)
(suc : β Ξ β Tm31 Ξ nat31 β Tm31 Ξ nat31)
(rec : β Ξ A β Tm31 Ξ nat31 β Tm31 Ξ (arr31 nat31 (arr31 A A)) β Tm31 Ξ A β Tm31 Ξ A)
β Tm31 Ξ A
var31 : β{Ξ A} β Var31 Ξ A β Tm31 Ξ A; var31
= Ξ» x Tm31 var31 lam app tt pair fst snd left right case zero suc rec β
var31 _ _ x
lam31 : β{Ξ A B} β Tm31 (snoc31 Ξ A) B β Tm31 Ξ (arr31 A B); lam31
= Ξ» t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec β
lam31 _ _ _ (t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec)
app31 : β{Ξ A B} β Tm31 Ξ (arr31 A B) β Tm31 Ξ A β Tm31 Ξ B; app31
= Ξ» t u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec β
app31 _ _ _ (t Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
tt31 : β{Ξ} β Tm31 Ξ top31; tt31
= Ξ» Tm31 var31 lam31 app31 tt31 pair fst snd left right case zero suc rec β tt31 _
pair31 : β{Ξ A B} β Tm31 Ξ A β Tm31 Ξ B β Tm31 Ξ (prod31 A B); pair31
= Ξ» t u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec β
pair31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
fst31 : β{Ξ A B} β Tm31 Ξ (prod31 A B) β Tm31 Ξ A; fst31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec β
fst31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec)
snd31 : β{Ξ A B} β Tm31 Ξ (prod31 A B) β Tm31 Ξ B; snd31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec β
snd31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec)
left31 : β{Ξ A B} β Tm31 Ξ A β Tm31 Ξ (sum31 A B); left31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec β
left31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec)
right31 : β{Ξ A B} β Tm31 Ξ B β Tm31 Ξ (sum31 A B); right31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec β
right31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec)
case31 : β{Ξ A B C} β Tm31 Ξ (sum31 A B) β Tm31 Ξ (arr31 A C) β Tm31 Ξ (arr31 B C) β Tm31 Ξ C; case31
= Ξ» t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec β
case31 _ _ _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
zero31 : β{Ξ} β Tm31 Ξ nat31; zero31
= Ξ» Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc rec β zero31 _
suc31 : β{Ξ} β Tm31 Ξ nat31 β Tm31 Ξ nat31; suc31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec β
suc31 _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec)
rec31 : β{Ξ A} β Tm31 Ξ nat31 β Tm31 Ξ (arr31 nat31 (arr31 A A)) β Tm31 Ξ A β Tm31 Ξ A; rec31
= Ξ» t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31 β
rec31 _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
v031 : β{Ξ A} β Tm31 (snoc31 Ξ A) A; v031
= var31 vz31
v131 : β{Ξ A B} β Tm31 (snoc31 (snoc31 Ξ A) B) A; v131
= var31 (vs31 vz31)
v231 : β{Ξ A B C} β Tm31 (snoc31 (snoc31 (snoc31 Ξ A) B) C) A; v231
= var31 (vs31 (vs31 vz31))
v331 : β{Ξ A B C D} β Tm31 (snoc31 (snoc31 (snoc31 (snoc31 Ξ A) B) C) D) A; v331
= var31 (vs31 (vs31 (vs31 vz31)))
tbool31 : Ty31; tbool31
= sum31 top31 top31
true31 : β{Ξ} β Tm31 Ξ tbool31; true31
= left31 tt31
tfalse31 : β{Ξ} β Tm31 Ξ tbool31; tfalse31
= right31 tt31
ifthenelse31 : β{Ξ A} β Tm31 Ξ (arr31 tbool31 (arr31 A (arr31 A A))); ifthenelse31
= lam31 (lam31 (lam31 (case31 v231 (lam31 v231) (lam31 v131))))
times431 : β{Ξ A} β Tm31 Ξ (arr31 (arr31 A A) (arr31 A A)); times431
= lam31 (lam31 (app31 v131 (app31 v131 (app31 v131 (app31 v131 v031)))))
add31 : β{Ξ} β Tm31 Ξ (arr31 nat31 (arr31 nat31 nat31)); add31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (suc31 (app31 v131 v031)))))
(lam31 v031))
mul31 : β{Ξ} β Tm31 Ξ (arr31 nat31 (arr31 nat31 nat31)); mul31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (app31 (app31 add31 (app31 v131 v031)) v031))))
(lam31 zero31))
fact31 : β{Ξ} β Tm31 Ξ (arr31 nat31 nat31); fact31
= lam31 (rec31 v031 (lam31 (lam31 (app31 (app31 mul31 (suc31 v131)) v031)))
(suc31 zero31))
{-# OPTIONS --type-in-type #-}
Ty32 : Set
Ty32 =
(Ty32 : Set)
(nat top bot : Ty32)
(arr prod sum : Ty32 β Ty32 β Ty32)
β Ty32
nat32 : Ty32; nat32 = Ξ» _ nat32 _ _ _ _ _ β nat32
top32 : Ty32; top32 = Ξ» _ _ top32 _ _ _ _ β top32
bot32 : Ty32; bot32 = Ξ» _ _ _ bot32 _ _ _ β bot32
arr32 : Ty32 β Ty32 β Ty32; arr32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod sum β
arr32 (A Ty32 nat32 top32 bot32 arr32 prod sum) (B Ty32 nat32 top32 bot32 arr32 prod sum)
prod32 : Ty32 β Ty32 β Ty32; prod32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod32 sum β
prod32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum) (B Ty32 nat32 top32 bot32 arr32 prod32 sum)
sum32 : Ty32 β Ty32 β Ty32; sum32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod32 sum32 β
sum32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum32) (B Ty32 nat32 top32 bot32 arr32 prod32 sum32)
Con32 : Set; Con32
= (Con32 : Set)
(nil : Con32)
(snoc : Con32 β Ty32 β Con32)
β Con32
nil32 : Con32; nil32
= Ξ» Con32 nil32 snoc β nil32
snoc32 : Con32 β Ty32 β Con32; snoc32
= Ξ» Ξ A Con32 nil32 snoc32 β snoc32 (Ξ Con32 nil32 snoc32) A
Var32 : Con32 β Ty32 β Set; Var32
= Ξ» Ξ A β
(Var32 : Con32 β Ty32 β Set)
(vz : β Ξ A β Var32 (snoc32 Ξ A) A)
(vs : β Ξ B A β Var32 Ξ A β Var32 (snoc32 Ξ B) A)
β Var32 Ξ A
vz32 : β{Ξ A} β Var32 (snoc32 Ξ A) A; vz32
= Ξ» Var32 vz32 vs β vz32 _ _
vs32 : β{Ξ B A} β Var32 Ξ A β Var32 (snoc32 Ξ B) A; vs32
= Ξ» x Var32 vz32 vs32 β vs32 _ _ _ (x Var32 vz32 vs32)
Tm32 : Con32 β Ty32 β Set; Tm32
= Ξ» Ξ A β
(Tm32 : Con32 β Ty32 β Set)
(var : β Ξ A β Var32 Ξ A β Tm32 Ξ A)
(lam : β Ξ A B β Tm32 (snoc32 Ξ A) B β Tm32 Ξ (arr32 A B))
(app : β Ξ A B β Tm32 Ξ (arr32 A B) β Tm32 Ξ A β Tm32 Ξ B)
(tt : β Ξ β Tm32 Ξ top32)
(pair : β Ξ A B β Tm32 Ξ A β Tm32 Ξ B β Tm32 Ξ (prod32 A B))
(fst : β Ξ A B β Tm32 Ξ (prod32 A B) β Tm32 Ξ A)
(snd : β Ξ A B β Tm32 Ξ (prod32 A B) β Tm32 Ξ B)
(left : β Ξ A B β Tm32 Ξ A β Tm32 Ξ (sum32 A B))
(right : β Ξ A B β Tm32 Ξ B β Tm32 Ξ (sum32 A B))
(case : β Ξ A B C β Tm32 Ξ (sum32 A B) β Tm32 Ξ (arr32 A C) β Tm32 Ξ (arr32 B C) β Tm32 Ξ C)
(zero : β Ξ β Tm32 Ξ nat32)
(suc : β Ξ β Tm32 Ξ nat32 β Tm32 Ξ nat32)
(rec : β Ξ A β Tm32 Ξ nat32 β Tm32 Ξ (arr32 nat32 (arr32 A A)) β Tm32 Ξ A β Tm32 Ξ A)
β Tm32 Ξ A
var32 : β{Ξ A} β Var32 Ξ A β Tm32 Ξ A; var32
= Ξ» x Tm32 var32 lam app tt pair fst snd left right case zero suc rec β
var32 _ _ x
lam32 : β{Ξ A B} β Tm32 (snoc32 Ξ A) B β Tm32 Ξ (arr32 A B); lam32
= Ξ» t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec β
lam32 _ _ _ (t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec)
app32 : β{Ξ A B} β Tm32 Ξ (arr32 A B) β Tm32 Ξ A β Tm32 Ξ B; app32
= Ξ» t u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec β
app32 _ _ _ (t Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
tt32 : β{Ξ} β Tm32 Ξ top32; tt32
= Ξ» Tm32 var32 lam32 app32 tt32 pair fst snd left right case zero suc rec β tt32 _
pair32 : β{Ξ A B} β Tm32 Ξ A β Tm32 Ξ B β Tm32 Ξ (prod32 A B); pair32
= Ξ» t u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec β
pair32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
fst32 : β{Ξ A B} β Tm32 Ξ (prod32 A B) β Tm32 Ξ A; fst32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec β
fst32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec)
snd32 : β{Ξ A B} β Tm32 Ξ (prod32 A B) β Tm32 Ξ B; snd32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec β
snd32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec)
left32 : β{Ξ A B} β Tm32 Ξ A β Tm32 Ξ (sum32 A B); left32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec β
left32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec)
right32 : β{Ξ A B} β Tm32 Ξ B β Tm32 Ξ (sum32 A B); right32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec β
right32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec)
case32 : β{Ξ A B C} β Tm32 Ξ (sum32 A B) β Tm32 Ξ (arr32 A C) β Tm32 Ξ (arr32 B C) β Tm32 Ξ C; case32
= Ξ» t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec β
case32 _ _ _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
zero32 : β{Ξ} β Tm32 Ξ nat32; zero32
= Ξ» Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc rec β zero32 _
suc32 : β{Ξ} β Tm32 Ξ nat32 β Tm32 Ξ nat32; suc32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec β
suc32 _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec)
rec32 : β{Ξ A} β Tm32 Ξ nat32 β Tm32 Ξ (arr32 nat32 (arr32 A A)) β Tm32 Ξ A β Tm32 Ξ A; rec32
= Ξ» t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32 β
rec32 _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
v032 : β{Ξ A} β Tm32 (snoc32 Ξ A) A; v032
= var32 vz32
v132 : β{Ξ A B} β Tm32 (snoc32 (snoc32 Ξ A) B) A; v132
= var32 (vs32 vz32)
v232 : β{Ξ A B C} β Tm32 (snoc32 (snoc32 (snoc32 Ξ A) B) C) A; v232
= var32 (vs32 (vs32 vz32))
v332 : β{Ξ A B C D} β Tm32 (snoc32 (snoc32 (snoc32 (snoc32 Ξ A) B) C) D) A; v332
= var32 (vs32 (vs32 (vs32 vz32)))
tbool32 : Ty32; tbool32
= sum32 top32 top32
true32 : β{Ξ} β Tm32 Ξ tbool32; true32
= left32 tt32
tfalse32 : β{Ξ} β Tm32 Ξ tbool32; tfalse32
= right32 tt32
ifthenelse32 : β{Ξ A} β Tm32 Ξ (arr32 tbool32 (arr32 A (arr32 A A))); ifthenelse32
= lam32 (lam32 (lam32 (case32 v232 (lam32 v232) (lam32 v132))))
times432 : β{Ξ A} β Tm32 Ξ (arr32 (arr32 A A) (arr32 A A)); times432
= lam32 (lam32 (app32 v132 (app32 v132 (app32 v132 (app32 v132 v032)))))
add32 : β{Ξ} β Tm32 Ξ (arr32 nat32 (arr32 nat32 nat32)); add32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (suc32 (app32 v132 v032)))))
(lam32 v032))
mul32 : β{Ξ} β Tm32 Ξ (arr32 nat32 (arr32 nat32 nat32)); mul32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (app32 (app32 add32 (app32 v132 v032)) v032))))
(lam32 zero32))
fact32 : β{Ξ} β Tm32 Ξ (arr32 nat32 nat32); fact32
= lam32 (rec32 v032 (lam32 (lam32 (app32 (app32 mul32 (suc32 v132)) v032)))
(suc32 zero32))
{-# OPTIONS --type-in-type #-}
Ty33 : Set
Ty33 =
(Ty33 : Set)
(nat top bot : Ty33)
(arr prod sum : Ty33 β Ty33 β Ty33)
β Ty33
nat33 : Ty33; nat33 = Ξ» _ nat33 _ _ _ _ _ β nat33
top33 : Ty33; top33 = Ξ» _ _ top33 _ _ _ _ β top33
bot33 : Ty33; bot33 = Ξ» _ _ _ bot33 _ _ _ β bot33
arr33 : Ty33 β Ty33 β Ty33; arr33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod sum β
arr33 (A Ty33 nat33 top33 bot33 arr33 prod sum) (B Ty33 nat33 top33 bot33 arr33 prod sum)
prod33 : Ty33 β Ty33 β Ty33; prod33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod33 sum β
prod33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum) (B Ty33 nat33 top33 bot33 arr33 prod33 sum)
sum33 : Ty33 β Ty33 β Ty33; sum33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod33 sum33 β
sum33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum33) (B Ty33 nat33 top33 bot33 arr33 prod33 sum33)
Con33 : Set; Con33
= (Con33 : Set)
(nil : Con33)
(snoc : Con33 β Ty33 β Con33)
β Con33
nil33 : Con33; nil33
= Ξ» Con33 nil33 snoc β nil33
snoc33 : Con33 β Ty33 β Con33; snoc33
= Ξ» Ξ A Con33 nil33 snoc33 β snoc33 (Ξ Con33 nil33 snoc33) A
Var33 : Con33 β Ty33 β Set; Var33
= Ξ» Ξ A β
(Var33 : Con33 β Ty33 β Set)
(vz : β Ξ A β Var33 (snoc33 Ξ A) A)
(vs : β Ξ B A β Var33 Ξ A β Var33 (snoc33 Ξ B) A)
β Var33 Ξ A
vz33 : β{Ξ A} β Var33 (snoc33 Ξ A) A; vz33
= Ξ» Var33 vz33 vs β vz33 _ _
vs33 : β{Ξ B A} β Var33 Ξ A β Var33 (snoc33 Ξ B) A; vs33
= Ξ» x Var33 vz33 vs33 β vs33 _ _ _ (x Var33 vz33 vs33)
Tm33 : Con33 β Ty33 β Set; Tm33
= Ξ» Ξ A β
(Tm33 : Con33 β Ty33 β Set)
(var : β Ξ A β Var33 Ξ A β Tm33 Ξ A)
(lam : β Ξ A B β Tm33 (snoc33 Ξ A) B β Tm33 Ξ (arr33 A B))
(app : β Ξ A B β Tm33 Ξ (arr33 A B) β Tm33 Ξ A β Tm33 Ξ B)
(tt : β Ξ β Tm33 Ξ top33)
(pair : β Ξ A B β Tm33 Ξ A β Tm33 Ξ B β Tm33 Ξ (prod33 A B))
(fst : β Ξ A B β Tm33 Ξ (prod33 A B) β Tm33 Ξ A)
(snd : β Ξ A B β Tm33 Ξ (prod33 A B) β Tm33 Ξ B)
(left : β Ξ A B β Tm33 Ξ A β Tm33 Ξ (sum33 A B))
(right : β Ξ A B β Tm33 Ξ B β Tm33 Ξ (sum33 A B))
(case : β Ξ A B C β Tm33 Ξ (sum33 A B) β Tm33 Ξ (arr33 A C) β Tm33 Ξ (arr33 B C) β Tm33 Ξ C)
(zero : β Ξ β Tm33 Ξ nat33)
(suc : β Ξ β Tm33 Ξ nat33 β Tm33 Ξ nat33)
(rec : β Ξ A β Tm33 Ξ nat33 β Tm33 Ξ (arr33 nat33 (arr33 A A)) β Tm33 Ξ A β Tm33 Ξ A)
β Tm33 Ξ A
var33 : β{Ξ A} β Var33 Ξ A β Tm33 Ξ A; var33
= Ξ» x Tm33 var33 lam app tt pair fst snd left right case zero suc rec β
var33 _ _ x
lam33 : β{Ξ A B} β Tm33 (snoc33 Ξ A) B β Tm33 Ξ (arr33 A B); lam33
= Ξ» t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec β
lam33 _ _ _ (t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec)
app33 : β{Ξ A B} β Tm33 Ξ (arr33 A B) β Tm33 Ξ A β Tm33 Ξ B; app33
= Ξ» t u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec β
app33 _ _ _ (t Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
tt33 : β{Ξ} β Tm33 Ξ top33; tt33
= Ξ» Tm33 var33 lam33 app33 tt33 pair fst snd left right case zero suc rec β tt33 _
pair33 : β{Ξ A B} β Tm33 Ξ A β Tm33 Ξ B β Tm33 Ξ (prod33 A B); pair33
= Ξ» t u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec β
pair33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
fst33 : β{Ξ A B} β Tm33 Ξ (prod33 A B) β Tm33 Ξ A; fst33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec β
fst33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec)
snd33 : β{Ξ A B} β Tm33 Ξ (prod33 A B) β Tm33 Ξ B; snd33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec β
snd33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec)
left33 : β{Ξ A B} β Tm33 Ξ A β Tm33 Ξ (sum33 A B); left33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec β
left33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec)
right33 : β{Ξ A B} β Tm33 Ξ B β Tm33 Ξ (sum33 A B); right33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec β
right33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec)
case33 : β{Ξ A B C} β Tm33 Ξ (sum33 A B) β Tm33 Ξ (arr33 A C) β Tm33 Ξ (arr33 B C) β Tm33 Ξ C; case33
= Ξ» t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec β
case33 _ _ _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
zero33 : β{Ξ} β Tm33 Ξ nat33; zero33
= Ξ» Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc rec β zero33 _
suc33 : β{Ξ} β Tm33 Ξ nat33 β Tm33 Ξ nat33; suc33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec β
suc33 _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec)
rec33 : β{Ξ A} β Tm33 Ξ nat33 β Tm33 Ξ (arr33 nat33 (arr33 A A)) β Tm33 Ξ A β Tm33 Ξ A; rec33
= Ξ» t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33 β
rec33 _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
v033 : β{Ξ A} β Tm33 (snoc33 Ξ A) A; v033
= var33 vz33
v133 : β{Ξ A B} β Tm33 (snoc33 (snoc33 Ξ A) B) A; v133
= var33 (vs33 vz33)
v233 : β{Ξ A B C} β Tm33 (snoc33 (snoc33 (snoc33 Ξ A) B) C) A; v233
= var33 (vs33 (vs33 vz33))
v333 : β{Ξ A B C D} β Tm33 (snoc33 (snoc33 (snoc33 (snoc33 Ξ A) B) C) D) A; v333
= var33 (vs33 (vs33 (vs33 vz33)))
tbool33 : Ty33; tbool33
= sum33 top33 top33
true33 : β{Ξ} β Tm33 Ξ tbool33; true33
= left33 tt33
tfalse33 : β{Ξ} β Tm33 Ξ tbool33; tfalse33
= right33 tt33
ifthenelse33 : β{Ξ A} β Tm33 Ξ (arr33 tbool33 (arr33 A (arr33 A A))); ifthenelse33
= lam33 (lam33 (lam33 (case33 v233 (lam33 v233) (lam33 v133))))
times433 : β{Ξ A} β Tm33 Ξ (arr33 (arr33 A A) (arr33 A A)); times433
= lam33 (lam33 (app33 v133 (app33 v133 (app33 v133 (app33 v133 v033)))))
add33 : β{Ξ} β Tm33 Ξ (arr33 nat33 (arr33 nat33 nat33)); add33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (suc33 (app33 v133 v033)))))
(lam33 v033))
mul33 : β{Ξ} β Tm33 Ξ (arr33 nat33 (arr33 nat33 nat33)); mul33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (app33 (app33 add33 (app33 v133 v033)) v033))))
(lam33 zero33))
fact33 : β{Ξ} β Tm33 Ξ (arr33 nat33 nat33); fact33
= lam33 (rec33 v033 (lam33 (lam33 (app33 (app33 mul33 (suc33 v133)) v033)))
(suc33 zero33))
{-# OPTIONS --type-in-type #-}
Ty34 : Set
Ty34 =
(Ty34 : Set)
(nat top bot : Ty34)
(arr prod sum : Ty34 β Ty34 β Ty34)
β Ty34
nat34 : Ty34; nat34 = Ξ» _ nat34 _ _ _ _ _ β nat34
top34 : Ty34; top34 = Ξ» _ _ top34 _ _ _ _ β top34
bot34 : Ty34; bot34 = Ξ» _ _ _ bot34 _ _ _ β bot34
arr34 : Ty34 β Ty34 β Ty34; arr34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod sum β
arr34 (A Ty34 nat34 top34 bot34 arr34 prod sum) (B Ty34 nat34 top34 bot34 arr34 prod sum)
prod34 : Ty34 β Ty34 β Ty34; prod34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod34 sum β
prod34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum) (B Ty34 nat34 top34 bot34 arr34 prod34 sum)
sum34 : Ty34 β Ty34 β Ty34; sum34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod34 sum34 β
sum34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum34) (B Ty34 nat34 top34 bot34 arr34 prod34 sum34)
Con34 : Set; Con34
= (Con34 : Set)
(nil : Con34)
(snoc : Con34 β Ty34 β Con34)
β Con34
nil34 : Con34; nil34
= Ξ» Con34 nil34 snoc β nil34
snoc34 : Con34 β Ty34 β Con34; snoc34
= Ξ» Ξ A Con34 nil34 snoc34 β snoc34 (Ξ Con34 nil34 snoc34) A
Var34 : Con34 β Ty34 β Set; Var34
= Ξ» Ξ A β
(Var34 : Con34 β Ty34 β Set)
(vz : β Ξ A β Var34 (snoc34 Ξ A) A)
(vs : β Ξ B A β Var34 Ξ A β Var34 (snoc34 Ξ B) A)
β Var34 Ξ A
vz34 : β{Ξ A} β Var34 (snoc34 Ξ A) A; vz34
= Ξ» Var34 vz34 vs β vz34 _ _
vs34 : β{Ξ B A} β Var34 Ξ A β Var34 (snoc34 Ξ B) A; vs34
= Ξ» x Var34 vz34 vs34 β vs34 _ _ _ (x Var34 vz34 vs34)
Tm34 : Con34 β Ty34 β Set; Tm34
= Ξ» Ξ A β
(Tm34 : Con34 β Ty34 β Set)
(var : β Ξ A β Var34 Ξ A β Tm34 Ξ A)
(lam : β Ξ A B β Tm34 (snoc34 Ξ A) B β Tm34 Ξ (arr34 A B))
(app : β Ξ A B β Tm34 Ξ (arr34 A B) β Tm34 Ξ A β Tm34 Ξ B)
(tt : β Ξ β Tm34 Ξ top34)
(pair : β Ξ A B β Tm34 Ξ A β Tm34 Ξ B β Tm34 Ξ (prod34 A B))
(fst : β Ξ A B β Tm34 Ξ (prod34 A B) β Tm34 Ξ A)
(snd : β Ξ A B β Tm34 Ξ (prod34 A B) β Tm34 Ξ B)
(left : β Ξ A B β Tm34 Ξ A β Tm34 Ξ (sum34 A B))
(right : β Ξ A B β Tm34 Ξ B β Tm34 Ξ (sum34 A B))
(case : β Ξ A B C β Tm34 Ξ (sum34 A B) β Tm34 Ξ (arr34 A C) β Tm34 Ξ (arr34 B C) β Tm34 Ξ C)
(zero : β Ξ β Tm34 Ξ nat34)
(suc : β Ξ β Tm34 Ξ nat34 β Tm34 Ξ nat34)
(rec : β Ξ A β Tm34 Ξ nat34 β Tm34 Ξ (arr34 nat34 (arr34 A A)) β Tm34 Ξ A β Tm34 Ξ A)
β Tm34 Ξ A
var34 : β{Ξ A} β Var34 Ξ A β Tm34 Ξ A; var34
= Ξ» x Tm34 var34 lam app tt pair fst snd left right case zero suc rec β
var34 _ _ x
lam34 : β{Ξ A B} β Tm34 (snoc34 Ξ A) B β Tm34 Ξ (arr34 A B); lam34
= Ξ» t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec β
lam34 _ _ _ (t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec)
app34 : β{Ξ A B} β Tm34 Ξ (arr34 A B) β Tm34 Ξ A β Tm34 Ξ B; app34
= Ξ» t u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec β
app34 _ _ _ (t Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
tt34 : β{Ξ} β Tm34 Ξ top34; tt34
= Ξ» Tm34 var34 lam34 app34 tt34 pair fst snd left right case zero suc rec β tt34 _
pair34 : β{Ξ A B} β Tm34 Ξ A β Tm34 Ξ B β Tm34 Ξ (prod34 A B); pair34
= Ξ» t u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec β
pair34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
fst34 : β{Ξ A B} β Tm34 Ξ (prod34 A B) β Tm34 Ξ A; fst34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec β
fst34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec)
snd34 : β{Ξ A B} β Tm34 Ξ (prod34 A B) β Tm34 Ξ B; snd34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec β
snd34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec)
left34 : β{Ξ A B} β Tm34 Ξ A β Tm34 Ξ (sum34 A B); left34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec β
left34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec)
right34 : β{Ξ A B} β Tm34 Ξ B β Tm34 Ξ (sum34 A B); right34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec β
right34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec)
case34 : β{Ξ A B C} β Tm34 Ξ (sum34 A B) β Tm34 Ξ (arr34 A C) β Tm34 Ξ (arr34 B C) β Tm34 Ξ C; case34
= Ξ» t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec β
case34 _ _ _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
zero34 : β{Ξ} β Tm34 Ξ nat34; zero34
= Ξ» Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc rec β zero34 _
suc34 : β{Ξ} β Tm34 Ξ nat34 β Tm34 Ξ nat34; suc34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec β
suc34 _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec)
rec34 : β{Ξ A} β Tm34 Ξ nat34 β Tm34 Ξ (arr34 nat34 (arr34 A A)) β Tm34 Ξ A β Tm34 Ξ A; rec34
= Ξ» t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34 β
rec34 _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
v034 : β{Ξ A} β Tm34 (snoc34 Ξ A) A; v034
= var34 vz34
v134 : β{Ξ A B} β Tm34 (snoc34 (snoc34 Ξ A) B) A; v134
= var34 (vs34 vz34)
v234 : β{Ξ A B C} β Tm34 (snoc34 (snoc34 (snoc34 Ξ A) B) C) A; v234
= var34 (vs34 (vs34 vz34))
v334 : β{Ξ A B C D} β Tm34 (snoc34 (snoc34 (snoc34 (snoc34 Ξ A) B) C) D) A; v334
= var34 (vs34 (vs34 (vs34 vz34)))
tbool34 : Ty34; tbool34
= sum34 top34 top34
true34 : β{Ξ} β Tm34 Ξ tbool34; true34
= left34 tt34
tfalse34 : β{Ξ} β Tm34 Ξ tbool34; tfalse34
= right34 tt34
ifthenelse34 : β{Ξ A} β Tm34 Ξ (arr34 tbool34 (arr34 A (arr34 A A))); ifthenelse34
= lam34 (lam34 (lam34 (case34 v234 (lam34 v234) (lam34 v134))))
times434 : β{Ξ A} β Tm34 Ξ (arr34 (arr34 A A) (arr34 A A)); times434
= lam34 (lam34 (app34 v134 (app34 v134 (app34 v134 (app34 v134 v034)))))
add34 : β{Ξ} β Tm34 Ξ (arr34 nat34 (arr34 nat34 nat34)); add34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (suc34 (app34 v134 v034)))))
(lam34 v034))
mul34 : β{Ξ} β Tm34 Ξ (arr34 nat34 (arr34 nat34 nat34)); mul34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (app34 (app34 add34 (app34 v134 v034)) v034))))
(lam34 zero34))
fact34 : β{Ξ} β Tm34 Ξ (arr34 nat34 nat34); fact34
= lam34 (rec34 v034 (lam34 (lam34 (app34 (app34 mul34 (suc34 v134)) v034)))
(suc34 zero34))
{-# OPTIONS --type-in-type #-}
Ty35 : Set
Ty35 =
(Ty35 : Set)
(nat top bot : Ty35)
(arr prod sum : Ty35 β Ty35 β Ty35)
β Ty35
nat35 : Ty35; nat35 = Ξ» _ nat35 _ _ _ _ _ β nat35
top35 : Ty35; top35 = Ξ» _ _ top35 _ _ _ _ β top35
bot35 : Ty35; bot35 = Ξ» _ _ _ bot35 _ _ _ β bot35
arr35 : Ty35 β Ty35 β Ty35; arr35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod sum β
arr35 (A Ty35 nat35 top35 bot35 arr35 prod sum) (B Ty35 nat35 top35 bot35 arr35 prod sum)
prod35 : Ty35 β Ty35 β Ty35; prod35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod35 sum β
prod35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum) (B Ty35 nat35 top35 bot35 arr35 prod35 sum)
sum35 : Ty35 β Ty35 β Ty35; sum35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod35 sum35 β
sum35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum35) (B Ty35 nat35 top35 bot35 arr35 prod35 sum35)
Con35 : Set; Con35
= (Con35 : Set)
(nil : Con35)
(snoc : Con35 β Ty35 β Con35)
β Con35
nil35 : Con35; nil35
= Ξ» Con35 nil35 snoc β nil35
snoc35 : Con35 β Ty35 β Con35; snoc35
= Ξ» Ξ A Con35 nil35 snoc35 β snoc35 (Ξ Con35 nil35 snoc35) A
Var35 : Con35 β Ty35 β Set; Var35
= Ξ» Ξ A β
(Var35 : Con35 β Ty35 β Set)
(vz : β Ξ A β Var35 (snoc35 Ξ A) A)
(vs : β Ξ B A β Var35 Ξ A β Var35 (snoc35 Ξ B) A)
β Var35 Ξ A
vz35 : β{Ξ A} β Var35 (snoc35 Ξ A) A; vz35
= Ξ» Var35 vz35 vs β vz35 _ _
vs35 : β{Ξ B A} β Var35 Ξ A β Var35 (snoc35 Ξ B) A; vs35
= Ξ» x Var35 vz35 vs35 β vs35 _ _ _ (x Var35 vz35 vs35)
Tm35 : Con35 β Ty35 β Set; Tm35
= Ξ» Ξ A β
(Tm35 : Con35 β Ty35 β Set)
(var : β Ξ A β Var35 Ξ A β Tm35 Ξ A)
(lam : β Ξ A B β Tm35 (snoc35 Ξ A) B β Tm35 Ξ (arr35 A B))
(app : β Ξ A B β Tm35 Ξ (arr35 A B) β Tm35 Ξ A β Tm35 Ξ B)
(tt : β Ξ β Tm35 Ξ top35)
(pair : β Ξ A B β Tm35 Ξ A β Tm35 Ξ B β Tm35 Ξ (prod35 A B))
(fst : β Ξ A B β Tm35 Ξ (prod35 A B) β Tm35 Ξ A)
(snd : β Ξ A B β Tm35 Ξ (prod35 A B) β Tm35 Ξ B)
(left : β Ξ A B β Tm35 Ξ A β Tm35 Ξ (sum35 A B))
(right : β Ξ A B β Tm35 Ξ B β Tm35 Ξ (sum35 A B))
(case : β Ξ A B C β Tm35 Ξ (sum35 A B) β Tm35 Ξ (arr35 A C) β Tm35 Ξ (arr35 B C) β Tm35 Ξ C)
(zero : β Ξ β Tm35 Ξ nat35)
(suc : β Ξ β Tm35 Ξ nat35 β Tm35 Ξ nat35)
(rec : β Ξ A β Tm35 Ξ nat35 β Tm35 Ξ (arr35 nat35 (arr35 A A)) β Tm35 Ξ A β Tm35 Ξ A)
β Tm35 Ξ A
var35 : β{Ξ A} β Var35 Ξ A β Tm35 Ξ A; var35
= Ξ» x Tm35 var35 lam app tt pair fst snd left right case zero suc rec β
var35 _ _ x
lam35 : β{Ξ A B} β Tm35 (snoc35 Ξ A) B β Tm35 Ξ (arr35 A B); lam35
= Ξ» t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec β
lam35 _ _ _ (t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec)
app35 : β{Ξ A B} β Tm35 Ξ (arr35 A B) β Tm35 Ξ A β Tm35 Ξ B; app35
= Ξ» t u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec β
app35 _ _ _ (t Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
tt35 : β{Ξ} β Tm35 Ξ top35; tt35
= Ξ» Tm35 var35 lam35 app35 tt35 pair fst snd left right case zero suc rec β tt35 _
pair35 : β{Ξ A B} β Tm35 Ξ A β Tm35 Ξ B β Tm35 Ξ (prod35 A B); pair35
= Ξ» t u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec β
pair35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
fst35 : β{Ξ A B} β Tm35 Ξ (prod35 A B) β Tm35 Ξ A; fst35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec β
fst35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec)
snd35 : β{Ξ A B} β Tm35 Ξ (prod35 A B) β Tm35 Ξ B; snd35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec β
snd35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec)
left35 : β{Ξ A B} β Tm35 Ξ A β Tm35 Ξ (sum35 A B); left35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec β
left35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec)
right35 : β{Ξ A B} β Tm35 Ξ B β Tm35 Ξ (sum35 A B); right35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec β
right35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec)
case35 : β{Ξ A B C} β Tm35 Ξ (sum35 A B) β Tm35 Ξ (arr35 A C) β Tm35 Ξ (arr35 B C) β Tm35 Ξ C; case35
= Ξ» t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec β
case35 _ _ _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
zero35 : β{Ξ} β Tm35 Ξ nat35; zero35
= Ξ» Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc rec β zero35 _
suc35 : β{Ξ} β Tm35 Ξ nat35 β Tm35 Ξ nat35; suc35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec β
suc35 _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec)
rec35 : β{Ξ A} β Tm35 Ξ nat35 β Tm35 Ξ (arr35 nat35 (arr35 A A)) β Tm35 Ξ A β Tm35 Ξ A; rec35
= Ξ» t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35 β
rec35 _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
v035 : β{Ξ A} β Tm35 (snoc35 Ξ A) A; v035
= var35 vz35
v135 : β{Ξ A B} β Tm35 (snoc35 (snoc35 Ξ A) B) A; v135
= var35 (vs35 vz35)
v235 : β{Ξ A B C} β Tm35 (snoc35 (snoc35 (snoc35 Ξ A) B) C) A; v235
= var35 (vs35 (vs35 vz35))
v335 : β{Ξ A B C D} β Tm35 (snoc35 (snoc35 (snoc35 (snoc35 Ξ A) B) C) D) A; v335
= var35 (vs35 (vs35 (vs35 vz35)))
tbool35 : Ty35; tbool35
= sum35 top35 top35
true35 : β{Ξ} β Tm35 Ξ tbool35; true35
= left35 tt35
tfalse35 : β{Ξ} β Tm35 Ξ tbool35; tfalse35
= right35 tt35
ifthenelse35 : β{Ξ A} β Tm35 Ξ (arr35 tbool35 (arr35 A (arr35 A A))); ifthenelse35
= lam35 (lam35 (lam35 (case35 v235 (lam35 v235) (lam35 v135))))
times435 : β{Ξ A} β Tm35 Ξ (arr35 (arr35 A A) (arr35 A A)); times435
= lam35 (lam35 (app35 v135 (app35 v135 (app35 v135 (app35 v135 v035)))))
add35 : β{Ξ} β Tm35 Ξ (arr35 nat35 (arr35 nat35 nat35)); add35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (suc35 (app35 v135 v035)))))
(lam35 v035))
mul35 : β{Ξ} β Tm35 Ξ (arr35 nat35 (arr35 nat35 nat35)); mul35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (app35 (app35 add35 (app35 v135 v035)) v035))))
(lam35 zero35))
fact35 : β{Ξ} β Tm35 Ξ (arr35 nat35 nat35); fact35
= lam35 (rec35 v035 (lam35 (lam35 (app35 (app35 mul35 (suc35 v135)) v035)))
(suc35 zero35))
{-# OPTIONS --type-in-type #-}
Ty36 : Set
Ty36 =
(Ty36 : Set)
(nat top bot : Ty36)
(arr prod sum : Ty36 β Ty36 β Ty36)
β Ty36
nat36 : Ty36; nat36 = Ξ» _ nat36 _ _ _ _ _ β nat36
top36 : Ty36; top36 = Ξ» _ _ top36 _ _ _ _ β top36
bot36 : Ty36; bot36 = Ξ» _ _ _ bot36 _ _ _ β bot36
arr36 : Ty36 β Ty36 β Ty36; arr36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod sum β
arr36 (A Ty36 nat36 top36 bot36 arr36 prod sum) (B Ty36 nat36 top36 bot36 arr36 prod sum)
prod36 : Ty36 β Ty36 β Ty36; prod36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod36 sum β
prod36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum) (B Ty36 nat36 top36 bot36 arr36 prod36 sum)
sum36 : Ty36 β Ty36 β Ty36; sum36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod36 sum36 β
sum36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum36) (B Ty36 nat36 top36 bot36 arr36 prod36 sum36)
Con36 : Set; Con36
= (Con36 : Set)
(nil : Con36)
(snoc : Con36 β Ty36 β Con36)
β Con36
nil36 : Con36; nil36
= Ξ» Con36 nil36 snoc β nil36
snoc36 : Con36 β Ty36 β Con36; snoc36
= Ξ» Ξ A Con36 nil36 snoc36 β snoc36 (Ξ Con36 nil36 snoc36) A
Var36 : Con36 β Ty36 β Set; Var36
= Ξ» Ξ A β
(Var36 : Con36 β Ty36 β Set)
(vz : β Ξ A β Var36 (snoc36 Ξ A) A)
(vs : β Ξ B A β Var36 Ξ A β Var36 (snoc36 Ξ B) A)
β Var36 Ξ A
vz36 : β{Ξ A} β Var36 (snoc36 Ξ A) A; vz36
= Ξ» Var36 vz36 vs β vz36 _ _
vs36 : β{Ξ B A} β Var36 Ξ A β Var36 (snoc36 Ξ B) A; vs36
= Ξ» x Var36 vz36 vs36 β vs36 _ _ _ (x Var36 vz36 vs36)
Tm36 : Con36 β Ty36 β Set; Tm36
= Ξ» Ξ A β
(Tm36 : Con36 β Ty36 β Set)
(var : β Ξ A β Var36 Ξ A β Tm36 Ξ A)
(lam : β Ξ A B β Tm36 (snoc36 Ξ A) B β Tm36 Ξ (arr36 A B))
(app : β Ξ A B β Tm36 Ξ (arr36 A B) β Tm36 Ξ A β Tm36 Ξ B)
(tt : β Ξ β Tm36 Ξ top36)
(pair : β Ξ A B β Tm36 Ξ A β Tm36 Ξ B β Tm36 Ξ (prod36 A B))
(fst : β Ξ A B β Tm36 Ξ (prod36 A B) β Tm36 Ξ A)
(snd : β Ξ A B β Tm36 Ξ (prod36 A B) β Tm36 Ξ B)
(left : β Ξ A B β Tm36 Ξ A β Tm36 Ξ (sum36 A B))
(right : β Ξ A B β Tm36 Ξ B β Tm36 Ξ (sum36 A B))
(case : β Ξ A B C β Tm36 Ξ (sum36 A B) β Tm36 Ξ (arr36 A C) β Tm36 Ξ (arr36 B C) β Tm36 Ξ C)
(zero : β Ξ β Tm36 Ξ nat36)
(suc : β Ξ β Tm36 Ξ nat36 β Tm36 Ξ nat36)
(rec : β Ξ A β Tm36 Ξ nat36 β Tm36 Ξ (arr36 nat36 (arr36 A A)) β Tm36 Ξ A β Tm36 Ξ A)
β Tm36 Ξ A
var36 : β{Ξ A} β Var36 Ξ A β Tm36 Ξ A; var36
= Ξ» x Tm36 var36 lam app tt pair fst snd left right case zero suc rec β
var36 _ _ x
lam36 : β{Ξ A B} β Tm36 (snoc36 Ξ A) B β Tm36 Ξ (arr36 A B); lam36
= Ξ» t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec β
lam36 _ _ _ (t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec)
app36 : β{Ξ A B} β Tm36 Ξ (arr36 A B) β Tm36 Ξ A β Tm36 Ξ B; app36
= Ξ» t u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec β
app36 _ _ _ (t Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
tt36 : β{Ξ} β Tm36 Ξ top36; tt36
= Ξ» Tm36 var36 lam36 app36 tt36 pair fst snd left right case zero suc rec β tt36 _
pair36 : β{Ξ A B} β Tm36 Ξ A β Tm36 Ξ B β Tm36 Ξ (prod36 A B); pair36
= Ξ» t u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec β
pair36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
fst36 : β{Ξ A B} β Tm36 Ξ (prod36 A B) β Tm36 Ξ A; fst36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec β
fst36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec)
snd36 : β{Ξ A B} β Tm36 Ξ (prod36 A B) β Tm36 Ξ B; snd36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec β
snd36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec)
left36 : β{Ξ A B} β Tm36 Ξ A β Tm36 Ξ (sum36 A B); left36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec β
left36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec)
right36 : β{Ξ A B} β Tm36 Ξ B β Tm36 Ξ (sum36 A B); right36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec β
right36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec)
case36 : β{Ξ A B C} β Tm36 Ξ (sum36 A B) β Tm36 Ξ (arr36 A C) β Tm36 Ξ (arr36 B C) β Tm36 Ξ C; case36
= Ξ» t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec β
case36 _ _ _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
zero36 : β{Ξ} β Tm36 Ξ nat36; zero36
= Ξ» Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc rec β zero36 _
suc36 : β{Ξ} β Tm36 Ξ nat36 β Tm36 Ξ nat36; suc36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec β
suc36 _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec)
rec36 : β{Ξ A} β Tm36 Ξ nat36 β Tm36 Ξ (arr36 nat36 (arr36 A A)) β Tm36 Ξ A β Tm36 Ξ A; rec36
= Ξ» t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36 β
rec36 _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
v036 : β{Ξ A} β Tm36 (snoc36 Ξ A) A; v036
= var36 vz36
v136 : β{Ξ A B} β Tm36 (snoc36 (snoc36 Ξ A) B) A; v136
= var36 (vs36 vz36)
v236 : β{Ξ A B C} β Tm36 (snoc36 (snoc36 (snoc36 Ξ A) B) C) A; v236
= var36 (vs36 (vs36 vz36))
v336 : β{Ξ A B C D} β Tm36 (snoc36 (snoc36 (snoc36 (snoc36 Ξ A) B) C) D) A; v336
= var36 (vs36 (vs36 (vs36 vz36)))
tbool36 : Ty36; tbool36
= sum36 top36 top36
true36 : β{Ξ} β Tm36 Ξ tbool36; true36
= left36 tt36
tfalse36 : β{Ξ} β Tm36 Ξ tbool36; tfalse36
= right36 tt36
ifthenelse36 : β{Ξ A} β Tm36 Ξ (arr36 tbool36 (arr36 A (arr36 A A))); ifthenelse36
= lam36 (lam36 (lam36 (case36 v236 (lam36 v236) (lam36 v136))))
times436 : β{Ξ A} β Tm36 Ξ (arr36 (arr36 A A) (arr36 A A)); times436
= lam36 (lam36 (app36 v136 (app36 v136 (app36 v136 (app36 v136 v036)))))
add36 : β{Ξ} β Tm36 Ξ (arr36 nat36 (arr36 nat36 nat36)); add36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (suc36 (app36 v136 v036)))))
(lam36 v036))
mul36 : β{Ξ} β Tm36 Ξ (arr36 nat36 (arr36 nat36 nat36)); mul36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (app36 (app36 add36 (app36 v136 v036)) v036))))
(lam36 zero36))
fact36 : β{Ξ} β Tm36 Ξ (arr36 nat36 nat36); fact36
= lam36 (rec36 v036 (lam36 (lam36 (app36 (app36 mul36 (suc36 v136)) v036)))
(suc36 zero36))
{-# OPTIONS --type-in-type #-}
Ty37 : Set
Ty37 =
(Ty37 : Set)
(nat top bot : Ty37)
(arr prod sum : Ty37 β Ty37 β Ty37)
β Ty37
nat37 : Ty37; nat37 = Ξ» _ nat37 _ _ _ _ _ β nat37
top37 : Ty37; top37 = Ξ» _ _ top37 _ _ _ _ β top37
bot37 : Ty37; bot37 = Ξ» _ _ _ bot37 _ _ _ β bot37
arr37 : Ty37 β Ty37 β Ty37; arr37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod sum β
arr37 (A Ty37 nat37 top37 bot37 arr37 prod sum) (B Ty37 nat37 top37 bot37 arr37 prod sum)
prod37 : Ty37 β Ty37 β Ty37; prod37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod37 sum β
prod37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum) (B Ty37 nat37 top37 bot37 arr37 prod37 sum)
sum37 : Ty37 β Ty37 β Ty37; sum37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod37 sum37 β
sum37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum37) (B Ty37 nat37 top37 bot37 arr37 prod37 sum37)
Con37 : Set; Con37
= (Con37 : Set)
(nil : Con37)
(snoc : Con37 β Ty37 β Con37)
β Con37
nil37 : Con37; nil37
= Ξ» Con37 nil37 snoc β nil37
snoc37 : Con37 β Ty37 β Con37; snoc37
= Ξ» Ξ A Con37 nil37 snoc37 β snoc37 (Ξ Con37 nil37 snoc37) A
Var37 : Con37 β Ty37 β Set; Var37
= Ξ» Ξ A β
(Var37 : Con37 β Ty37 β Set)
(vz : β Ξ A β Var37 (snoc37 Ξ A) A)
(vs : β Ξ B A β Var37 Ξ A β Var37 (snoc37 Ξ B) A)
β Var37 Ξ A
vz37 : β{Ξ A} β Var37 (snoc37 Ξ A) A; vz37
= Ξ» Var37 vz37 vs β vz37 _ _
vs37 : β{Ξ B A} β Var37 Ξ A β Var37 (snoc37 Ξ B) A; vs37
= Ξ» x Var37 vz37 vs37 β vs37 _ _ _ (x Var37 vz37 vs37)
Tm37 : Con37 β Ty37 β Set; Tm37
= Ξ» Ξ A β
(Tm37 : Con37 β Ty37 β Set)
(var : β Ξ A β Var37 Ξ A β Tm37 Ξ A)
(lam : β Ξ A B β Tm37 (snoc37 Ξ A) B β Tm37 Ξ (arr37 A B))
(app : β Ξ A B β Tm37 Ξ (arr37 A B) β Tm37 Ξ A β Tm37 Ξ B)
(tt : β Ξ β Tm37 Ξ top37)
(pair : β Ξ A B β Tm37 Ξ A β Tm37 Ξ B β Tm37 Ξ (prod37 A B))
(fst : β Ξ A B β Tm37 Ξ (prod37 A B) β Tm37 Ξ A)
(snd : β Ξ A B β Tm37 Ξ (prod37 A B) β Tm37 Ξ B)
(left : β Ξ A B β Tm37 Ξ A β Tm37 Ξ (sum37 A B))
(right : β Ξ A B β Tm37 Ξ B β Tm37 Ξ (sum37 A B))
(case : β Ξ A B C β Tm37 Ξ (sum37 A B) β Tm37 Ξ (arr37 A C) β Tm37 Ξ (arr37 B C) β Tm37 Ξ C)
(zero : β Ξ β Tm37 Ξ nat37)
(suc : β Ξ β Tm37 Ξ nat37 β Tm37 Ξ nat37)
(rec : β Ξ A β Tm37 Ξ nat37 β Tm37 Ξ (arr37 nat37 (arr37 A A)) β Tm37 Ξ A β Tm37 Ξ A)
β Tm37 Ξ A
var37 : β{Ξ A} β Var37 Ξ A β Tm37 Ξ A; var37
= Ξ» x Tm37 var37 lam app tt pair fst snd left right case zero suc rec β
var37 _ _ x
lam37 : β{Ξ A B} β Tm37 (snoc37 Ξ A) B β Tm37 Ξ (arr37 A B); lam37
= Ξ» t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec β
lam37 _ _ _ (t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec)
app37 : β{Ξ A B} β Tm37 Ξ (arr37 A B) β Tm37 Ξ A β Tm37 Ξ B; app37
= Ξ» t u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec β
app37 _ _ _ (t Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
tt37 : β{Ξ} β Tm37 Ξ top37; tt37
= Ξ» Tm37 var37 lam37 app37 tt37 pair fst snd left right case zero suc rec β tt37 _
pair37 : β{Ξ A B} β Tm37 Ξ A β Tm37 Ξ B β Tm37 Ξ (prod37 A B); pair37
= Ξ» t u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec β
pair37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
fst37 : β{Ξ A B} β Tm37 Ξ (prod37 A B) β Tm37 Ξ A; fst37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec β
fst37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec)
snd37 : β{Ξ A B} β Tm37 Ξ (prod37 A B) β Tm37 Ξ B; snd37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec β
snd37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec)
left37 : β{Ξ A B} β Tm37 Ξ A β Tm37 Ξ (sum37 A B); left37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec β
left37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec)
right37 : β{Ξ A B} β Tm37 Ξ B β Tm37 Ξ (sum37 A B); right37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec β
right37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec)
case37 : β{Ξ A B C} β Tm37 Ξ (sum37 A B) β Tm37 Ξ (arr37 A C) β Tm37 Ξ (arr37 B C) β Tm37 Ξ C; case37
= Ξ» t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec β
case37 _ _ _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
zero37 : β{Ξ} β Tm37 Ξ nat37; zero37
= Ξ» Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc rec β zero37 _
suc37 : β{Ξ} β Tm37 Ξ nat37 β Tm37 Ξ nat37; suc37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec β
suc37 _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec)
rec37 : β{Ξ A} β Tm37 Ξ nat37 β Tm37 Ξ (arr37 nat37 (arr37 A A)) β Tm37 Ξ A β Tm37 Ξ A; rec37
= Ξ» t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37 β
rec37 _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
v037 : β{Ξ A} β Tm37 (snoc37 Ξ A) A; v037
= var37 vz37
v137 : β{Ξ A B} β Tm37 (snoc37 (snoc37 Ξ A) B) A; v137
= var37 (vs37 vz37)
v237 : β{Ξ A B C} β Tm37 (snoc37 (snoc37 (snoc37 Ξ A) B) C) A; v237
= var37 (vs37 (vs37 vz37))
v337 : β{Ξ A B C D} β Tm37 (snoc37 (snoc37 (snoc37 (snoc37 Ξ A) B) C) D) A; v337
= var37 (vs37 (vs37 (vs37 vz37)))
tbool37 : Ty37; tbool37
= sum37 top37 top37
true37 : β{Ξ} β Tm37 Ξ tbool37; true37
= left37 tt37
tfalse37 : β{Ξ} β Tm37 Ξ tbool37; tfalse37
= right37 tt37
ifthenelse37 : β{Ξ A} β Tm37 Ξ (arr37 tbool37 (arr37 A (arr37 A A))); ifthenelse37
= lam37 (lam37 (lam37 (case37 v237 (lam37 v237) (lam37 v137))))
times437 : β{Ξ A} β Tm37 Ξ (arr37 (arr37 A A) (arr37 A A)); times437
= lam37 (lam37 (app37 v137 (app37 v137 (app37 v137 (app37 v137 v037)))))
add37 : β{Ξ} β Tm37 Ξ (arr37 nat37 (arr37 nat37 nat37)); add37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (suc37 (app37 v137 v037)))))
(lam37 v037))
mul37 : β{Ξ} β Tm37 Ξ (arr37 nat37 (arr37 nat37 nat37)); mul37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (app37 (app37 add37 (app37 v137 v037)) v037))))
(lam37 zero37))
fact37 : β{Ξ} β Tm37 Ξ (arr37 nat37 nat37); fact37
= lam37 (rec37 v037 (lam37 (lam37 (app37 (app37 mul37 (suc37 v137)) v037)))
(suc37 zero37))
{-# OPTIONS --type-in-type #-}
Ty38 : Set
Ty38 =
(Ty38 : Set)
(nat top bot : Ty38)
(arr prod sum : Ty38 β Ty38 β Ty38)
β Ty38
nat38 : Ty38; nat38 = Ξ» _ nat38 _ _ _ _ _ β nat38
top38 : Ty38; top38 = Ξ» _ _ top38 _ _ _ _ β top38
bot38 : Ty38; bot38 = Ξ» _ _ _ bot38 _ _ _ β bot38
arr38 : Ty38 β Ty38 β Ty38; arr38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod sum β
arr38 (A Ty38 nat38 top38 bot38 arr38 prod sum) (B Ty38 nat38 top38 bot38 arr38 prod sum)
prod38 : Ty38 β Ty38 β Ty38; prod38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod38 sum β
prod38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum) (B Ty38 nat38 top38 bot38 arr38 prod38 sum)
sum38 : Ty38 β Ty38 β Ty38; sum38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod38 sum38 β
sum38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum38) (B Ty38 nat38 top38 bot38 arr38 prod38 sum38)
Con38 : Set; Con38
= (Con38 : Set)
(nil : Con38)
(snoc : Con38 β Ty38 β Con38)
β Con38
nil38 : Con38; nil38
= Ξ» Con38 nil38 snoc β nil38
snoc38 : Con38 β Ty38 β Con38; snoc38
= Ξ» Ξ A Con38 nil38 snoc38 β snoc38 (Ξ Con38 nil38 snoc38) A
Var38 : Con38 β Ty38 β Set; Var38
= Ξ» Ξ A β
(Var38 : Con38 β Ty38 β Set)
(vz : β Ξ A β Var38 (snoc38 Ξ A) A)
(vs : β Ξ B A β Var38 Ξ A β Var38 (snoc38 Ξ B) A)
β Var38 Ξ A
vz38 : β{Ξ A} β Var38 (snoc38 Ξ A) A; vz38
= Ξ» Var38 vz38 vs β vz38 _ _
vs38 : β{Ξ B A} β Var38 Ξ A β Var38 (snoc38 Ξ B) A; vs38
= Ξ» x Var38 vz38 vs38 β vs38 _ _ _ (x Var38 vz38 vs38)
Tm38 : Con38 β Ty38 β Set; Tm38
= Ξ» Ξ A β
(Tm38 : Con38 β Ty38 β Set)
(var : β Ξ A β Var38 Ξ A β Tm38 Ξ A)
(lam : β Ξ A B β Tm38 (snoc38 Ξ A) B β Tm38 Ξ (arr38 A B))
(app : β Ξ A B β Tm38 Ξ (arr38 A B) β Tm38 Ξ A β Tm38 Ξ B)
(tt : β Ξ β Tm38 Ξ top38)
(pair : β Ξ A B β Tm38 Ξ A β Tm38 Ξ B β Tm38 Ξ (prod38 A B))
(fst : β Ξ A B β Tm38 Ξ (prod38 A B) β Tm38 Ξ A)
(snd : β Ξ A B β Tm38 Ξ (prod38 A B) β Tm38 Ξ B)
(left : β Ξ A B β Tm38 Ξ A β Tm38 Ξ (sum38 A B))
(right : β Ξ A B β Tm38 Ξ B β Tm38 Ξ (sum38 A B))
(case : β Ξ A B C β Tm38 Ξ (sum38 A B) β Tm38 Ξ (arr38 A C) β Tm38 Ξ (arr38 B C) β Tm38 Ξ C)
(zero : β Ξ β Tm38 Ξ nat38)
(suc : β Ξ β Tm38 Ξ nat38 β Tm38 Ξ nat38)
(rec : β Ξ A β Tm38 Ξ nat38 β Tm38 Ξ (arr38 nat38 (arr38 A A)) β Tm38 Ξ A β Tm38 Ξ A)
β Tm38 Ξ A
var38 : β{Ξ A} β Var38 Ξ A β Tm38 Ξ A; var38
= Ξ» x Tm38 var38 lam app tt pair fst snd left right case zero suc rec β
var38 _ _ x
lam38 : β{Ξ A B} β Tm38 (snoc38 Ξ A) B β Tm38 Ξ (arr38 A B); lam38
= Ξ» t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec β
lam38 _ _ _ (t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec)
app38 : β{Ξ A B} β Tm38 Ξ (arr38 A B) β Tm38 Ξ A β Tm38 Ξ B; app38
= Ξ» t u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec β
app38 _ _ _ (t Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
tt38 : β{Ξ} β Tm38 Ξ top38; tt38
= Ξ» Tm38 var38 lam38 app38 tt38 pair fst snd left right case zero suc rec β tt38 _
pair38 : β{Ξ A B} β Tm38 Ξ A β Tm38 Ξ B β Tm38 Ξ (prod38 A B); pair38
= Ξ» t u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec β
pair38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
fst38 : β{Ξ A B} β Tm38 Ξ (prod38 A B) β Tm38 Ξ A; fst38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec β
fst38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec)
snd38 : β{Ξ A B} β Tm38 Ξ (prod38 A B) β Tm38 Ξ B; snd38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec β
snd38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec)
left38 : β{Ξ A B} β Tm38 Ξ A β Tm38 Ξ (sum38 A B); left38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec β
left38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec)
right38 : β{Ξ A B} β Tm38 Ξ B β Tm38 Ξ (sum38 A B); right38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec β
right38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec)
case38 : β{Ξ A B C} β Tm38 Ξ (sum38 A B) β Tm38 Ξ (arr38 A C) β Tm38 Ξ (arr38 B C) β Tm38 Ξ C; case38
= Ξ» t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec β
case38 _ _ _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
zero38 : β{Ξ} β Tm38 Ξ nat38; zero38
= Ξ» Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc rec β zero38 _
suc38 : β{Ξ} β Tm38 Ξ nat38 β Tm38 Ξ nat38; suc38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec β
suc38 _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec)
rec38 : β{Ξ A} β Tm38 Ξ nat38 β Tm38 Ξ (arr38 nat38 (arr38 A A)) β Tm38 Ξ A β Tm38 Ξ A; rec38
= Ξ» t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38 β
rec38 _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
v038 : β{Ξ A} β Tm38 (snoc38 Ξ A) A; v038
= var38 vz38
v138 : β{Ξ A B} β Tm38 (snoc38 (snoc38 Ξ A) B) A; v138
= var38 (vs38 vz38)
v238 : β{Ξ A B C} β Tm38 (snoc38 (snoc38 (snoc38 Ξ A) B) C) A; v238
= var38 (vs38 (vs38 vz38))
v338 : β{Ξ A B C D} β Tm38 (snoc38 (snoc38 (snoc38 (snoc38 Ξ A) B) C) D) A; v338
= var38 (vs38 (vs38 (vs38 vz38)))
tbool38 : Ty38; tbool38
= sum38 top38 top38
true38 : β{Ξ} β Tm38 Ξ tbool38; true38
= left38 tt38
tfalse38 : β{Ξ} β Tm38 Ξ tbool38; tfalse38
= right38 tt38
ifthenelse38 : β{Ξ A} β Tm38 Ξ (arr38 tbool38 (arr38 A (arr38 A A))); ifthenelse38
= lam38 (lam38 (lam38 (case38 v238 (lam38 v238) (lam38 v138))))
times438 : β{Ξ A} β Tm38 Ξ (arr38 (arr38 A A) (arr38 A A)); times438
= lam38 (lam38 (app38 v138 (app38 v138 (app38 v138 (app38 v138 v038)))))
add38 : β{Ξ} β Tm38 Ξ (arr38 nat38 (arr38 nat38 nat38)); add38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (suc38 (app38 v138 v038)))))
(lam38 v038))
mul38 : β{Ξ} β Tm38 Ξ (arr38 nat38 (arr38 nat38 nat38)); mul38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (app38 (app38 add38 (app38 v138 v038)) v038))))
(lam38 zero38))
fact38 : β{Ξ} β Tm38 Ξ (arr38 nat38 nat38); fact38
= lam38 (rec38 v038 (lam38 (lam38 (app38 (app38 mul38 (suc38 v138)) v038)))
(suc38 zero38))
{-# OPTIONS --type-in-type #-}
Ty39 : Set
Ty39 =
(Ty39 : Set)
(nat top bot : Ty39)
(arr prod sum : Ty39 β Ty39 β Ty39)
β Ty39
nat39 : Ty39; nat39 = Ξ» _ nat39 _ _ _ _ _ β nat39
top39 : Ty39; top39 = Ξ» _ _ top39 _ _ _ _ β top39
bot39 : Ty39; bot39 = Ξ» _ _ _ bot39 _ _ _ β bot39
arr39 : Ty39 β Ty39 β Ty39; arr39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod sum β
arr39 (A Ty39 nat39 top39 bot39 arr39 prod sum) (B Ty39 nat39 top39 bot39 arr39 prod sum)
prod39 : Ty39 β Ty39 β Ty39; prod39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod39 sum β
prod39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum) (B Ty39 nat39 top39 bot39 arr39 prod39 sum)
sum39 : Ty39 β Ty39 β Ty39; sum39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod39 sum39 β
sum39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum39) (B Ty39 nat39 top39 bot39 arr39 prod39 sum39)
Con39 : Set; Con39
= (Con39 : Set)
(nil : Con39)
(snoc : Con39 β Ty39 β Con39)
β Con39
nil39 : Con39; nil39
= Ξ» Con39 nil39 snoc β nil39
snoc39 : Con39 β Ty39 β Con39; snoc39
= Ξ» Ξ A Con39 nil39 snoc39 β snoc39 (Ξ Con39 nil39 snoc39) A
Var39 : Con39 β Ty39 β Set; Var39
= Ξ» Ξ A β
(Var39 : Con39 β Ty39 β Set)
(vz : β Ξ A β Var39 (snoc39 Ξ A) A)
(vs : β Ξ B A β Var39 Ξ A β Var39 (snoc39 Ξ B) A)
β Var39 Ξ A
vz39 : β{Ξ A} β Var39 (snoc39 Ξ A) A; vz39
= Ξ» Var39 vz39 vs β vz39 _ _
vs39 : β{Ξ B A} β Var39 Ξ A β Var39 (snoc39 Ξ B) A; vs39
= Ξ» x Var39 vz39 vs39 β vs39 _ _ _ (x Var39 vz39 vs39)
Tm39 : Con39 β Ty39 β Set; Tm39
= Ξ» Ξ A β
(Tm39 : Con39 β Ty39 β Set)
(var : β Ξ A β Var39 Ξ A β Tm39 Ξ A)
(lam : β Ξ A B β Tm39 (snoc39 Ξ A) B β Tm39 Ξ (arr39 A B))
(app : β Ξ A B β Tm39 Ξ (arr39 A B) β Tm39 Ξ A β Tm39 Ξ B)
(tt : β Ξ β Tm39 Ξ top39)
(pair : β Ξ A B β Tm39 Ξ A β Tm39 Ξ B β Tm39 Ξ (prod39 A B))
(fst : β Ξ A B β Tm39 Ξ (prod39 A B) β Tm39 Ξ A)
(snd : β Ξ A B β Tm39 Ξ (prod39 A B) β Tm39 Ξ B)
(left : β Ξ A B β Tm39 Ξ A β Tm39 Ξ (sum39 A B))
(right : β Ξ A B β Tm39 Ξ B β Tm39 Ξ (sum39 A B))
(case : β Ξ A B C β Tm39 Ξ (sum39 A B) β Tm39 Ξ (arr39 A C) β Tm39 Ξ (arr39 B C) β Tm39 Ξ C)
(zero : β Ξ β Tm39 Ξ nat39)
(suc : β Ξ β Tm39 Ξ nat39 β Tm39 Ξ nat39)
(rec : β Ξ A β Tm39 Ξ nat39 β Tm39 Ξ (arr39 nat39 (arr39 A A)) β Tm39 Ξ A β Tm39 Ξ A)
β Tm39 Ξ A
var39 : β{Ξ A} β Var39 Ξ A β Tm39 Ξ A; var39
= Ξ» x Tm39 var39 lam app tt pair fst snd left right case zero suc rec β
var39 _ _ x
lam39 : β{Ξ A B} β Tm39 (snoc39 Ξ A) B β Tm39 Ξ (arr39 A B); lam39
= Ξ» t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec β
lam39 _ _ _ (t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec)
app39 : β{Ξ A B} β Tm39 Ξ (arr39 A B) β Tm39 Ξ A β Tm39 Ξ B; app39
= Ξ» t u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec β
app39 _ _ _ (t Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
tt39 : β{Ξ} β Tm39 Ξ top39; tt39
= Ξ» Tm39 var39 lam39 app39 tt39 pair fst snd left right case zero suc rec β tt39 _
pair39 : β{Ξ A B} β Tm39 Ξ A β Tm39 Ξ B β Tm39 Ξ (prod39 A B); pair39
= Ξ» t u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec β
pair39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
fst39 : β{Ξ A B} β Tm39 Ξ (prod39 A B) β Tm39 Ξ A; fst39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec β
fst39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec)
snd39 : β{Ξ A B} β Tm39 Ξ (prod39 A B) β Tm39 Ξ B; snd39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec β
snd39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec)
left39 : β{Ξ A B} β Tm39 Ξ A β Tm39 Ξ (sum39 A B); left39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec β
left39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec)
right39 : β{Ξ A B} β Tm39 Ξ B β Tm39 Ξ (sum39 A B); right39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec β
right39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec)
case39 : β{Ξ A B C} β Tm39 Ξ (sum39 A B) β Tm39 Ξ (arr39 A C) β Tm39 Ξ (arr39 B C) β Tm39 Ξ C; case39
= Ξ» t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec β
case39 _ _ _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
zero39 : β{Ξ} β Tm39 Ξ nat39; zero39
= Ξ» Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc rec β zero39 _
suc39 : β{Ξ} β Tm39 Ξ nat39 β Tm39 Ξ nat39; suc39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec β
suc39 _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec)
rec39 : β{Ξ A} β Tm39 Ξ nat39 β Tm39 Ξ (arr39 nat39 (arr39 A A)) β Tm39 Ξ A β Tm39 Ξ A; rec39
= Ξ» t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39 β
rec39 _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
v039 : β{Ξ A} β Tm39 (snoc39 Ξ A) A; v039
= var39 vz39
v139 : β{Ξ A B} β Tm39 (snoc39 (snoc39 Ξ A) B) A; v139
= var39 (vs39 vz39)
v239 : β{Ξ A B C} β Tm39 (snoc39 (snoc39 (snoc39 Ξ A) B) C) A; v239
= var39 (vs39 (vs39 vz39))
v339 : β{Ξ A B C D} β Tm39 (snoc39 (snoc39 (snoc39 (snoc39 Ξ A) B) C) D) A; v339
= var39 (vs39 (vs39 (vs39 vz39)))
tbool39 : Ty39; tbool39
= sum39 top39 top39
true39 : β{Ξ} β Tm39 Ξ tbool39; true39
= left39 tt39
tfalse39 : β{Ξ} β Tm39 Ξ tbool39; tfalse39
= right39 tt39
ifthenelse39 : β{Ξ A} β Tm39 Ξ (arr39 tbool39 (arr39 A (arr39 A A))); ifthenelse39
= lam39 (lam39 (lam39 (case39 v239 (lam39 v239) (lam39 v139))))
times439 : β{Ξ A} β Tm39 Ξ (arr39 (arr39 A A) (arr39 A A)); times439
= lam39 (lam39 (app39 v139 (app39 v139 (app39 v139 (app39 v139 v039)))))
add39 : β{Ξ} β Tm39 Ξ (arr39 nat39 (arr39 nat39 nat39)); add39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (suc39 (app39 v139 v039)))))
(lam39 v039))
mul39 : β{Ξ} β Tm39 Ξ (arr39 nat39 (arr39 nat39 nat39)); mul39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (app39 (app39 add39 (app39 v139 v039)) v039))))
(lam39 zero39))
fact39 : β{Ξ} β Tm39 Ξ (arr39 nat39 nat39); fact39
= lam39 (rec39 v039 (lam39 (lam39 (app39 (app39 mul39 (suc39 v139)) v039)))
(suc39 zero39))
{-# OPTIONS --type-in-type #-}
Ty40 : Set
Ty40 =
(Ty40 : Set)
(nat top bot : Ty40)
(arr prod sum : Ty40 β Ty40 β Ty40)
β Ty40
nat40 : Ty40; nat40 = Ξ» _ nat40 _ _ _ _ _ β nat40
top40 : Ty40; top40 = Ξ» _ _ top40 _ _ _ _ β top40
bot40 : Ty40; bot40 = Ξ» _ _ _ bot40 _ _ _ β bot40
arr40 : Ty40 β Ty40 β Ty40; arr40
= Ξ» A B Ty40 nat40 top40 bot40 arr40 prod sum β
arr40 (A Ty40 nat40 top40 bot40 arr40 prod sum) (B Ty40 nat40 top40 bot40 arr40 prod sum)
prod40 : Ty40 β Ty40 β Ty40; prod40
= Ξ» A B Ty40 nat40 top40 bot40 arr40 prod40 sum β
prod40 (A Ty40 nat40 top40 bot40 arr40 prod40 sum) (B Ty40 nat40 top40 bot40 arr40 prod40 sum)
sum40 : Ty40 β Ty40 β Ty40; sum40
= Ξ» A B Ty40 nat40 top40 bot40 arr40 prod40 sum40 β
sum40 (A Ty40 nat40 top40 bot40 arr40 prod40 sum40) (B Ty40 nat40 top40 bot40 arr40 prod40 sum40)
Con40 : Set; Con40
= (Con40 : Set)
(nil : Con40)
(snoc : Con40 β Ty40 β Con40)
β Con40
nil40 : Con40; nil40
= Ξ» Con40 nil40 snoc β nil40
snoc40 : Con40 β Ty40 β Con40; snoc40
= Ξ» Ξ A Con40 nil40 snoc40 β snoc40 (Ξ Con40 nil40 snoc40) A
Var40 : Con40 β Ty40 β Set; Var40
= Ξ» Ξ A β
(Var40 : Con40 β Ty40 β Set)
(vz : β Ξ A β Var40 (snoc40 Ξ A) A)
(vs : β Ξ B A β Var40 Ξ A β Var40 (snoc40 Ξ B) A)
β Var40 Ξ A
vz40 : β{Ξ A} β Var40 (snoc40 Ξ A) A; vz40
= Ξ» Var40 vz40 vs β vz40 _ _
vs40 : β{Ξ B A} β Var40 Ξ A β Var40 (snoc40 Ξ B) A; vs40
= Ξ» x Var40 vz40 vs40 β vs40 _ _ _ (x Var40 vz40 vs40)
Tm40 : Con40 β Ty40 β Set; Tm40
= Ξ» Ξ A β
(Tm40 : Con40 β Ty40 β Set)
(var : β Ξ A β Var40 Ξ A β Tm40 Ξ A)
(lam : β Ξ A B β Tm40 (snoc40 Ξ A) B β Tm40 Ξ (arr40 A B))
(app : β Ξ A B β Tm40 Ξ (arr40 A B) β Tm40 Ξ A β Tm40 Ξ B)
(tt : β Ξ β Tm40 Ξ top40)
(pair : β Ξ A B β Tm40 Ξ A β Tm40 Ξ B β Tm40 Ξ (prod40 A B))
(fst : β Ξ A B β Tm40 Ξ (prod40 A B) β Tm40 Ξ A)
(snd : β Ξ A B β Tm40 Ξ (prod40 A B) β Tm40 Ξ B)
(left : β Ξ A B β Tm40 Ξ A β Tm40 Ξ (sum40 A B))
(right : β Ξ A B β Tm40 Ξ B β Tm40 Ξ (sum40 A B))
(case : β Ξ A B C β Tm40 Ξ (sum40 A B) β Tm40 Ξ (arr40 A C) β Tm40 Ξ (arr40 B C) β Tm40 Ξ C)
(zero : β Ξ β Tm40 Ξ nat40)
(suc : β Ξ β Tm40 Ξ nat40 β Tm40 Ξ nat40)
(rec : β Ξ A β Tm40 Ξ nat40 β Tm40 Ξ (arr40 nat40 (arr40 A A)) β Tm40 Ξ A β Tm40 Ξ A)
β Tm40 Ξ A
var40 : β{Ξ A} β Var40 Ξ A β Tm40 Ξ A; var40
= Ξ» x Tm40 var40 lam app tt pair fst snd left right case zero suc rec β
var40 _ _ x
lam40 : β{Ξ A B} β Tm40 (snoc40 Ξ A) B β Tm40 Ξ (arr40 A B); lam40
= Ξ» t Tm40 var40 lam40 app tt pair fst snd left right case zero suc rec β
lam40 _ _ _ (t Tm40 var40 lam40 app tt pair fst snd left right case zero suc rec)
app40 : β{Ξ A B} β Tm40 Ξ (arr40 A B) β Tm40 Ξ A β Tm40 Ξ B; app40
= Ξ» t u Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec β
app40 _ _ _ (t Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec)
(u Tm40 var40 lam40 app40 tt pair fst snd left right case zero suc rec)
tt40 : β{Ξ} β Tm40 Ξ top40; tt40
= Ξ» Tm40 var40 lam40 app40 tt40 pair fst snd left right case zero suc rec β tt40 _
pair40 : β{Ξ A B} β Tm40 Ξ A β Tm40 Ξ B β Tm40 Ξ (prod40 A B); pair40
= Ξ» t u Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec β
pair40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec)
(u Tm40 var40 lam40 app40 tt40 pair40 fst snd left right case zero suc rec)
fst40 : β{Ξ A B} β Tm40 Ξ (prod40 A B) β Tm40 Ξ A; fst40
= Ξ» t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd left right case zero suc rec β
fst40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd left right case zero suc rec)
snd40 : β{Ξ A B} β Tm40 Ξ (prod40 A B) β Tm40 Ξ B; snd40
= Ξ» t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left right case zero suc rec β
snd40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left right case zero suc rec)
left40 : β{Ξ A B} β Tm40 Ξ A β Tm40 Ξ (sum40 A B); left40
= Ξ» t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right case zero suc rec β
left40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right case zero suc rec)
right40 : β{Ξ A B} β Tm40 Ξ B β Tm40 Ξ (sum40 A B); right40
= Ξ» t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case zero suc rec β
right40 _ _ _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case zero suc rec)
case40 : β{Ξ A B C} β Tm40 Ξ (sum40 A B) β Tm40 Ξ (arr40 A C) β Tm40 Ξ (arr40 B C) β Tm40 Ξ C; case40
= Ξ» t u v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec β
case40 _ _ _ _
(t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
(u Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
(v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero suc rec)
zero40 : β{Ξ} β Tm40 Ξ nat40; zero40
= Ξ» Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc rec β zero40 _
suc40 : β{Ξ} β Tm40 Ξ nat40 β Tm40 Ξ nat40; suc40
= Ξ» t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec β
suc40 _ (t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec)
rec40 : β{Ξ A} β Tm40 Ξ nat40 β Tm40 Ξ (arr40 nat40 (arr40 A A)) β Tm40 Ξ A β Tm40 Ξ A; rec40
= Ξ» t u v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40 β
rec40 _ _
(t Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
(u Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
(v Tm40 var40 lam40 app40 tt40 pair40 fst40 snd40 left40 right40 case40 zero40 suc40 rec40)
v040 : β{Ξ A} β Tm40 (snoc40 Ξ A) A; v040
= var40 vz40
v140 : β{Ξ A B} β Tm40 (snoc40 (snoc40 Ξ A) B) A; v140
= var40 (vs40 vz40)
v240 : β{Ξ A B C} β Tm40 (snoc40 (snoc40 (snoc40 Ξ A) B) C) A; v240
= var40 (vs40 (vs40 vz40))
v340 : β{Ξ A B C D} β Tm40 (snoc40 (snoc40 (snoc40 (snoc40 Ξ A) B) C) D) A; v340
= var40 (vs40 (vs40 (vs40 vz40)))
tbool40 : Ty40; tbool40
= sum40 top40 top40
true40 : β{Ξ} β Tm40 Ξ tbool40; true40
= left40 tt40
tfalse40 : β{Ξ} β Tm40 Ξ tbool40; tfalse40
= right40 tt40
ifthenelse40 : β{Ξ A} β Tm40 Ξ (arr40 tbool40 (arr40 A (arr40 A A))); ifthenelse40
= lam40 (lam40 (lam40 (case40 v240 (lam40 v240) (lam40 v140))))
times440 : β{Ξ A} β Tm40 Ξ (arr40 (arr40 A A) (arr40 A A)); times440
= lam40 (lam40 (app40 v140 (app40 v140 (app40 v140 (app40 v140 v040)))))
add40 : β{Ξ} β Tm40 Ξ (arr40 nat40 (arr40 nat40 nat40)); add40
= lam40 (rec40 v040
(lam40 (lam40 (lam40 (suc40 (app40 v140 v040)))))
(lam40 v040))
mul40 : β{Ξ} β Tm40 Ξ (arr40 nat40 (arr40 nat40 nat40)); mul40
= lam40 (rec40 v040
(lam40 (lam40 (lam40 (app40 (app40 add40 (app40 v140 v040)) v040))))
(lam40 zero40))
fact40 : β{Ξ} β Tm40 Ξ (arr40 nat40 nat40); fact40
= lam40 (rec40 v040 (lam40 (lam40 (app40 (app40 mul40 (suc40 v140)) v040)))
(suc40 zero40))
{-# OPTIONS --type-in-type #-}
Ty41 : Set
Ty41 =
(Ty41 : Set)
(nat top bot : Ty41)
(arr prod sum : Ty41 β Ty41 β Ty41)
β Ty41
nat41 : Ty41; nat41 = Ξ» _ nat41 _ _ _ _ _ β nat41
top41 : Ty41; top41 = Ξ» _ _ top41 _ _ _ _ β top41
bot41 : Ty41; bot41 = Ξ» _ _ _ bot41 _ _ _ β bot41
arr41 : Ty41 β Ty41 β Ty41; arr41
= Ξ» A B Ty41 nat41 top41 bot41 arr41 prod sum β
arr41 (A Ty41 nat41 top41 bot41 arr41 prod sum) (B Ty41 nat41 top41 bot41 arr41 prod sum)
prod41 : Ty41 β Ty41 β Ty41; prod41
= Ξ» A B Ty41 nat41 top41 bot41 arr41 prod41 sum β
prod41 (A Ty41 nat41 top41 bot41 arr41 prod41 sum) (B Ty41 nat41 top41 bot41 arr41 prod41 sum)
sum41 : Ty41 β Ty41 β Ty41; sum41
= Ξ» A B Ty41 nat41 top41 bot41 arr41 prod41 sum41 β
sum41 (A Ty41 nat41 top41 bot41 arr41 prod41 sum41) (B Ty41 nat41 top41 bot41 arr41 prod41 sum41)
Con41 : Set; Con41
= (Con41 : Set)
(nil : Con41)
(snoc : Con41 β Ty41 β Con41)
β Con41
nil41 : Con41; nil41
= Ξ» Con41 nil41 snoc β nil41
snoc41 : Con41 β Ty41 β Con41; snoc41
= Ξ» Ξ A Con41 nil41 snoc41 β snoc41 (Ξ Con41 nil41 snoc41) A
Var41 : Con41 β Ty41 β Set; Var41
= Ξ» Ξ A β
(Var41 : Con41 β Ty41 β Set)
(vz : β Ξ A β Var41 (snoc41 Ξ A) A)
(vs : β Ξ B A β Var41 Ξ A β Var41 (snoc41 Ξ B) A)
β Var41 Ξ A
vz41 : β{Ξ A} β Var41 (snoc41 Ξ A) A; vz41
= Ξ» Var41 vz41 vs β vz41 _ _
vs41 : β{Ξ B A} β Var41 Ξ A β Var41 (snoc41 Ξ B) A; vs41
= Ξ» x Var41 vz41 vs41 β vs41 _ _ _ (x Var41 vz41 vs41)
Tm41 : Con41 β Ty41 β Set; Tm41
= Ξ» Ξ A β
(Tm41 : Con41 β Ty41 β Set)
(var : β Ξ A β Var41 Ξ A β Tm41 Ξ A)
(lam : β Ξ A B β Tm41 (snoc41 Ξ A) B β Tm41 Ξ (arr41 A B))
(app : β Ξ A B β Tm41 Ξ (arr41 A B) β Tm41 Ξ A β Tm41 Ξ B)
(tt : β Ξ β Tm41 Ξ top41)
(pair : β Ξ A B β Tm41 Ξ A β Tm41 Ξ B β Tm41 Ξ (prod41 A B))
(fst : β Ξ A B β Tm41 Ξ (prod41 A B) β Tm41 Ξ A)
(snd : β Ξ A B β Tm41 Ξ (prod41 A B) β Tm41 Ξ B)
(left : β Ξ A B β Tm41 Ξ A β Tm41 Ξ (sum41 A B))
(right : β Ξ A B β Tm41 Ξ B β Tm41 Ξ (sum41 A B))
(case : β Ξ A B C β Tm41 Ξ (sum41 A B) β Tm41 Ξ (arr41 A C) β Tm41 Ξ (arr41 B C) β Tm41 Ξ C)
(zero : β Ξ β Tm41 Ξ nat41)
(suc : β Ξ β Tm41 Ξ nat41 β Tm41 Ξ nat41)
(rec : β Ξ A β Tm41 Ξ nat41 β Tm41 Ξ (arr41 nat41 (arr41 A A)) β Tm41 Ξ A β Tm41 Ξ A)
β Tm41 Ξ A
var41 : β{Ξ A} β Var41 Ξ A β Tm41 Ξ A; var41
= Ξ» x Tm41 var41 lam app tt pair fst snd left right case zero suc rec β
var41 _ _ x
lam41 : β{Ξ A B} β Tm41 (snoc41 Ξ A) B β Tm41 Ξ (arr41 A B); lam41
= Ξ» t Tm41 var41 lam41 app tt pair fst snd left right case zero suc rec β
lam41 _ _ _ (t Tm41 var41 lam41 app tt pair fst snd left right case zero suc rec)
app41 : β{Ξ A B} β Tm41 Ξ (arr41 A B) β Tm41 Ξ A β Tm41 Ξ B; app41
= Ξ» t u Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec β
app41 _ _ _ (t Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec)
(u Tm41 var41 lam41 app41 tt pair fst snd left right case zero suc rec)
tt41 : β{Ξ} β Tm41 Ξ top41; tt41
= Ξ» Tm41 var41 lam41 app41 tt41 pair fst snd left right case zero suc rec β tt41 _
pair41 : β{Ξ A B} β Tm41 Ξ A β Tm41 Ξ B β Tm41 Ξ (prod41 A B); pair41
= Ξ» t u Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec β
pair41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec)
(u Tm41 var41 lam41 app41 tt41 pair41 fst snd left right case zero suc rec)
fst41 : β{Ξ A B} β Tm41 Ξ (prod41 A B) β Tm41 Ξ A; fst41
= Ξ» t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd left right case zero suc rec β
fst41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd left right case zero suc rec)
snd41 : β{Ξ A B} β Tm41 Ξ (prod41 A B) β Tm41 Ξ B; snd41
= Ξ» t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left right case zero suc rec β
snd41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left right case zero suc rec)
left41 : β{Ξ A B} β Tm41 Ξ A β Tm41 Ξ (sum41 A B); left41
= Ξ» t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right case zero suc rec β
left41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right case zero suc rec)
right41 : β{Ξ A B} β Tm41 Ξ B β Tm41 Ξ (sum41 A B); right41
= Ξ» t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case zero suc rec β
right41 _ _ _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case zero suc rec)
case41 : β{Ξ A B C} β Tm41 Ξ (sum41 A B) β Tm41 Ξ (arr41 A C) β Tm41 Ξ (arr41 B C) β Tm41 Ξ C; case41
= Ξ» t u v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec β
case41 _ _ _ _
(t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
(u Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
(v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero suc rec)
zero41 : β{Ξ} β Tm41 Ξ nat41; zero41
= Ξ» Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc rec β zero41 _
suc41 : β{Ξ} β Tm41 Ξ nat41 β Tm41 Ξ nat41; suc41
= Ξ» t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec β
suc41 _ (t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec)
rec41 : β{Ξ A} β Tm41 Ξ nat41 β Tm41 Ξ (arr41 nat41 (arr41 A A)) β Tm41 Ξ A β Tm41 Ξ A; rec41
= Ξ» t u v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41 β
rec41 _ _
(t Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
(u Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
(v Tm41 var41 lam41 app41 tt41 pair41 fst41 snd41 left41 right41 case41 zero41 suc41 rec41)
v041 : β{Ξ A} β Tm41 (snoc41 Ξ A) A; v041
= var41 vz41
v141 : β{Ξ A B} β Tm41 (snoc41 (snoc41 Ξ A) B) A; v141
= var41 (vs41 vz41)
v241 : β{Ξ A B C} β Tm41 (snoc41 (snoc41 (snoc41 Ξ A) B) C) A; v241
= var41 (vs41 (vs41 vz41))
v341 : β{Ξ A B C D} β Tm41 (snoc41 (snoc41 (snoc41 (snoc41 Ξ A) B) C) D) A; v341
= var41 (vs41 (vs41 (vs41 vz41)))
tbool41 : Ty41; tbool41
= sum41 top41 top41
true41 : β{Ξ} β Tm41 Ξ tbool41; true41
= left41 tt41
tfalse41 : β{Ξ} β Tm41 Ξ tbool41; tfalse41
= right41 tt41
ifthenelse41 : β{Ξ A} β Tm41 Ξ (arr41 tbool41 (arr41 A (arr41 A A))); ifthenelse41
= lam41 (lam41 (lam41 (case41 v241 (lam41 v241) (lam41 v141))))
times441 : β{Ξ A} β Tm41 Ξ (arr41 (arr41 A A) (arr41 A A)); times441
= lam41 (lam41 (app41 v141 (app41 v141 (app41 v141 (app41 v141 v041)))))
add41 : β{Ξ} β Tm41 Ξ (arr41 nat41 (arr41 nat41 nat41)); add41
= lam41 (rec41 v041
(lam41 (lam41 (lam41 (suc41 (app41 v141 v041)))))
(lam41 v041))
mul41 : β{Ξ} β Tm41 Ξ (arr41 nat41 (arr41 nat41 nat41)); mul41
= lam41 (rec41 v041
(lam41 (lam41 (lam41 (app41 (app41 add41 (app41 v141 v041)) v041))))
(lam41 zero41))
fact41 : β{Ξ} β Tm41 Ξ (arr41 nat41 nat41); fact41
= lam41 (rec41 v041 (lam41 (lam41 (app41 (app41 mul41 (suc41 v141)) v041)))
(suc41 zero41))
{-# OPTIONS --type-in-type #-}
Ty42 : Set
Ty42 =
(Ty42 : Set)
(nat top bot : Ty42)
(arr prod sum : Ty42 β Ty42 β Ty42)
β Ty42
nat42 : Ty42; nat42 = Ξ» _ nat42 _ _ _ _ _ β nat42
top42 : Ty42; top42 = Ξ» _ _ top42 _ _ _ _ β top42
bot42 : Ty42; bot42 = Ξ» _ _ _ bot42 _ _ _ β bot42
arr42 : Ty42 β Ty42 β Ty42; arr42
= Ξ» A B Ty42 nat42 top42 bot42 arr42 prod sum β
arr42 (A Ty42 nat42 top42 bot42 arr42 prod sum) (B Ty42 nat42 top42 bot42 arr42 prod sum)
prod42 : Ty42 β Ty42 β Ty42; prod42
= Ξ» A B Ty42 nat42 top42 bot42 arr42 prod42 sum β
prod42 (A Ty42 nat42 top42 bot42 arr42 prod42 sum) (B Ty42 nat42 top42 bot42 arr42 prod42 sum)
sum42 : Ty42 β Ty42 β Ty42; sum42
= Ξ» A B Ty42 nat42 top42 bot42 arr42 prod42 sum42 β
sum42 (A Ty42 nat42 top42 bot42 arr42 prod42 sum42) (B Ty42 nat42 top42 bot42 arr42 prod42 sum42)
Con42 : Set; Con42
= (Con42 : Set)
(nil : Con42)
(snoc : Con42 β Ty42 β Con42)
β Con42
nil42 : Con42; nil42
= Ξ» Con42 nil42 snoc β nil42
snoc42 : Con42 β Ty42 β Con42; snoc42
= Ξ» Ξ A Con42 nil42 snoc42 β snoc42 (Ξ Con42 nil42 snoc42) A
Var42 : Con42 β Ty42 β Set; Var42
= Ξ» Ξ A β
(Var42 : Con42 β Ty42 β Set)
(vz : β Ξ A β Var42 (snoc42 Ξ A) A)
(vs : β Ξ B A β Var42 Ξ A β Var42 (snoc42 Ξ B) A)
β Var42 Ξ A
vz42 : β{Ξ A} β Var42 (snoc42 Ξ A) A; vz42
= Ξ» Var42 vz42 vs β vz42 _ _
vs42 : β{Ξ B A} β Var42 Ξ A β Var42 (snoc42 Ξ B) A; vs42
= Ξ» x Var42 vz42 vs42 β vs42 _ _ _ (x Var42 vz42 vs42)
Tm42 : Con42 β Ty42 β Set; Tm42
= Ξ» Ξ A β
(Tm42 : Con42 β Ty42 β Set)
(var : β Ξ A β Var42 Ξ A β Tm42 Ξ A)
(lam : β Ξ A B β Tm42 (snoc42 Ξ A) B β Tm42 Ξ (arr42 A B))
(app : β Ξ A B β Tm42 Ξ (arr42 A B) β Tm42 Ξ A β Tm42 Ξ B)
(tt : β Ξ β Tm42 Ξ top42)
(pair : β Ξ A B β Tm42 Ξ A β Tm42 Ξ B β Tm42 Ξ (prod42 A B))
(fst : β Ξ A B β Tm42 Ξ (prod42 A B) β Tm42 Ξ A)
(snd : β Ξ A B β Tm42 Ξ (prod42 A B) β Tm42 Ξ B)
(left : β Ξ A B β Tm42 Ξ A β Tm42 Ξ (sum42 A B))
(right : β Ξ A B β Tm42 Ξ B β Tm42 Ξ (sum42 A B))
(case : β Ξ A B C β Tm42 Ξ (sum42 A B) β Tm42 Ξ (arr42 A C) β Tm42 Ξ (arr42 B C) β Tm42 Ξ C)
(zero : β Ξ β Tm42 Ξ nat42)
(suc : β Ξ β Tm42 Ξ nat42 β Tm42 Ξ nat42)
(rec : β Ξ A β Tm42 Ξ nat42 β Tm42 Ξ (arr42 nat42 (arr42 A A)) β Tm42 Ξ A β Tm42 Ξ A)
β Tm42 Ξ A
var42 : β{Ξ A} β Var42 Ξ A β Tm42 Ξ A; var42
= Ξ» x Tm42 var42 lam app tt pair fst snd left right case zero suc rec β
var42 _ _ x
lam42 : β{Ξ A B} β Tm42 (snoc42 Ξ A) B β Tm42 Ξ (arr42 A B); lam42
= Ξ» t Tm42 var42 lam42 app tt pair fst snd left right case zero suc rec β
lam42 _ _ _ (t Tm42 var42 lam42 app tt pair fst snd left right case zero suc rec)
app42 : β{Ξ A B} β Tm42 Ξ (arr42 A B) β Tm42 Ξ A β Tm42 Ξ B; app42
= Ξ» t u Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec β
app42 _ _ _ (t Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec)
(u Tm42 var42 lam42 app42 tt pair fst snd left right case zero suc rec)
tt42 : β{Ξ} β Tm42 Ξ top42; tt42
= Ξ» Tm42 var42 lam42 app42 tt42 pair fst snd left right case zero suc rec β tt42 _
pair42 : β{Ξ A B} β Tm42 Ξ A β Tm42 Ξ B β Tm42 Ξ (prod42 A B); pair42
= Ξ» t u Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec β
pair42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec)
(u Tm42 var42 lam42 app42 tt42 pair42 fst snd left right case zero suc rec)
fst42 : β{Ξ A B} β Tm42 Ξ (prod42 A B) β Tm42 Ξ A; fst42
= Ξ» t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd left right case zero suc rec β
fst42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd left right case zero suc rec)
snd42 : β{Ξ A B} β Tm42 Ξ (prod42 A B) β Tm42 Ξ B; snd42
= Ξ» t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left right case zero suc rec β
snd42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left right case zero suc rec)
left42 : β{Ξ A B} β Tm42 Ξ A β Tm42 Ξ (sum42 A B); left42
= Ξ» t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right case zero suc rec β
left42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right case zero suc rec)
right42 : β{Ξ A B} β Tm42 Ξ B β Tm42 Ξ (sum42 A B); right42
= Ξ» t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case zero suc rec β
right42 _ _ _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case zero suc rec)
case42 : β{Ξ A B C} β Tm42 Ξ (sum42 A B) β Tm42 Ξ (arr42 A C) β Tm42 Ξ (arr42 B C) β Tm42 Ξ C; case42
= Ξ» t u v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec β
case42 _ _ _ _
(t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
(u Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
(v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero suc rec)
zero42 : β{Ξ} β Tm42 Ξ nat42; zero42
= Ξ» Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc rec β zero42 _
suc42 : β{Ξ} β Tm42 Ξ nat42 β Tm42 Ξ nat42; suc42
= Ξ» t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec β
suc42 _ (t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec)
rec42 : β{Ξ A} β Tm42 Ξ nat42 β Tm42 Ξ (arr42 nat42 (arr42 A A)) β Tm42 Ξ A β Tm42 Ξ A; rec42
= Ξ» t u v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42 β
rec42 _ _
(t Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
(u Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
(v Tm42 var42 lam42 app42 tt42 pair42 fst42 snd42 left42 right42 case42 zero42 suc42 rec42)
v042 : β{Ξ A} β Tm42 (snoc42 Ξ A) A; v042
= var42 vz42
v142 : β{Ξ A B} β Tm42 (snoc42 (snoc42 Ξ A) B) A; v142
= var42 (vs42 vz42)
v242 : β{Ξ A B C} β Tm42 (snoc42 (snoc42 (snoc42 Ξ A) B) C) A; v242
= var42 (vs42 (vs42 vz42))
v342 : β{Ξ A B C D} β Tm42 (snoc42 (snoc42 (snoc42 (snoc42 Ξ A) B) C) D) A; v342
= var42 (vs42 (vs42 (vs42 vz42)))
tbool42 : Ty42; tbool42
= sum42 top42 top42
true42 : β{Ξ} β Tm42 Ξ tbool42; true42
= left42 tt42
tfalse42 : β{Ξ} β Tm42 Ξ tbool42; tfalse42
= right42 tt42
ifthenelse42 : β{Ξ A} β Tm42 Ξ (arr42 tbool42 (arr42 A (arr42 A A))); ifthenelse42
= lam42 (lam42 (lam42 (case42 v242 (lam42 v242) (lam42 v142))))
times442 : β{Ξ A} β Tm42 Ξ (arr42 (arr42 A A) (arr42 A A)); times442
= lam42 (lam42 (app42 v142 (app42 v142 (app42 v142 (app42 v142 v042)))))
add42 : β{Ξ} β Tm42 Ξ (arr42 nat42 (arr42 nat42 nat42)); add42
= lam42 (rec42 v042
(lam42 (lam42 (lam42 (suc42 (app42 v142 v042)))))
(lam42 v042))
mul42 : β{Ξ} β Tm42 Ξ (arr42 nat42 (arr42 nat42 nat42)); mul42
= lam42 (rec42 v042
(lam42 (lam42 (lam42 (app42 (app42 add42 (app42 v142 v042)) v042))))
(lam42 zero42))
fact42 : β{Ξ} β Tm42 Ξ (arr42 nat42 nat42); fact42
= lam42 (rec42 v042 (lam42 (lam42 (app42 (app42 mul42 (suc42 v142)) v042)))
(suc42 zero42))
{-# OPTIONS --type-in-type #-}
Ty43 : Set
Ty43 =
(Ty43 : Set)
(nat top bot : Ty43)
(arr prod sum : Ty43 β Ty43 β Ty43)
β Ty43
nat43 : Ty43; nat43 = Ξ» _ nat43 _ _ _ _ _ β nat43
top43 : Ty43; top43 = Ξ» _ _ top43 _ _ _ _ β top43
bot43 : Ty43; bot43 = Ξ» _ _ _ bot43 _ _ _ β bot43
arr43 : Ty43 β Ty43 β Ty43; arr43
= Ξ» A B Ty43 nat43 top43 bot43 arr43 prod sum β
arr43 (A Ty43 nat43 top43 bot43 arr43 prod sum) (B Ty43 nat43 top43 bot43 arr43 prod sum)
prod43 : Ty43 β Ty43 β Ty43; prod43
= Ξ» A B Ty43 nat43 top43 bot43 arr43 prod43 sum β
prod43 (A Ty43 nat43 top43 bot43 arr43 prod43 sum) (B Ty43 nat43 top43 bot43 arr43 prod43 sum)
sum43 : Ty43 β Ty43 β Ty43; sum43
= Ξ» A B Ty43 nat43 top43 bot43 arr43 prod43 sum43 β
sum43 (A Ty43 nat43 top43 bot43 arr43 prod43 sum43) (B Ty43 nat43 top43 bot43 arr43 prod43 sum43)
Con43 : Set; Con43
= (Con43 : Set)
(nil : Con43)
(snoc : Con43 β Ty43 β Con43)
β Con43
nil43 : Con43; nil43
= Ξ» Con43 nil43 snoc β nil43
snoc43 : Con43 β Ty43 β Con43; snoc43
= Ξ» Ξ A Con43 nil43 snoc43 β snoc43 (Ξ Con43 nil43 snoc43) A
Var43 : Con43 β Ty43 β Set; Var43
= Ξ» Ξ A β
(Var43 : Con43 β Ty43 β Set)
(vz : β Ξ A β Var43 (snoc43 Ξ A) A)
(vs : β Ξ B A β Var43 Ξ A β Var43 (snoc43 Ξ B) A)
β Var43 Ξ A
vz43 : β{Ξ A} β Var43 (snoc43 Ξ A) A; vz43
= Ξ» Var43 vz43 vs β vz43 _ _
vs43 : β{Ξ B A} β Var43 Ξ A β Var43 (snoc43 Ξ B) A; vs43
= Ξ» x Var43 vz43 vs43 β vs43 _ _ _ (x Var43 vz43 vs43)
Tm43 : Con43 β Ty43 β Set; Tm43
= Ξ» Ξ A β
(Tm43 : Con43 β Ty43 β Set)
(var : β Ξ A β Var43 Ξ A β Tm43 Ξ A)
(lam : β Ξ A B β Tm43 (snoc43 Ξ A) B β Tm43 Ξ (arr43 A B))
(app : β Ξ A B β Tm43 Ξ (arr43 A B) β Tm43 Ξ A β Tm43 Ξ B)
(tt : β Ξ β Tm43 Ξ top43)
(pair : β Ξ A B β Tm43 Ξ A β Tm43 Ξ B β Tm43 Ξ (prod43 A B))
(fst : β Ξ A B β Tm43 Ξ (prod43 A B) β Tm43 Ξ A)
(snd : β Ξ A B β Tm43 Ξ (prod43 A B) β Tm43 Ξ B)
(left : β Ξ A B β Tm43 Ξ A β Tm43 Ξ (sum43 A B))
(right : β Ξ A B β Tm43 Ξ B β Tm43 Ξ (sum43 A B))
(case : β Ξ A B C β Tm43 Ξ (sum43 A B) β Tm43 Ξ (arr43 A C) β Tm43 Ξ (arr43 B C) β Tm43 Ξ C)
(zero : β Ξ β Tm43 Ξ nat43)
(suc : β Ξ β Tm43 Ξ nat43 β Tm43 Ξ nat43)
(rec : β Ξ A β Tm43 Ξ nat43 β Tm43 Ξ (arr43 nat43 (arr43 A A)) β Tm43 Ξ A β Tm43 Ξ A)
β Tm43 Ξ A
var43 : β{Ξ A} β Var43 Ξ A β Tm43 Ξ A; var43
= Ξ» x Tm43 var43 lam app tt pair fst snd left right case zero suc rec β
var43 _ _ x
lam43 : β{Ξ A B} β Tm43 (snoc43 Ξ A) B β Tm43 Ξ (arr43 A B); lam43
= Ξ» t Tm43 var43 lam43 app tt pair fst snd left right case zero suc rec β
lam43 _ _ _ (t Tm43 var43 lam43 app tt pair fst snd left right case zero suc rec)
app43 : β{Ξ A B} β Tm43 Ξ (arr43 A B) β Tm43 Ξ A β Tm43 Ξ B; app43
= Ξ» t u Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec β
app43 _ _ _ (t Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec)
(u Tm43 var43 lam43 app43 tt pair fst snd left right case zero suc rec)
tt43 : β{Ξ} β Tm43 Ξ top43; tt43
= Ξ» Tm43 var43 lam43 app43 tt43 pair fst snd left right case zero suc rec β tt43 _
pair43 : β{Ξ A B} β Tm43 Ξ A β Tm43 Ξ B β Tm43 Ξ (prod43 A B); pair43
= Ξ» t u Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec β
pair43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec)
(u Tm43 var43 lam43 app43 tt43 pair43 fst snd left right case zero suc rec)
fst43 : β{Ξ A B} β Tm43 Ξ (prod43 A B) β Tm43 Ξ A; fst43
= Ξ» t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd left right case zero suc rec β
fst43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd left right case zero suc rec)
snd43 : β{Ξ A B} β Tm43 Ξ (prod43 A B) β Tm43 Ξ B; snd43
= Ξ» t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left right case zero suc rec β
snd43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left right case zero suc rec)
left43 : β{Ξ A B} β Tm43 Ξ A β Tm43 Ξ (sum43 A B); left43
= Ξ» t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right case zero suc rec β
left43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right case zero suc rec)
right43 : β{Ξ A B} β Tm43 Ξ B β Tm43 Ξ (sum43 A B); right43
= Ξ» t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case zero suc rec β
right43 _ _ _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case zero suc rec)
case43 : β{Ξ A B C} β Tm43 Ξ (sum43 A B) β Tm43 Ξ (arr43 A C) β Tm43 Ξ (arr43 B C) β Tm43 Ξ C; case43
= Ξ» t u v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec β
case43 _ _ _ _
(t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
(u Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
(v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero suc rec)
zero43 : β{Ξ} β Tm43 Ξ nat43; zero43
= Ξ» Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc rec β zero43 _
suc43 : β{Ξ} β Tm43 Ξ nat43 β Tm43 Ξ nat43; suc43
= Ξ» t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec β
suc43 _ (t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec)
rec43 : β{Ξ A} β Tm43 Ξ nat43 β Tm43 Ξ (arr43 nat43 (arr43 A A)) β Tm43 Ξ A β Tm43 Ξ A; rec43
= Ξ» t u v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43 β
rec43 _ _
(t Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
(u Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
(v Tm43 var43 lam43 app43 tt43 pair43 fst43 snd43 left43 right43 case43 zero43 suc43 rec43)
v043 : β{Ξ A} β Tm43 (snoc43 Ξ A) A; v043
= var43 vz43
v143 : β{Ξ A B} β Tm43 (snoc43 (snoc43 Ξ A) B) A; v143
= var43 (vs43 vz43)
v243 : β{Ξ A B C} β Tm43 (snoc43 (snoc43 (snoc43 Ξ A) B) C) A; v243
= var43 (vs43 (vs43 vz43))
v343 : β{Ξ A B C D} β Tm43 (snoc43 (snoc43 (snoc43 (snoc43 Ξ A) B) C) D) A; v343
= var43 (vs43 (vs43 (vs43 vz43)))
tbool43 : Ty43; tbool43
= sum43 top43 top43
true43 : β{Ξ} β Tm43 Ξ tbool43; true43
= left43 tt43
tfalse43 : β{Ξ} β Tm43 Ξ tbool43; tfalse43
= right43 tt43
ifthenelse43 : β{Ξ A} β Tm43 Ξ (arr43 tbool43 (arr43 A (arr43 A A))); ifthenelse43
= lam43 (lam43 (lam43 (case43 v243 (lam43 v243) (lam43 v143))))
times443 : β{Ξ A} β Tm43 Ξ (arr43 (arr43 A A) (arr43 A A)); times443
= lam43 (lam43 (app43 v143 (app43 v143 (app43 v143 (app43 v143 v043)))))
add43 : β{Ξ} β Tm43 Ξ (arr43 nat43 (arr43 nat43 nat43)); add43
= lam43 (rec43 v043
(lam43 (lam43 (lam43 (suc43 (app43 v143 v043)))))
(lam43 v043))
mul43 : β{Ξ} β Tm43 Ξ (arr43 nat43 (arr43 nat43 nat43)); mul43
= lam43 (rec43 v043
(lam43 (lam43 (lam43 (app43 (app43 add43 (app43 v143 v043)) v043))))
(lam43 zero43))
fact43 : β{Ξ} β Tm43 Ξ (arr43 nat43 nat43); fact43
= lam43 (rec43 v043 (lam43 (lam43 (app43 (app43 mul43 (suc43 v143)) v043)))
(suc43 zero43))
{-# OPTIONS --type-in-type #-}
Ty44 : Set
Ty44 =
(Ty44 : Set)
(nat top bot : Ty44)
(arr prod sum : Ty44 β Ty44 β Ty44)
β Ty44
nat44 : Ty44; nat44 = Ξ» _ nat44 _ _ _ _ _ β nat44
top44 : Ty44; top44 = Ξ» _ _ top44 _ _ _ _ β top44
bot44 : Ty44; bot44 = Ξ» _ _ _ bot44 _ _ _ β bot44
arr44 : Ty44 β Ty44 β Ty44; arr44
= Ξ» A B Ty44 nat44 top44 bot44 arr44 prod sum β
arr44 (A Ty44 nat44 top44 bot44 arr44 prod sum) (B Ty44 nat44 top44 bot44 arr44 prod sum)
prod44 : Ty44 β Ty44 β Ty44; prod44
= Ξ» A B Ty44 nat44 top44 bot44 arr44 prod44 sum β
prod44 (A Ty44 nat44 top44 bot44 arr44 prod44 sum) (B Ty44 nat44 top44 bot44 arr44 prod44 sum)
sum44 : Ty44 β Ty44 β Ty44; sum44
= Ξ» A B Ty44 nat44 top44 bot44 arr44 prod44 sum44 β
sum44 (A Ty44 nat44 top44 bot44 arr44 prod44 sum44) (B Ty44 nat44 top44 bot44 arr44 prod44 sum44)
Con44 : Set; Con44
= (Con44 : Set)
(nil : Con44)
(snoc : Con44 β Ty44 β Con44)
β Con44
nil44 : Con44; nil44
= Ξ» Con44 nil44 snoc β nil44
snoc44 : Con44 β Ty44 β Con44; snoc44
= Ξ» Ξ A Con44 nil44 snoc44 β snoc44 (Ξ Con44 nil44 snoc44) A
Var44 : Con44 β Ty44 β Set; Var44
= Ξ» Ξ A β
(Var44 : Con44 β Ty44 β Set)
(vz : β Ξ A β Var44 (snoc44 Ξ A) A)
(vs : β Ξ B A β Var44 Ξ A β Var44 (snoc44 Ξ B) A)
β Var44 Ξ A
vz44 : β{Ξ A} β Var44 (snoc44 Ξ A) A; vz44
= Ξ» Var44 vz44 vs β vz44 _ _
vs44 : β{Ξ B A} β Var44 Ξ A β Var44 (snoc44 Ξ B) A; vs44
= Ξ» x Var44 vz44 vs44 β vs44 _ _ _ (x Var44 vz44 vs44)
Tm44 : Con44 β Ty44 β Set; Tm44
= Ξ» Ξ A β
(Tm44 : Con44 β Ty44 β Set)
(var : β Ξ A β Var44 Ξ A β Tm44 Ξ A)
(lam : β Ξ A B β Tm44 (snoc44 Ξ A) B β Tm44 Ξ (arr44 A B))
(app : β Ξ A B β Tm44 Ξ (arr44 A B) β Tm44 Ξ A β Tm44 Ξ B)
(tt : β Ξ β Tm44 Ξ top44)
(pair : β Ξ A B β Tm44 Ξ A β Tm44 Ξ B β Tm44 Ξ (prod44 A B))
(fst : β Ξ A B β Tm44 Ξ (prod44 A B) β Tm44 Ξ A)
(snd : β Ξ A B β Tm44 Ξ (prod44 A B) β Tm44 Ξ B)
(left : β Ξ A B β Tm44 Ξ A β Tm44 Ξ (sum44 A B))
(right : β Ξ A B β Tm44 Ξ B β Tm44 Ξ (sum44 A B))
(case : β Ξ A B C β Tm44 Ξ (sum44 A B) β Tm44 Ξ (arr44 A C) β Tm44 Ξ (arr44 B C) β Tm44 Ξ C)
(zero : β Ξ β Tm44 Ξ nat44)
(suc : β Ξ β Tm44 Ξ nat44 β Tm44 Ξ nat44)
(rec : β Ξ A β Tm44 Ξ nat44 β Tm44 Ξ (arr44 nat44 (arr44 A A)) β Tm44 Ξ A β Tm44 Ξ A)
β Tm44 Ξ A
var44 : β{Ξ A} β Var44 Ξ A β Tm44 Ξ A; var44
= Ξ» x Tm44 var44 lam app tt pair fst snd left right case zero suc rec β
var44 _ _ x
lam44 : β{Ξ A B} β Tm44 (snoc44 Ξ A) B β Tm44 Ξ (arr44 A B); lam44
= Ξ» t Tm44 var44 lam44 app tt pair fst snd left right case zero suc rec β
lam44 _ _ _ (t Tm44 var44 lam44 app tt pair fst snd left right case zero suc rec)
app44 : β{Ξ A B} β Tm44 Ξ (arr44 A B) β Tm44 Ξ A β Tm44 Ξ B; app44
= Ξ» t u Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec β
app44 _ _ _ (t Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec)
(u Tm44 var44 lam44 app44 tt pair fst snd left right case zero suc rec)
tt44 : β{Ξ} β Tm44 Ξ top44; tt44
= Ξ» Tm44 var44 lam44 app44 tt44 pair fst snd left right case zero suc rec β tt44 _
pair44 : β{Ξ A B} β Tm44 Ξ A β Tm44 Ξ B β Tm44 Ξ (prod44 A B); pair44
= Ξ» t u Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec β
pair44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec)
(u Tm44 var44 lam44 app44 tt44 pair44 fst snd left right case zero suc rec)
fst44 : β{Ξ A B} β Tm44 Ξ (prod44 A B) β Tm44 Ξ A; fst44
= Ξ» t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd left right case zero suc rec β
fst44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd left right case zero suc rec)
snd44 : β{Ξ A B} β Tm44 Ξ (prod44 A B) β Tm44 Ξ B; snd44
= Ξ» t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left right case zero suc rec β
snd44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left right case zero suc rec)
left44 : β{Ξ A B} β Tm44 Ξ A β Tm44 Ξ (sum44 A B); left44
= Ξ» t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right case zero suc rec β
left44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right case zero suc rec)
right44 : β{Ξ A B} β Tm44 Ξ B β Tm44 Ξ (sum44 A B); right44
= Ξ» t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case zero suc rec β
right44 _ _ _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case zero suc rec)
case44 : β{Ξ A B C} β Tm44 Ξ (sum44 A B) β Tm44 Ξ (arr44 A C) β Tm44 Ξ (arr44 B C) β Tm44 Ξ C; case44
= Ξ» t u v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec β
case44 _ _ _ _
(t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
(u Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
(v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero suc rec)
zero44 : β{Ξ} β Tm44 Ξ nat44; zero44
= Ξ» Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc rec β zero44 _
suc44 : β{Ξ} β Tm44 Ξ nat44 β Tm44 Ξ nat44; suc44
= Ξ» t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec β
suc44 _ (t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec)
rec44 : β{Ξ A} β Tm44 Ξ nat44 β Tm44 Ξ (arr44 nat44 (arr44 A A)) β Tm44 Ξ A β Tm44 Ξ A; rec44
= Ξ» t u v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44 β
rec44 _ _
(t Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
(u Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
(v Tm44 var44 lam44 app44 tt44 pair44 fst44 snd44 left44 right44 case44 zero44 suc44 rec44)
v044 : β{Ξ A} β Tm44 (snoc44 Ξ A) A; v044
= var44 vz44
v144 : β{Ξ A B} β Tm44 (snoc44 (snoc44 Ξ A) B) A; v144
= var44 (vs44 vz44)
v244 : β{Ξ A B C} β Tm44 (snoc44 (snoc44 (snoc44 Ξ A) B) C) A; v244
= var44 (vs44 (vs44 vz44))
v344 : β{Ξ A B C D} β Tm44 (snoc44 (snoc44 (snoc44 (snoc44 Ξ A) B) C) D) A; v344
= var44 (vs44 (vs44 (vs44 vz44)))
tbool44 : Ty44; tbool44
= sum44 top44 top44
true44 : β{Ξ} β Tm44 Ξ tbool44; true44
= left44 tt44
tfalse44 : β{Ξ} β Tm44 Ξ tbool44; tfalse44
= right44 tt44
ifthenelse44 : β{Ξ A} β Tm44 Ξ (arr44 tbool44 (arr44 A (arr44 A A))); ifthenelse44
= lam44 (lam44 (lam44 (case44 v244 (lam44 v244) (lam44 v144))))
times444 : β{Ξ A} β Tm44 Ξ (arr44 (arr44 A A) (arr44 A A)); times444
= lam44 (lam44 (app44 v144 (app44 v144 (app44 v144 (app44 v144 v044)))))
add44 : β{Ξ} β Tm44 Ξ (arr44 nat44 (arr44 nat44 nat44)); add44
= lam44 (rec44 v044
(lam44 (lam44 (lam44 (suc44 (app44 v144 v044)))))
(lam44 v044))
mul44 : β{Ξ} β Tm44 Ξ (arr44 nat44 (arr44 nat44 nat44)); mul44
= lam44 (rec44 v044
(lam44 (lam44 (lam44 (app44 (app44 add44 (app44 v144 v044)) v044))))
(lam44 zero44))
fact44 : β{Ξ} β Tm44 Ξ (arr44 nat44 nat44); fact44
= lam44 (rec44 v044 (lam44 (lam44 (app44 (app44 mul44 (suc44 v144)) v044)))
(suc44 zero44))
{-# OPTIONS --type-in-type #-}
Ty45 : Set
Ty45 =
(Ty45 : Set)
(nat top bot : Ty45)
(arr prod sum : Ty45 β Ty45 β Ty45)
β Ty45
nat45 : Ty45; nat45 = Ξ» _ nat45 _ _ _ _ _ β nat45
top45 : Ty45; top45 = Ξ» _ _ top45 _ _ _ _ β top45
bot45 : Ty45; bot45 = Ξ» _ _ _ bot45 _ _ _ β bot45
arr45 : Ty45 β Ty45 β Ty45; arr45
= Ξ» A B Ty45 nat45 top45 bot45 arr45 prod sum β
arr45 (A Ty45 nat45 top45 bot45 arr45 prod sum) (B Ty45 nat45 top45 bot45 arr45 prod sum)
prod45 : Ty45 β Ty45 β Ty45; prod45
= Ξ» A B Ty45 nat45 top45 bot45 arr45 prod45 sum β
prod45 (A Ty45 nat45 top45 bot45 arr45 prod45 sum) (B Ty45 nat45 top45 bot45 arr45 prod45 sum)
sum45 : Ty45 β Ty45 β Ty45; sum45
= Ξ» A B Ty45 nat45 top45 bot45 arr45 prod45 sum45 β
sum45 (A Ty45 nat45 top45 bot45 arr45 prod45 sum45) (B Ty45 nat45 top45 bot45 arr45 prod45 sum45)
Con45 : Set; Con45
= (Con45 : Set)
(nil : Con45)
(snoc : Con45 β Ty45 β Con45)
β Con45
nil45 : Con45; nil45
= Ξ» Con45 nil45 snoc β nil45
snoc45 : Con45 β Ty45 β Con45; snoc45
= Ξ» Ξ A Con45 nil45 snoc45 β snoc45 (Ξ Con45 nil45 snoc45) A
Var45 : Con45 β Ty45 β Set; Var45
= Ξ» Ξ A β
(Var45 : Con45 β Ty45 β Set)
(vz : β Ξ A β Var45 (snoc45 Ξ A) A)
(vs : β Ξ B A β Var45 Ξ A β Var45 (snoc45 Ξ B) A)
β Var45 Ξ A
vz45 : β{Ξ A} β Var45 (snoc45 Ξ A) A; vz45
= Ξ» Var45 vz45 vs β vz45 _ _
vs45 : β{Ξ B A} β Var45 Ξ A β Var45 (snoc45 Ξ B) A; vs45
= Ξ» x Var45 vz45 vs45 β vs45 _ _ _ (x Var45 vz45 vs45)
Tm45 : Con45 β Ty45 β Set; Tm45
= Ξ» Ξ A β
(Tm45 : Con45 β Ty45 β Set)
(var : β Ξ A β Var45 Ξ A β Tm45 Ξ A)
(lam : β Ξ A B β Tm45 (snoc45 Ξ A) B β Tm45 Ξ (arr45 A B))
(app : β Ξ A B β Tm45 Ξ (arr45 A B) β Tm45 Ξ A β Tm45 Ξ B)
(tt : β Ξ β Tm45 Ξ top45)
(pair : β Ξ A B β Tm45 Ξ A β Tm45 Ξ B β Tm45 Ξ (prod45 A B))
(fst : β Ξ A B β Tm45 Ξ (prod45 A B) β Tm45 Ξ A)
(snd : β Ξ A B β Tm45 Ξ (prod45 A B) β Tm45 Ξ B)
(left : β Ξ A B β Tm45 Ξ A β Tm45 Ξ (sum45 A B))
(right : β Ξ A B β Tm45 Ξ B β Tm45 Ξ (sum45 A B))
(case : β Ξ A B C β Tm45 Ξ (sum45 A B) β Tm45 Ξ (arr45 A C) β Tm45 Ξ (arr45 B C) β Tm45 Ξ C)
(zero : β Ξ β Tm45 Ξ nat45)
(suc : β Ξ β Tm45 Ξ nat45 β Tm45 Ξ nat45)
(rec : β Ξ A β Tm45 Ξ nat45 β Tm45 Ξ (arr45 nat45 (arr45 A A)) β Tm45 Ξ A β Tm45 Ξ A)
β Tm45 Ξ A
var45 : β{Ξ A} β Var45 Ξ A β Tm45 Ξ A; var45
= Ξ» x Tm45 var45 lam app tt pair fst snd left right case zero suc rec β
var45 _ _ x
lam45 : β{Ξ A B} β Tm45 (snoc45 Ξ A) B β Tm45 Ξ (arr45 A B); lam45
= Ξ» t Tm45 var45 lam45 app tt pair fst snd left right case zero suc rec β
lam45 _ _ _ (t Tm45 var45 lam45 app tt pair fst snd left right case zero suc rec)
app45 : β{Ξ A B} β Tm45 Ξ (arr45 A B) β Tm45 Ξ A β Tm45 Ξ B; app45
= Ξ» t u Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec β
app45 _ _ _ (t Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec)
(u Tm45 var45 lam45 app45 tt pair fst snd left right case zero suc rec)
tt45 : β{Ξ} β Tm45 Ξ top45; tt45
= Ξ» Tm45 var45 lam45 app45 tt45 pair fst snd left right case zero suc rec β tt45 _
pair45 : β{Ξ A B} β Tm45 Ξ A β Tm45 Ξ B β Tm45 Ξ (prod45 A B); pair45
= Ξ» t u Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec β
pair45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec)
(u Tm45 var45 lam45 app45 tt45 pair45 fst snd left right case zero suc rec)
fst45 : β{Ξ A B} β Tm45 Ξ (prod45 A B) β Tm45 Ξ A; fst45
= Ξ» t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd left right case zero suc rec β
fst45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd left right case zero suc rec)
snd45 : β{Ξ A B} β Tm45 Ξ (prod45 A B) β Tm45 Ξ B; snd45
= Ξ» t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left right case zero suc rec β
snd45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left right case zero suc rec)
left45 : β{Ξ A B} β Tm45 Ξ A β Tm45 Ξ (sum45 A B); left45
= Ξ» t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right case zero suc rec β
left45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right case zero suc rec)
right45 : β{Ξ A B} β Tm45 Ξ B β Tm45 Ξ (sum45 A B); right45
= Ξ» t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case zero suc rec β
right45 _ _ _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case zero suc rec)
case45 : β{Ξ A B C} β Tm45 Ξ (sum45 A B) β Tm45 Ξ (arr45 A C) β Tm45 Ξ (arr45 B C) β Tm45 Ξ C; case45
= Ξ» t u v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec β
case45 _ _ _ _
(t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
(u Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
(v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero suc rec)
zero45 : β{Ξ} β Tm45 Ξ nat45; zero45
= Ξ» Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc rec β zero45 _
suc45 : β{Ξ} β Tm45 Ξ nat45 β Tm45 Ξ nat45; suc45
= Ξ» t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec β
suc45 _ (t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec)
rec45 : β{Ξ A} β Tm45 Ξ nat45 β Tm45 Ξ (arr45 nat45 (arr45 A A)) β Tm45 Ξ A β Tm45 Ξ A; rec45
= Ξ» t u v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45 β
rec45 _ _
(t Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
(u Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
(v Tm45 var45 lam45 app45 tt45 pair45 fst45 snd45 left45 right45 case45 zero45 suc45 rec45)
v045 : β{Ξ A} β Tm45 (snoc45 Ξ A) A; v045
= var45 vz45
v145 : β{Ξ A B} β Tm45 (snoc45 (snoc45 Ξ A) B) A; v145
= var45 (vs45 vz45)
v245 : β{Ξ A B C} β Tm45 (snoc45 (snoc45 (snoc45 Ξ A) B) C) A; v245
= var45 (vs45 (vs45 vz45))
v345 : β{Ξ A B C D} β Tm45 (snoc45 (snoc45 (snoc45 (snoc45 Ξ A) B) C) D) A; v345
= var45 (vs45 (vs45 (vs45 vz45)))
tbool45 : Ty45; tbool45
= sum45 top45 top45
true45 : β{Ξ} β Tm45 Ξ tbool45; true45
= left45 tt45
tfalse45 : β{Ξ} β Tm45 Ξ tbool45; tfalse45
= right45 tt45
ifthenelse45 : β{Ξ A} β Tm45 Ξ (arr45 tbool45 (arr45 A (arr45 A A))); ifthenelse45
= lam45 (lam45 (lam45 (case45 v245 (lam45 v245) (lam45 v145))))
times445 : β{Ξ A} β Tm45 Ξ (arr45 (arr45 A A) (arr45 A A)); times445
= lam45 (lam45 (app45 v145 (app45 v145 (app45 v145 (app45 v145 v045)))))
add45 : β{Ξ} β Tm45 Ξ (arr45 nat45 (arr45 nat45 nat45)); add45
= lam45 (rec45 v045
(lam45 (lam45 (lam45 (suc45 (app45 v145 v045)))))
(lam45 v045))
mul45 : β{Ξ} β Tm45 Ξ (arr45 nat45 (arr45 nat45 nat45)); mul45
= lam45 (rec45 v045
(lam45 (lam45 (lam45 (app45 (app45 add45 (app45 v145 v045)) v045))))
(lam45 zero45))
fact45 : β{Ξ} β Tm45 Ξ (arr45 nat45 nat45); fact45
= lam45 (rec45 v045 (lam45 (lam45 (app45 (app45 mul45 (suc45 v145)) v045)))
(suc45 zero45))
{-# OPTIONS --type-in-type #-}
Ty46 : Set
Ty46 =
(Ty46 : Set)
(nat top bot : Ty46)
(arr prod sum : Ty46 β Ty46 β Ty46)
β Ty46
nat46 : Ty46; nat46 = Ξ» _ nat46 _ _ _ _ _ β nat46
top46 : Ty46; top46 = Ξ» _ _ top46 _ _ _ _ β top46
bot46 : Ty46; bot46 = Ξ» _ _ _ bot46 _ _ _ β bot46
arr46 : Ty46 β Ty46 β Ty46; arr46
= Ξ» A B Ty46 nat46 top46 bot46 arr46 prod sum β
arr46 (A Ty46 nat46 top46 bot46 arr46 prod sum) (B Ty46 nat46 top46 bot46 arr46 prod sum)
prod46 : Ty46 β Ty46 β Ty46; prod46
= Ξ» A B Ty46 nat46 top46 bot46 arr46 prod46 sum β
prod46 (A Ty46 nat46 top46 bot46 arr46 prod46 sum) (B Ty46 nat46 top46 bot46 arr46 prod46 sum)
sum46 : Ty46 β Ty46 β Ty46; sum46
= Ξ» A B Ty46 nat46 top46 bot46 arr46 prod46 sum46 β
sum46 (A Ty46 nat46 top46 bot46 arr46 prod46 sum46) (B Ty46 nat46 top46 bot46 arr46 prod46 sum46)
Con46 : Set; Con46
= (Con46 : Set)
(nil : Con46)
(snoc : Con46 β Ty46 β Con46)
β Con46
nil46 : Con46; nil46
= Ξ» Con46 nil46 snoc β nil46
snoc46 : Con46 β Ty46 β Con46; snoc46
= Ξ» Ξ A Con46 nil46 snoc46 β snoc46 (Ξ Con46 nil46 snoc46) A
Var46 : Con46 β Ty46 β Set; Var46
= Ξ» Ξ A β
(Var46 : Con46 β Ty46 β Set)
(vz : β Ξ A β Var46 (snoc46 Ξ A) A)
(vs : β Ξ B A β Var46 Ξ A β Var46 (snoc46 Ξ B) A)
β Var46 Ξ A
vz46 : β{Ξ A} β Var46 (snoc46 Ξ A) A; vz46
= Ξ» Var46 vz46 vs β vz46 _ _
vs46 : β{Ξ B A} β Var46 Ξ A β Var46 (snoc46 Ξ B) A; vs46
= Ξ» x Var46 vz46 vs46 β vs46 _ _ _ (x Var46 vz46 vs46)
Tm46 : Con46 β Ty46 β Set; Tm46
= Ξ» Ξ A β
(Tm46 : Con46 β Ty46 β Set)
(var : β Ξ A β Var46 Ξ A β Tm46 Ξ A)
(lam : β Ξ A B β Tm46 (snoc46 Ξ A) B β Tm46 Ξ (arr46 A B))
(app : β Ξ A B β Tm46 Ξ (arr46 A B) β Tm46 Ξ A β Tm46 Ξ B)
(tt : β Ξ β Tm46 Ξ top46)
(pair : β Ξ A B β Tm46 Ξ A β Tm46 Ξ B β Tm46 Ξ (prod46 A B))
(fst : β Ξ A B β Tm46 Ξ (prod46 A B) β Tm46 Ξ A)
(snd : β Ξ A B β Tm46 Ξ (prod46 A B) β Tm46 Ξ B)
(left : β Ξ A B β Tm46 Ξ A β Tm46 Ξ (sum46 A B))
(right : β Ξ A B β Tm46 Ξ B β Tm46 Ξ (sum46 A B))
(case : β Ξ A B C β Tm46 Ξ (sum46 A B) β Tm46 Ξ (arr46 A C) β Tm46 Ξ (arr46 B C) β Tm46 Ξ C)
(zero : β Ξ β Tm46 Ξ nat46)
(suc : β Ξ β Tm46 Ξ nat46 β Tm46 Ξ nat46)
(rec : β Ξ A β Tm46 Ξ nat46 β Tm46 Ξ (arr46 nat46 (arr46 A A)) β Tm46 Ξ A β Tm46 Ξ A)
β Tm46 Ξ A
var46 : β{Ξ A} β Var46 Ξ A β Tm46 Ξ A; var46
= Ξ» x Tm46 var46 lam app tt pair fst snd left right case zero suc rec β
var46 _ _ x
lam46 : β{Ξ A B} β Tm46 (snoc46 Ξ A) B β Tm46 Ξ (arr46 A B); lam46
= Ξ» t Tm46 var46 lam46 app tt pair fst snd left right case zero suc rec β
lam46 _ _ _ (t Tm46 var46 lam46 app tt pair fst snd left right case zero suc rec)
app46 : β{Ξ A B} β Tm46 Ξ (arr46 A B) β Tm46 Ξ A β Tm46 Ξ B; app46
= Ξ» t u Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec β
app46 _ _ _ (t Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec)
(u Tm46 var46 lam46 app46 tt pair fst snd left right case zero suc rec)
tt46 : β{Ξ} β Tm46 Ξ top46; tt46
= Ξ» Tm46 var46 lam46 app46 tt46 pair fst snd left right case zero suc rec β tt46 _
pair46 : β{Ξ A B} β Tm46 Ξ A β Tm46 Ξ B β Tm46 Ξ (prod46 A B); pair46
= Ξ» t u Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec β
pair46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec)
(u Tm46 var46 lam46 app46 tt46 pair46 fst snd left right case zero suc rec)
fst46 : β{Ξ A B} β Tm46 Ξ (prod46 A B) β Tm46 Ξ A; fst46
= Ξ» t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd left right case zero suc rec β
fst46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd left right case zero suc rec)
snd46 : β{Ξ A B} β Tm46 Ξ (prod46 A B) β Tm46 Ξ B; snd46
= Ξ» t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left right case zero suc rec β
snd46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left right case zero suc rec)
left46 : β{Ξ A B} β Tm46 Ξ A β Tm46 Ξ (sum46 A B); left46
= Ξ» t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right case zero suc rec β
left46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right case zero suc rec)
right46 : β{Ξ A B} β Tm46 Ξ B β Tm46 Ξ (sum46 A B); right46
= Ξ» t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case zero suc rec β
right46 _ _ _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case zero suc rec)
case46 : β{Ξ A B C} β Tm46 Ξ (sum46 A B) β Tm46 Ξ (arr46 A C) β Tm46 Ξ (arr46 B C) β Tm46 Ξ C; case46
= Ξ» t u v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec β
case46 _ _ _ _
(t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
(u Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
(v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero suc rec)
zero46 : β{Ξ} β Tm46 Ξ nat46; zero46
= Ξ» Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc rec β zero46 _
suc46 : β{Ξ} β Tm46 Ξ nat46 β Tm46 Ξ nat46; suc46
= Ξ» t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec β
suc46 _ (t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec)
rec46 : β{Ξ A} β Tm46 Ξ nat46 β Tm46 Ξ (arr46 nat46 (arr46 A A)) β Tm46 Ξ A β Tm46 Ξ A; rec46
= Ξ» t u v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46 β
rec46 _ _
(t Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
(u Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
(v Tm46 var46 lam46 app46 tt46 pair46 fst46 snd46 left46 right46 case46 zero46 suc46 rec46)
v046 : β{Ξ A} β Tm46 (snoc46 Ξ A) A; v046
= var46 vz46
v146 : β{Ξ A B} β Tm46 (snoc46 (snoc46 Ξ A) B) A; v146
= var46 (vs46 vz46)
v246 : β{Ξ A B C} β Tm46 (snoc46 (snoc46 (snoc46 Ξ A) B) C) A; v246
= var46 (vs46 (vs46 vz46))
v346 : β{Ξ A B C D} β Tm46 (snoc46 (snoc46 (snoc46 (snoc46 Ξ A) B) C) D) A; v346
= var46 (vs46 (vs46 (vs46 vz46)))
tbool46 : Ty46; tbool46
= sum46 top46 top46
true46 : β{Ξ} β Tm46 Ξ tbool46; true46
= left46 tt46
tfalse46 : β{Ξ} β Tm46 Ξ tbool46; tfalse46
= right46 tt46
ifthenelse46 : β{Ξ A} β Tm46 Ξ (arr46 tbool46 (arr46 A (arr46 A A))); ifthenelse46
= lam46 (lam46 (lam46 (case46 v246 (lam46 v246) (lam46 v146))))
times446 : β{Ξ A} β Tm46 Ξ (arr46 (arr46 A A) (arr46 A A)); times446
= lam46 (lam46 (app46 v146 (app46 v146 (app46 v146 (app46 v146 v046)))))
add46 : β{Ξ} β Tm46 Ξ (arr46 nat46 (arr46 nat46 nat46)); add46
= lam46 (rec46 v046
(lam46 (lam46 (lam46 (suc46 (app46 v146 v046)))))
(lam46 v046))
mul46 : β{Ξ} β Tm46 Ξ (arr46 nat46 (arr46 nat46 nat46)); mul46
= lam46 (rec46 v046
(lam46 (lam46 (lam46 (app46 (app46 add46 (app46 v146 v046)) v046))))
(lam46 zero46))
fact46 : β{Ξ} β Tm46 Ξ (arr46 nat46 nat46); fact46
= lam46 (rec46 v046 (lam46 (lam46 (app46 (app46 mul46 (suc46 v146)) v046)))
(suc46 zero46))
{-# OPTIONS --type-in-type #-}
Ty47 : Set
Ty47 =
(Ty47 : Set)
(nat top bot : Ty47)
(arr prod sum : Ty47 β Ty47 β Ty47)
β Ty47
nat47 : Ty47; nat47 = Ξ» _ nat47 _ _ _ _ _ β nat47
top47 : Ty47; top47 = Ξ» _ _ top47 _ _ _ _ β top47
bot47 : Ty47; bot47 = Ξ» _ _ _ bot47 _ _ _ β bot47
arr47 : Ty47 β Ty47 β Ty47; arr47
= Ξ» A B Ty47 nat47 top47 bot47 arr47 prod sum β
arr47 (A Ty47 nat47 top47 bot47 arr47 prod sum) (B Ty47 nat47 top47 bot47 arr47 prod sum)
prod47 : Ty47 β Ty47 β Ty47; prod47
= Ξ» A B Ty47 nat47 top47 bot47 arr47 prod47 sum β
prod47 (A Ty47 nat47 top47 bot47 arr47 prod47 sum) (B Ty47 nat47 top47 bot47 arr47 prod47 sum)
sum47 : Ty47 β Ty47 β Ty47; sum47
= Ξ» A B Ty47 nat47 top47 bot47 arr47 prod47 sum47 β
sum47 (A Ty47 nat47 top47 bot47 arr47 prod47 sum47) (B Ty47 nat47 top47 bot47 arr47 prod47 sum47)
Con47 : Set; Con47
= (Con47 : Set)
(nil : Con47)
(snoc : Con47 β Ty47 β Con47)
β Con47
nil47 : Con47; nil47
= Ξ» Con47 nil47 snoc β nil47
snoc47 : Con47 β Ty47 β Con47; snoc47
= Ξ» Ξ A Con47 nil47 snoc47 β snoc47 (Ξ Con47 nil47 snoc47) A
Var47 : Con47 β Ty47 β Set; Var47
= Ξ» Ξ A β
(Var47 : Con47 β Ty47 β Set)
(vz : β Ξ A β Var47 (snoc47 Ξ A) A)
(vs : β Ξ B A β Var47 Ξ A β Var47 (snoc47 Ξ B) A)
β Var47 Ξ A
vz47 : β{Ξ A} β Var47 (snoc47 Ξ A) A; vz47
= Ξ» Var47 vz47 vs β vz47 _ _
vs47 : β{Ξ B A} β Var47 Ξ A β Var47 (snoc47 Ξ B) A; vs47
= Ξ» x Var47 vz47 vs47 β vs47 _ _ _ (x Var47 vz47 vs47)
Tm47 : Con47 β Ty47 β Set; Tm47
= Ξ» Ξ A β
(Tm47 : Con47 β Ty47 β Set)
(var : β Ξ A β Var47 Ξ A β Tm47 Ξ A)
(lam : β Ξ A B β Tm47 (snoc47 Ξ A) B β Tm47 Ξ (arr47 A B))
(app : β Ξ A B β Tm47 Ξ (arr47 A B) β Tm47 Ξ A β Tm47 Ξ B)
(tt : β Ξ β Tm47 Ξ top47)
(pair : β Ξ A B β Tm47 Ξ A β Tm47 Ξ B β Tm47 Ξ (prod47 A B))
(fst : β Ξ A B β Tm47 Ξ (prod47 A B) β Tm47 Ξ A)
(snd : β Ξ A B β Tm47 Ξ (prod47 A B) β Tm47 Ξ B)
(left : β Ξ A B β Tm47 Ξ A β Tm47 Ξ (sum47 A B))
(right : β Ξ A B β Tm47 Ξ B β Tm47 Ξ (sum47 A B))
(case : β Ξ A B C β Tm47 Ξ (sum47 A B) β Tm47 Ξ (arr47 A C) β Tm47 Ξ (arr47 B C) β Tm47 Ξ C)
(zero : β Ξ β Tm47 Ξ nat47)
(suc : β Ξ β Tm47 Ξ nat47 β Tm47 Ξ nat47)
(rec : β Ξ A β Tm47 Ξ nat47 β Tm47 Ξ (arr47 nat47 (arr47 A A)) β Tm47 Ξ A β Tm47 Ξ A)
β Tm47 Ξ A
var47 : β{Ξ A} β Var47 Ξ A β Tm47 Ξ A; var47
= Ξ» x Tm47 var47 lam app tt pair fst snd left right case zero suc rec β
var47 _ _ x
lam47 : β{Ξ A B} β Tm47 (snoc47 Ξ A) B β Tm47 Ξ (arr47 A B); lam47
= Ξ» t Tm47 var47 lam47 app tt pair fst snd left right case zero suc rec β
lam47 _ _ _ (t Tm47 var47 lam47 app tt pair fst snd left right case zero suc rec)
app47 : β{Ξ A B} β Tm47 Ξ (arr47 A B) β Tm47 Ξ A β Tm47 Ξ B; app47
= Ξ» t u Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec β
app47 _ _ _ (t Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec)
(u Tm47 var47 lam47 app47 tt pair fst snd left right case zero suc rec)
tt47 : β{Ξ} β Tm47 Ξ top47; tt47
= Ξ» Tm47 var47 lam47 app47 tt47 pair fst snd left right case zero suc rec β tt47 _
pair47 : β{Ξ A B} β Tm47 Ξ A β Tm47 Ξ B β Tm47 Ξ (prod47 A B); pair47
= Ξ» t u Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec β
pair47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec)
(u Tm47 var47 lam47 app47 tt47 pair47 fst snd left right case zero suc rec)
fst47 : β{Ξ A B} β Tm47 Ξ (prod47 A B) β Tm47 Ξ A; fst47
= Ξ» t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd left right case zero suc rec β
fst47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd left right case zero suc rec)
snd47 : β{Ξ A B} β Tm47 Ξ (prod47 A B) β Tm47 Ξ B; snd47
= Ξ» t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left right case zero suc rec β
snd47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left right case zero suc rec)
left47 : β{Ξ A B} β Tm47 Ξ A β Tm47 Ξ (sum47 A B); left47
= Ξ» t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right case zero suc rec β
left47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right case zero suc rec)
right47 : β{Ξ A B} β Tm47 Ξ B β Tm47 Ξ (sum47 A B); right47
= Ξ» t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case zero suc rec β
right47 _ _ _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case zero suc rec)
case47 : β{Ξ A B C} β Tm47 Ξ (sum47 A B) β Tm47 Ξ (arr47 A C) β Tm47 Ξ (arr47 B C) β Tm47 Ξ C; case47
= Ξ» t u v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec β
case47 _ _ _ _
(t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
(u Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
(v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero suc rec)
zero47 : β{Ξ} β Tm47 Ξ nat47; zero47
= Ξ» Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc rec β zero47 _
suc47 : β{Ξ} β Tm47 Ξ nat47 β Tm47 Ξ nat47; suc47
= Ξ» t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec β
suc47 _ (t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec)
rec47 : β{Ξ A} β Tm47 Ξ nat47 β Tm47 Ξ (arr47 nat47 (arr47 A A)) β Tm47 Ξ A β Tm47 Ξ A; rec47
= Ξ» t u v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47 β
rec47 _ _
(t Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
(u Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
(v Tm47 var47 lam47 app47 tt47 pair47 fst47 snd47 left47 right47 case47 zero47 suc47 rec47)
v047 : β{Ξ A} β Tm47 (snoc47 Ξ A) A; v047
= var47 vz47
v147 : β{Ξ A B} β Tm47 (snoc47 (snoc47 Ξ A) B) A; v147
= var47 (vs47 vz47)
v247 : β{Ξ A B C} β Tm47 (snoc47 (snoc47 (snoc47 Ξ A) B) C) A; v247
= var47 (vs47 (vs47 vz47))
v347 : β{Ξ A B C D} β Tm47 (snoc47 (snoc47 (snoc47 (snoc47 Ξ A) B) C) D) A; v347
= var47 (vs47 (vs47 (vs47 vz47)))
tbool47 : Ty47; tbool47
= sum47 top47 top47
true47 : β{Ξ} β Tm47 Ξ tbool47; true47
= left47 tt47
tfalse47 : β{Ξ} β Tm47 Ξ tbool47; tfalse47
= right47 tt47
ifthenelse47 : β{Ξ A} β Tm47 Ξ (arr47 tbool47 (arr47 A (arr47 A A))); ifthenelse47
= lam47 (lam47 (lam47 (case47 v247 (lam47 v247) (lam47 v147))))
times447 : β{Ξ A} β Tm47 Ξ (arr47 (arr47 A A) (arr47 A A)); times447
= lam47 (lam47 (app47 v147 (app47 v147 (app47 v147 (app47 v147 v047)))))
add47 : β{Ξ} β Tm47 Ξ (arr47 nat47 (arr47 nat47 nat47)); add47
= lam47 (rec47 v047
(lam47 (lam47 (lam47 (suc47 (app47 v147 v047)))))
(lam47 v047))
mul47 : β{Ξ} β Tm47 Ξ (arr47 nat47 (arr47 nat47 nat47)); mul47
= lam47 (rec47 v047
(lam47 (lam47 (lam47 (app47 (app47 add47 (app47 v147 v047)) v047))))
(lam47 zero47))
fact47 : β{Ξ} β Tm47 Ξ (arr47 nat47 nat47); fact47
= lam47 (rec47 v047 (lam47 (lam47 (app47 (app47 mul47 (suc47 v147)) v047)))
(suc47 zero47))
{-# OPTIONS --type-in-type #-}
Ty48 : Set
Ty48 =
(Ty48 : Set)
(nat top bot : Ty48)
(arr prod sum : Ty48 β Ty48 β Ty48)
β Ty48
nat48 : Ty48; nat48 = Ξ» _ nat48 _ _ _ _ _ β nat48
top48 : Ty48; top48 = Ξ» _ _ top48 _ _ _ _ β top48
bot48 : Ty48; bot48 = Ξ» _ _ _ bot48 _ _ _ β bot48
arr48 : Ty48 β Ty48 β Ty48; arr48
= Ξ» A B Ty48 nat48 top48 bot48 arr48 prod sum β
arr48 (A Ty48 nat48 top48 bot48 arr48 prod sum) (B Ty48 nat48 top48 bot48 arr48 prod sum)
prod48 : Ty48 β Ty48 β Ty48; prod48
= Ξ» A B Ty48 nat48 top48 bot48 arr48 prod48 sum β
prod48 (A Ty48 nat48 top48 bot48 arr48 prod48 sum) (B Ty48 nat48 top48 bot48 arr48 prod48 sum)
sum48 : Ty48 β Ty48 β Ty48; sum48
= Ξ» A B Ty48 nat48 top48 bot48 arr48 prod48 sum48 β
sum48 (A Ty48 nat48 top48 bot48 arr48 prod48 sum48) (B Ty48 nat48 top48 bot48 arr48 prod48 sum48)
Con48 : Set; Con48
= (Con48 : Set)
(nil : Con48)
(snoc : Con48 β Ty48 β Con48)
β Con48
nil48 : Con48; nil48
= Ξ» Con48 nil48 snoc β nil48
snoc48 : Con48 β Ty48 β Con48; snoc48
= Ξ» Ξ A Con48 nil48 snoc48 β snoc48 (Ξ Con48 nil48 snoc48) A
Var48 : Con48 β Ty48 β Set; Var48
= Ξ» Ξ A β
(Var48 : Con48 β Ty48 β Set)
(vz : β Ξ A β Var48 (snoc48 Ξ A) A)
(vs : β Ξ B A β Var48 Ξ A β Var48 (snoc48 Ξ B) A)
β Var48 Ξ A
vz48 : β{Ξ A} β Var48 (snoc48 Ξ A) A; vz48
= Ξ» Var48 vz48 vs β vz48 _ _
vs48 : β{Ξ B A} β Var48 Ξ A β Var48 (snoc48 Ξ B) A; vs48
= Ξ» x Var48 vz48 vs48 β vs48 _ _ _ (x Var48 vz48 vs48)
Tm48 : Con48 β Ty48 β Set; Tm48
= Ξ» Ξ A β
(Tm48 : Con48 β Ty48 β Set)
(var : β Ξ A β Var48 Ξ A β Tm48 Ξ A)
(lam : β Ξ A B β Tm48 (snoc48 Ξ A) B β Tm48 Ξ (arr48 A B))
(app : β Ξ A B β Tm48 Ξ (arr48 A B) β Tm48 Ξ A β Tm48 Ξ B)
(tt : β Ξ β Tm48 Ξ top48)
(pair : β Ξ A B β Tm48 Ξ A β Tm48 Ξ B β Tm48 Ξ (prod48 A B))
(fst : β Ξ A B β Tm48 Ξ (prod48 A B) β Tm48 Ξ A)
(snd : β Ξ A B β Tm48 Ξ (prod48 A B) β Tm48 Ξ B)
(left : β Ξ A B β Tm48 Ξ A β Tm48 Ξ (sum48 A B))
(right : β Ξ A B β Tm48 Ξ B β Tm48 Ξ (sum48 A B))
(case : β Ξ A B C β Tm48 Ξ (sum48 A B) β Tm48 Ξ (arr48 A C) β Tm48 Ξ (arr48 B C) β Tm48 Ξ C)
(zero : β Ξ β Tm48 Ξ nat48)
(suc : β Ξ β Tm48 Ξ nat48 β Tm48 Ξ nat48)
(rec : β Ξ A β Tm48 Ξ nat48 β Tm48 Ξ (arr48 nat48 (arr48 A A)) β Tm48 Ξ A β Tm48 Ξ A)
β Tm48 Ξ A
var48 : β{Ξ A} β Var48 Ξ A β Tm48 Ξ A; var48
= Ξ» x Tm48 var48 lam app tt pair fst snd left right case zero suc rec β
var48 _ _ x
lam48 : β{Ξ A B} β Tm48 (snoc48 Ξ A) B β Tm48 Ξ (arr48 A B); lam48
= Ξ» t Tm48 var48 lam48 app tt pair fst snd left right case zero suc rec β
lam48 _ _ _ (t Tm48 var48 lam48 app tt pair fst snd left right case zero suc rec)
app48 : β{Ξ A B} β Tm48 Ξ (arr48 A B) β Tm48 Ξ A β Tm48 Ξ B; app48
= Ξ» t u Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec β
app48 _ _ _ (t Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec)
(u Tm48 var48 lam48 app48 tt pair fst snd left right case zero suc rec)
tt48 : β{Ξ} β Tm48 Ξ top48; tt48
= Ξ» Tm48 var48 lam48 app48 tt48 pair fst snd left right case zero suc rec β tt48 _
pair48 : β{Ξ A B} β Tm48 Ξ A β Tm48 Ξ B β Tm48 Ξ (prod48 A B); pair48
= Ξ» t u Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec β
pair48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec)
(u Tm48 var48 lam48 app48 tt48 pair48 fst snd left right case zero suc rec)
fst48 : β{Ξ A B} β Tm48 Ξ (prod48 A B) β Tm48 Ξ A; fst48
= Ξ» t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd left right case zero suc rec β
fst48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd left right case zero suc rec)
snd48 : β{Ξ A B} β Tm48 Ξ (prod48 A B) β Tm48 Ξ B; snd48
= Ξ» t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left right case zero suc rec β
snd48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left right case zero suc rec)
left48 : β{Ξ A B} β Tm48 Ξ A β Tm48 Ξ (sum48 A B); left48
= Ξ» t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right case zero suc rec β
left48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right case zero suc rec)
right48 : β{Ξ A B} β Tm48 Ξ B β Tm48 Ξ (sum48 A B); right48
= Ξ» t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case zero suc rec β
right48 _ _ _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case zero suc rec)
case48 : β{Ξ A B C} β Tm48 Ξ (sum48 A B) β Tm48 Ξ (arr48 A C) β Tm48 Ξ (arr48 B C) β Tm48 Ξ C; case48
= Ξ» t u v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec β
case48 _ _ _ _
(t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
(u Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
(v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero suc rec)
zero48 : β{Ξ} β Tm48 Ξ nat48; zero48
= Ξ» Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc rec β zero48 _
suc48 : β{Ξ} β Tm48 Ξ nat48 β Tm48 Ξ nat48; suc48
= Ξ» t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec β
suc48 _ (t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec)
rec48 : β{Ξ A} β Tm48 Ξ nat48 β Tm48 Ξ (arr48 nat48 (arr48 A A)) β Tm48 Ξ A β Tm48 Ξ A; rec48
= Ξ» t u v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48 β
rec48 _ _
(t Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
(u Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
(v Tm48 var48 lam48 app48 tt48 pair48 fst48 snd48 left48 right48 case48 zero48 suc48 rec48)
v048 : β{Ξ A} β Tm48 (snoc48 Ξ A) A; v048
= var48 vz48
v148 : β{Ξ A B} β Tm48 (snoc48 (snoc48 Ξ A) B) A; v148
= var48 (vs48 vz48)
v248 : β{Ξ A B C} β Tm48 (snoc48 (snoc48 (snoc48 Ξ A) B) C) A; v248
= var48 (vs48 (vs48 vz48))
v348 : β{Ξ A B C D} β Tm48 (snoc48 (snoc48 (snoc48 (snoc48 Ξ A) B) C) D) A; v348
= var48 (vs48 (vs48 (vs48 vz48)))
tbool48 : Ty48; tbool48
= sum48 top48 top48
true48 : β{Ξ} β Tm48 Ξ tbool48; true48
= left48 tt48
tfalse48 : β{Ξ} β Tm48 Ξ tbool48; tfalse48
= right48 tt48
ifthenelse48 : β{Ξ A} β Tm48 Ξ (arr48 tbool48 (arr48 A (arr48 A A))); ifthenelse48
= lam48 (lam48 (lam48 (case48 v248 (lam48 v248) (lam48 v148))))
times448 : β{Ξ A} β Tm48 Ξ (arr48 (arr48 A A) (arr48 A A)); times448
= lam48 (lam48 (app48 v148 (app48 v148 (app48 v148 (app48 v148 v048)))))
add48 : β{Ξ} β Tm48 Ξ (arr48 nat48 (arr48 nat48 nat48)); add48
= lam48 (rec48 v048
(lam48 (lam48 (lam48 (suc48 (app48 v148 v048)))))
(lam48 v048))
mul48 : β{Ξ} β Tm48 Ξ (arr48 nat48 (arr48 nat48 nat48)); mul48
= lam48 (rec48 v048
(lam48 (lam48 (lam48 (app48 (app48 add48 (app48 v148 v048)) v048))))
(lam48 zero48))
fact48 : β{Ξ} β Tm48 Ξ (arr48 nat48 nat48); fact48
= lam48 (rec48 v048 (lam48 (lam48 (app48 (app48 mul48 (suc48 v148)) v048)))
(suc48 zero48))
{-# OPTIONS --type-in-type #-}
Ty49 : Set
Ty49 =
(Ty49 : Set)
(nat top bot : Ty49)
(arr prod sum : Ty49 β Ty49 β Ty49)
β Ty49
nat49 : Ty49; nat49 = Ξ» _ nat49 _ _ _ _ _ β nat49
top49 : Ty49; top49 = Ξ» _ _ top49 _ _ _ _ β top49
bot49 : Ty49; bot49 = Ξ» _ _ _ bot49 _ _ _ β bot49
arr49 : Ty49 β Ty49 β Ty49; arr49
= Ξ» A B Ty49 nat49 top49 bot49 arr49 prod sum β
arr49 (A Ty49 nat49 top49 bot49 arr49 prod sum) (B Ty49 nat49 top49 bot49 arr49 prod sum)
prod49 : Ty49 β Ty49 β Ty49; prod49
= Ξ» A B Ty49 nat49 top49 bot49 arr49 prod49 sum β
prod49 (A Ty49 nat49 top49 bot49 arr49 prod49 sum) (B Ty49 nat49 top49 bot49 arr49 prod49 sum)
sum49 : Ty49 β Ty49 β Ty49; sum49
= Ξ» A B Ty49 nat49 top49 bot49 arr49 prod49 sum49 β
sum49 (A Ty49 nat49 top49 bot49 arr49 prod49 sum49) (B Ty49 nat49 top49 bot49 arr49 prod49 sum49)
Con49 : Set; Con49
= (Con49 : Set)
(nil : Con49)
(snoc : Con49 β Ty49 β Con49)
β Con49
nil49 : Con49; nil49
= Ξ» Con49 nil49 snoc β nil49
snoc49 : Con49 β Ty49 β Con49; snoc49
= Ξ» Ξ A Con49 nil49 snoc49 β snoc49 (Ξ Con49 nil49 snoc49) A
Var49 : Con49 β Ty49 β Set; Var49
= Ξ» Ξ A β
(Var49 : Con49 β Ty49 β Set)
(vz : β Ξ A β Var49 (snoc49 Ξ A) A)
(vs : β Ξ B A β Var49 Ξ A β Var49 (snoc49 Ξ B) A)
β Var49 Ξ A
vz49 : β{Ξ A} β Var49 (snoc49 Ξ A) A; vz49
= Ξ» Var49 vz49 vs β vz49 _ _
vs49 : β{Ξ B A} β Var49 Ξ A β Var49 (snoc49 Ξ B) A; vs49
= Ξ» x Var49 vz49 vs49 β vs49 _ _ _ (x Var49 vz49 vs49)
Tm49 : Con49 β Ty49 β Set; Tm49
= Ξ» Ξ A β
(Tm49 : Con49 β Ty49 β Set)
(var : β Ξ A β Var49 Ξ A β Tm49 Ξ A)
(lam : β Ξ A B β Tm49 (snoc49 Ξ A) B β Tm49 Ξ (arr49 A B))
(app : β Ξ A B β Tm49 Ξ (arr49 A B) β Tm49 Ξ A β Tm49 Ξ B)
(tt : β Ξ β Tm49 Ξ top49)
(pair : β Ξ A B β Tm49 Ξ A β Tm49 Ξ B β Tm49 Ξ (prod49 A B))
(fst : β Ξ A B β Tm49 Ξ (prod49 A B) β Tm49 Ξ A)
(snd : β Ξ A B β Tm49 Ξ (prod49 A B) β Tm49 Ξ B)
(left : β Ξ A B β Tm49 Ξ A β Tm49 Ξ (sum49 A B))
(right : β Ξ A B β Tm49 Ξ B β Tm49 Ξ (sum49 A B))
(case : β Ξ A B C β Tm49 Ξ (sum49 A B) β Tm49 Ξ (arr49 A C) β Tm49 Ξ (arr49 B C) β Tm49 Ξ C)
(zero : β Ξ β Tm49 Ξ nat49)
(suc : β Ξ β Tm49 Ξ nat49 β Tm49 Ξ nat49)
(rec : β Ξ A β Tm49 Ξ nat49 β Tm49 Ξ (arr49 nat49 (arr49 A A)) β Tm49 Ξ A β Tm49 Ξ A)
β Tm49 Ξ A
var49 : β{Ξ A} β Var49 Ξ A β Tm49 Ξ A; var49
= Ξ» x Tm49 var49 lam app tt pair fst snd left right case zero suc rec β
var49 _ _ x
lam49 : β{Ξ A B} β Tm49 (snoc49 Ξ A) B β Tm49 Ξ (arr49 A B); lam49
= Ξ» t Tm49 var49 lam49 app tt pair fst snd left right case zero suc rec β
lam49 _ _ _ (t Tm49 var49 lam49 app tt pair fst snd left right case zero suc rec)
app49 : β{Ξ A B} β Tm49 Ξ (arr49 A B) β Tm49 Ξ A β Tm49 Ξ B; app49
= Ξ» t u Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec β
app49 _ _ _ (t Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec)
(u Tm49 var49 lam49 app49 tt pair fst snd left right case zero suc rec)
tt49 : β{Ξ} β Tm49 Ξ top49; tt49
= Ξ» Tm49 var49 lam49 app49 tt49 pair fst snd left right case zero suc rec β tt49 _
pair49 : β{Ξ A B} β Tm49 Ξ A β Tm49 Ξ B β Tm49 Ξ (prod49 A B); pair49
= Ξ» t u Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec β
pair49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec)
(u Tm49 var49 lam49 app49 tt49 pair49 fst snd left right case zero suc rec)
fst49 : β{Ξ A B} β Tm49 Ξ (prod49 A B) β Tm49 Ξ A; fst49
= Ξ» t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd left right case zero suc rec β
fst49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd left right case zero suc rec)
snd49 : β{Ξ A B} β Tm49 Ξ (prod49 A B) β Tm49 Ξ B; snd49
= Ξ» t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left right case zero suc rec β
snd49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left right case zero suc rec)
left49 : β{Ξ A B} β Tm49 Ξ A β Tm49 Ξ (sum49 A B); left49
= Ξ» t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right case zero suc rec β
left49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right case zero suc rec)
right49 : β{Ξ A B} β Tm49 Ξ B β Tm49 Ξ (sum49 A B); right49
= Ξ» t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case zero suc rec β
right49 _ _ _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case zero suc rec)
case49 : β{Ξ A B C} β Tm49 Ξ (sum49 A B) β Tm49 Ξ (arr49 A C) β Tm49 Ξ (arr49 B C) β Tm49 Ξ C; case49
= Ξ» t u v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec β
case49 _ _ _ _
(t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
(u Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
(v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero suc rec)
zero49 : β{Ξ} β Tm49 Ξ nat49; zero49
= Ξ» Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc rec β zero49 _
suc49 : β{Ξ} β Tm49 Ξ nat49 β Tm49 Ξ nat49; suc49
= Ξ» t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec β
suc49 _ (t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec)
rec49 : β{Ξ A} β Tm49 Ξ nat49 β Tm49 Ξ (arr49 nat49 (arr49 A A)) β Tm49 Ξ A β Tm49 Ξ A; rec49
= Ξ» t u v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49 β
rec49 _ _
(t Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
(u Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
(v Tm49 var49 lam49 app49 tt49 pair49 fst49 snd49 left49 right49 case49 zero49 suc49 rec49)
v049 : β{Ξ A} β Tm49 (snoc49 Ξ A) A; v049
= var49 vz49
v149 : β{Ξ A B} β Tm49 (snoc49 (snoc49 Ξ A) B) A; v149
= var49 (vs49 vz49)
v249 : β{Ξ A B C} β Tm49 (snoc49 (snoc49 (snoc49 Ξ A) B) C) A; v249
= var49 (vs49 (vs49 vz49))
v349 : β{Ξ A B C D} β Tm49 (snoc49 (snoc49 (snoc49 (snoc49 Ξ A) B) C) D) A; v349
= var49 (vs49 (vs49 (vs49 vz49)))
tbool49 : Ty49; tbool49
= sum49 top49 top49
true49 : β{Ξ} β Tm49 Ξ tbool49; true49
= left49 tt49
tfalse49 : β{Ξ} β Tm49 Ξ tbool49; tfalse49
= right49 tt49
ifthenelse49 : β{Ξ A} β Tm49 Ξ (arr49 tbool49 (arr49 A (arr49 A A))); ifthenelse49
= lam49 (lam49 (lam49 (case49 v249 (lam49 v249) (lam49 v149))))
times449 : β{Ξ A} β Tm49 Ξ (arr49 (arr49 A A) (arr49 A A)); times449
= lam49 (lam49 (app49 v149 (app49 v149 (app49 v149 (app49 v149 v049)))))
add49 : β{Ξ} β Tm49 Ξ (arr49 nat49 (arr49 nat49 nat49)); add49
= lam49 (rec49 v049
(lam49 (lam49 (lam49 (suc49 (app49 v149 v049)))))
(lam49 v049))
mul49 : β{Ξ} β Tm49 Ξ (arr49 nat49 (arr49 nat49 nat49)); mul49
= lam49 (rec49 v049
(lam49 (lam49 (lam49 (app49 (app49 add49 (app49 v149 v049)) v049))))
(lam49 zero49))
fact49 : β{Ξ} β Tm49 Ξ (arr49 nat49 nat49); fact49
= lam49 (rec49 v049 (lam49 (lam49 (app49 (app49 mul49 (suc49 v149)) v049)))
(suc49 zero49))
{-# OPTIONS --type-in-type #-}
Ty50 : Set
Ty50 =
(Ty50 : Set)
(nat top bot : Ty50)
(arr prod sum : Ty50 β Ty50 β Ty50)
β Ty50
nat50 : Ty50; nat50 = Ξ» _ nat50 _ _ _ _ _ β nat50
top50 : Ty50; top50 = Ξ» _ _ top50 _ _ _ _ β top50
bot50 : Ty50; bot50 = Ξ» _ _ _ bot50 _ _ _ β bot50
arr50 : Ty50 β Ty50 β Ty50; arr50
= Ξ» A B Ty50 nat50 top50 bot50 arr50 prod sum β
arr50 (A Ty50 nat50 top50 bot50 arr50 prod sum) (B Ty50 nat50 top50 bot50 arr50 prod sum)
prod50 : Ty50 β Ty50 β Ty50; prod50
= Ξ» A B Ty50 nat50 top50 bot50 arr50 prod50 sum β
prod50 (A Ty50 nat50 top50 bot50 arr50 prod50 sum) (B Ty50 nat50 top50 bot50 arr50 prod50 sum)
sum50 : Ty50 β Ty50 β Ty50; sum50
= Ξ» A B Ty50 nat50 top50 bot50 arr50 prod50 sum50 β
sum50 (A Ty50 nat50 top50 bot50 arr50 prod50 sum50) (B Ty50 nat50 top50 bot50 arr50 prod50 sum50)
Con50 : Set; Con50
= (Con50 : Set)
(nil : Con50)
(snoc : Con50 β Ty50 β Con50)
β Con50
nil50 : Con50; nil50
= Ξ» Con50 nil50 snoc β nil50
snoc50 : Con50 β Ty50 β Con50; snoc50
= Ξ» Ξ A Con50 nil50 snoc50 β snoc50 (Ξ Con50 nil50 snoc50) A
Var50 : Con50 β Ty50 β Set; Var50
= Ξ» Ξ A β
(Var50 : Con50 β Ty50 β Set)
(vz : β Ξ A β Var50 (snoc50 Ξ A) A)
(vs : β Ξ B A β Var50 Ξ A β Var50 (snoc50 Ξ B) A)
β Var50 Ξ A
vz50 : β{Ξ A} β Var50 (snoc50 Ξ A) A; vz50
= Ξ» Var50 vz50 vs β vz50 _ _
vs50 : β{Ξ B A} β Var50 Ξ A β Var50 (snoc50 Ξ B) A; vs50
= Ξ» x Var50 vz50 vs50 β vs50 _ _ _ (x Var50 vz50 vs50)
Tm50 : Con50 β Ty50 β Set; Tm50
= Ξ» Ξ A β
(Tm50 : Con50 β Ty50 β Set)
(var : β Ξ A β Var50 Ξ A β Tm50 Ξ A)
(lam : β Ξ A B β Tm50 (snoc50 Ξ A) B β Tm50 Ξ (arr50 A B))
(app : β Ξ A B β Tm50 Ξ (arr50 A B) β Tm50 Ξ A β Tm50 Ξ B)
(tt : β Ξ β Tm50 Ξ top50)
(pair : β Ξ A B β Tm50 Ξ A β Tm50 Ξ B β Tm50 Ξ (prod50 A B))
(fst : β Ξ A B β Tm50 Ξ (prod50 A B) β Tm50 Ξ A)
(snd : β Ξ A B β Tm50 Ξ (prod50 A B) β Tm50 Ξ B)
(left : β Ξ A B β Tm50 Ξ A β Tm50 Ξ (sum50 A B))
(right : β Ξ A B β Tm50 Ξ B β Tm50 Ξ (sum50 A B))
(case : β Ξ A B C β Tm50 Ξ (sum50 A B) β Tm50 Ξ (arr50 A C) β Tm50 Ξ (arr50 B C) β Tm50 Ξ C)
(zero : β Ξ β Tm50 Ξ nat50)
(suc : β Ξ β Tm50 Ξ nat50 β Tm50 Ξ nat50)
(rec : β Ξ A β Tm50 Ξ nat50 β Tm50 Ξ (arr50 nat50 (arr50 A A)) β Tm50 Ξ A β Tm50 Ξ A)
β Tm50 Ξ A
var50 : β{Ξ A} β Var50 Ξ A β Tm50 Ξ A; var50
= Ξ» x Tm50 var50 lam app tt pair fst snd left right case zero suc rec β
var50 _ _ x
lam50 : β{Ξ A B} β Tm50 (snoc50 Ξ A) B β Tm50 Ξ (arr50 A B); lam50
= Ξ» t Tm50 var50 lam50 app tt pair fst snd left right case zero suc rec β
lam50 _ _ _ (t Tm50 var50 lam50 app tt pair fst snd left right case zero suc rec)
app50 : β{Ξ A B} β Tm50 Ξ (arr50 A B) β Tm50 Ξ A β Tm50 Ξ B; app50
= Ξ» t u Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec β
app50 _ _ _ (t Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec)
(u Tm50 var50 lam50 app50 tt pair fst snd left right case zero suc rec)
tt50 : β{Ξ} β Tm50 Ξ top50; tt50
= Ξ» Tm50 var50 lam50 app50 tt50 pair fst snd left right case zero suc rec β tt50 _
pair50 : β{Ξ A B} β Tm50 Ξ A β Tm50 Ξ B β Tm50 Ξ (prod50 A B); pair50
= Ξ» t u Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec β
pair50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec)
(u Tm50 var50 lam50 app50 tt50 pair50 fst snd left right case zero suc rec)
fst50 : β{Ξ A B} β Tm50 Ξ (prod50 A B) β Tm50 Ξ A; fst50
= Ξ» t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd left right case zero suc rec β
fst50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd left right case zero suc rec)
snd50 : β{Ξ A B} β Tm50 Ξ (prod50 A B) β Tm50 Ξ B; snd50
= Ξ» t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left right case zero suc rec β
snd50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left right case zero suc rec)
left50 : β{Ξ A B} β Tm50 Ξ A β Tm50 Ξ (sum50 A B); left50
= Ξ» t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right case zero suc rec β
left50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right case zero suc rec)
right50 : β{Ξ A B} β Tm50 Ξ B β Tm50 Ξ (sum50 A B); right50
= Ξ» t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case zero suc rec β
right50 _ _ _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case zero suc rec)
case50 : β{Ξ A B C} β Tm50 Ξ (sum50 A B) β Tm50 Ξ (arr50 A C) β Tm50 Ξ (arr50 B C) β Tm50 Ξ C; case50
= Ξ» t u v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec β
case50 _ _ _ _
(t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
(u Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
(v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero suc rec)
zero50 : β{Ξ} β Tm50 Ξ nat50; zero50
= Ξ» Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc rec β zero50 _
suc50 : β{Ξ} β Tm50 Ξ nat50 β Tm50 Ξ nat50; suc50
= Ξ» t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec β
suc50 _ (t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec)
rec50 : β{Ξ A} β Tm50 Ξ nat50 β Tm50 Ξ (arr50 nat50 (arr50 A A)) β Tm50 Ξ A β Tm50 Ξ A; rec50
= Ξ» t u v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50 β
rec50 _ _
(t Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
(u Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
(v Tm50 var50 lam50 app50 tt50 pair50 fst50 snd50 left50 right50 case50 zero50 suc50 rec50)
v050 : β{Ξ A} β Tm50 (snoc50 Ξ A) A; v050
= var50 vz50
v150 : β{Ξ A B} β Tm50 (snoc50 (snoc50 Ξ A) B) A; v150
= var50 (vs50 vz50)
v250 : β{Ξ A B C} β Tm50 (snoc50 (snoc50 (snoc50 Ξ A) B) C) A; v250
= var50 (vs50 (vs50 vz50))
v350 : β{Ξ A B C D} β Tm50 (snoc50 (snoc50 (snoc50 (snoc50 Ξ A) B) C) D) A; v350
= var50 (vs50 (vs50 (vs50 vz50)))
tbool50 : Ty50; tbool50
= sum50 top50 top50
true50 : β{Ξ} β Tm50 Ξ tbool50; true50
= left50 tt50
tfalse50 : β{Ξ} β Tm50 Ξ tbool50; tfalse50
= right50 tt50
ifthenelse50 : β{Ξ A} β Tm50 Ξ (arr50 tbool50 (arr50 A (arr50 A A))); ifthenelse50
= lam50 (lam50 (lam50 (case50 v250 (lam50 v250) (lam50 v150))))
times450 : β{Ξ A} β Tm50 Ξ (arr50 (arr50 A A) (arr50 A A)); times450
= lam50 (lam50 (app50 v150 (app50 v150 (app50 v150 (app50 v150 v050)))))
add50 : β{Ξ} β Tm50 Ξ (arr50 nat50 (arr50 nat50 nat50)); add50
= lam50 (rec50 v050
(lam50 (lam50 (lam50 (suc50 (app50 v150 v050)))))
(lam50 v050))
mul50 : β{Ξ} β Tm50 Ξ (arr50 nat50 (arr50 nat50 nat50)); mul50
= lam50 (rec50 v050
(lam50 (lam50 (lam50 (app50 (app50 add50 (app50 v150 v050)) v050))))
(lam50 zero50))
fact50 : β{Ξ} β Tm50 Ξ (arr50 nat50 nat50); fact50
= lam50 (rec50 v050 (lam50 (lam50 (app50 (app50 mul50 (suc50 v150)) v050)))
(suc50 zero50))
{-# OPTIONS --type-in-type #-}
Ty51 : Set
Ty51 =
(Ty51 : Set)
(nat top bot : Ty51)
(arr prod sum : Ty51 β Ty51 β Ty51)
β Ty51
nat51 : Ty51; nat51 = Ξ» _ nat51 _ _ _ _ _ β nat51
top51 : Ty51; top51 = Ξ» _ _ top51 _ _ _ _ β top51
bot51 : Ty51; bot51 = Ξ» _ _ _ bot51 _ _ _ β bot51
arr51 : Ty51 β Ty51 β Ty51; arr51
= Ξ» A B Ty51 nat51 top51 bot51 arr51 prod sum β
arr51 (A Ty51 nat51 top51 bot51 arr51 prod sum) (B Ty51 nat51 top51 bot51 arr51 prod sum)
prod51 : Ty51 β Ty51 β Ty51; prod51
= Ξ» A B Ty51 nat51 top51 bot51 arr51 prod51 sum β
prod51 (A Ty51 nat51 top51 bot51 arr51 prod51 sum) (B Ty51 nat51 top51 bot51 arr51 prod51 sum)
sum51 : Ty51 β Ty51 β Ty51; sum51
= Ξ» A B Ty51 nat51 top51 bot51 arr51 prod51 sum51 β
sum51 (A Ty51 nat51 top51 bot51 arr51 prod51 sum51) (B Ty51 nat51 top51 bot51 arr51 prod51 sum51)
Con51 : Set; Con51
= (Con51 : Set)
(nil : Con51)
(snoc : Con51 β Ty51 β Con51)
β Con51
nil51 : Con51; nil51
= Ξ» Con51 nil51 snoc β nil51
snoc51 : Con51 β Ty51 β Con51; snoc51
= Ξ» Ξ A Con51 nil51 snoc51 β snoc51 (Ξ Con51 nil51 snoc51) A
Var51 : Con51 β Ty51 β Set; Var51
= Ξ» Ξ A β
(Var51 : Con51 β Ty51 β Set)
(vz : β Ξ A β Var51 (snoc51 Ξ A) A)
(vs : β Ξ B A β Var51 Ξ A β Var51 (snoc51 Ξ B) A)
β Var51 Ξ A
vz51 : β{Ξ A} β Var51 (snoc51 Ξ A) A; vz51
= Ξ» Var51 vz51 vs β vz51 _ _
vs51 : β{Ξ B A} β Var51 Ξ A β Var51 (snoc51 Ξ B) A; vs51
= Ξ» x Var51 vz51 vs51 β vs51 _ _ _ (x Var51 vz51 vs51)
Tm51 : Con51 β Ty51 β Set; Tm51
= Ξ» Ξ A β
(Tm51 : Con51 β Ty51 β Set)
(var : β Ξ A β Var51 Ξ A β Tm51 Ξ A)
(lam : β Ξ A B β Tm51 (snoc51 Ξ A) B β Tm51 Ξ (arr51 A B))
(app : β Ξ A B β Tm51 Ξ (arr51 A B) β Tm51 Ξ A β Tm51 Ξ B)
(tt : β Ξ β Tm51 Ξ top51)
(pair : β Ξ A B β Tm51 Ξ A β Tm51 Ξ B β Tm51 Ξ (prod51 A B))
(fst : β Ξ A B β Tm51 Ξ (prod51 A B) β Tm51 Ξ A)
(snd : β Ξ A B β Tm51 Ξ (prod51 A B) β Tm51 Ξ B)
(left : β Ξ A B β Tm51 Ξ A β Tm51 Ξ (sum51 A B))
(right : β Ξ A B β Tm51 Ξ B β Tm51 Ξ (sum51 A B))
(case : β Ξ A B C β Tm51 Ξ (sum51 A B) β Tm51 Ξ (arr51 A C) β Tm51 Ξ (arr51 B C) β Tm51 Ξ C)
(zero : β Ξ β Tm51 Ξ nat51)
(suc : β Ξ β Tm51 Ξ nat51 β Tm51 Ξ nat51)
(rec : β Ξ A β Tm51 Ξ nat51 β Tm51 Ξ (arr51 nat51 (arr51 A A)) β Tm51 Ξ A β Tm51 Ξ A)
β Tm51 Ξ A
var51 : β{Ξ A} β Var51 Ξ A β Tm51 Ξ A; var51
= Ξ» x Tm51 var51 lam app tt pair fst snd left right case zero suc rec β
var51 _ _ x
lam51 : β{Ξ A B} β Tm51 (snoc51 Ξ A) B β Tm51 Ξ (arr51 A B); lam51
= Ξ» t Tm51 var51 lam51 app tt pair fst snd left right case zero suc rec β
lam51 _ _ _ (t Tm51 var51 lam51 app tt pair fst snd left right case zero suc rec)
app51 : β{Ξ A B} β Tm51 Ξ (arr51 A B) β Tm51 Ξ A β Tm51 Ξ B; app51
= Ξ» t u Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec β
app51 _ _ _ (t Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec)
(u Tm51 var51 lam51 app51 tt pair fst snd left right case zero suc rec)
tt51 : β{Ξ} β Tm51 Ξ top51; tt51
= Ξ» Tm51 var51 lam51 app51 tt51 pair fst snd left right case zero suc rec β tt51 _
pair51 : β{Ξ A B} β Tm51 Ξ A β Tm51 Ξ B β Tm51 Ξ (prod51 A B); pair51
= Ξ» t u Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec β
pair51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec)
(u Tm51 var51 lam51 app51 tt51 pair51 fst snd left right case zero suc rec)
fst51 : β{Ξ A B} β Tm51 Ξ (prod51 A B) β Tm51 Ξ A; fst51
= Ξ» t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd left right case zero suc rec β
fst51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd left right case zero suc rec)
snd51 : β{Ξ A B} β Tm51 Ξ (prod51 A B) β Tm51 Ξ B; snd51
= Ξ» t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left right case zero suc rec β
snd51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left right case zero suc rec)
left51 : β{Ξ A B} β Tm51 Ξ A β Tm51 Ξ (sum51 A B); left51
= Ξ» t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right case zero suc rec β
left51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right case zero suc rec)
right51 : β{Ξ A B} β Tm51 Ξ B β Tm51 Ξ (sum51 A B); right51
= Ξ» t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case zero suc rec β
right51 _ _ _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case zero suc rec)
case51 : β{Ξ A B C} β Tm51 Ξ (sum51 A B) β Tm51 Ξ (arr51 A C) β Tm51 Ξ (arr51 B C) β Tm51 Ξ C; case51
= Ξ» t u v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec β
case51 _ _ _ _
(t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
(u Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
(v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero suc rec)
zero51 : β{Ξ} β Tm51 Ξ nat51; zero51
= Ξ» Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc rec β zero51 _
suc51 : β{Ξ} β Tm51 Ξ nat51 β Tm51 Ξ nat51; suc51
= Ξ» t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec β
suc51 _ (t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec)
rec51 : β{Ξ A} β Tm51 Ξ nat51 β Tm51 Ξ (arr51 nat51 (arr51 A A)) β Tm51 Ξ A β Tm51 Ξ A; rec51
= Ξ» t u v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51 β
rec51 _ _
(t Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
(u Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
(v Tm51 var51 lam51 app51 tt51 pair51 fst51 snd51 left51 right51 case51 zero51 suc51 rec51)
v051 : β{Ξ A} β Tm51 (snoc51 Ξ A) A; v051
= var51 vz51
v151 : β{Ξ A B} β Tm51 (snoc51 (snoc51 Ξ A) B) A; v151
= var51 (vs51 vz51)
v251 : β{Ξ A B C} β Tm51 (snoc51 (snoc51 (snoc51 Ξ A) B) C) A; v251
= var51 (vs51 (vs51 vz51))
v351 : β{Ξ A B C D} β Tm51 (snoc51 (snoc51 (snoc51 (snoc51 Ξ A) B) C) D) A; v351
= var51 (vs51 (vs51 (vs51 vz51)))
tbool51 : Ty51; tbool51
= sum51 top51 top51
true51 : β{Ξ} β Tm51 Ξ tbool51; true51
= left51 tt51
tfalse51 : β{Ξ} β Tm51 Ξ tbool51; tfalse51
= right51 tt51
ifthenelse51 : β{Ξ A} β Tm51 Ξ (arr51 tbool51 (arr51 A (arr51 A A))); ifthenelse51
= lam51 (lam51 (lam51 (case51 v251 (lam51 v251) (lam51 v151))))
times451 : β{Ξ A} β Tm51 Ξ (arr51 (arr51 A A) (arr51 A A)); times451
= lam51 (lam51 (app51 v151 (app51 v151 (app51 v151 (app51 v151 v051)))))
add51 : β{Ξ} β Tm51 Ξ (arr51 nat51 (arr51 nat51 nat51)); add51
= lam51 (rec51 v051
(lam51 (lam51 (lam51 (suc51 (app51 v151 v051)))))
(lam51 v051))
mul51 : β{Ξ} β Tm51 Ξ (arr51 nat51 (arr51 nat51 nat51)); mul51
= lam51 (rec51 v051
(lam51 (lam51 (lam51 (app51 (app51 add51 (app51 v151 v051)) v051))))
(lam51 zero51))
fact51 : β{Ξ} β Tm51 Ξ (arr51 nat51 nat51); fact51
= lam51 (rec51 v051 (lam51 (lam51 (app51 (app51 mul51 (suc51 v151)) v051)))
(suc51 zero51))
{-# OPTIONS --type-in-type #-}
Ty52 : Set
Ty52 =
(Ty52 : Set)
(nat top bot : Ty52)
(arr prod sum : Ty52 β Ty52 β Ty52)
β Ty52
nat52 : Ty52; nat52 = Ξ» _ nat52 _ _ _ _ _ β nat52
top52 : Ty52; top52 = Ξ» _ _ top52 _ _ _ _ β top52
bot52 : Ty52; bot52 = Ξ» _ _ _ bot52 _ _ _ β bot52
arr52 : Ty52 β Ty52 β Ty52; arr52
= Ξ» A B Ty52 nat52 top52 bot52 arr52 prod sum β
arr52 (A Ty52 nat52 top52 bot52 arr52 prod sum) (B Ty52 nat52 top52 bot52 arr52 prod sum)
prod52 : Ty52 β Ty52 β Ty52; prod52
= Ξ» A B Ty52 nat52 top52 bot52 arr52 prod52 sum β
prod52 (A Ty52 nat52 top52 bot52 arr52 prod52 sum) (B Ty52 nat52 top52 bot52 arr52 prod52 sum)
sum52 : Ty52 β Ty52 β Ty52; sum52
= Ξ» A B Ty52 nat52 top52 bot52 arr52 prod52 sum52 β
sum52 (A Ty52 nat52 top52 bot52 arr52 prod52 sum52) (B Ty52 nat52 top52 bot52 arr52 prod52 sum52)
Con52 : Set; Con52
= (Con52 : Set)
(nil : Con52)
(snoc : Con52 β Ty52 β Con52)
β Con52
nil52 : Con52; nil52
= Ξ» Con52 nil52 snoc β nil52
snoc52 : Con52 β Ty52 β Con52; snoc52
= Ξ» Ξ A Con52 nil52 snoc52 β snoc52 (Ξ Con52 nil52 snoc52) A
Var52 : Con52 β Ty52 β Set; Var52
= Ξ» Ξ A β
(Var52 : Con52 β Ty52 β Set)
(vz : β Ξ A β Var52 (snoc52 Ξ A) A)
(vs : β Ξ B A β Var52 Ξ A β Var52 (snoc52 Ξ B) A)
β Var52 Ξ A
vz52 : β{Ξ A} β Var52 (snoc52 Ξ A) A; vz52
= Ξ» Var52 vz52 vs β vz52 _ _
vs52 : β{Ξ B A} β Var52 Ξ A β Var52 (snoc52 Ξ B) A; vs52
= Ξ» x Var52 vz52 vs52 β vs52 _ _ _ (x Var52 vz52 vs52)
Tm52 : Con52 β Ty52 β Set; Tm52
= Ξ» Ξ A β
(Tm52 : Con52 β Ty52 β Set)
(var : β Ξ A β Var52 Ξ A β Tm52 Ξ A)
(lam : β Ξ A B β Tm52 (snoc52 Ξ A) B β Tm52 Ξ (arr52 A B))
(app : β Ξ A B β Tm52 Ξ (arr52 A B) β Tm52 Ξ A β Tm52 Ξ B)
(tt : β Ξ β Tm52 Ξ top52)
(pair : β Ξ A B β Tm52 Ξ A β Tm52 Ξ B β Tm52 Ξ (prod52 A B))
(fst : β Ξ A B β Tm52 Ξ (prod52 A B) β Tm52 Ξ A)
(snd : β Ξ A B β Tm52 Ξ (prod52 A B) β Tm52 Ξ B)
(left : β Ξ A B β Tm52 Ξ A β Tm52 Ξ (sum52 A B))
(right : β Ξ A B β Tm52 Ξ B β Tm52 Ξ (sum52 A B))
(case : β Ξ A B C β Tm52 Ξ (sum52 A B) β Tm52 Ξ (arr52 A C) β Tm52 Ξ (arr52 B C) β Tm52 Ξ C)
(zero : β Ξ β Tm52 Ξ nat52)
(suc : β Ξ β Tm52 Ξ nat52 β Tm52 Ξ nat52)
(rec : β Ξ A β Tm52 Ξ nat52 β Tm52 Ξ (arr52 nat52 (arr52 A A)) β Tm52 Ξ A β Tm52 Ξ A)
β Tm52 Ξ A
var52 : β{Ξ A} β Var52 Ξ A β Tm52 Ξ A; var52
= Ξ» x Tm52 var52 lam app tt pair fst snd left right case zero suc rec β
var52 _ _ x
lam52 : β{Ξ A B} β Tm52 (snoc52 Ξ A) B β Tm52 Ξ (arr52 A B); lam52
= Ξ» t Tm52 var52 lam52 app tt pair fst snd left right case zero suc rec β
lam52 _ _ _ (t Tm52 var52 lam52 app tt pair fst snd left right case zero suc rec)
app52 : β{Ξ A B} β Tm52 Ξ (arr52 A B) β Tm52 Ξ A β Tm52 Ξ B; app52
= Ξ» t u Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec β
app52 _ _ _ (t Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec)
(u Tm52 var52 lam52 app52 tt pair fst snd left right case zero suc rec)
tt52 : β{Ξ} β Tm52 Ξ top52; tt52
= Ξ» Tm52 var52 lam52 app52 tt52 pair fst snd left right case zero suc rec β tt52 _
pair52 : β{Ξ A B} β Tm52 Ξ A β Tm52 Ξ B β Tm52 Ξ (prod52 A B); pair52
= Ξ» t u Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec β
pair52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec)
(u Tm52 var52 lam52 app52 tt52 pair52 fst snd left right case zero suc rec)
fst52 : β{Ξ A B} β Tm52 Ξ (prod52 A B) β Tm52 Ξ A; fst52
= Ξ» t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd left right case zero suc rec β
fst52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd left right case zero suc rec)
snd52 : β{Ξ A B} β Tm52 Ξ (prod52 A B) β Tm52 Ξ B; snd52
= Ξ» t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left right case zero suc rec β
snd52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left right case zero suc rec)
left52 : β{Ξ A B} β Tm52 Ξ A β Tm52 Ξ (sum52 A B); left52
= Ξ» t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right case zero suc rec β
left52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right case zero suc rec)
right52 : β{Ξ A B} β Tm52 Ξ B β Tm52 Ξ (sum52 A B); right52
= Ξ» t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case zero suc rec β
right52 _ _ _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case zero suc rec)
case52 : β{Ξ A B C} β Tm52 Ξ (sum52 A B) β Tm52 Ξ (arr52 A C) β Tm52 Ξ (arr52 B C) β Tm52 Ξ C; case52
= Ξ» t u v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec β
case52 _ _ _ _
(t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
(u Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
(v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero suc rec)
zero52 : β{Ξ} β Tm52 Ξ nat52; zero52
= Ξ» Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc rec β zero52 _
suc52 : β{Ξ} β Tm52 Ξ nat52 β Tm52 Ξ nat52; suc52
= Ξ» t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec β
suc52 _ (t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec)
rec52 : β{Ξ A} β Tm52 Ξ nat52 β Tm52 Ξ (arr52 nat52 (arr52 A A)) β Tm52 Ξ A β Tm52 Ξ A; rec52
= Ξ» t u v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52 β
rec52 _ _
(t Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
(u Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
(v Tm52 var52 lam52 app52 tt52 pair52 fst52 snd52 left52 right52 case52 zero52 suc52 rec52)
v052 : β{Ξ A} β Tm52 (snoc52 Ξ A) A; v052
= var52 vz52
v152 : β{Ξ A B} β Tm52 (snoc52 (snoc52 Ξ A) B) A; v152
= var52 (vs52 vz52)
v252 : β{Ξ A B C} β Tm52 (snoc52 (snoc52 (snoc52 Ξ A) B) C) A; v252
= var52 (vs52 (vs52 vz52))
v352 : β{Ξ A B C D} β Tm52 (snoc52 (snoc52 (snoc52 (snoc52 Ξ A) B) C) D) A; v352
= var52 (vs52 (vs52 (vs52 vz52)))
tbool52 : Ty52; tbool52
= sum52 top52 top52
true52 : β{Ξ} β Tm52 Ξ tbool52; true52
= left52 tt52
tfalse52 : β{Ξ} β Tm52 Ξ tbool52; tfalse52
= right52 tt52
ifthenelse52 : β{Ξ A} β Tm52 Ξ (arr52 tbool52 (arr52 A (arr52 A A))); ifthenelse52
= lam52 (lam52 (lam52 (case52 v252 (lam52 v252) (lam52 v152))))
times452 : β{Ξ A} β Tm52 Ξ (arr52 (arr52 A A) (arr52 A A)); times452
= lam52 (lam52 (app52 v152 (app52 v152 (app52 v152 (app52 v152 v052)))))
add52 : β{Ξ} β Tm52 Ξ (arr52 nat52 (arr52 nat52 nat52)); add52
= lam52 (rec52 v052
(lam52 (lam52 (lam52 (suc52 (app52 v152 v052)))))
(lam52 v052))
mul52 : β{Ξ} β Tm52 Ξ (arr52 nat52 (arr52 nat52 nat52)); mul52
= lam52 (rec52 v052
(lam52 (lam52 (lam52 (app52 (app52 add52 (app52 v152 v052)) v052))))
(lam52 zero52))
fact52 : β{Ξ} β Tm52 Ξ (arr52 nat52 nat52); fact52
= lam52 (rec52 v052 (lam52 (lam52 (app52 (app52 mul52 (suc52 v152)) v052)))
(suc52 zero52))
{-# OPTIONS --type-in-type #-}
Ty53 : Set
Ty53 =
(Ty53 : Set)
(nat top bot : Ty53)
(arr prod sum : Ty53 β Ty53 β Ty53)
β Ty53
nat53 : Ty53; nat53 = Ξ» _ nat53 _ _ _ _ _ β nat53
top53 : Ty53; top53 = Ξ» _ _ top53 _ _ _ _ β top53
bot53 : Ty53; bot53 = Ξ» _ _ _ bot53 _ _ _ β bot53
arr53 : Ty53 β Ty53 β Ty53; arr53
= Ξ» A B Ty53 nat53 top53 bot53 arr53 prod sum β
arr53 (A Ty53 nat53 top53 bot53 arr53 prod sum) (B Ty53 nat53 top53 bot53 arr53 prod sum)
prod53 : Ty53 β Ty53 β Ty53; prod53
= Ξ» A B Ty53 nat53 top53 bot53 arr53 prod53 sum β
prod53 (A Ty53 nat53 top53 bot53 arr53 prod53 sum) (B Ty53 nat53 top53 bot53 arr53 prod53 sum)
sum53 : Ty53 β Ty53 β Ty53; sum53
= Ξ» A B Ty53 nat53 top53 bot53 arr53 prod53 sum53 β
sum53 (A Ty53 nat53 top53 bot53 arr53 prod53 sum53) (B Ty53 nat53 top53 bot53 arr53 prod53 sum53)
Con53 : Set; Con53
= (Con53 : Set)
(nil : Con53)
(snoc : Con53 β Ty53 β Con53)
β Con53
nil53 : Con53; nil53
= Ξ» Con53 nil53 snoc β nil53
snoc53 : Con53 β Ty53 β Con53; snoc53
= Ξ» Ξ A Con53 nil53 snoc53 β snoc53 (Ξ Con53 nil53 snoc53) A
Var53 : Con53 β Ty53 β Set; Var53
= Ξ» Ξ A β
(Var53 : Con53 β Ty53 β Set)
(vz : β Ξ A β Var53 (snoc53 Ξ A) A)
(vs : β Ξ B A β Var53 Ξ A β Var53 (snoc53 Ξ B) A)
β Var53 Ξ A
vz53 : β{Ξ A} β Var53 (snoc53 Ξ A) A; vz53
= Ξ» Var53 vz53 vs β vz53 _ _
vs53 : β{Ξ B A} β Var53 Ξ A β Var53 (snoc53 Ξ B) A; vs53
= Ξ» x Var53 vz53 vs53 β vs53 _ _ _ (x Var53 vz53 vs53)
Tm53 : Con53 β Ty53 β Set; Tm53
= Ξ» Ξ A β
(Tm53 : Con53 β Ty53 β Set)
(var : β Ξ A β Var53 Ξ A β Tm53 Ξ A)
(lam : β Ξ A B β Tm53 (snoc53 Ξ A) B β Tm53 Ξ (arr53 A B))
(app : β Ξ A B β Tm53 Ξ (arr53 A B) β Tm53 Ξ A β Tm53 Ξ B)
(tt : β Ξ β Tm53 Ξ top53)
(pair : β Ξ A B β Tm53 Ξ A β Tm53 Ξ B β Tm53 Ξ (prod53 A B))
(fst : β Ξ A B β Tm53 Ξ (prod53 A B) β Tm53 Ξ A)
(snd : β Ξ A B β Tm53 Ξ (prod53 A B) β Tm53 Ξ B)
(left : β Ξ A B β Tm53 Ξ A β Tm53 Ξ (sum53 A B))
(right : β Ξ A B β Tm53 Ξ B β Tm53 Ξ (sum53 A B))
(case : β Ξ A B C β Tm53 Ξ (sum53 A B) β Tm53 Ξ (arr53 A C) β Tm53 Ξ (arr53 B C) β Tm53 Ξ C)
(zero : β Ξ β Tm53 Ξ nat53)
(suc : β Ξ β Tm53 Ξ nat53 β Tm53 Ξ nat53)
(rec : β Ξ A β Tm53 Ξ nat53 β Tm53 Ξ (arr53 nat53 (arr53 A A)) β Tm53 Ξ A β Tm53 Ξ A)
β Tm53 Ξ A
var53 : β{Ξ A} β Var53 Ξ A β Tm53 Ξ A; var53
= Ξ» x Tm53 var53 lam app tt pair fst snd left right case zero suc rec β
var53 _ _ x
lam53 : β{Ξ A B} β Tm53 (snoc53 Ξ A) B β Tm53 Ξ (arr53 A B); lam53
= Ξ» t Tm53 var53 lam53 app tt pair fst snd left right case zero suc rec β
lam53 _ _ _ (t Tm53 var53 lam53 app tt pair fst snd left right case zero suc rec)
app53 : β{Ξ A B} β Tm53 Ξ (arr53 A B) β Tm53 Ξ A β Tm53 Ξ B; app53
= Ξ» t u Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec β
app53 _ _ _ (t Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec)
(u Tm53 var53 lam53 app53 tt pair fst snd left right case zero suc rec)
tt53 : β{Ξ} β Tm53 Ξ top53; tt53
= Ξ» Tm53 var53 lam53 app53 tt53 pair fst snd left right case zero suc rec β tt53 _
pair53 : β{Ξ A B} β Tm53 Ξ A β Tm53 Ξ B β Tm53 Ξ (prod53 A B); pair53
= Ξ» t u Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec β
pair53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec)
(u Tm53 var53 lam53 app53 tt53 pair53 fst snd left right case zero suc rec)
fst53 : β{Ξ A B} β Tm53 Ξ (prod53 A B) β Tm53 Ξ A; fst53
= Ξ» t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd left right case zero suc rec β
fst53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd left right case zero suc rec)
snd53 : β{Ξ A B} β Tm53 Ξ (prod53 A B) β Tm53 Ξ B; snd53
= Ξ» t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left right case zero suc rec β
snd53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left right case zero suc rec)
left53 : β{Ξ A B} β Tm53 Ξ A β Tm53 Ξ (sum53 A B); left53
= Ξ» t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right case zero suc rec β
left53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right case zero suc rec)
right53 : β{Ξ A B} β Tm53 Ξ B β Tm53 Ξ (sum53 A B); right53
= Ξ» t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case zero suc rec β
right53 _ _ _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case zero suc rec)
case53 : β{Ξ A B C} β Tm53 Ξ (sum53 A B) β Tm53 Ξ (arr53 A C) β Tm53 Ξ (arr53 B C) β Tm53 Ξ C; case53
= Ξ» t u v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec β
case53 _ _ _ _
(t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
(u Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
(v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero suc rec)
zero53 : β{Ξ} β Tm53 Ξ nat53; zero53
= Ξ» Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc rec β zero53 _
suc53 : β{Ξ} β Tm53 Ξ nat53 β Tm53 Ξ nat53; suc53
= Ξ» t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec β
suc53 _ (t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec)
rec53 : β{Ξ A} β Tm53 Ξ nat53 β Tm53 Ξ (arr53 nat53 (arr53 A A)) β Tm53 Ξ A β Tm53 Ξ A; rec53
= Ξ» t u v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53 β
rec53 _ _
(t Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
(u Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
(v Tm53 var53 lam53 app53 tt53 pair53 fst53 snd53 left53 right53 case53 zero53 suc53 rec53)
v053 : β{Ξ A} β Tm53 (snoc53 Ξ A) A; v053
= var53 vz53
v153 : β{Ξ A B} β Tm53 (snoc53 (snoc53 Ξ A) B) A; v153
= var53 (vs53 vz53)
v253 : β{Ξ A B C} β Tm53 (snoc53 (snoc53 (snoc53 Ξ A) B) C) A; v253
= var53 (vs53 (vs53 vz53))
v353 : β{Ξ A B C D} β Tm53 (snoc53 (snoc53 (snoc53 (snoc53 Ξ A) B) C) D) A; v353
= var53 (vs53 (vs53 (vs53 vz53)))
tbool53 : Ty53; tbool53
= sum53 top53 top53
true53 : β{Ξ} β Tm53 Ξ tbool53; true53
= left53 tt53
tfalse53 : β{Ξ} β Tm53 Ξ tbool53; tfalse53
= right53 tt53
ifthenelse53 : β{Ξ A} β Tm53 Ξ (arr53 tbool53 (arr53 A (arr53 A A))); ifthenelse53
= lam53 (lam53 (lam53 (case53 v253 (lam53 v253) (lam53 v153))))
times453 : β{Ξ A} β Tm53 Ξ (arr53 (arr53 A A) (arr53 A A)); times453
= lam53 (lam53 (app53 v153 (app53 v153 (app53 v153 (app53 v153 v053)))))
add53 : β{Ξ} β Tm53 Ξ (arr53 nat53 (arr53 nat53 nat53)); add53
= lam53 (rec53 v053
(lam53 (lam53 (lam53 (suc53 (app53 v153 v053)))))
(lam53 v053))
mul53 : β{Ξ} β Tm53 Ξ (arr53 nat53 (arr53 nat53 nat53)); mul53
= lam53 (rec53 v053
(lam53 (lam53 (lam53 (app53 (app53 add53 (app53 v153 v053)) v053))))
(lam53 zero53))
fact53 : β{Ξ} β Tm53 Ξ (arr53 nat53 nat53); fact53
= lam53 (rec53 v053 (lam53 (lam53 (app53 (app53 mul53 (suc53 v153)) v053)))
(suc53 zero53))
{-# OPTIONS --type-in-type #-}
Ty54 : Set
Ty54 =
(Ty54 : Set)
(nat top bot : Ty54)
(arr prod sum : Ty54 β Ty54 β Ty54)
β Ty54
nat54 : Ty54; nat54 = Ξ» _ nat54 _ _ _ _ _ β nat54
top54 : Ty54; top54 = Ξ» _ _ top54 _ _ _ _ β top54
bot54 : Ty54; bot54 = Ξ» _ _ _ bot54 _ _ _ β bot54
arr54 : Ty54 β Ty54 β Ty54; arr54
= Ξ» A B Ty54 nat54 top54 bot54 arr54 prod sum β
arr54 (A Ty54 nat54 top54 bot54 arr54 prod sum) (B Ty54 nat54 top54 bot54 arr54 prod sum)
prod54 : Ty54 β Ty54 β Ty54; prod54
= Ξ» A B Ty54 nat54 top54 bot54 arr54 prod54 sum β
prod54 (A Ty54 nat54 top54 bot54 arr54 prod54 sum) (B Ty54 nat54 top54 bot54 arr54 prod54 sum)
sum54 : Ty54 β Ty54 β Ty54; sum54
= Ξ» A B Ty54 nat54 top54 bot54 arr54 prod54 sum54 β
sum54 (A Ty54 nat54 top54 bot54 arr54 prod54 sum54) (B Ty54 nat54 top54 bot54 arr54 prod54 sum54)
Con54 : Set; Con54
= (Con54 : Set)
(nil : Con54)
(snoc : Con54 β Ty54 β Con54)
β Con54
nil54 : Con54; nil54
= Ξ» Con54 nil54 snoc β nil54
snoc54 : Con54 β Ty54 β Con54; snoc54
= Ξ» Ξ A Con54 nil54 snoc54 β snoc54 (Ξ Con54 nil54 snoc54) A
Var54 : Con54 β Ty54 β Set; Var54
= Ξ» Ξ A β
(Var54 : Con54 β Ty54 β Set)
(vz : β Ξ A β Var54 (snoc54 Ξ A) A)
(vs : β Ξ B A β Var54 Ξ A β Var54 (snoc54 Ξ B) A)
β Var54 Ξ A
vz54 : β{Ξ A} β Var54 (snoc54 Ξ A) A; vz54
= Ξ» Var54 vz54 vs β vz54 _ _
vs54 : β{Ξ B A} β Var54 Ξ A β Var54 (snoc54 Ξ B) A; vs54
= Ξ» x Var54 vz54 vs54 β vs54 _ _ _ (x Var54 vz54 vs54)
Tm54 : Con54 β Ty54 β Set; Tm54
= Ξ» Ξ A β
(Tm54 : Con54 β Ty54 β Set)
(var : β Ξ A β Var54 Ξ A β Tm54 Ξ A)
(lam : β Ξ A B β Tm54 (snoc54 Ξ A) B β Tm54 Ξ (arr54 A B))
(app : β Ξ A B β Tm54 Ξ (arr54 A B) β Tm54 Ξ A β Tm54 Ξ B)
(tt : β Ξ β Tm54 Ξ top54)
(pair : β Ξ A B β Tm54 Ξ A β Tm54 Ξ B β Tm54 Ξ (prod54 A B))
(fst : β Ξ A B β Tm54 Ξ (prod54 A B) β Tm54 Ξ A)
(snd : β Ξ A B β Tm54 Ξ (prod54 A B) β Tm54 Ξ B)
(left : β Ξ A B β Tm54 Ξ A β Tm54 Ξ (sum54 A B))
(right : β Ξ A B β Tm54 Ξ B β Tm54 Ξ (sum54 A B))
(case : β Ξ A B C β Tm54 Ξ (sum54 A B) β Tm54 Ξ (arr54 A C) β Tm54 Ξ (arr54 B C) β Tm54 Ξ C)
(zero : β Ξ β Tm54 Ξ nat54)
(suc : β Ξ β Tm54 Ξ nat54 β Tm54 Ξ nat54)
(rec : β Ξ A β Tm54 Ξ nat54 β Tm54 Ξ (arr54 nat54 (arr54 A A)) β Tm54 Ξ A β Tm54 Ξ A)
β Tm54 Ξ A
var54 : β{Ξ A} β Var54 Ξ A β Tm54 Ξ A; var54
= Ξ» x Tm54 var54 lam app tt pair fst snd left right case zero suc rec β
var54 _ _ x
lam54 : β{Ξ A B} β Tm54 (snoc54 Ξ A) B β Tm54 Ξ (arr54 A B); lam54
= Ξ» t Tm54 var54 lam54 app tt pair fst snd left right case zero suc rec β
lam54 _ _ _ (t Tm54 var54 lam54 app tt pair fst snd left right case zero suc rec)
app54 : β{Ξ A B} β Tm54 Ξ (arr54 A B) β Tm54 Ξ A β Tm54 Ξ B; app54
= Ξ» t u Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec β
app54 _ _ _ (t Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec)
(u Tm54 var54 lam54 app54 tt pair fst snd left right case zero suc rec)
tt54 : β{Ξ} β Tm54 Ξ top54; tt54
= Ξ» Tm54 var54 lam54 app54 tt54 pair fst snd left right case zero suc rec β tt54 _
pair54 : β{Ξ A B} β Tm54 Ξ A β Tm54 Ξ B β Tm54 Ξ (prod54 A B); pair54
= Ξ» t u Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec β
pair54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec)
(u Tm54 var54 lam54 app54 tt54 pair54 fst snd left right case zero suc rec)
fst54 : β{Ξ A B} β Tm54 Ξ (prod54 A B) β Tm54 Ξ A; fst54
= Ξ» t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd left right case zero suc rec β
fst54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd left right case zero suc rec)
snd54 : β{Ξ A B} β Tm54 Ξ (prod54 A B) β Tm54 Ξ B; snd54
= Ξ» t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left right case zero suc rec β
snd54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left right case zero suc rec)
left54 : β{Ξ A B} β Tm54 Ξ A β Tm54 Ξ (sum54 A B); left54
= Ξ» t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right case zero suc rec β
left54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right case zero suc rec)
right54 : β{Ξ A B} β Tm54 Ξ B β Tm54 Ξ (sum54 A B); right54
= Ξ» t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case zero suc rec β
right54 _ _ _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case zero suc rec)
case54 : β{Ξ A B C} β Tm54 Ξ (sum54 A B) β Tm54 Ξ (arr54 A C) β Tm54 Ξ (arr54 B C) β Tm54 Ξ C; case54
= Ξ» t u v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec β
case54 _ _ _ _
(t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
(u Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
(v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero suc rec)
zero54 : β{Ξ} β Tm54 Ξ nat54; zero54
= Ξ» Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc rec β zero54 _
suc54 : β{Ξ} β Tm54 Ξ nat54 β Tm54 Ξ nat54; suc54
= Ξ» t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec β
suc54 _ (t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec)
rec54 : β{Ξ A} β Tm54 Ξ nat54 β Tm54 Ξ (arr54 nat54 (arr54 A A)) β Tm54 Ξ A β Tm54 Ξ A; rec54
= Ξ» t u v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54 β
rec54 _ _
(t Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
(u Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
(v Tm54 var54 lam54 app54 tt54 pair54 fst54 snd54 left54 right54 case54 zero54 suc54 rec54)
v054 : β{Ξ A} β Tm54 (snoc54 Ξ A) A; v054
= var54 vz54
v154 : β{Ξ A B} β Tm54 (snoc54 (snoc54 Ξ A) B) A; v154
= var54 (vs54 vz54)
v254 : β{Ξ A B C} β Tm54 (snoc54 (snoc54 (snoc54 Ξ A) B) C) A; v254
= var54 (vs54 (vs54 vz54))
v354 : β{Ξ A B C D} β Tm54 (snoc54 (snoc54 (snoc54 (snoc54 Ξ A) B) C) D) A; v354
= var54 (vs54 (vs54 (vs54 vz54)))
tbool54 : Ty54; tbool54
= sum54 top54 top54
true54 : β{Ξ} β Tm54 Ξ tbool54; true54
= left54 tt54
tfalse54 : β{Ξ} β Tm54 Ξ tbool54; tfalse54
= right54 tt54
ifthenelse54 : β{Ξ A} β Tm54 Ξ (arr54 tbool54 (arr54 A (arr54 A A))); ifthenelse54
= lam54 (lam54 (lam54 (case54 v254 (lam54 v254) (lam54 v154))))
times454 : β{Ξ A} β Tm54 Ξ (arr54 (arr54 A A) (arr54 A A)); times454
= lam54 (lam54 (app54 v154 (app54 v154 (app54 v154 (app54 v154 v054)))))
add54 : β{Ξ} β Tm54 Ξ (arr54 nat54 (arr54 nat54 nat54)); add54
= lam54 (rec54 v054
(lam54 (lam54 (lam54 (suc54 (app54 v154 v054)))))
(lam54 v054))
mul54 : β{Ξ} β Tm54 Ξ (arr54 nat54 (arr54 nat54 nat54)); mul54
= lam54 (rec54 v054
(lam54 (lam54 (lam54 (app54 (app54 add54 (app54 v154 v054)) v054))))
(lam54 zero54))
fact54 : β{Ξ} β Tm54 Ξ (arr54 nat54 nat54); fact54
= lam54 (rec54 v054 (lam54 (lam54 (app54 (app54 mul54 (suc54 v154)) v054)))
(suc54 zero54))
{-# OPTIONS --type-in-type #-}
Ty55 : Set
Ty55 =
(Ty55 : Set)
(nat top bot : Ty55)
(arr prod sum : Ty55 β Ty55 β Ty55)
β Ty55
nat55 : Ty55; nat55 = Ξ» _ nat55 _ _ _ _ _ β nat55
top55 : Ty55; top55 = Ξ» _ _ top55 _ _ _ _ β top55
bot55 : Ty55; bot55 = Ξ» _ _ _ bot55 _ _ _ β bot55
arr55 : Ty55 β Ty55 β Ty55; arr55
= Ξ» A B Ty55 nat55 top55 bot55 arr55 prod sum β
arr55 (A Ty55 nat55 top55 bot55 arr55 prod sum) (B Ty55 nat55 top55 bot55 arr55 prod sum)
prod55 : Ty55 β Ty55 β Ty55; prod55
= Ξ» A B Ty55 nat55 top55 bot55 arr55 prod55 sum β
prod55 (A Ty55 nat55 top55 bot55 arr55 prod55 sum) (B Ty55 nat55 top55 bot55 arr55 prod55 sum)
sum55 : Ty55 β Ty55 β Ty55; sum55
= Ξ» A B Ty55 nat55 top55 bot55 arr55 prod55 sum55 β
sum55 (A Ty55 nat55 top55 bot55 arr55 prod55 sum55) (B Ty55 nat55 top55 bot55 arr55 prod55 sum55)
Con55 : Set; Con55
= (Con55 : Set)
(nil : Con55)
(snoc : Con55 β Ty55 β Con55)
β Con55
nil55 : Con55; nil55
= Ξ» Con55 nil55 snoc β nil55
snoc55 : Con55 β Ty55 β Con55; snoc55
= Ξ» Ξ A Con55 nil55 snoc55 β snoc55 (Ξ Con55 nil55 snoc55) A
Var55 : Con55 β Ty55 β Set; Var55
= Ξ» Ξ A β
(Var55 : Con55 β Ty55 β Set)
(vz : β Ξ A β Var55 (snoc55 Ξ A) A)
(vs : β Ξ B A β Var55 Ξ A β Var55 (snoc55 Ξ B) A)
β Var55 Ξ A
vz55 : β{Ξ A} β Var55 (snoc55 Ξ A) A; vz55
= Ξ» Var55 vz55 vs β vz55 _ _
vs55 : β{Ξ B A} β Var55 Ξ A β Var55 (snoc55 Ξ B) A; vs55
= Ξ» x Var55 vz55 vs55 β vs55 _ _ _ (x Var55 vz55 vs55)
Tm55 : Con55 β Ty55 β Set; Tm55
= Ξ» Ξ A β
(Tm55 : Con55 β Ty55 β Set)
(var : β Ξ A β Var55 Ξ A β Tm55 Ξ A)
(lam : β Ξ A B β Tm55 (snoc55 Ξ A) B β Tm55 Ξ (arr55 A B))
(app : β Ξ A B β Tm55 Ξ (arr55 A B) β Tm55 Ξ A β Tm55 Ξ B)
(tt : β Ξ β Tm55 Ξ top55)
(pair : β Ξ A B β Tm55 Ξ A β Tm55 Ξ B β Tm55 Ξ (prod55 A B))
(fst : β Ξ A B β Tm55 Ξ (prod55 A B) β Tm55 Ξ A)
(snd : β Ξ A B β Tm55 Ξ (prod55 A B) β Tm55 Ξ B)
(left : β Ξ A B β Tm55 Ξ A β Tm55 Ξ (sum55 A B))
(right : β Ξ A B β Tm55 Ξ B β Tm55 Ξ (sum55 A B))
(case : β Ξ A B C β Tm55 Ξ (sum55 A B) β Tm55 Ξ (arr55 A C) β Tm55 Ξ (arr55 B C) β Tm55 Ξ C)
(zero : β Ξ β Tm55 Ξ nat55)
(suc : β Ξ β Tm55 Ξ nat55 β Tm55 Ξ nat55)
(rec : β Ξ A β Tm55 Ξ nat55 β Tm55 Ξ (arr55 nat55 (arr55 A A)) β Tm55 Ξ A β Tm55 Ξ A)
β Tm55 Ξ A
var55 : β{Ξ A} β Var55 Ξ A β Tm55 Ξ A; var55
= Ξ» x Tm55 var55 lam app tt pair fst snd left right case zero suc rec β
var55 _ _ x
lam55 : β{Ξ A B} β Tm55 (snoc55 Ξ A) B β Tm55 Ξ (arr55 A B); lam55
= Ξ» t Tm55 var55 lam55 app tt pair fst snd left right case zero suc rec β
lam55 _ _ _ (t Tm55 var55 lam55 app tt pair fst snd left right case zero suc rec)
app55 : β{Ξ A B} β Tm55 Ξ (arr55 A B) β Tm55 Ξ A β Tm55 Ξ B; app55
= Ξ» t u Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec β
app55 _ _ _ (t Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec)
(u Tm55 var55 lam55 app55 tt pair fst snd left right case zero suc rec)
tt55 : β{Ξ} β Tm55 Ξ top55; tt55
= Ξ» Tm55 var55 lam55 app55 tt55 pair fst snd left right case zero suc rec β tt55 _
pair55 : β{Ξ A B} β Tm55 Ξ A β Tm55 Ξ B β Tm55 Ξ (prod55 A B); pair55
= Ξ» t u Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec β
pair55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec)
(u Tm55 var55 lam55 app55 tt55 pair55 fst snd left right case zero suc rec)
fst55 : β{Ξ A B} β Tm55 Ξ (prod55 A B) β Tm55 Ξ A; fst55
= Ξ» t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd left right case zero suc rec β
fst55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd left right case zero suc rec)
snd55 : β{Ξ A B} β Tm55 Ξ (prod55 A B) β Tm55 Ξ B; snd55
= Ξ» t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left right case zero suc rec β
snd55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left right case zero suc rec)
left55 : β{Ξ A B} β Tm55 Ξ A β Tm55 Ξ (sum55 A B); left55
= Ξ» t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right case zero suc rec β
left55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right case zero suc rec)
right55 : β{Ξ A B} β Tm55 Ξ B β Tm55 Ξ (sum55 A B); right55
= Ξ» t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case zero suc rec β
right55 _ _ _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case zero suc rec)
case55 : β{Ξ A B C} β Tm55 Ξ (sum55 A B) β Tm55 Ξ (arr55 A C) β Tm55 Ξ (arr55 B C) β Tm55 Ξ C; case55
= Ξ» t u v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec β
case55 _ _ _ _
(t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
(u Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
(v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero suc rec)
zero55 : β{Ξ} β Tm55 Ξ nat55; zero55
= Ξ» Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc rec β zero55 _
suc55 : β{Ξ} β Tm55 Ξ nat55 β Tm55 Ξ nat55; suc55
= Ξ» t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec β
suc55 _ (t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec)
rec55 : β{Ξ A} β Tm55 Ξ nat55 β Tm55 Ξ (arr55 nat55 (arr55 A A)) β Tm55 Ξ A β Tm55 Ξ A; rec55
= Ξ» t u v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55 β
rec55 _ _
(t Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
(u Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
(v Tm55 var55 lam55 app55 tt55 pair55 fst55 snd55 left55 right55 case55 zero55 suc55 rec55)
v055 : β{Ξ A} β Tm55 (snoc55 Ξ A) A; v055
= var55 vz55
v155 : β{Ξ A B} β Tm55 (snoc55 (snoc55 Ξ A) B) A; v155
= var55 (vs55 vz55)
v255 : β{Ξ A B C} β Tm55 (snoc55 (snoc55 (snoc55 Ξ A) B) C) A; v255
= var55 (vs55 (vs55 vz55))
v355 : β{Ξ A B C D} β Tm55 (snoc55 (snoc55 (snoc55 (snoc55 Ξ A) B) C) D) A; v355
= var55 (vs55 (vs55 (vs55 vz55)))
tbool55 : Ty55; tbool55
= sum55 top55 top55
true55 : β{Ξ} β Tm55 Ξ tbool55; true55
= left55 tt55
tfalse55 : β{Ξ} β Tm55 Ξ tbool55; tfalse55
= right55 tt55
ifthenelse55 : β{Ξ A} β Tm55 Ξ (arr55 tbool55 (arr55 A (arr55 A A))); ifthenelse55
= lam55 (lam55 (lam55 (case55 v255 (lam55 v255) (lam55 v155))))
times455 : β{Ξ A} β Tm55 Ξ (arr55 (arr55 A A) (arr55 A A)); times455
= lam55 (lam55 (app55 v155 (app55 v155 (app55 v155 (app55 v155 v055)))))
add55 : β{Ξ} β Tm55 Ξ (arr55 nat55 (arr55 nat55 nat55)); add55
= lam55 (rec55 v055
(lam55 (lam55 (lam55 (suc55 (app55 v155 v055)))))
(lam55 v055))
mul55 : β{Ξ} β Tm55 Ξ (arr55 nat55 (arr55 nat55 nat55)); mul55
= lam55 (rec55 v055
(lam55 (lam55 (lam55 (app55 (app55 add55 (app55 v155 v055)) v055))))
(lam55 zero55))
fact55 : β{Ξ} β Tm55 Ξ (arr55 nat55 nat55); fact55
= lam55 (rec55 v055 (lam55 (lam55 (app55 (app55 mul55 (suc55 v155)) v055)))
(suc55 zero55))
{-# OPTIONS --type-in-type #-}
Ty56 : Set
Ty56 =
(Ty56 : Set)
(nat top bot : Ty56)
(arr prod sum : Ty56 β Ty56 β Ty56)
β Ty56
nat56 : Ty56; nat56 = Ξ» _ nat56 _ _ _ _ _ β nat56
top56 : Ty56; top56 = Ξ» _ _ top56 _ _ _ _ β top56
bot56 : Ty56; bot56 = Ξ» _ _ _ bot56 _ _ _ β bot56
arr56 : Ty56 β Ty56 β Ty56; arr56
= Ξ» A B Ty56 nat56 top56 bot56 arr56 prod sum β
arr56 (A Ty56 nat56 top56 bot56 arr56 prod sum) (B Ty56 nat56 top56 bot56 arr56 prod sum)
prod56 : Ty56 β Ty56 β Ty56; prod56
= Ξ» A B Ty56 nat56 top56 bot56 arr56 prod56 sum β
prod56 (A Ty56 nat56 top56 bot56 arr56 prod56 sum) (B Ty56 nat56 top56 bot56 arr56 prod56 sum)
sum56 : Ty56 β Ty56 β Ty56; sum56
= Ξ» A B Ty56 nat56 top56 bot56 arr56 prod56 sum56 β
sum56 (A Ty56 nat56 top56 bot56 arr56 prod56 sum56) (B Ty56 nat56 top56 bot56 arr56 prod56 sum56)
Con56 : Set; Con56
= (Con56 : Set)
(nil : Con56)
(snoc : Con56 β Ty56 β Con56)
β Con56
nil56 : Con56; nil56
= Ξ» Con56 nil56 snoc β nil56
snoc56 : Con56 β Ty56 β Con56; snoc56
= Ξ» Ξ A Con56 nil56 snoc56 β snoc56 (Ξ Con56 nil56 snoc56) A
Var56 : Con56 β Ty56 β Set; Var56
= Ξ» Ξ A β
(Var56 : Con56 β Ty56 β Set)
(vz : β Ξ A β Var56 (snoc56 Ξ A) A)
(vs : β Ξ B A β Var56 Ξ A β Var56 (snoc56 Ξ B) A)
β Var56 Ξ A
vz56 : β{Ξ A} β Var56 (snoc56 Ξ A) A; vz56
= Ξ» Var56 vz56 vs β vz56 _ _
vs56 : β{Ξ B A} β Var56 Ξ A β Var56 (snoc56 Ξ B) A; vs56
= Ξ» x Var56 vz56 vs56 β vs56 _ _ _ (x Var56 vz56 vs56)
Tm56 : Con56 β Ty56 β Set; Tm56
= Ξ» Ξ A β
(Tm56 : Con56 β Ty56 β Set)
(var : β Ξ A β Var56 Ξ A β Tm56 Ξ A)
(lam : β Ξ A B β Tm56 (snoc56 Ξ A) B β Tm56 Ξ (arr56 A B))
(app : β Ξ A B β Tm56 Ξ (arr56 A B) β Tm56 Ξ A β Tm56 Ξ B)
(tt : β Ξ β Tm56 Ξ top56)
(pair : β Ξ A B β Tm56 Ξ A β Tm56 Ξ B β Tm56 Ξ (prod56 A B))
(fst : β Ξ A B β Tm56 Ξ (prod56 A B) β Tm56 Ξ A)
(snd : β Ξ A B β Tm56 Ξ (prod56 A B) β Tm56 Ξ B)
(left : β Ξ A B β Tm56 Ξ A β Tm56 Ξ (sum56 A B))
(right : β Ξ A B β Tm56 Ξ B β Tm56 Ξ (sum56 A B))
(case : β Ξ A B C β Tm56 Ξ (sum56 A B) β Tm56 Ξ (arr56 A C) β Tm56 Ξ (arr56 B C) β Tm56 Ξ C)
(zero : β Ξ β Tm56 Ξ nat56)
(suc : β Ξ β Tm56 Ξ nat56 β Tm56 Ξ nat56)
(rec : β Ξ A β Tm56 Ξ nat56 β Tm56 Ξ (arr56 nat56 (arr56 A A)) β Tm56 Ξ A β Tm56 Ξ A)
β Tm56 Ξ A
var56 : β{Ξ A} β Var56 Ξ A β Tm56 Ξ A; var56
= Ξ» x Tm56 var56 lam app tt pair fst snd left right case zero suc rec β
var56 _ _ x
lam56 : β{Ξ A B} β Tm56 (snoc56 Ξ A) B β Tm56 Ξ (arr56 A B); lam56
= Ξ» t Tm56 var56 lam56 app tt pair fst snd left right case zero suc rec β
lam56 _ _ _ (t Tm56 var56 lam56 app tt pair fst snd left right case zero suc rec)
app56 : β{Ξ A B} β Tm56 Ξ (arr56 A B) β Tm56 Ξ A β Tm56 Ξ B; app56
= Ξ» t u Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec β
app56 _ _ _ (t Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec)
(u Tm56 var56 lam56 app56 tt pair fst snd left right case zero suc rec)
tt56 : β{Ξ} β Tm56 Ξ top56; tt56
= Ξ» Tm56 var56 lam56 app56 tt56 pair fst snd left right case zero suc rec β tt56 _
pair56 : β{Ξ A B} β Tm56 Ξ A β Tm56 Ξ B β Tm56 Ξ (prod56 A B); pair56
= Ξ» t u Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec β
pair56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec)
(u Tm56 var56 lam56 app56 tt56 pair56 fst snd left right case zero suc rec)
fst56 : β{Ξ A B} β Tm56 Ξ (prod56 A B) β Tm56 Ξ A; fst56
= Ξ» t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd left right case zero suc rec β
fst56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd left right case zero suc rec)
snd56 : β{Ξ A B} β Tm56 Ξ (prod56 A B) β Tm56 Ξ B; snd56
= Ξ» t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left right case zero suc rec β
snd56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left right case zero suc rec)
left56 : β{Ξ A B} β Tm56 Ξ A β Tm56 Ξ (sum56 A B); left56
= Ξ» t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right case zero suc rec β
left56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right case zero suc rec)
right56 : β{Ξ A B} β Tm56 Ξ B β Tm56 Ξ (sum56 A B); right56
= Ξ» t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case zero suc rec β
right56 _ _ _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case zero suc rec)
case56 : β{Ξ A B C} β Tm56 Ξ (sum56 A B) β Tm56 Ξ (arr56 A C) β Tm56 Ξ (arr56 B C) β Tm56 Ξ C; case56
= Ξ» t u v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec β
case56 _ _ _ _
(t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
(u Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
(v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero suc rec)
zero56 : β{Ξ} β Tm56 Ξ nat56; zero56
= Ξ» Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc rec β zero56 _
suc56 : β{Ξ} β Tm56 Ξ nat56 β Tm56 Ξ nat56; suc56
= Ξ» t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec β
suc56 _ (t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec)
rec56 : β{Ξ A} β Tm56 Ξ nat56 β Tm56 Ξ (arr56 nat56 (arr56 A A)) β Tm56 Ξ A β Tm56 Ξ A; rec56
= Ξ» t u v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56 β
rec56 _ _
(t Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
(u Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
(v Tm56 var56 lam56 app56 tt56 pair56 fst56 snd56 left56 right56 case56 zero56 suc56 rec56)
v056 : β{Ξ A} β Tm56 (snoc56 Ξ A) A; v056
= var56 vz56
v156 : β{Ξ A B} β Tm56 (snoc56 (snoc56 Ξ A) B) A; v156
= var56 (vs56 vz56)
v256 : β{Ξ A B C} β Tm56 (snoc56 (snoc56 (snoc56 Ξ A) B) C) A; v256
= var56 (vs56 (vs56 vz56))
v356 : β{Ξ A B C D} β Tm56 (snoc56 (snoc56 (snoc56 (snoc56 Ξ A) B) C) D) A; v356
= var56 (vs56 (vs56 (vs56 vz56)))
tbool56 : Ty56; tbool56
= sum56 top56 top56
true56 : β{Ξ} β Tm56 Ξ tbool56; true56
= left56 tt56
tfalse56 : β{Ξ} β Tm56 Ξ tbool56; tfalse56
= right56 tt56
ifthenelse56 : β{Ξ A} β Tm56 Ξ (arr56 tbool56 (arr56 A (arr56 A A))); ifthenelse56
= lam56 (lam56 (lam56 (case56 v256 (lam56 v256) (lam56 v156))))
times456 : β{Ξ A} β Tm56 Ξ (arr56 (arr56 A A) (arr56 A A)); times456
= lam56 (lam56 (app56 v156 (app56 v156 (app56 v156 (app56 v156 v056)))))
add56 : β{Ξ} β Tm56 Ξ (arr56 nat56 (arr56 nat56 nat56)); add56
= lam56 (rec56 v056
(lam56 (lam56 (lam56 (suc56 (app56 v156 v056)))))
(lam56 v056))
mul56 : β{Ξ} β Tm56 Ξ (arr56 nat56 (arr56 nat56 nat56)); mul56
= lam56 (rec56 v056
(lam56 (lam56 (lam56 (app56 (app56 add56 (app56 v156 v056)) v056))))
(lam56 zero56))
fact56 : β{Ξ} β Tm56 Ξ (arr56 nat56 nat56); fact56
= lam56 (rec56 v056 (lam56 (lam56 (app56 (app56 mul56 (suc56 v156)) v056)))
(suc56 zero56))
{-# OPTIONS --type-in-type #-}
Ty57 : Set
Ty57 =
(Ty57 : Set)
(nat top bot : Ty57)
(arr prod sum : Ty57 β Ty57 β Ty57)
β Ty57
nat57 : Ty57; nat57 = Ξ» _ nat57 _ _ _ _ _ β nat57
top57 : Ty57; top57 = Ξ» _ _ top57 _ _ _ _ β top57
bot57 : Ty57; bot57 = Ξ» _ _ _ bot57 _ _ _ β bot57
arr57 : Ty57 β Ty57 β Ty57; arr57
= Ξ» A B Ty57 nat57 top57 bot57 arr57 prod sum β
arr57 (A Ty57 nat57 top57 bot57 arr57 prod sum) (B Ty57 nat57 top57 bot57 arr57 prod sum)
prod57 : Ty57 β Ty57 β Ty57; prod57
= Ξ» A B Ty57 nat57 top57 bot57 arr57 prod57 sum β
prod57 (A Ty57 nat57 top57 bot57 arr57 prod57 sum) (B Ty57 nat57 top57 bot57 arr57 prod57 sum)
sum57 : Ty57 β Ty57 β Ty57; sum57
= Ξ» A B Ty57 nat57 top57 bot57 arr57 prod57 sum57 β
sum57 (A Ty57 nat57 top57 bot57 arr57 prod57 sum57) (B Ty57 nat57 top57 bot57 arr57 prod57 sum57)
Con57 : Set; Con57
= (Con57 : Set)
(nil : Con57)
(snoc : Con57 β Ty57 β Con57)
β Con57
nil57 : Con57; nil57
= Ξ» Con57 nil57 snoc β nil57
snoc57 : Con57 β Ty57 β Con57; snoc57
= Ξ» Ξ A Con57 nil57 snoc57 β snoc57 (Ξ Con57 nil57 snoc57) A
Var57 : Con57 β Ty57 β Set; Var57
= Ξ» Ξ A β
(Var57 : Con57 β Ty57 β Set)
(vz : β Ξ A β Var57 (snoc57 Ξ A) A)
(vs : β Ξ B A β Var57 Ξ A β Var57 (snoc57 Ξ B) A)
β Var57 Ξ A
vz57 : β{Ξ A} β Var57 (snoc57 Ξ A) A; vz57
= Ξ» Var57 vz57 vs β vz57 _ _
vs57 : β{Ξ B A} β Var57 Ξ A β Var57 (snoc57 Ξ B) A; vs57
= Ξ» x Var57 vz57 vs57 β vs57 _ _ _ (x Var57 vz57 vs57)
Tm57 : Con57 β Ty57 β Set; Tm57
= Ξ» Ξ A β
(Tm57 : Con57 β Ty57 β Set)
(var : β Ξ A β Var57 Ξ A β Tm57 Ξ A)
(lam : β Ξ A B β Tm57 (snoc57 Ξ A) B β Tm57 Ξ (arr57 A B))
(app : β Ξ A B β Tm57 Ξ (arr57 A B) β Tm57 Ξ A β Tm57 Ξ B)
(tt : β Ξ β Tm57 Ξ top57)
(pair : β Ξ A B β Tm57 Ξ A β Tm57 Ξ B β Tm57 Ξ (prod57 A B))
(fst : β Ξ A B β Tm57 Ξ (prod57 A B) β Tm57 Ξ A)
(snd : β Ξ A B β Tm57 Ξ (prod57 A B) β Tm57 Ξ B)
(left : β Ξ A B β Tm57 Ξ A β Tm57 Ξ (sum57 A B))
(right : β Ξ A B β Tm57 Ξ B β Tm57 Ξ (sum57 A B))
(case : β Ξ A B C β Tm57 Ξ (sum57 A B) β Tm57 Ξ (arr57 A C) β Tm57 Ξ (arr57 B C) β Tm57 Ξ C)
(zero : β Ξ β Tm57 Ξ nat57)
(suc : β Ξ β Tm57 Ξ nat57 β Tm57 Ξ nat57)
(rec : β Ξ A β Tm57 Ξ nat57 β Tm57 Ξ (arr57 nat57 (arr57 A A)) β Tm57 Ξ A β Tm57 Ξ A)
β Tm57 Ξ A
var57 : β{Ξ A} β Var57 Ξ A β Tm57 Ξ A; var57
= Ξ» x Tm57 var57 lam app tt pair fst snd left right case zero suc rec β
var57 _ _ x
lam57 : β{Ξ A B} β Tm57 (snoc57 Ξ A) B β Tm57 Ξ (arr57 A B); lam57
= Ξ» t Tm57 var57 lam57 app tt pair fst snd left right case zero suc rec β
lam57 _ _ _ (t Tm57 var57 lam57 app tt pair fst snd left right case zero suc rec)
app57 : β{Ξ A B} β Tm57 Ξ (arr57 A B) β Tm57 Ξ A β Tm57 Ξ B; app57
= Ξ» t u Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec β
app57 _ _ _ (t Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec)
(u Tm57 var57 lam57 app57 tt pair fst snd left right case zero suc rec)
tt57 : β{Ξ} β Tm57 Ξ top57; tt57
= Ξ» Tm57 var57 lam57 app57 tt57 pair fst snd left right case zero suc rec β tt57 _
pair57 : β{Ξ A B} β Tm57 Ξ A β Tm57 Ξ B β Tm57 Ξ (prod57 A B); pair57
= Ξ» t u Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec β
pair57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec)
(u Tm57 var57 lam57 app57 tt57 pair57 fst snd left right case zero suc rec)
fst57 : β{Ξ A B} β Tm57 Ξ (prod57 A B) β Tm57 Ξ A; fst57
= Ξ» t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd left right case zero suc rec β
fst57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd left right case zero suc rec)
snd57 : β{Ξ A B} β Tm57 Ξ (prod57 A B) β Tm57 Ξ B; snd57
= Ξ» t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left right case zero suc rec β
snd57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left right case zero suc rec)
left57 : β{Ξ A B} β Tm57 Ξ A β Tm57 Ξ (sum57 A B); left57
= Ξ» t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right case zero suc rec β
left57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right case zero suc rec)
right57 : β{Ξ A B} β Tm57 Ξ B β Tm57 Ξ (sum57 A B); right57
= Ξ» t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case zero suc rec β
right57 _ _ _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case zero suc rec)
case57 : β{Ξ A B C} β Tm57 Ξ (sum57 A B) β Tm57 Ξ (arr57 A C) β Tm57 Ξ (arr57 B C) β Tm57 Ξ C; case57
= Ξ» t u v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec β
case57 _ _ _ _
(t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
(u Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
(v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero suc rec)
zero57 : β{Ξ} β Tm57 Ξ nat57; zero57
= Ξ» Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc rec β zero57 _
suc57 : β{Ξ} β Tm57 Ξ nat57 β Tm57 Ξ nat57; suc57
= Ξ» t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec β
suc57 _ (t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec)
rec57 : β{Ξ A} β Tm57 Ξ nat57 β Tm57 Ξ (arr57 nat57 (arr57 A A)) β Tm57 Ξ A β Tm57 Ξ A; rec57
= Ξ» t u v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57 β
rec57 _ _
(t Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
(u Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
(v Tm57 var57 lam57 app57 tt57 pair57 fst57 snd57 left57 right57 case57 zero57 suc57 rec57)
v057 : β{Ξ A} β Tm57 (snoc57 Ξ A) A; v057
= var57 vz57
v157 : β{Ξ A B} β Tm57 (snoc57 (snoc57 Ξ A) B) A; v157
= var57 (vs57 vz57)
v257 : β{Ξ A B C} β Tm57 (snoc57 (snoc57 (snoc57 Ξ A) B) C) A; v257
= var57 (vs57 (vs57 vz57))
v357 : β{Ξ A B C D} β Tm57 (snoc57 (snoc57 (snoc57 (snoc57 Ξ A) B) C) D) A; v357
= var57 (vs57 (vs57 (vs57 vz57)))
tbool57 : Ty57; tbool57
= sum57 top57 top57
true57 : β{Ξ} β Tm57 Ξ tbool57; true57
= left57 tt57
tfalse57 : β{Ξ} β Tm57 Ξ tbool57; tfalse57
= right57 tt57
ifthenelse57 : β{Ξ A} β Tm57 Ξ (arr57 tbool57 (arr57 A (arr57 A A))); ifthenelse57
= lam57 (lam57 (lam57 (case57 v257 (lam57 v257) (lam57 v157))))
times457 : β{Ξ A} β Tm57 Ξ (arr57 (arr57 A A) (arr57 A A)); times457
= lam57 (lam57 (app57 v157 (app57 v157 (app57 v157 (app57 v157 v057)))))
add57 : β{Ξ} β Tm57 Ξ (arr57 nat57 (arr57 nat57 nat57)); add57
= lam57 (rec57 v057
(lam57 (lam57 (lam57 (suc57 (app57 v157 v057)))))
(lam57 v057))
mul57 : β{Ξ} β Tm57 Ξ (arr57 nat57 (arr57 nat57 nat57)); mul57
= lam57 (rec57 v057
(lam57 (lam57 (lam57 (app57 (app57 add57 (app57 v157 v057)) v057))))
(lam57 zero57))
fact57 : β{Ξ} β Tm57 Ξ (arr57 nat57 nat57); fact57
= lam57 (rec57 v057 (lam57 (lam57 (app57 (app57 mul57 (suc57 v157)) v057)))
(suc57 zero57))
{-# OPTIONS --type-in-type #-}
Ty58 : Set
Ty58 =
(Ty58 : Set)
(nat top bot : Ty58)
(arr prod sum : Ty58 β Ty58 β Ty58)
β Ty58
nat58 : Ty58; nat58 = Ξ» _ nat58 _ _ _ _ _ β nat58
top58 : Ty58; top58 = Ξ» _ _ top58 _ _ _ _ β top58
bot58 : Ty58; bot58 = Ξ» _ _ _ bot58 _ _ _ β bot58
arr58 : Ty58 β Ty58 β Ty58; arr58
= Ξ» A B Ty58 nat58 top58 bot58 arr58 prod sum β
arr58 (A Ty58 nat58 top58 bot58 arr58 prod sum) (B Ty58 nat58 top58 bot58 arr58 prod sum)
prod58 : Ty58 β Ty58 β Ty58; prod58
= Ξ» A B Ty58 nat58 top58 bot58 arr58 prod58 sum β
prod58 (A Ty58 nat58 top58 bot58 arr58 prod58 sum) (B Ty58 nat58 top58 bot58 arr58 prod58 sum)
sum58 : Ty58 β Ty58 β Ty58; sum58
= Ξ» A B Ty58 nat58 top58 bot58 arr58 prod58 sum58 β
sum58 (A Ty58 nat58 top58 bot58 arr58 prod58 sum58) (B Ty58 nat58 top58 bot58 arr58 prod58 sum58)
Con58 : Set; Con58
= (Con58 : Set)
(nil : Con58)
(snoc : Con58 β Ty58 β Con58)
β Con58
nil58 : Con58; nil58
= Ξ» Con58 nil58 snoc β nil58
snoc58 : Con58 β Ty58 β Con58; snoc58
= Ξ» Ξ A Con58 nil58 snoc58 β snoc58 (Ξ Con58 nil58 snoc58) A
Var58 : Con58 β Ty58 β Set; Var58
= Ξ» Ξ A β
(Var58 : Con58 β Ty58 β Set)
(vz : β Ξ A β Var58 (snoc58 Ξ A) A)
(vs : β Ξ B A β Var58 Ξ A β Var58 (snoc58 Ξ B) A)
β Var58 Ξ A
vz58 : β{Ξ A} β Var58 (snoc58 Ξ A) A; vz58
= Ξ» Var58 vz58 vs β vz58 _ _
vs58 : β{Ξ B A} β Var58 Ξ A β Var58 (snoc58 Ξ B) A; vs58
= Ξ» x Var58 vz58 vs58 β vs58 _ _ _ (x Var58 vz58 vs58)
Tm58 : Con58 β Ty58 β Set; Tm58
= Ξ» Ξ A β
(Tm58 : Con58 β Ty58 β Set)
(var : β Ξ A β Var58 Ξ A β Tm58 Ξ A)
(lam : β Ξ A B β Tm58 (snoc58 Ξ A) B β Tm58 Ξ (arr58 A B))
(app : β Ξ A B β Tm58 Ξ (arr58 A B) β Tm58 Ξ A β Tm58 Ξ B)
(tt : β Ξ β Tm58 Ξ top58)
(pair : β Ξ A B β Tm58 Ξ A β Tm58 Ξ B β Tm58 Ξ (prod58 A B))
(fst : β Ξ A B β Tm58 Ξ (prod58 A B) β Tm58 Ξ A)
(snd : β Ξ A B β Tm58 Ξ (prod58 A B) β Tm58 Ξ B)
(left : β Ξ A B β Tm58 Ξ A β Tm58 Ξ (sum58 A B))
(right : β Ξ A B β Tm58 Ξ B β Tm58 Ξ (sum58 A B))
(case : β Ξ A B C β Tm58 Ξ (sum58 A B) β Tm58 Ξ (arr58 A C) β Tm58 Ξ (arr58 B C) β Tm58 Ξ C)
(zero : β Ξ β Tm58 Ξ nat58)
(suc : β Ξ β Tm58 Ξ nat58 β Tm58 Ξ nat58)
(rec : β Ξ A β Tm58 Ξ nat58 β Tm58 Ξ (arr58 nat58 (arr58 A A)) β Tm58 Ξ A β Tm58 Ξ A)
β Tm58 Ξ A
var58 : β{Ξ A} β Var58 Ξ A β Tm58 Ξ A; var58
= Ξ» x Tm58 var58 lam app tt pair fst snd left right case zero suc rec β
var58 _ _ x
lam58 : β{Ξ A B} β Tm58 (snoc58 Ξ A) B β Tm58 Ξ (arr58 A B); lam58
= Ξ» t Tm58 var58 lam58 app tt pair fst snd left right case zero suc rec β
lam58 _ _ _ (t Tm58 var58 lam58 app tt pair fst snd left right case zero suc rec)
app58 : β{Ξ A B} β Tm58 Ξ (arr58 A B) β Tm58 Ξ A β Tm58 Ξ B; app58
= Ξ» t u Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec β
app58 _ _ _ (t Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec)
(u Tm58 var58 lam58 app58 tt pair fst snd left right case zero suc rec)
tt58 : β{Ξ} β Tm58 Ξ top58; tt58
= Ξ» Tm58 var58 lam58 app58 tt58 pair fst snd left right case zero suc rec β tt58 _
pair58 : β{Ξ A B} β Tm58 Ξ A β Tm58 Ξ B β Tm58 Ξ (prod58 A B); pair58
= Ξ» t u Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec β
pair58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec)
(u Tm58 var58 lam58 app58 tt58 pair58 fst snd left right case zero suc rec)
fst58 : β{Ξ A B} β Tm58 Ξ (prod58 A B) β Tm58 Ξ A; fst58
= Ξ» t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd left right case zero suc rec β
fst58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd left right case zero suc rec)
snd58 : β{Ξ A B} β Tm58 Ξ (prod58 A B) β Tm58 Ξ B; snd58
= Ξ» t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left right case zero suc rec β
snd58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left right case zero suc rec)
left58 : β{Ξ A B} β Tm58 Ξ A β Tm58 Ξ (sum58 A B); left58
= Ξ» t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right case zero suc rec β
left58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right case zero suc rec)
right58 : β{Ξ A B} β Tm58 Ξ B β Tm58 Ξ (sum58 A B); right58
= Ξ» t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case zero suc rec β
right58 _ _ _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case zero suc rec)
case58 : β{Ξ A B C} β Tm58 Ξ (sum58 A B) β Tm58 Ξ (arr58 A C) β Tm58 Ξ (arr58 B C) β Tm58 Ξ C; case58
= Ξ» t u v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec β
case58 _ _ _ _
(t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
(u Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
(v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero suc rec)
zero58 : β{Ξ} β Tm58 Ξ nat58; zero58
= Ξ» Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc rec β zero58 _
suc58 : β{Ξ} β Tm58 Ξ nat58 β Tm58 Ξ nat58; suc58
= Ξ» t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec β
suc58 _ (t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec)
rec58 : β{Ξ A} β Tm58 Ξ nat58 β Tm58 Ξ (arr58 nat58 (arr58 A A)) β Tm58 Ξ A β Tm58 Ξ A; rec58
= Ξ» t u v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58 β
rec58 _ _
(t Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
(u Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
(v Tm58 var58 lam58 app58 tt58 pair58 fst58 snd58 left58 right58 case58 zero58 suc58 rec58)
v058 : β{Ξ A} β Tm58 (snoc58 Ξ A) A; v058
= var58 vz58
v158 : β{Ξ A B} β Tm58 (snoc58 (snoc58 Ξ A) B) A; v158
= var58 (vs58 vz58)
v258 : β{Ξ A B C} β Tm58 (snoc58 (snoc58 (snoc58 Ξ A) B) C) A; v258
= var58 (vs58 (vs58 vz58))
v358 : β{Ξ A B C D} β Tm58 (snoc58 (snoc58 (snoc58 (snoc58 Ξ A) B) C) D) A; v358
= var58 (vs58 (vs58 (vs58 vz58)))
tbool58 : Ty58; tbool58
= sum58 top58 top58
true58 : β{Ξ} β Tm58 Ξ tbool58; true58
= left58 tt58
tfalse58 : β{Ξ} β Tm58 Ξ tbool58; tfalse58
= right58 tt58
ifthenelse58 : β{Ξ A} β Tm58 Ξ (arr58 tbool58 (arr58 A (arr58 A A))); ifthenelse58
= lam58 (lam58 (lam58 (case58 v258 (lam58 v258) (lam58 v158))))
times458 : β{Ξ A} β Tm58 Ξ (arr58 (arr58 A A) (arr58 A A)); times458
= lam58 (lam58 (app58 v158 (app58 v158 (app58 v158 (app58 v158 v058)))))
add58 : β{Ξ} β Tm58 Ξ (arr58 nat58 (arr58 nat58 nat58)); add58
= lam58 (rec58 v058
(lam58 (lam58 (lam58 (suc58 (app58 v158 v058)))))
(lam58 v058))
mul58 : β{Ξ} β Tm58 Ξ (arr58 nat58 (arr58 nat58 nat58)); mul58
= lam58 (rec58 v058
(lam58 (lam58 (lam58 (app58 (app58 add58 (app58 v158 v058)) v058))))
(lam58 zero58))
fact58 : β{Ξ} β Tm58 Ξ (arr58 nat58 nat58); fact58
= lam58 (rec58 v058 (lam58 (lam58 (app58 (app58 mul58 (suc58 v158)) v058)))
(suc58 zero58))
{-# OPTIONS --type-in-type #-}
Ty59 : Set
Ty59 =
(Ty59 : Set)
(nat top bot : Ty59)
(arr prod sum : Ty59 β Ty59 β Ty59)
β Ty59
nat59 : Ty59; nat59 = Ξ» _ nat59 _ _ _ _ _ β nat59
top59 : Ty59; top59 = Ξ» _ _ top59 _ _ _ _ β top59
bot59 : Ty59; bot59 = Ξ» _ _ _ bot59 _ _ _ β bot59
arr59 : Ty59 β Ty59 β Ty59; arr59
= Ξ» A B Ty59 nat59 top59 bot59 arr59 prod sum β
arr59 (A Ty59 nat59 top59 bot59 arr59 prod sum) (B Ty59 nat59 top59 bot59 arr59 prod sum)
prod59 : Ty59 β Ty59 β Ty59; prod59
= Ξ» A B Ty59 nat59 top59 bot59 arr59 prod59 sum β
prod59 (A Ty59 nat59 top59 bot59 arr59 prod59 sum) (B Ty59 nat59 top59 bot59 arr59 prod59 sum)
sum59 : Ty59 β Ty59 β Ty59; sum59
= Ξ» A B Ty59 nat59 top59 bot59 arr59 prod59 sum59 β
sum59 (A Ty59 nat59 top59 bot59 arr59 prod59 sum59) (B Ty59 nat59 top59 bot59 arr59 prod59 sum59)
Con59 : Set; Con59
= (Con59 : Set)
(nil : Con59)
(snoc : Con59 β Ty59 β Con59)
β Con59
nil59 : Con59; nil59
= Ξ» Con59 nil59 snoc β nil59
snoc59 : Con59 β Ty59 β Con59; snoc59
= Ξ» Ξ A Con59 nil59 snoc59 β snoc59 (Ξ Con59 nil59 snoc59) A
Var59 : Con59 β Ty59 β Set; Var59
= Ξ» Ξ A β
(Var59 : Con59 β Ty59 β Set)
(vz : β Ξ A β Var59 (snoc59 Ξ A) A)
(vs : β Ξ B A β Var59 Ξ A β Var59 (snoc59 Ξ B) A)
β Var59 Ξ A
vz59 : β{Ξ A} β Var59 (snoc59 Ξ A) A; vz59
= Ξ» Var59 vz59 vs β vz59 _ _
vs59 : β{Ξ B A} β Var59 Ξ A β Var59 (snoc59 Ξ B) A; vs59
= Ξ» x Var59 vz59 vs59 β vs59 _ _ _ (x Var59 vz59 vs59)
Tm59 : Con59 β Ty59 β Set; Tm59
= Ξ» Ξ A β
(Tm59 : Con59 β Ty59 β Set)
(var : β Ξ A β Var59 Ξ A β Tm59 Ξ A)
(lam : β Ξ A B β Tm59 (snoc59 Ξ A) B β Tm59 Ξ (arr59 A B))
(app : β Ξ A B β Tm59 Ξ (arr59 A B) β Tm59 Ξ A β Tm59 Ξ B)
(tt : β Ξ β Tm59 Ξ top59)
(pair : β Ξ A B β Tm59 Ξ A β Tm59 Ξ B β Tm59 Ξ (prod59 A B))
(fst : β Ξ A B β Tm59 Ξ (prod59 A B) β Tm59 Ξ A)
(snd : β Ξ A B β Tm59 Ξ (prod59 A B) β Tm59 Ξ B)
(left : β Ξ A B β Tm59 Ξ A β Tm59 Ξ (sum59 A B))
(right : β Ξ A B β Tm59 Ξ B β Tm59 Ξ (sum59 A B))
(case : β Ξ A B C β Tm59 Ξ (sum59 A B) β Tm59 Ξ (arr59 A C) β Tm59 Ξ (arr59 B C) β Tm59 Ξ C)
(zero : β Ξ β Tm59 Ξ nat59)
(suc : β Ξ β Tm59 Ξ nat59 β Tm59 Ξ nat59)
(rec : β Ξ A β Tm59 Ξ nat59 β Tm59 Ξ (arr59 nat59 (arr59 A A)) β Tm59 Ξ A β Tm59 Ξ A)
β Tm59 Ξ A
var59 : β{Ξ A} β Var59 Ξ A β Tm59 Ξ A; var59
= Ξ» x Tm59 var59 lam app tt pair fst snd left right case zero suc rec β
var59 _ _ x
lam59 : β{Ξ A B} β Tm59 (snoc59 Ξ A) B β Tm59 Ξ (arr59 A B); lam59
= Ξ» t Tm59 var59 lam59 app tt pair fst snd left right case zero suc rec β
lam59 _ _ _ (t Tm59 var59 lam59 app tt pair fst snd left right case zero suc rec)
app59 : β{Ξ A B} β Tm59 Ξ (arr59 A B) β Tm59 Ξ A β Tm59 Ξ B; app59
= Ξ» t u Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec β
app59 _ _ _ (t Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec)
(u Tm59 var59 lam59 app59 tt pair fst snd left right case zero suc rec)
tt59 : β{Ξ} β Tm59 Ξ top59; tt59
= Ξ» Tm59 var59 lam59 app59 tt59 pair fst snd left right case zero suc rec β tt59 _
pair59 : β{Ξ A B} β Tm59 Ξ A β Tm59 Ξ B β Tm59 Ξ (prod59 A B); pair59
= Ξ» t u Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec β
pair59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec)
(u Tm59 var59 lam59 app59 tt59 pair59 fst snd left right case zero suc rec)
fst59 : β{Ξ A B} β Tm59 Ξ (prod59 A B) β Tm59 Ξ A; fst59
= Ξ» t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd left right case zero suc rec β
fst59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd left right case zero suc rec)
snd59 : β{Ξ A B} β Tm59 Ξ (prod59 A B) β Tm59 Ξ B; snd59
= Ξ» t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left right case zero suc rec β
snd59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left right case zero suc rec)
left59 : β{Ξ A B} β Tm59 Ξ A β Tm59 Ξ (sum59 A B); left59
= Ξ» t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right case zero suc rec β
left59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right case zero suc rec)
right59 : β{Ξ A B} β Tm59 Ξ B β Tm59 Ξ (sum59 A B); right59
= Ξ» t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case zero suc rec β
right59 _ _ _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case zero suc rec)
case59 : β{Ξ A B C} β Tm59 Ξ (sum59 A B) β Tm59 Ξ (arr59 A C) β Tm59 Ξ (arr59 B C) β Tm59 Ξ C; case59
= Ξ» t u v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec β
case59 _ _ _ _
(t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
(u Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
(v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero suc rec)
zero59 : β{Ξ} β Tm59 Ξ nat59; zero59
= Ξ» Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc rec β zero59 _
suc59 : β{Ξ} β Tm59 Ξ nat59 β Tm59 Ξ nat59; suc59
= Ξ» t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec β
suc59 _ (t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec)
rec59 : β{Ξ A} β Tm59 Ξ nat59 β Tm59 Ξ (arr59 nat59 (arr59 A A)) β Tm59 Ξ A β Tm59 Ξ A; rec59
= Ξ» t u v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59 β
rec59 _ _
(t Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
(u Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
(v Tm59 var59 lam59 app59 tt59 pair59 fst59 snd59 left59 right59 case59 zero59 suc59 rec59)
v059 : β{Ξ A} β Tm59 (snoc59 Ξ A) A; v059
= var59 vz59
v159 : β{Ξ A B} β Tm59 (snoc59 (snoc59 Ξ A) B) A; v159
= var59 (vs59 vz59)
v259 : β{Ξ A B C} β Tm59 (snoc59 (snoc59 (snoc59 Ξ A) B) C) A; v259
= var59 (vs59 (vs59 vz59))
v359 : β{Ξ A B C D} β Tm59 (snoc59 (snoc59 (snoc59 (snoc59 Ξ A) B) C) D) A; v359
= var59 (vs59 (vs59 (vs59 vz59)))
tbool59 : Ty59; tbool59
= sum59 top59 top59
true59 : β{Ξ} β Tm59 Ξ tbool59; true59
= left59 tt59
tfalse59 : β{Ξ} β Tm59 Ξ tbool59; tfalse59
= right59 tt59
ifthenelse59 : β{Ξ A} β Tm59 Ξ (arr59 tbool59 (arr59 A (arr59 A A))); ifthenelse59
= lam59 (lam59 (lam59 (case59 v259 (lam59 v259) (lam59 v159))))
times459 : β{Ξ A} β Tm59 Ξ (arr59 (arr59 A A) (arr59 A A)); times459
= lam59 (lam59 (app59 v159 (app59 v159 (app59 v159 (app59 v159 v059)))))
add59 : β{Ξ} β Tm59 Ξ (arr59 nat59 (arr59 nat59 nat59)); add59
= lam59 (rec59 v059
(lam59 (lam59 (lam59 (suc59 (app59 v159 v059)))))
(lam59 v059))
mul59 : β{Ξ} β Tm59 Ξ (arr59 nat59 (arr59 nat59 nat59)); mul59
= lam59 (rec59 v059
(lam59 (lam59 (lam59 (app59 (app59 add59 (app59 v159 v059)) v059))))
(lam59 zero59))
fact59 : β{Ξ} β Tm59 Ξ (arr59 nat59 nat59); fact59
= lam59 (rec59 v059 (lam59 (lam59 (app59 (app59 mul59 (suc59 v159)) v059)))
(suc59 zero59))
{-# OPTIONS --type-in-type #-}
Ty60 : Set
Ty60 =
(Ty60 : Set)
(nat top bot : Ty60)
(arr prod sum : Ty60 β Ty60 β Ty60)
β Ty60
nat60 : Ty60; nat60 = Ξ» _ nat60 _ _ _ _ _ β nat60
top60 : Ty60; top60 = Ξ» _ _ top60 _ _ _ _ β top60
bot60 : Ty60; bot60 = Ξ» _ _ _ bot60 _ _ _ β bot60
arr60 : Ty60 β Ty60 β Ty60; arr60
= Ξ» A B Ty60 nat60 top60 bot60 arr60 prod sum β
arr60 (A Ty60 nat60 top60 bot60 arr60 prod sum) (B Ty60 nat60 top60 bot60 arr60 prod sum)
prod60 : Ty60 β Ty60 β Ty60; prod60
= Ξ» A B Ty60 nat60 top60 bot60 arr60 prod60 sum β
prod60 (A Ty60 nat60 top60 bot60 arr60 prod60 sum) (B Ty60 nat60 top60 bot60 arr60 prod60 sum)
sum60 : Ty60 β Ty60 β Ty60; sum60
= Ξ» A B Ty60 nat60 top60 bot60 arr60 prod60 sum60 β
sum60 (A Ty60 nat60 top60 bot60 arr60 prod60 sum60) (B Ty60 nat60 top60 bot60 arr60 prod60 sum60)
Con60 : Set; Con60
= (Con60 : Set)
(nil : Con60)
(snoc : Con60 β Ty60 β Con60)
β Con60
nil60 : Con60; nil60
= Ξ» Con60 nil60 snoc β nil60
snoc60 : Con60 β Ty60 β Con60; snoc60
= Ξ» Ξ A Con60 nil60 snoc60 β snoc60 (Ξ Con60 nil60 snoc60) A
Var60 : Con60 β Ty60 β Set; Var60
= Ξ» Ξ A β
(Var60 : Con60 β Ty60 β Set)
(vz : β Ξ A β Var60 (snoc60 Ξ A) A)
(vs : β Ξ B A β Var60 Ξ A β Var60 (snoc60 Ξ B) A)
β Var60 Ξ A
vz60 : β{Ξ A} β Var60 (snoc60 Ξ A) A; vz60
= Ξ» Var60 vz60 vs β vz60 _ _
vs60 : β{Ξ B A} β Var60 Ξ A β Var60 (snoc60 Ξ B) A; vs60
= Ξ» x Var60 vz60 vs60 β vs60 _ _ _ (x Var60 vz60 vs60)
Tm60 : Con60 β Ty60 β Set; Tm60
= Ξ» Ξ A β
(Tm60 : Con60 β Ty60 β Set)
(var : β Ξ A β Var60 Ξ A β Tm60 Ξ A)
(lam : β Ξ A B β Tm60 (snoc60 Ξ A) B β Tm60 Ξ (arr60 A B))
(app : β Ξ A B β Tm60 Ξ (arr60 A B) β Tm60 Ξ A β Tm60 Ξ B)
(tt : β Ξ β Tm60 Ξ top60)
(pair : β Ξ A B β Tm60 Ξ A β Tm60 Ξ B β Tm60 Ξ (prod60 A B))
(fst : β Ξ A B β Tm60 Ξ (prod60 A B) β Tm60 Ξ A)
(snd : β Ξ A B β Tm60 Ξ (prod60 A B) β Tm60 Ξ B)
(left : β Ξ A B β Tm60 Ξ A β Tm60 Ξ (sum60 A B))
(right : β Ξ A B β Tm60 Ξ B β Tm60 Ξ (sum60 A B))
(case : β Ξ A B C β Tm60 Ξ (sum60 A B) β Tm60 Ξ (arr60 A C) β Tm60 Ξ (arr60 B C) β Tm60 Ξ C)
(zero : β Ξ β Tm60 Ξ nat60)
(suc : β Ξ β Tm60 Ξ nat60 β Tm60 Ξ nat60)
(rec : β Ξ A β Tm60 Ξ nat60 β Tm60 Ξ (arr60 nat60 (arr60 A A)) β Tm60 Ξ A β Tm60 Ξ A)
β Tm60 Ξ A
var60 : β{Ξ A} β Var60 Ξ A β Tm60 Ξ A; var60
= Ξ» x Tm60 var60 lam app tt pair fst snd left right case zero suc rec β
var60 _ _ x
lam60 : β{Ξ A B} β Tm60 (snoc60 Ξ A) B β Tm60 Ξ (arr60 A B); lam60
= Ξ» t Tm60 var60 lam60 app tt pair fst snd left right case zero suc rec β
lam60 _ _ _ (t Tm60 var60 lam60 app tt pair fst snd left right case zero suc rec)
app60 : β{Ξ A B} β Tm60 Ξ (arr60 A B) β Tm60 Ξ A β Tm60 Ξ B; app60
= Ξ» t u Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec β
app60 _ _ _ (t Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec)
(u Tm60 var60 lam60 app60 tt pair fst snd left right case zero suc rec)
tt60 : β{Ξ} β Tm60 Ξ top60; tt60
= Ξ» Tm60 var60 lam60 app60 tt60 pair fst snd left right case zero suc rec β tt60 _
pair60 : β{Ξ A B} β Tm60 Ξ A β Tm60 Ξ B β Tm60 Ξ (prod60 A B); pair60
= Ξ» t u Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec β
pair60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec)
(u Tm60 var60 lam60 app60 tt60 pair60 fst snd left right case zero suc rec)
fst60 : β{Ξ A B} β Tm60 Ξ (prod60 A B) β Tm60 Ξ A; fst60
= Ξ» t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd left right case zero suc rec β
fst60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd left right case zero suc rec)
snd60 : β{Ξ A B} β Tm60 Ξ (prod60 A B) β Tm60 Ξ B; snd60
= Ξ» t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left right case zero suc rec β
snd60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left right case zero suc rec)
left60 : β{Ξ A B} β Tm60 Ξ A β Tm60 Ξ (sum60 A B); left60
= Ξ» t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right case zero suc rec β
left60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right case zero suc rec)
right60 : β{Ξ A B} β Tm60 Ξ B β Tm60 Ξ (sum60 A B); right60
= Ξ» t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case zero suc rec β
right60 _ _ _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case zero suc rec)
case60 : β{Ξ A B C} β Tm60 Ξ (sum60 A B) β Tm60 Ξ (arr60 A C) β Tm60 Ξ (arr60 B C) β Tm60 Ξ C; case60
= Ξ» t u v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec β
case60 _ _ _ _
(t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
(u Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
(v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero suc rec)
zero60 : β{Ξ} β Tm60 Ξ nat60; zero60
= Ξ» Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc rec β zero60 _
suc60 : β{Ξ} β Tm60 Ξ nat60 β Tm60 Ξ nat60; suc60
= Ξ» t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec β
suc60 _ (t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec)
rec60 : β{Ξ A} β Tm60 Ξ nat60 β Tm60 Ξ (arr60 nat60 (arr60 A A)) β Tm60 Ξ A β Tm60 Ξ A; rec60
= Ξ» t u v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60 β
rec60 _ _
(t Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
(u Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
(v Tm60 var60 lam60 app60 tt60 pair60 fst60 snd60 left60 right60 case60 zero60 suc60 rec60)
v060 : β{Ξ A} β Tm60 (snoc60 Ξ A) A; v060
= var60 vz60
v160 : β{Ξ A B} β Tm60 (snoc60 (snoc60 Ξ A) B) A; v160
= var60 (vs60 vz60)
v260 : β{Ξ A B C} β Tm60 (snoc60 (snoc60 (snoc60 Ξ A) B) C) A; v260
= var60 (vs60 (vs60 vz60))
v360 : β{Ξ A B C D} β Tm60 (snoc60 (snoc60 (snoc60 (snoc60 Ξ A) B) C) D) A; v360
= var60 (vs60 (vs60 (vs60 vz60)))
tbool60 : Ty60; tbool60
= sum60 top60 top60
true60 : β{Ξ} β Tm60 Ξ tbool60; true60
= left60 tt60
tfalse60 : β{Ξ} β Tm60 Ξ tbool60; tfalse60
= right60 tt60
ifthenelse60 : β{Ξ A} β Tm60 Ξ (arr60 tbool60 (arr60 A (arr60 A A))); ifthenelse60
= lam60 (lam60 (lam60 (case60 v260 (lam60 v260) (lam60 v160))))
times460 : β{Ξ A} β Tm60 Ξ (arr60 (arr60 A A) (arr60 A A)); times460
= lam60 (lam60 (app60 v160 (app60 v160 (app60 v160 (app60 v160 v060)))))
add60 : β{Ξ} β Tm60 Ξ (arr60 nat60 (arr60 nat60 nat60)); add60
= lam60 (rec60 v060
(lam60 (lam60 (lam60 (suc60 (app60 v160 v060)))))
(lam60 v060))
mul60 : β{Ξ} β Tm60 Ξ (arr60 nat60 (arr60 nat60 nat60)); mul60
= lam60 (rec60 v060
(lam60 (lam60 (lam60 (app60 (app60 add60 (app60 v160 v060)) v060))))
(lam60 zero60))
fact60 : β{Ξ} β Tm60 Ξ (arr60 nat60 nat60); fact60
= lam60 (rec60 v060 (lam60 (lam60 (app60 (app60 mul60 (suc60 v160)) v060)))
(suc60 zero60))
{-# OPTIONS --type-in-type #-}
Ty61 : Set
Ty61 =
(Ty61 : Set)
(nat top bot : Ty61)
(arr prod sum : Ty61 β Ty61 β Ty61)
β Ty61
nat61 : Ty61; nat61 = Ξ» _ nat61 _ _ _ _ _ β nat61
top61 : Ty61; top61 = Ξ» _ _ top61 _ _ _ _ β top61
bot61 : Ty61; bot61 = Ξ» _ _ _ bot61 _ _ _ β bot61
arr61 : Ty61 β Ty61 β Ty61; arr61
= Ξ» A B Ty61 nat61 top61 bot61 arr61 prod sum β
arr61 (A Ty61 nat61 top61 bot61 arr61 prod sum) (B Ty61 nat61 top61 bot61 arr61 prod sum)
prod61 : Ty61 β Ty61 β Ty61; prod61
= Ξ» A B Ty61 nat61 top61 bot61 arr61 prod61 sum β
prod61 (A Ty61 nat61 top61 bot61 arr61 prod61 sum) (B Ty61 nat61 top61 bot61 arr61 prod61 sum)
sum61 : Ty61 β Ty61 β Ty61; sum61
= Ξ» A B Ty61 nat61 top61 bot61 arr61 prod61 sum61 β
sum61 (A Ty61 nat61 top61 bot61 arr61 prod61 sum61) (B Ty61 nat61 top61 bot61 arr61 prod61 sum61)
Con61 : Set; Con61
= (Con61 : Set)
(nil : Con61)
(snoc : Con61 β Ty61 β Con61)
β Con61
nil61 : Con61; nil61
= Ξ» Con61 nil61 snoc β nil61
snoc61 : Con61 β Ty61 β Con61; snoc61
= Ξ» Ξ A Con61 nil61 snoc61 β snoc61 (Ξ Con61 nil61 snoc61) A
Var61 : Con61 β Ty61 β Set; Var61
= Ξ» Ξ A β
(Var61 : Con61 β Ty61 β Set)
(vz : β Ξ A β Var61 (snoc61 Ξ A) A)
(vs : β Ξ B A β Var61 Ξ A β Var61 (snoc61 Ξ B) A)
β Var61 Ξ A
vz61 : β{Ξ A} β Var61 (snoc61 Ξ A) A; vz61
= Ξ» Var61 vz61 vs β vz61 _ _
vs61 : β{Ξ B A} β Var61 Ξ A β Var61 (snoc61 Ξ B) A; vs61
= Ξ» x Var61 vz61 vs61 β vs61 _ _ _ (x Var61 vz61 vs61)
Tm61 : Con61 β Ty61 β Set; Tm61
= Ξ» Ξ A β
(Tm61 : Con61 β Ty61 β Set)
(var : β Ξ A β Var61 Ξ A β Tm61 Ξ A)
(lam : β Ξ A B β Tm61 (snoc61 Ξ A) B β Tm61 Ξ (arr61 A B))
(app : β Ξ A B β Tm61 Ξ (arr61 A B) β Tm61 Ξ A β Tm61 Ξ B)
(tt : β Ξ β Tm61 Ξ top61)
(pair : β Ξ A B β Tm61 Ξ A β Tm61 Ξ B β Tm61 Ξ (prod61 A B))
(fst : β Ξ A B β Tm61 Ξ (prod61 A B) β Tm61 Ξ A)
(snd : β Ξ A B β Tm61 Ξ (prod61 A B) β Tm61 Ξ B)
(left : β Ξ A B β Tm61 Ξ A β Tm61 Ξ (sum61 A B))
(right : β Ξ A B β Tm61 Ξ B β Tm61 Ξ (sum61 A B))
(case : β Ξ A B C β Tm61 Ξ (sum61 A B) β Tm61 Ξ (arr61 A C) β Tm61 Ξ (arr61 B C) β Tm61 Ξ C)
(zero : β Ξ β Tm61 Ξ nat61)
(suc : β Ξ β Tm61 Ξ nat61 β Tm61 Ξ nat61)
(rec : β Ξ A β Tm61 Ξ nat61 β Tm61 Ξ (arr61 nat61 (arr61 A A)) β Tm61 Ξ A β Tm61 Ξ A)
β Tm61 Ξ A
var61 : β{Ξ A} β Var61 Ξ A β Tm61 Ξ A; var61
= Ξ» x Tm61 var61 lam app tt pair fst snd left right case zero suc rec β
var61 _ _ x
lam61 : β{Ξ A B} β Tm61 (snoc61 Ξ A) B β Tm61 Ξ (arr61 A B); lam61
= Ξ» t Tm61 var61 lam61 app tt pair fst snd left right case zero suc rec β
lam61 _ _ _ (t Tm61 var61 lam61 app tt pair fst snd left right case zero suc rec)
app61 : β{Ξ A B} β Tm61 Ξ (arr61 A B) β Tm61 Ξ A β Tm61 Ξ B; app61
= Ξ» t u Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec β
app61 _ _ _ (t Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec)
(u Tm61 var61 lam61 app61 tt pair fst snd left right case zero suc rec)
tt61 : β{Ξ} β Tm61 Ξ top61; tt61
= Ξ» Tm61 var61 lam61 app61 tt61 pair fst snd left right case zero suc rec β tt61 _
pair61 : β{Ξ A B} β Tm61 Ξ A β Tm61 Ξ B β Tm61 Ξ (prod61 A B); pair61
= Ξ» t u Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec β
pair61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec)
(u Tm61 var61 lam61 app61 tt61 pair61 fst snd left right case zero suc rec)
fst61 : β{Ξ A B} β Tm61 Ξ (prod61 A B) β Tm61 Ξ A; fst61
= Ξ» t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd left right case zero suc rec β
fst61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd left right case zero suc rec)
snd61 : β{Ξ A B} β Tm61 Ξ (prod61 A B) β Tm61 Ξ B; snd61
= Ξ» t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left right case zero suc rec β
snd61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left right case zero suc rec)
left61 : β{Ξ A B} β Tm61 Ξ A β Tm61 Ξ (sum61 A B); left61
= Ξ» t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right case zero suc rec β
left61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right case zero suc rec)
right61 : β{Ξ A B} β Tm61 Ξ B β Tm61 Ξ (sum61 A B); right61
= Ξ» t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case zero suc rec β
right61 _ _ _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case zero suc rec)
case61 : β{Ξ A B C} β Tm61 Ξ (sum61 A B) β Tm61 Ξ (arr61 A C) β Tm61 Ξ (arr61 B C) β Tm61 Ξ C; case61
= Ξ» t u v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec β
case61 _ _ _ _
(t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
(u Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
(v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero suc rec)
zero61 : β{Ξ} β Tm61 Ξ nat61; zero61
= Ξ» Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc rec β zero61 _
suc61 : β{Ξ} β Tm61 Ξ nat61 β Tm61 Ξ nat61; suc61
= Ξ» t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec β
suc61 _ (t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec)
rec61 : β{Ξ A} β Tm61 Ξ nat61 β Tm61 Ξ (arr61 nat61 (arr61 A A)) β Tm61 Ξ A β Tm61 Ξ A; rec61
= Ξ» t u v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61 β
rec61 _ _
(t Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
(u Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
(v Tm61 var61 lam61 app61 tt61 pair61 fst61 snd61 left61 right61 case61 zero61 suc61 rec61)
v061 : β{Ξ A} β Tm61 (snoc61 Ξ A) A; v061
= var61 vz61
v161 : β{Ξ A B} β Tm61 (snoc61 (snoc61 Ξ A) B) A; v161
= var61 (vs61 vz61)
v261 : β{Ξ A B C} β Tm61 (snoc61 (snoc61 (snoc61 Ξ A) B) C) A; v261
= var61 (vs61 (vs61 vz61))
v361 : β{Ξ A B C D} β Tm61 (snoc61 (snoc61 (snoc61 (snoc61 Ξ A) B) C) D) A; v361
= var61 (vs61 (vs61 (vs61 vz61)))
tbool61 : Ty61; tbool61
= sum61 top61 top61
true61 : β{Ξ} β Tm61 Ξ tbool61; true61
= left61 tt61
tfalse61 : β{Ξ} β Tm61 Ξ tbool61; tfalse61
= right61 tt61
ifthenelse61 : β{Ξ A} β Tm61 Ξ (arr61 tbool61 (arr61 A (arr61 A A))); ifthenelse61
= lam61 (lam61 (lam61 (case61 v261 (lam61 v261) (lam61 v161))))
times461 : β{Ξ A} β Tm61 Ξ (arr61 (arr61 A A) (arr61 A A)); times461
= lam61 (lam61 (app61 v161 (app61 v161 (app61 v161 (app61 v161 v061)))))
add61 : β{Ξ} β Tm61 Ξ (arr61 nat61 (arr61 nat61 nat61)); add61
= lam61 (rec61 v061
(lam61 (lam61 (lam61 (suc61 (app61 v161 v061)))))
(lam61 v061))
mul61 : β{Ξ} β Tm61 Ξ (arr61 nat61 (arr61 nat61 nat61)); mul61
= lam61 (rec61 v061
(lam61 (lam61 (lam61 (app61 (app61 add61 (app61 v161 v061)) v061))))
(lam61 zero61))
fact61 : β{Ξ} β Tm61 Ξ (arr61 nat61 nat61); fact61
= lam61 (rec61 v061 (lam61 (lam61 (app61 (app61 mul61 (suc61 v161)) v061)))
(suc61 zero61))
{-# OPTIONS --type-in-type #-}
Ty62 : Set
Ty62 =
(Ty62 : Set)
(nat top bot : Ty62)
(arr prod sum : Ty62 β Ty62 β Ty62)
β Ty62
nat62 : Ty62; nat62 = Ξ» _ nat62 _ _ _ _ _ β nat62
top62 : Ty62; top62 = Ξ» _ _ top62 _ _ _ _ β top62
bot62 : Ty62; bot62 = Ξ» _ _ _ bot62 _ _ _ β bot62
arr62 : Ty62 β Ty62 β Ty62; arr62
= Ξ» A B Ty62 nat62 top62 bot62 arr62 prod sum β
arr62 (A Ty62 nat62 top62 bot62 arr62 prod sum) (B Ty62 nat62 top62 bot62 arr62 prod sum)
prod62 : Ty62 β Ty62 β Ty62; prod62
= Ξ» A B Ty62 nat62 top62 bot62 arr62 prod62 sum β
prod62 (A Ty62 nat62 top62 bot62 arr62 prod62 sum) (B Ty62 nat62 top62 bot62 arr62 prod62 sum)
sum62 : Ty62 β Ty62 β Ty62; sum62
= Ξ» A B Ty62 nat62 top62 bot62 arr62 prod62 sum62 β
sum62 (A Ty62 nat62 top62 bot62 arr62 prod62 sum62) (B Ty62 nat62 top62 bot62 arr62 prod62 sum62)
Con62 : Set; Con62
= (Con62 : Set)
(nil : Con62)
(snoc : Con62 β Ty62 β Con62)
β Con62
nil62 : Con62; nil62
= Ξ» Con62 nil62 snoc β nil62
snoc62 : Con62 β Ty62 β Con62; snoc62
= Ξ» Ξ A Con62 nil62 snoc62 β snoc62 (Ξ Con62 nil62 snoc62) A
Var62 : Con62 β Ty62 β Set; Var62
= Ξ» Ξ A β
(Var62 : Con62 β Ty62 β Set)
(vz : β Ξ A β Var62 (snoc62 Ξ A) A)
(vs : β Ξ B A β Var62 Ξ A β Var62 (snoc62 Ξ B) A)
β Var62 Ξ A
vz62 : β{Ξ A} β Var62 (snoc62 Ξ A) A; vz62
= Ξ» Var62 vz62 vs β vz62 _ _
vs62 : β{Ξ B A} β Var62 Ξ A β Var62 (snoc62 Ξ B) A; vs62
= Ξ» x Var62 vz62 vs62 β vs62 _ _ _ (x Var62 vz62 vs62)
Tm62 : Con62 β Ty62 β Set; Tm62
= Ξ» Ξ A β
(Tm62 : Con62 β Ty62 β Set)
(var : β Ξ A β Var62 Ξ A β Tm62 Ξ A)
(lam : β Ξ A B β Tm62 (snoc62 Ξ A) B β Tm62 Ξ (arr62 A B))
(app : β Ξ A B β Tm62 Ξ (arr62 A B) β Tm62 Ξ A β Tm62 Ξ B)
(tt : β Ξ β Tm62 Ξ top62)
(pair : β Ξ A B β Tm62 Ξ A β Tm62 Ξ B β Tm62 Ξ (prod62 A B))
(fst : β Ξ A B β Tm62 Ξ (prod62 A B) β Tm62 Ξ A)
(snd : β Ξ A B β Tm62 Ξ (prod62 A B) β Tm62 Ξ B)
(left : β Ξ A B β Tm62 Ξ A β Tm62 Ξ (sum62 A B))
(right : β Ξ A B β Tm62 Ξ B β Tm62 Ξ (sum62 A B))
(case : β Ξ A B C β Tm62 Ξ (sum62 A B) β Tm62 Ξ (arr62 A C) β Tm62 Ξ (arr62 B C) β Tm62 Ξ C)
(zero : β Ξ β Tm62 Ξ nat62)
(suc : β Ξ β Tm62 Ξ nat62 β Tm62 Ξ nat62)
(rec : β Ξ A β Tm62 Ξ nat62 β Tm62 Ξ (arr62 nat62 (arr62 A A)) β Tm62 Ξ A β Tm62 Ξ A)
β Tm62 Ξ A
var62 : β{Ξ A} β Var62 Ξ A β Tm62 Ξ A; var62
= Ξ» x Tm62 var62 lam app tt pair fst snd left right case zero suc rec β
var62 _ _ x
lam62 : β{Ξ A B} β Tm62 (snoc62 Ξ A) B β Tm62 Ξ (arr62 A B); lam62
= Ξ» t Tm62 var62 lam62 app tt pair fst snd left right case zero suc rec β
lam62 _ _ _ (t Tm62 var62 lam62 app tt pair fst snd left right case zero suc rec)
app62 : β{Ξ A B} β Tm62 Ξ (arr62 A B) β Tm62 Ξ A β Tm62 Ξ B; app62
= Ξ» t u Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec β
app62 _ _ _ (t Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec)
(u Tm62 var62 lam62 app62 tt pair fst snd left right case zero suc rec)
tt62 : β{Ξ} β Tm62 Ξ top62; tt62
= Ξ» Tm62 var62 lam62 app62 tt62 pair fst snd left right case zero suc rec β tt62 _
pair62 : β{Ξ A B} β Tm62 Ξ A β Tm62 Ξ B β Tm62 Ξ (prod62 A B); pair62
= Ξ» t u Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec β
pair62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec)
(u Tm62 var62 lam62 app62 tt62 pair62 fst snd left right case zero suc rec)
fst62 : β{Ξ A B} β Tm62 Ξ (prod62 A B) β Tm62 Ξ A; fst62
= Ξ» t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd left right case zero suc rec β
fst62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd left right case zero suc rec)
snd62 : β{Ξ A B} β Tm62 Ξ (prod62 A B) β Tm62 Ξ B; snd62
= Ξ» t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left right case zero suc rec β
snd62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left right case zero suc rec)
left62 : β{Ξ A B} β Tm62 Ξ A β Tm62 Ξ (sum62 A B); left62
= Ξ» t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right case zero suc rec β
left62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right case zero suc rec)
right62 : β{Ξ A B} β Tm62 Ξ B β Tm62 Ξ (sum62 A B); right62
= Ξ» t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case zero suc rec β
right62 _ _ _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case zero suc rec)
case62 : β{Ξ A B C} β Tm62 Ξ (sum62 A B) β Tm62 Ξ (arr62 A C) β Tm62 Ξ (arr62 B C) β Tm62 Ξ C; case62
= Ξ» t u v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec β
case62 _ _ _ _
(t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
(u Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
(v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero suc rec)
zero62 : β{Ξ} β Tm62 Ξ nat62; zero62
= Ξ» Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc rec β zero62 _
suc62 : β{Ξ} β Tm62 Ξ nat62 β Tm62 Ξ nat62; suc62
= Ξ» t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec β
suc62 _ (t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec)
rec62 : β{Ξ A} β Tm62 Ξ nat62 β Tm62 Ξ (arr62 nat62 (arr62 A A)) β Tm62 Ξ A β Tm62 Ξ A; rec62
= Ξ» t u v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62 β
rec62 _ _
(t Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
(u Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
(v Tm62 var62 lam62 app62 tt62 pair62 fst62 snd62 left62 right62 case62 zero62 suc62 rec62)
v062 : β{Ξ A} β Tm62 (snoc62 Ξ A) A; v062
= var62 vz62
v162 : β{Ξ A B} β Tm62 (snoc62 (snoc62 Ξ A) B) A; v162
= var62 (vs62 vz62)
v262 : β{Ξ A B C} β Tm62 (snoc62 (snoc62 (snoc62 Ξ A) B) C) A; v262
= var62 (vs62 (vs62 vz62))
v362 : β{Ξ A B C D} β Tm62 (snoc62 (snoc62 (snoc62 (snoc62 Ξ A) B) C) D) A; v362
= var62 (vs62 (vs62 (vs62 vz62)))
tbool62 : Ty62; tbool62
= sum62 top62 top62
true62 : β{Ξ} β Tm62 Ξ tbool62; true62
= left62 tt62
tfalse62 : β{Ξ} β Tm62 Ξ tbool62; tfalse62
= right62 tt62
ifthenelse62 : β{Ξ A} β Tm62 Ξ (arr62 tbool62 (arr62 A (arr62 A A))); ifthenelse62
= lam62 (lam62 (lam62 (case62 v262 (lam62 v262) (lam62 v162))))
times462 : β{Ξ A} β Tm62 Ξ (arr62 (arr62 A A) (arr62 A A)); times462
= lam62 (lam62 (app62 v162 (app62 v162 (app62 v162 (app62 v162 v062)))))
add62 : β{Ξ} β Tm62 Ξ (arr62 nat62 (arr62 nat62 nat62)); add62
= lam62 (rec62 v062
(lam62 (lam62 (lam62 (suc62 (app62 v162 v062)))))
(lam62 v062))
mul62 : β{Ξ} β Tm62 Ξ (arr62 nat62 (arr62 nat62 nat62)); mul62
= lam62 (rec62 v062
(lam62 (lam62 (lam62 (app62 (app62 add62 (app62 v162 v062)) v062))))
(lam62 zero62))
fact62 : β{Ξ} β Tm62 Ξ (arr62 nat62 nat62); fact62
= lam62 (rec62 v062 (lam62 (lam62 (app62 (app62 mul62 (suc62 v162)) v062)))
(suc62 zero62))
{-# OPTIONS --type-in-type #-}
Ty63 : Set
Ty63 =
(Ty63 : Set)
(nat top bot : Ty63)
(arr prod sum : Ty63 β Ty63 β Ty63)
β Ty63
nat63 : Ty63; nat63 = Ξ» _ nat63 _ _ _ _ _ β nat63
top63 : Ty63; top63 = Ξ» _ _ top63 _ _ _ _ β top63
bot63 : Ty63; bot63 = Ξ» _ _ _ bot63 _ _ _ β bot63
arr63 : Ty63 β Ty63 β Ty63; arr63
= Ξ» A B Ty63 nat63 top63 bot63 arr63 prod sum β
arr63 (A Ty63 nat63 top63 bot63 arr63 prod sum) (B Ty63 nat63 top63 bot63 arr63 prod sum)
prod63 : Ty63 β Ty63 β Ty63; prod63
= Ξ» A B Ty63 nat63 top63 bot63 arr63 prod63 sum β
prod63 (A Ty63 nat63 top63 bot63 arr63 prod63 sum) (B Ty63 nat63 top63 bot63 arr63 prod63 sum)
sum63 : Ty63 β Ty63 β Ty63; sum63
= Ξ» A B Ty63 nat63 top63 bot63 arr63 prod63 sum63 β
sum63 (A Ty63 nat63 top63 bot63 arr63 prod63 sum63) (B Ty63 nat63 top63 bot63 arr63 prod63 sum63)
Con63 : Set; Con63
= (Con63 : Set)
(nil : Con63)
(snoc : Con63 β Ty63 β Con63)
β Con63
nil63 : Con63; nil63
= Ξ» Con63 nil63 snoc β nil63
snoc63 : Con63 β Ty63 β Con63; snoc63
= Ξ» Ξ A Con63 nil63 snoc63 β snoc63 (Ξ Con63 nil63 snoc63) A
Var63 : Con63 β Ty63 β Set; Var63
= Ξ» Ξ A β
(Var63 : Con63 β Ty63 β Set)
(vz : β Ξ A β Var63 (snoc63 Ξ A) A)
(vs : β Ξ B A β Var63 Ξ A β Var63 (snoc63 Ξ B) A)
β Var63 Ξ A
vz63 : β{Ξ A} β Var63 (snoc63 Ξ A) A; vz63
= Ξ» Var63 vz63 vs β vz63 _ _
vs63 : β{Ξ B A} β Var63 Ξ A β Var63 (snoc63 Ξ B) A; vs63
= Ξ» x Var63 vz63 vs63 β vs63 _ _ _ (x Var63 vz63 vs63)
Tm63 : Con63 β Ty63 β Set; Tm63
= Ξ» Ξ A β
(Tm63 : Con63 β Ty63 β Set)
(var : β Ξ A β Var63 Ξ A β Tm63 Ξ A)
(lam : β Ξ A B β Tm63 (snoc63 Ξ A) B β Tm63 Ξ (arr63 A B))
(app : β Ξ A B β Tm63 Ξ (arr63 A B) β Tm63 Ξ A β Tm63 Ξ B)
(tt : β Ξ β Tm63 Ξ top63)
(pair : β Ξ A B β Tm63 Ξ A β Tm63 Ξ B β Tm63 Ξ (prod63 A B))
(fst : β Ξ A B β Tm63 Ξ (prod63 A B) β Tm63 Ξ A)
(snd : β Ξ A B β Tm63 Ξ (prod63 A B) β Tm63 Ξ B)
(left : β Ξ A B β Tm63 Ξ A β Tm63 Ξ (sum63 A B))
(right : β Ξ A B β Tm63 Ξ B β Tm63 Ξ (sum63 A B))
(case : β Ξ A B C β Tm63 Ξ (sum63 A B) β Tm63 Ξ (arr63 A C) β Tm63 Ξ (arr63 B C) β Tm63 Ξ C)
(zero : β Ξ β Tm63 Ξ nat63)
(suc : β Ξ β Tm63 Ξ nat63 β Tm63 Ξ nat63)
(rec : β Ξ A β Tm63 Ξ nat63 β Tm63 Ξ (arr63 nat63 (arr63 A A)) β Tm63 Ξ A β Tm63 Ξ A)
β Tm63 Ξ A
var63 : β{Ξ A} β Var63 Ξ A β Tm63 Ξ A; var63
= Ξ» x Tm63 var63 lam app tt pair fst snd left right case zero suc rec β
var63 _ _ x
lam63 : β{Ξ A B} β Tm63 (snoc63 Ξ A) B β Tm63 Ξ (arr63 A B); lam63
= Ξ» t Tm63 var63 lam63 app tt pair fst snd left right case zero suc rec β
lam63 _ _ _ (t Tm63 var63 lam63 app tt pair fst snd left right case zero suc rec)
app63 : β{Ξ A B} β Tm63 Ξ (arr63 A B) β Tm63 Ξ A β Tm63 Ξ B; app63
= Ξ» t u Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec β
app63 _ _ _ (t Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec)
(u Tm63 var63 lam63 app63 tt pair fst snd left right case zero suc rec)
tt63 : β{Ξ} β Tm63 Ξ top63; tt63
= Ξ» Tm63 var63 lam63 app63 tt63 pair fst snd left right case zero suc rec β tt63 _
pair63 : β{Ξ A B} β Tm63 Ξ A β Tm63 Ξ B β Tm63 Ξ (prod63 A B); pair63
= Ξ» t u Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec β
pair63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec)
(u Tm63 var63 lam63 app63 tt63 pair63 fst snd left right case zero suc rec)
fst63 : β{Ξ A B} β Tm63 Ξ (prod63 A B) β Tm63 Ξ A; fst63
= Ξ» t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd left right case zero suc rec β
fst63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd left right case zero suc rec)
snd63 : β{Ξ A B} β Tm63 Ξ (prod63 A B) β Tm63 Ξ B; snd63
= Ξ» t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left right case zero suc rec β
snd63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left right case zero suc rec)
left63 : β{Ξ A B} β Tm63 Ξ A β Tm63 Ξ (sum63 A B); left63
= Ξ» t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right case zero suc rec β
left63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right case zero suc rec)
right63 : β{Ξ A B} β Tm63 Ξ B β Tm63 Ξ (sum63 A B); right63
= Ξ» t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case zero suc rec β
right63 _ _ _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case zero suc rec)
case63 : β{Ξ A B C} β Tm63 Ξ (sum63 A B) β Tm63 Ξ (arr63 A C) β Tm63 Ξ (arr63 B C) β Tm63 Ξ C; case63
= Ξ» t u v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec β
case63 _ _ _ _
(t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
(u Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
(v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero suc rec)
zero63 : β{Ξ} β Tm63 Ξ nat63; zero63
= Ξ» Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc rec β zero63 _
suc63 : β{Ξ} β Tm63 Ξ nat63 β Tm63 Ξ nat63; suc63
= Ξ» t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec β
suc63 _ (t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec)
rec63 : β{Ξ A} β Tm63 Ξ nat63 β Tm63 Ξ (arr63 nat63 (arr63 A A)) β Tm63 Ξ A β Tm63 Ξ A; rec63
= Ξ» t u v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63 β
rec63 _ _
(t Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
(u Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
(v Tm63 var63 lam63 app63 tt63 pair63 fst63 snd63 left63 right63 case63 zero63 suc63 rec63)
v063 : β{Ξ A} β Tm63 (snoc63 Ξ A) A; v063
= var63 vz63
v163 : β{Ξ A B} β Tm63 (snoc63 (snoc63 Ξ A) B) A; v163
= var63 (vs63 vz63)
v263 : β{Ξ A B C} β Tm63 (snoc63 (snoc63 (snoc63 Ξ A) B) C) A; v263
= var63 (vs63 (vs63 vz63))
v363 : β{Ξ A B C D} β Tm63 (snoc63 (snoc63 (snoc63 (snoc63 Ξ A) B) C) D) A; v363
= var63 (vs63 (vs63 (vs63 vz63)))
tbool63 : Ty63; tbool63
= sum63 top63 top63
true63 : β{Ξ} β Tm63 Ξ tbool63; true63
= left63 tt63
tfalse63 : β{Ξ} β Tm63 Ξ tbool63; tfalse63
= right63 tt63
ifthenelse63 : β{Ξ A} β Tm63 Ξ (arr63 tbool63 (arr63 A (arr63 A A))); ifthenelse63
= lam63 (lam63 (lam63 (case63 v263 (lam63 v263) (lam63 v163))))
times463 : β{Ξ A} β Tm63 Ξ (arr63 (arr63 A A) (arr63 A A)); times463
= lam63 (lam63 (app63 v163 (app63 v163 (app63 v163 (app63 v163 v063)))))
add63 : β{Ξ} β Tm63 Ξ (arr63 nat63 (arr63 nat63 nat63)); add63
= lam63 (rec63 v063
(lam63 (lam63 (lam63 (suc63 (app63 v163 v063)))))
(lam63 v063))
mul63 : β{Ξ} β Tm63 Ξ (arr63 nat63 (arr63 nat63 nat63)); mul63
= lam63 (rec63 v063
(lam63 (lam63 (lam63 (app63 (app63 add63 (app63 v163 v063)) v063))))
(lam63 zero63))
fact63 : β{Ξ} β Tm63 Ξ (arr63 nat63 nat63); fact63
= lam63 (rec63 v063 (lam63 (lam63 (app63 (app63 mul63 (suc63 v163)) v063)))
(suc63 zero63))
{-# OPTIONS --type-in-type #-}
Ty64 : Set
Ty64 =
(Ty64 : Set)
(nat top bot : Ty64)
(arr prod sum : Ty64 β Ty64 β Ty64)
β Ty64
nat64 : Ty64; nat64 = Ξ» _ nat64 _ _ _ _ _ β nat64
top64 : Ty64; top64 = Ξ» _ _ top64 _ _ _ _ β top64
bot64 : Ty64; bot64 = Ξ» _ _ _ bot64 _ _ _ β bot64
arr64 : Ty64 β Ty64 β Ty64; arr64
= Ξ» A B Ty64 nat64 top64 bot64 arr64 prod sum β
arr64 (A Ty64 nat64 top64 bot64 arr64 prod sum) (B Ty64 nat64 top64 bot64 arr64 prod sum)
prod64 : Ty64 β Ty64 β Ty64; prod64
= Ξ» A B Ty64 nat64 top64 bot64 arr64 prod64 sum β
prod64 (A Ty64 nat64 top64 bot64 arr64 prod64 sum) (B Ty64 nat64 top64 bot64 arr64 prod64 sum)
sum64 : Ty64 β Ty64 β Ty64; sum64
= Ξ» A B Ty64 nat64 top64 bot64 arr64 prod64 sum64 β
sum64 (A Ty64 nat64 top64 bot64 arr64 prod64 sum64) (B Ty64 nat64 top64 bot64 arr64 prod64 sum64)
Con64 : Set; Con64
= (Con64 : Set)
(nil : Con64)
(snoc : Con64 β Ty64 β Con64)
β Con64
nil64 : Con64; nil64
= Ξ» Con64 nil64 snoc β nil64
snoc64 : Con64 β Ty64 β Con64; snoc64
= Ξ» Ξ A Con64 nil64 snoc64 β snoc64 (Ξ Con64 nil64 snoc64) A
Var64 : Con64 β Ty64 β Set; Var64
= Ξ» Ξ A β
(Var64 : Con64 β Ty64 β Set)
(vz : β Ξ A β Var64 (snoc64 Ξ A) A)
(vs : β Ξ B A β Var64 Ξ A β Var64 (snoc64 Ξ B) A)
β Var64 Ξ A
vz64 : β{Ξ A} β Var64 (snoc64 Ξ A) A; vz64
= Ξ» Var64 vz64 vs β vz64 _ _
vs64 : β{Ξ B A} β Var64 Ξ A β Var64 (snoc64 Ξ B) A; vs64
= Ξ» x Var64 vz64 vs64 β vs64 _ _ _ (x Var64 vz64 vs64)
Tm64 : Con64 β Ty64 β Set; Tm64
= Ξ» Ξ A β
(Tm64 : Con64 β Ty64 β Set)
(var : β Ξ A β Var64 Ξ A β Tm64 Ξ A)
(lam : β Ξ A B β Tm64 (snoc64 Ξ A) B β Tm64 Ξ (arr64 A B))
(app : β Ξ A B β Tm64 Ξ (arr64 A B) β Tm64 Ξ A β Tm64 Ξ B)
(tt : β Ξ β Tm64 Ξ top64)
(pair : β Ξ A B β Tm64 Ξ A β Tm64 Ξ B β Tm64 Ξ (prod64 A B))
(fst : β Ξ A B β Tm64 Ξ (prod64 A B) β Tm64 Ξ A)
(snd : β Ξ A B β Tm64 Ξ (prod64 A B) β Tm64 Ξ B)
(left : β Ξ A B β Tm64 Ξ A β Tm64 Ξ (sum64 A B))
(right : β Ξ A B β Tm64 Ξ B β Tm64 Ξ (sum64 A B))
(case : β Ξ A B C β Tm64 Ξ (sum64 A B) β Tm64 Ξ (arr64 A C) β Tm64 Ξ (arr64 B C) β Tm64 Ξ C)
(zero : β Ξ β Tm64 Ξ nat64)
(suc : β Ξ β Tm64 Ξ nat64 β Tm64 Ξ nat64)
(rec : β Ξ A β Tm64 Ξ nat64 β Tm64 Ξ (arr64 nat64 (arr64 A A)) β Tm64 Ξ A β Tm64 Ξ A)
β Tm64 Ξ A
var64 : β{Ξ A} β Var64 Ξ A β Tm64 Ξ A; var64
= Ξ» x Tm64 var64 lam app tt pair fst snd left right case zero suc rec β
var64 _ _ x
lam64 : β{Ξ A B} β Tm64 (snoc64 Ξ A) B β Tm64 Ξ (arr64 A B); lam64
= Ξ» t Tm64 var64 lam64 app tt pair fst snd left right case zero suc rec β
lam64 _ _ _ (t Tm64 var64 lam64 app tt pair fst snd left right case zero suc rec)
app64 : β{Ξ A B} β Tm64 Ξ (arr64 A B) β Tm64 Ξ A β Tm64 Ξ B; app64
= Ξ» t u Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec β
app64 _ _ _ (t Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec)
(u Tm64 var64 lam64 app64 tt pair fst snd left right case zero suc rec)
tt64 : β{Ξ} β Tm64 Ξ top64; tt64
= Ξ» Tm64 var64 lam64 app64 tt64 pair fst snd left right case zero suc rec β tt64 _
pair64 : β{Ξ A B} β Tm64 Ξ A β Tm64 Ξ B β Tm64 Ξ (prod64 A B); pair64
= Ξ» t u Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec β
pair64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec)
(u Tm64 var64 lam64 app64 tt64 pair64 fst snd left right case zero suc rec)
fst64 : β{Ξ A B} β Tm64 Ξ (prod64 A B) β Tm64 Ξ A; fst64
= Ξ» t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd left right case zero suc rec β
fst64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd left right case zero suc rec)
snd64 : β{Ξ A B} β Tm64 Ξ (prod64 A B) β Tm64 Ξ B; snd64
= Ξ» t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left right case zero suc rec β
snd64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left right case zero suc rec)
left64 : β{Ξ A B} β Tm64 Ξ A β Tm64 Ξ (sum64 A B); left64
= Ξ» t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right case zero suc rec β
left64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right case zero suc rec)
right64 : β{Ξ A B} β Tm64 Ξ B β Tm64 Ξ (sum64 A B); right64
= Ξ» t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case zero suc rec β
right64 _ _ _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case zero suc rec)
case64 : β{Ξ A B C} β Tm64 Ξ (sum64 A B) β Tm64 Ξ (arr64 A C) β Tm64 Ξ (arr64 B C) β Tm64 Ξ C; case64
= Ξ» t u v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec β
case64 _ _ _ _
(t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
(u Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
(v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero suc rec)
zero64 : β{Ξ} β Tm64 Ξ nat64; zero64
= Ξ» Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc rec β zero64 _
suc64 : β{Ξ} β Tm64 Ξ nat64 β Tm64 Ξ nat64; suc64
= Ξ» t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec β
suc64 _ (t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec)
rec64 : β{Ξ A} β Tm64 Ξ nat64 β Tm64 Ξ (arr64 nat64 (arr64 A A)) β Tm64 Ξ A β Tm64 Ξ A; rec64
= Ξ» t u v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64 β
rec64 _ _
(t Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
(u Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
(v Tm64 var64 lam64 app64 tt64 pair64 fst64 snd64 left64 right64 case64 zero64 suc64 rec64)
v064 : β{Ξ A} β Tm64 (snoc64 Ξ A) A; v064
= var64 vz64
v164 : β{Ξ A B} β Tm64 (snoc64 (snoc64 Ξ A) B) A; v164
= var64 (vs64 vz64)
v264 : β{Ξ A B C} β Tm64 (snoc64 (snoc64 (snoc64 Ξ A) B) C) A; v264
= var64 (vs64 (vs64 vz64))
v364 : β{Ξ A B C D} β Tm64 (snoc64 (snoc64 (snoc64 (snoc64 Ξ A) B) C) D) A; v364
= var64 (vs64 (vs64 (vs64 vz64)))
tbool64 : Ty64; tbool64
= sum64 top64 top64
true64 : β{Ξ} β Tm64 Ξ tbool64; true64
= left64 tt64
tfalse64 : β{Ξ} β Tm64 Ξ tbool64; tfalse64
= right64 tt64
ifthenelse64 : β{Ξ A} β Tm64 Ξ (arr64 tbool64 (arr64 A (arr64 A A))); ifthenelse64
= lam64 (lam64 (lam64 (case64 v264 (lam64 v264) (lam64 v164))))
times464 : β{Ξ A} β Tm64 Ξ (arr64 (arr64 A A) (arr64 A A)); times464
= lam64 (lam64 (app64 v164 (app64 v164 (app64 v164 (app64 v164 v064)))))
add64 : β{Ξ} β Tm64 Ξ (arr64 nat64 (arr64 nat64 nat64)); add64
= lam64 (rec64 v064
(lam64 (lam64 (lam64 (suc64 (app64 v164 v064)))))
(lam64 v064))
mul64 : β{Ξ} β Tm64 Ξ (arr64 nat64 (arr64 nat64 nat64)); mul64
= lam64 (rec64 v064
(lam64 (lam64 (lam64 (app64 (app64 add64 (app64 v164 v064)) v064))))
(lam64 zero64))
fact64 : β{Ξ} β Tm64 Ξ (arr64 nat64 nat64); fact64
= lam64 (rec64 v064 (lam64 (lam64 (app64 (app64 mul64 (suc64 v164)) v064)))
(suc64 zero64))
{-# OPTIONS --type-in-type #-}
Ty65 : Set
Ty65 =
(Ty65 : Set)
(nat top bot : Ty65)
(arr prod sum : Ty65 β Ty65 β Ty65)
β Ty65
nat65 : Ty65; nat65 = Ξ» _ nat65 _ _ _ _ _ β nat65
top65 : Ty65; top65 = Ξ» _ _ top65 _ _ _ _ β top65
bot65 : Ty65; bot65 = Ξ» _ _ _ bot65 _ _ _ β bot65
arr65 : Ty65 β Ty65 β Ty65; arr65
= Ξ» A B Ty65 nat65 top65 bot65 arr65 prod sum β
arr65 (A Ty65 nat65 top65 bot65 arr65 prod sum) (B Ty65 nat65 top65 bot65 arr65 prod sum)
prod65 : Ty65 β Ty65 β Ty65; prod65
= Ξ» A B Ty65 nat65 top65 bot65 arr65 prod65 sum β
prod65 (A Ty65 nat65 top65 bot65 arr65 prod65 sum) (B Ty65 nat65 top65 bot65 arr65 prod65 sum)
sum65 : Ty65 β Ty65 β Ty65; sum65
= Ξ» A B Ty65 nat65 top65 bot65 arr65 prod65 sum65 β
sum65 (A Ty65 nat65 top65 bot65 arr65 prod65 sum65) (B Ty65 nat65 top65 bot65 arr65 prod65 sum65)
Con65 : Set; Con65
= (Con65 : Set)
(nil : Con65)
(snoc : Con65 β Ty65 β Con65)
β Con65
nil65 : Con65; nil65
= Ξ» Con65 nil65 snoc β nil65
snoc65 : Con65 β Ty65 β Con65; snoc65
= Ξ» Ξ A Con65 nil65 snoc65 β snoc65 (Ξ Con65 nil65 snoc65) A
Var65 : Con65 β Ty65 β Set; Var65
= Ξ» Ξ A β
(Var65 : Con65 β Ty65 β Set)
(vz : β Ξ A β Var65 (snoc65 Ξ A) A)
(vs : β Ξ B A β Var65 Ξ A β Var65 (snoc65 Ξ B) A)
β Var65 Ξ A
vz65 : β{Ξ A} β Var65 (snoc65 Ξ A) A; vz65
= Ξ» Var65 vz65 vs β vz65 _ _
vs65 : β{Ξ B A} β Var65 Ξ A β Var65 (snoc65 Ξ B) A; vs65
= Ξ» x Var65 vz65 vs65 β vs65 _ _ _ (x Var65 vz65 vs65)
Tm65 : Con65 β Ty65 β Set; Tm65
= Ξ» Ξ A β
(Tm65 : Con65 β Ty65 β Set)
(var : β Ξ A β Var65 Ξ A β Tm65 Ξ A)
(lam : β Ξ A B β Tm65 (snoc65 Ξ A) B β Tm65 Ξ (arr65 A B))
(app : β Ξ A B β Tm65 Ξ (arr65 A B) β Tm65 Ξ A β Tm65 Ξ B)
(tt : β Ξ β Tm65 Ξ top65)
(pair : β Ξ A B β Tm65 Ξ A β Tm65 Ξ B β Tm65 Ξ (prod65 A B))
(fst : β Ξ A B β Tm65 Ξ (prod65 A B) β Tm65 Ξ A)
(snd : β Ξ A B β Tm65 Ξ (prod65 A B) β Tm65 Ξ B)
(left : β Ξ A B β Tm65 Ξ A β Tm65 Ξ (sum65 A B))
(right : β Ξ A B β Tm65 Ξ B β Tm65 Ξ (sum65 A B))
(case : β Ξ A B C β Tm65 Ξ (sum65 A B) β Tm65 Ξ (arr65 A C) β Tm65 Ξ (arr65 B C) β Tm65 Ξ C)
(zero : β Ξ β Tm65 Ξ nat65)
(suc : β Ξ β Tm65 Ξ nat65 β Tm65 Ξ nat65)
(rec : β Ξ A β Tm65 Ξ nat65 β Tm65 Ξ (arr65 nat65 (arr65 A A)) β Tm65 Ξ A β Tm65 Ξ A)
β Tm65 Ξ A
var65 : β{Ξ A} β Var65 Ξ A β Tm65 Ξ A; var65
= Ξ» x Tm65 var65 lam app tt pair fst snd left right case zero suc rec β
var65 _ _ x
lam65 : β{Ξ A B} β Tm65 (snoc65 Ξ A) B β Tm65 Ξ (arr65 A B); lam65
= Ξ» t Tm65 var65 lam65 app tt pair fst snd left right case zero suc rec β
lam65 _ _ _ (t Tm65 var65 lam65 app tt pair fst snd left right case zero suc rec)
app65 : β{Ξ A B} β Tm65 Ξ (arr65 A B) β Tm65 Ξ A β Tm65 Ξ B; app65
= Ξ» t u Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec β
app65 _ _ _ (t Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec)
(u Tm65 var65 lam65 app65 tt pair fst snd left right case zero suc rec)
tt65 : β{Ξ} β Tm65 Ξ top65; tt65
= Ξ» Tm65 var65 lam65 app65 tt65 pair fst snd left right case zero suc rec β tt65 _
pair65 : β{Ξ A B} β Tm65 Ξ A β Tm65 Ξ B β Tm65 Ξ (prod65 A B); pair65
= Ξ» t u Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec β
pair65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec)
(u Tm65 var65 lam65 app65 tt65 pair65 fst snd left right case zero suc rec)
fst65 : β{Ξ A B} β Tm65 Ξ (prod65 A B) β Tm65 Ξ A; fst65
= Ξ» t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd left right case zero suc rec β
fst65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd left right case zero suc rec)
snd65 : β{Ξ A B} β Tm65 Ξ (prod65 A B) β Tm65 Ξ B; snd65
= Ξ» t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left right case zero suc rec β
snd65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left right case zero suc rec)
left65 : β{Ξ A B} β Tm65 Ξ A β Tm65 Ξ (sum65 A B); left65
= Ξ» t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right case zero suc rec β
left65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right case zero suc rec)
right65 : β{Ξ A B} β Tm65 Ξ B β Tm65 Ξ (sum65 A B); right65
= Ξ» t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case zero suc rec β
right65 _ _ _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case zero suc rec)
case65 : β{Ξ A B C} β Tm65 Ξ (sum65 A B) β Tm65 Ξ (arr65 A C) β Tm65 Ξ (arr65 B C) β Tm65 Ξ C; case65
= Ξ» t u v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec β
case65 _ _ _ _
(t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
(u Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
(v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero suc rec)
zero65 : β{Ξ} β Tm65 Ξ nat65; zero65
= Ξ» Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc rec β zero65 _
suc65 : β{Ξ} β Tm65 Ξ nat65 β Tm65 Ξ nat65; suc65
= Ξ» t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec β
suc65 _ (t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec)
rec65 : β{Ξ A} β Tm65 Ξ nat65 β Tm65 Ξ (arr65 nat65 (arr65 A A)) β Tm65 Ξ A β Tm65 Ξ A; rec65
= Ξ» t u v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65 β
rec65 _ _
(t Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
(u Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
(v Tm65 var65 lam65 app65 tt65 pair65 fst65 snd65 left65 right65 case65 zero65 suc65 rec65)
v065 : β{Ξ A} β Tm65 (snoc65 Ξ A) A; v065
= var65 vz65
v165 : β{Ξ A B} β Tm65 (snoc65 (snoc65 Ξ A) B) A; v165
= var65 (vs65 vz65)
v265 : β{Ξ A B C} β Tm65 (snoc65 (snoc65 (snoc65 Ξ A) B) C) A; v265
= var65 (vs65 (vs65 vz65))
v365 : β{Ξ A B C D} β Tm65 (snoc65 (snoc65 (snoc65 (snoc65 Ξ A) B) C) D) A; v365
= var65 (vs65 (vs65 (vs65 vz65)))
tbool65 : Ty65; tbool65
= sum65 top65 top65
true65 : β{Ξ} β Tm65 Ξ tbool65; true65
= left65 tt65
tfalse65 : β{Ξ} β Tm65 Ξ tbool65; tfalse65
= right65 tt65
ifthenelse65 : β{Ξ A} β Tm65 Ξ (arr65 tbool65 (arr65 A (arr65 A A))); ifthenelse65
= lam65 (lam65 (lam65 (case65 v265 (lam65 v265) (lam65 v165))))
times465 : β{Ξ A} β Tm65 Ξ (arr65 (arr65 A A) (arr65 A A)); times465
= lam65 (lam65 (app65 v165 (app65 v165 (app65 v165 (app65 v165 v065)))))
add65 : β{Ξ} β Tm65 Ξ (arr65 nat65 (arr65 nat65 nat65)); add65
= lam65 (rec65 v065
(lam65 (lam65 (lam65 (suc65 (app65 v165 v065)))))
(lam65 v065))
mul65 : β{Ξ} β Tm65 Ξ (arr65 nat65 (arr65 nat65 nat65)); mul65
= lam65 (rec65 v065
(lam65 (lam65 (lam65 (app65 (app65 add65 (app65 v165 v065)) v065))))
(lam65 zero65))
fact65 : β{Ξ} β Tm65 Ξ (arr65 nat65 nat65); fact65
= lam65 (rec65 v065 (lam65 (lam65 (app65 (app65 mul65 (suc65 v165)) v065)))
(suc65 zero65))
{-# OPTIONS --type-in-type #-}
Ty66 : Set
Ty66 =
(Ty66 : Set)
(nat top bot : Ty66)
(arr prod sum : Ty66 β Ty66 β Ty66)
β Ty66
nat66 : Ty66; nat66 = Ξ» _ nat66 _ _ _ _ _ β nat66
top66 : Ty66; top66 = Ξ» _ _ top66 _ _ _ _ β top66
bot66 : Ty66; bot66 = Ξ» _ _ _ bot66 _ _ _ β bot66
arr66 : Ty66 β Ty66 β Ty66; arr66
= Ξ» A B Ty66 nat66 top66 bot66 arr66 prod sum β
arr66 (A Ty66 nat66 top66 bot66 arr66 prod sum) (B Ty66 nat66 top66 bot66 arr66 prod sum)
prod66 : Ty66 β Ty66 β Ty66; prod66
= Ξ» A B Ty66 nat66 top66 bot66 arr66 prod66 sum β
prod66 (A Ty66 nat66 top66 bot66 arr66 prod66 sum) (B Ty66 nat66 top66 bot66 arr66 prod66 sum)
sum66 : Ty66 β Ty66 β Ty66; sum66
= Ξ» A B Ty66 nat66 top66 bot66 arr66 prod66 sum66 β
sum66 (A Ty66 nat66 top66 bot66 arr66 prod66 sum66) (B Ty66 nat66 top66 bot66 arr66 prod66 sum66)
Con66 : Set; Con66
= (Con66 : Set)
(nil : Con66)
(snoc : Con66 β Ty66 β Con66)
β Con66
nil66 : Con66; nil66
= Ξ» Con66 nil66 snoc β nil66
snoc66 : Con66 β Ty66 β Con66; snoc66
= Ξ» Ξ A Con66 nil66 snoc66 β snoc66 (Ξ Con66 nil66 snoc66) A
Var66 : Con66 β Ty66 β Set; Var66
= Ξ» Ξ A β
(Var66 : Con66 β Ty66 β Set)
(vz : β Ξ A β Var66 (snoc66 Ξ A) A)
(vs : β Ξ B A β Var66 Ξ A β Var66 (snoc66 Ξ B) A)
β Var66 Ξ A
vz66 : β{Ξ A} β Var66 (snoc66 Ξ A) A; vz66
= Ξ» Var66 vz66 vs β vz66 _ _
vs66 : β{Ξ B A} β Var66 Ξ A β Var66 (snoc66 Ξ B) A; vs66
= Ξ» x Var66 vz66 vs66 β vs66 _ _ _ (x Var66 vz66 vs66)
Tm66 : Con66 β Ty66 β Set; Tm66
= Ξ» Ξ A β
(Tm66 : Con66 β Ty66 β Set)
(var : β Ξ A β Var66 Ξ A β Tm66 Ξ A)
(lam : β Ξ A B β Tm66 (snoc66 Ξ A) B β Tm66 Ξ (arr66 A B))
(app : β Ξ A B β Tm66 Ξ (arr66 A B) β Tm66 Ξ A β Tm66 Ξ B)
(tt : β Ξ β Tm66 Ξ top66)
(pair : β Ξ A B β Tm66 Ξ A β Tm66 Ξ B β Tm66 Ξ (prod66 A B))
(fst : β Ξ A B β Tm66 Ξ (prod66 A B) β Tm66 Ξ A)
(snd : β Ξ A B β Tm66 Ξ (prod66 A B) β Tm66 Ξ B)
(left : β Ξ A B β Tm66 Ξ A β Tm66 Ξ (sum66 A B))
(right : β Ξ A B β Tm66 Ξ B β Tm66 Ξ (sum66 A B))
(case : β Ξ A B C β Tm66 Ξ (sum66 A B) β Tm66 Ξ (arr66 A C) β Tm66 Ξ (arr66 B C) β Tm66 Ξ C)
(zero : β Ξ β Tm66 Ξ nat66)
(suc : β Ξ β Tm66 Ξ nat66 β Tm66 Ξ nat66)
(rec : β Ξ A β Tm66 Ξ nat66 β Tm66 Ξ (arr66 nat66 (arr66 A A)) β Tm66 Ξ A β Tm66 Ξ A)
β Tm66 Ξ A
var66 : β{Ξ A} β Var66 Ξ A β Tm66 Ξ A; var66
= Ξ» x Tm66 var66 lam app tt pair fst snd left right case zero suc rec β
var66 _ _ x
lam66 : β{Ξ A B} β Tm66 (snoc66 Ξ A) B β Tm66 Ξ (arr66 A B); lam66
= Ξ» t Tm66 var66 lam66 app tt pair fst snd left right case zero suc rec β
lam66 _ _ _ (t Tm66 var66 lam66 app tt pair fst snd left right case zero suc rec)
app66 : β{Ξ A B} β Tm66 Ξ (arr66 A B) β Tm66 Ξ A β Tm66 Ξ B; app66
= Ξ» t u Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec β
app66 _ _ _ (t Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec)
(u Tm66 var66 lam66 app66 tt pair fst snd left right case zero suc rec)
tt66 : β{Ξ} β Tm66 Ξ top66; tt66
= Ξ» Tm66 var66 lam66 app66 tt66 pair fst snd left right case zero suc rec β tt66 _
pair66 : β{Ξ A B} β Tm66 Ξ A β Tm66 Ξ B β Tm66 Ξ (prod66 A B); pair66
= Ξ» t u Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec β
pair66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec)
(u Tm66 var66 lam66 app66 tt66 pair66 fst snd left right case zero suc rec)
fst66 : β{Ξ A B} β Tm66 Ξ (prod66 A B) β Tm66 Ξ A; fst66
= Ξ» t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd left right case zero suc rec β
fst66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd left right case zero suc rec)
snd66 : β{Ξ A B} β Tm66 Ξ (prod66 A B) β Tm66 Ξ B; snd66
= Ξ» t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left right case zero suc rec β
snd66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left right case zero suc rec)
left66 : β{Ξ A B} β Tm66 Ξ A β Tm66 Ξ (sum66 A B); left66
= Ξ» t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right case zero suc rec β
left66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right case zero suc rec)
right66 : β{Ξ A B} β Tm66 Ξ B β Tm66 Ξ (sum66 A B); right66
= Ξ» t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case zero suc rec β
right66 _ _ _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case zero suc rec)
case66 : β{Ξ A B C} β Tm66 Ξ (sum66 A B) β Tm66 Ξ (arr66 A C) β Tm66 Ξ (arr66 B C) β Tm66 Ξ C; case66
= Ξ» t u v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec β
case66 _ _ _ _
(t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
(u Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
(v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero suc rec)
zero66 : β{Ξ} β Tm66 Ξ nat66; zero66
= Ξ» Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc rec β zero66 _
suc66 : β{Ξ} β Tm66 Ξ nat66 β Tm66 Ξ nat66; suc66
= Ξ» t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec β
suc66 _ (t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec)
rec66 : β{Ξ A} β Tm66 Ξ nat66 β Tm66 Ξ (arr66 nat66 (arr66 A A)) β Tm66 Ξ A β Tm66 Ξ A; rec66
= Ξ» t u v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66 β
rec66 _ _
(t Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
(u Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
(v Tm66 var66 lam66 app66 tt66 pair66 fst66 snd66 left66 right66 case66 zero66 suc66 rec66)
v066 : β{Ξ A} β Tm66 (snoc66 Ξ A) A; v066
= var66 vz66
v166 : β{Ξ A B} β Tm66 (snoc66 (snoc66 Ξ A) B) A; v166
= var66 (vs66 vz66)
v266 : β{Ξ A B C} β Tm66 (snoc66 (snoc66 (snoc66 Ξ A) B) C) A; v266
= var66 (vs66 (vs66 vz66))
v366 : β{Ξ A B C D} β Tm66 (snoc66 (snoc66 (snoc66 (snoc66 Ξ A) B) C) D) A; v366
= var66 (vs66 (vs66 (vs66 vz66)))
tbool66 : Ty66; tbool66
= sum66 top66 top66
true66 : β{Ξ} β Tm66 Ξ tbool66; true66
= left66 tt66
tfalse66 : β{Ξ} β Tm66 Ξ tbool66; tfalse66
= right66 tt66
ifthenelse66 : β{Ξ A} β Tm66 Ξ (arr66 tbool66 (arr66 A (arr66 A A))); ifthenelse66
= lam66 (lam66 (lam66 (case66 v266 (lam66 v266) (lam66 v166))))
times466 : β{Ξ A} β Tm66 Ξ (arr66 (arr66 A A) (arr66 A A)); times466
= lam66 (lam66 (app66 v166 (app66 v166 (app66 v166 (app66 v166 v066)))))
add66 : β{Ξ} β Tm66 Ξ (arr66 nat66 (arr66 nat66 nat66)); add66
= lam66 (rec66 v066
(lam66 (lam66 (lam66 (suc66 (app66 v166 v066)))))
(lam66 v066))
mul66 : β{Ξ} β Tm66 Ξ (arr66 nat66 (arr66 nat66 nat66)); mul66
= lam66 (rec66 v066
(lam66 (lam66 (lam66 (app66 (app66 add66 (app66 v166 v066)) v066))))
(lam66 zero66))
fact66 : β{Ξ} β Tm66 Ξ (arr66 nat66 nat66); fact66
= lam66 (rec66 v066 (lam66 (lam66 (app66 (app66 mul66 (suc66 v166)) v066)))
(suc66 zero66))
{-# OPTIONS --type-in-type #-}
Ty67 : Set
Ty67 =
(Ty67 : Set)
(nat top bot : Ty67)
(arr prod sum : Ty67 β Ty67 β Ty67)
β Ty67
nat67 : Ty67; nat67 = Ξ» _ nat67 _ _ _ _ _ β nat67
top67 : Ty67; top67 = Ξ» _ _ top67 _ _ _ _ β top67
bot67 : Ty67; bot67 = Ξ» _ _ _ bot67 _ _ _ β bot67
arr67 : Ty67 β Ty67 β Ty67; arr67
= Ξ» A B Ty67 nat67 top67 bot67 arr67 prod sum β
arr67 (A Ty67 nat67 top67 bot67 arr67 prod sum) (B Ty67 nat67 top67 bot67 arr67 prod sum)
prod67 : Ty67 β Ty67 β Ty67; prod67
= Ξ» A B Ty67 nat67 top67 bot67 arr67 prod67 sum β
prod67 (A Ty67 nat67 top67 bot67 arr67 prod67 sum) (B Ty67 nat67 top67 bot67 arr67 prod67 sum)
sum67 : Ty67 β Ty67 β Ty67; sum67
= Ξ» A B Ty67 nat67 top67 bot67 arr67 prod67 sum67 β
sum67 (A Ty67 nat67 top67 bot67 arr67 prod67 sum67) (B Ty67 nat67 top67 bot67 arr67 prod67 sum67)
Con67 : Set; Con67
= (Con67 : Set)
(nil : Con67)
(snoc : Con67 β Ty67 β Con67)
β Con67
nil67 : Con67; nil67
= Ξ» Con67 nil67 snoc β nil67
snoc67 : Con67 β Ty67 β Con67; snoc67
= Ξ» Ξ A Con67 nil67 snoc67 β snoc67 (Ξ Con67 nil67 snoc67) A
Var67 : Con67 β Ty67 β Set; Var67
= Ξ» Ξ A β
(Var67 : Con67 β Ty67 β Set)
(vz : β Ξ A β Var67 (snoc67 Ξ A) A)
(vs : β Ξ B A β Var67 Ξ A β Var67 (snoc67 Ξ B) A)
β Var67 Ξ A
vz67 : β{Ξ A} β Var67 (snoc67 Ξ A) A; vz67
= Ξ» Var67 vz67 vs β vz67 _ _
vs67 : β{Ξ B A} β Var67 Ξ A β Var67 (snoc67 Ξ B) A; vs67
= Ξ» x Var67 vz67 vs67 β vs67 _ _ _ (x Var67 vz67 vs67)
Tm67 : Con67 β Ty67 β Set; Tm67
= Ξ» Ξ A β
(Tm67 : Con67 β Ty67 β Set)
(var : β Ξ A β Var67 Ξ A β Tm67 Ξ A)
(lam : β Ξ A B β Tm67 (snoc67 Ξ A) B β Tm67 Ξ (arr67 A B))
(app : β Ξ A B β Tm67 Ξ (arr67 A B) β Tm67 Ξ A β Tm67 Ξ B)
(tt : β Ξ β Tm67 Ξ top67)
(pair : β Ξ A B β Tm67 Ξ A β Tm67 Ξ B β Tm67 Ξ (prod67 A B))
(fst : β Ξ A B β Tm67 Ξ (prod67 A B) β Tm67 Ξ A)
(snd : β Ξ A B β Tm67 Ξ (prod67 A B) β Tm67 Ξ B)
(left : β Ξ A B β Tm67 Ξ A β Tm67 Ξ (sum67 A B))
(right : β Ξ A B β Tm67 Ξ B β Tm67 Ξ (sum67 A B))
(case : β Ξ A B C β Tm67 Ξ (sum67 A B) β Tm67 Ξ (arr67 A C) β Tm67 Ξ (arr67 B C) β Tm67 Ξ C)
(zero : β Ξ β Tm67 Ξ nat67)
(suc : β Ξ β Tm67 Ξ nat67 β Tm67 Ξ nat67)
(rec : β Ξ A β Tm67 Ξ nat67 β Tm67 Ξ (arr67 nat67 (arr67 A A)) β Tm67 Ξ A β Tm67 Ξ A)
β Tm67 Ξ A
var67 : β{Ξ A} β Var67 Ξ A β Tm67 Ξ A; var67
= Ξ» x Tm67 var67 lam app tt pair fst snd left right case zero suc rec β
var67 _ _ x
lam67 : β{Ξ A B} β Tm67 (snoc67 Ξ A) B β Tm67 Ξ (arr67 A B); lam67
= Ξ» t Tm67 var67 lam67 app tt pair fst snd left right case zero suc rec β
lam67 _ _ _ (t Tm67 var67 lam67 app tt pair fst snd left right case zero suc rec)
app67 : β{Ξ A B} β Tm67 Ξ (arr67 A B) β Tm67 Ξ A β Tm67 Ξ B; app67
= Ξ» t u Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec β
app67 _ _ _ (t Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec)
(u Tm67 var67 lam67 app67 tt pair fst snd left right case zero suc rec)
tt67 : β{Ξ} β Tm67 Ξ top67; tt67
= Ξ» Tm67 var67 lam67 app67 tt67 pair fst snd left right case zero suc rec β tt67 _
pair67 : β{Ξ A B} β Tm67 Ξ A β Tm67 Ξ B β Tm67 Ξ (prod67 A B); pair67
= Ξ» t u Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec β
pair67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec)
(u Tm67 var67 lam67 app67 tt67 pair67 fst snd left right case zero suc rec)
fst67 : β{Ξ A B} β Tm67 Ξ (prod67 A B) β Tm67 Ξ A; fst67
= Ξ» t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd left right case zero suc rec β
fst67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd left right case zero suc rec)
snd67 : β{Ξ A B} β Tm67 Ξ (prod67 A B) β Tm67 Ξ B; snd67
= Ξ» t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left right case zero suc rec β
snd67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left right case zero suc rec)
left67 : β{Ξ A B} β Tm67 Ξ A β Tm67 Ξ (sum67 A B); left67
= Ξ» t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right case zero suc rec β
left67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right case zero suc rec)
right67 : β{Ξ A B} β Tm67 Ξ B β Tm67 Ξ (sum67 A B); right67
= Ξ» t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case zero suc rec β
right67 _ _ _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case zero suc rec)
case67 : β{Ξ A B C} β Tm67 Ξ (sum67 A B) β Tm67 Ξ (arr67 A C) β Tm67 Ξ (arr67 B C) β Tm67 Ξ C; case67
= Ξ» t u v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec β
case67 _ _ _ _
(t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
(u Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
(v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero suc rec)
zero67 : β{Ξ} β Tm67 Ξ nat67; zero67
= Ξ» Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc rec β zero67 _
suc67 : β{Ξ} β Tm67 Ξ nat67 β Tm67 Ξ nat67; suc67
= Ξ» t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec β
suc67 _ (t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec)
rec67 : β{Ξ A} β Tm67 Ξ nat67 β Tm67 Ξ (arr67 nat67 (arr67 A A)) β Tm67 Ξ A β Tm67 Ξ A; rec67
= Ξ» t u v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67 β
rec67 _ _
(t Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
(u Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
(v Tm67 var67 lam67 app67 tt67 pair67 fst67 snd67 left67 right67 case67 zero67 suc67 rec67)
v067 : β{Ξ A} β Tm67 (snoc67 Ξ A) A; v067
= var67 vz67
v167 : β{Ξ A B} β Tm67 (snoc67 (snoc67 Ξ A) B) A; v167
= var67 (vs67 vz67)
v267 : β{Ξ A B C} β Tm67 (snoc67 (snoc67 (snoc67 Ξ A) B) C) A; v267
= var67 (vs67 (vs67 vz67))
v367 : β{Ξ A B C D} β Tm67 (snoc67 (snoc67 (snoc67 (snoc67 Ξ A) B) C) D) A; v367
= var67 (vs67 (vs67 (vs67 vz67)))
tbool67 : Ty67; tbool67
= sum67 top67 top67
true67 : β{Ξ} β Tm67 Ξ tbool67; true67
= left67 tt67
tfalse67 : β{Ξ} β Tm67 Ξ tbool67; tfalse67
= right67 tt67
ifthenelse67 : β{Ξ A} β Tm67 Ξ (arr67 tbool67 (arr67 A (arr67 A A))); ifthenelse67
= lam67 (lam67 (lam67 (case67 v267 (lam67 v267) (lam67 v167))))
times467 : β{Ξ A} β Tm67 Ξ (arr67 (arr67 A A) (arr67 A A)); times467
= lam67 (lam67 (app67 v167 (app67 v167 (app67 v167 (app67 v167 v067)))))
add67 : β{Ξ} β Tm67 Ξ (arr67 nat67 (arr67 nat67 nat67)); add67
= lam67 (rec67 v067
(lam67 (lam67 (lam67 (suc67 (app67 v167 v067)))))
(lam67 v067))
mul67 : β{Ξ} β Tm67 Ξ (arr67 nat67 (arr67 nat67 nat67)); mul67
= lam67 (rec67 v067
(lam67 (lam67 (lam67 (app67 (app67 add67 (app67 v167 v067)) v067))))
(lam67 zero67))
fact67 : β{Ξ} β Tm67 Ξ (arr67 nat67 nat67); fact67
= lam67 (rec67 v067 (lam67 (lam67 (app67 (app67 mul67 (suc67 v167)) v067)))
(suc67 zero67))
{-# OPTIONS --type-in-type #-}
Ty68 : Set
Ty68 =
(Ty68 : Set)
(nat top bot : Ty68)
(arr prod sum : Ty68 β Ty68 β Ty68)
β Ty68
nat68 : Ty68; nat68 = Ξ» _ nat68 _ _ _ _ _ β nat68
top68 : Ty68; top68 = Ξ» _ _ top68 _ _ _ _ β top68
bot68 : Ty68; bot68 = Ξ» _ _ _ bot68 _ _ _ β bot68
arr68 : Ty68 β Ty68 β Ty68; arr68
= Ξ» A B Ty68 nat68 top68 bot68 arr68 prod sum β
arr68 (A Ty68 nat68 top68 bot68 arr68 prod sum) (B Ty68 nat68 top68 bot68 arr68 prod sum)
prod68 : Ty68 β Ty68 β Ty68; prod68
= Ξ» A B Ty68 nat68 top68 bot68 arr68 prod68 sum β
prod68 (A Ty68 nat68 top68 bot68 arr68 prod68 sum) (B Ty68 nat68 top68 bot68 arr68 prod68 sum)
sum68 : Ty68 β Ty68 β Ty68; sum68
= Ξ» A B Ty68 nat68 top68 bot68 arr68 prod68 sum68 β
sum68 (A Ty68 nat68 top68 bot68 arr68 prod68 sum68) (B Ty68 nat68 top68 bot68 arr68 prod68 sum68)
Con68 : Set; Con68
= (Con68 : Set)
(nil : Con68)
(snoc : Con68 β Ty68 β Con68)
β Con68
nil68 : Con68; nil68
= Ξ» Con68 nil68 snoc β nil68
snoc68 : Con68 β Ty68 β Con68; snoc68
= Ξ» Ξ A Con68 nil68 snoc68 β snoc68 (Ξ Con68 nil68 snoc68) A
Var68 : Con68 β Ty68 β Set; Var68
= Ξ» Ξ A β
(Var68 : Con68 β Ty68 β Set)
(vz : β Ξ A β Var68 (snoc68 Ξ A) A)
(vs : β Ξ B A β Var68 Ξ A β Var68 (snoc68 Ξ B) A)
β Var68 Ξ A
vz68 : β{Ξ A} β Var68 (snoc68 Ξ A) A; vz68
= Ξ» Var68 vz68 vs β vz68 _ _
vs68 : β{Ξ B A} β Var68 Ξ A β Var68 (snoc68 Ξ B) A; vs68
= Ξ» x Var68 vz68 vs68 β vs68 _ _ _ (x Var68 vz68 vs68)
Tm68 : Con68 β Ty68 β Set; Tm68
= Ξ» Ξ A β
(Tm68 : Con68 β Ty68 β Set)
(var : β Ξ A β Var68 Ξ A β Tm68 Ξ A)
(lam : β Ξ A B β Tm68 (snoc68 Ξ A) B β Tm68 Ξ (arr68 A B))
(app : β Ξ A B β Tm68 Ξ (arr68 A B) β Tm68 Ξ A β Tm68 Ξ B)
(tt : β Ξ β Tm68 Ξ top68)
(pair : β Ξ A B β Tm68 Ξ A β Tm68 Ξ B β Tm68 Ξ (prod68 A B))
(fst : β Ξ A B β Tm68 Ξ (prod68 A B) β Tm68 Ξ A)
(snd : β Ξ A B β Tm68 Ξ (prod68 A B) β Tm68 Ξ B)
(left : β Ξ A B β Tm68 Ξ A β Tm68 Ξ (sum68 A B))
(right : β Ξ A B β Tm68 Ξ B β Tm68 Ξ (sum68 A B))
(case : β Ξ A B C β Tm68 Ξ (sum68 A B) β Tm68 Ξ (arr68 A C) β Tm68 Ξ (arr68 B C) β Tm68 Ξ C)
(zero : β Ξ β Tm68 Ξ nat68)
(suc : β Ξ β Tm68 Ξ nat68 β Tm68 Ξ nat68)
(rec : β Ξ A β Tm68 Ξ nat68 β Tm68 Ξ (arr68 nat68 (arr68 A A)) β Tm68 Ξ A β Tm68 Ξ A)
β Tm68 Ξ A
var68 : β{Ξ A} β Var68 Ξ A β Tm68 Ξ A; var68
= Ξ» x Tm68 var68 lam app tt pair fst snd left right case zero suc rec β
var68 _ _ x
lam68 : β{Ξ A B} β Tm68 (snoc68 Ξ A) B β Tm68 Ξ (arr68 A B); lam68
= Ξ» t Tm68 var68 lam68 app tt pair fst snd left right case zero suc rec β
lam68 _ _ _ (t Tm68 var68 lam68 app tt pair fst snd left right case zero suc rec)
app68 : β{Ξ A B} β Tm68 Ξ (arr68 A B) β Tm68 Ξ A β Tm68 Ξ B; app68
= Ξ» t u Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec β
app68 _ _ _ (t Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec)
(u Tm68 var68 lam68 app68 tt pair fst snd left right case zero suc rec)
tt68 : β{Ξ} β Tm68 Ξ top68; tt68
= Ξ» Tm68 var68 lam68 app68 tt68 pair fst snd left right case zero suc rec β tt68 _
pair68 : β{Ξ A B} β Tm68 Ξ A β Tm68 Ξ B β Tm68 Ξ (prod68 A B); pair68
= Ξ» t u Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec β
pair68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec)
(u Tm68 var68 lam68 app68 tt68 pair68 fst snd left right case zero suc rec)
fst68 : β{Ξ A B} β Tm68 Ξ (prod68 A B) β Tm68 Ξ A; fst68
= Ξ» t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd left right case zero suc rec β
fst68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd left right case zero suc rec)
snd68 : β{Ξ A B} β Tm68 Ξ (prod68 A B) β Tm68 Ξ B; snd68
= Ξ» t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left right case zero suc rec β
snd68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left right case zero suc rec)
left68 : β{Ξ A B} β Tm68 Ξ A β Tm68 Ξ (sum68 A B); left68
= Ξ» t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right case zero suc rec β
left68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right case zero suc rec)
right68 : β{Ξ A B} β Tm68 Ξ B β Tm68 Ξ (sum68 A B); right68
= Ξ» t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case zero suc rec β
right68 _ _ _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case zero suc rec)
case68 : β{Ξ A B C} β Tm68 Ξ (sum68 A B) β Tm68 Ξ (arr68 A C) β Tm68 Ξ (arr68 B C) β Tm68 Ξ C; case68
= Ξ» t u v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec β
case68 _ _ _ _
(t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
(u Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
(v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero suc rec)
zero68 : β{Ξ} β Tm68 Ξ nat68; zero68
= Ξ» Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc rec β zero68 _
suc68 : β{Ξ} β Tm68 Ξ nat68 β Tm68 Ξ nat68; suc68
= Ξ» t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec β
suc68 _ (t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec)
rec68 : β{Ξ A} β Tm68 Ξ nat68 β Tm68 Ξ (arr68 nat68 (arr68 A A)) β Tm68 Ξ A β Tm68 Ξ A; rec68
= Ξ» t u v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68 β
rec68 _ _
(t Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
(u Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
(v Tm68 var68 lam68 app68 tt68 pair68 fst68 snd68 left68 right68 case68 zero68 suc68 rec68)
v068 : β{Ξ A} β Tm68 (snoc68 Ξ A) A; v068
= var68 vz68
v168 : β{Ξ A B} β Tm68 (snoc68 (snoc68 Ξ A) B) A; v168
= var68 (vs68 vz68)
v268 : β{Ξ A B C} β Tm68 (snoc68 (snoc68 (snoc68 Ξ A) B) C) A; v268
= var68 (vs68 (vs68 vz68))
v368 : β{Ξ A B C D} β Tm68 (snoc68 (snoc68 (snoc68 (snoc68 Ξ A) B) C) D) A; v368
= var68 (vs68 (vs68 (vs68 vz68)))
tbool68 : Ty68; tbool68
= sum68 top68 top68
true68 : β{Ξ} β Tm68 Ξ tbool68; true68
= left68 tt68
tfalse68 : β{Ξ} β Tm68 Ξ tbool68; tfalse68
= right68 tt68
ifthenelse68 : β{Ξ A} β Tm68 Ξ (arr68 tbool68 (arr68 A (arr68 A A))); ifthenelse68
= lam68 (lam68 (lam68 (case68 v268 (lam68 v268) (lam68 v168))))
times468 : β{Ξ A} β Tm68 Ξ (arr68 (arr68 A A) (arr68 A A)); times468
= lam68 (lam68 (app68 v168 (app68 v168 (app68 v168 (app68 v168 v068)))))
add68 : β{Ξ} β Tm68 Ξ (arr68 nat68 (arr68 nat68 nat68)); add68
= lam68 (rec68 v068
(lam68 (lam68 (lam68 (suc68 (app68 v168 v068)))))
(lam68 v068))
mul68 : β{Ξ} β Tm68 Ξ (arr68 nat68 (arr68 nat68 nat68)); mul68
= lam68 (rec68 v068
(lam68 (lam68 (lam68 (app68 (app68 add68 (app68 v168 v068)) v068))))
(lam68 zero68))
fact68 : β{Ξ} β Tm68 Ξ (arr68 nat68 nat68); fact68
= lam68 (rec68 v068 (lam68 (lam68 (app68 (app68 mul68 (suc68 v168)) v068)))
(suc68 zero68))
{-# OPTIONS --type-in-type #-}
Ty69 : Set
Ty69 =
(Ty69 : Set)
(nat top bot : Ty69)
(arr prod sum : Ty69 β Ty69 β Ty69)
β Ty69
nat69 : Ty69; nat69 = Ξ» _ nat69 _ _ _ _ _ β nat69
top69 : Ty69; top69 = Ξ» _ _ top69 _ _ _ _ β top69
bot69 : Ty69; bot69 = Ξ» _ _ _ bot69 _ _ _ β bot69
arr69 : Ty69 β Ty69 β Ty69; arr69
= Ξ» A B Ty69 nat69 top69 bot69 arr69 prod sum β
arr69 (A Ty69 nat69 top69 bot69 arr69 prod sum) (B Ty69 nat69 top69 bot69 arr69 prod sum)
prod69 : Ty69 β Ty69 β Ty69; prod69
= Ξ» A B Ty69 nat69 top69 bot69 arr69 prod69 sum β
prod69 (A Ty69 nat69 top69 bot69 arr69 prod69 sum) (B Ty69 nat69 top69 bot69 arr69 prod69 sum)
sum69 : Ty69 β Ty69 β Ty69; sum69
= Ξ» A B Ty69 nat69 top69 bot69 arr69 prod69 sum69 β
sum69 (A Ty69 nat69 top69 bot69 arr69 prod69 sum69) (B Ty69 nat69 top69 bot69 arr69 prod69 sum69)
Con69 : Set; Con69
= (Con69 : Set)
(nil : Con69)
(snoc : Con69 β Ty69 β Con69)
β Con69
nil69 : Con69; nil69
= Ξ» Con69 nil69 snoc β nil69
snoc69 : Con69 β Ty69 β Con69; snoc69
= Ξ» Ξ A Con69 nil69 snoc69 β snoc69 (Ξ Con69 nil69 snoc69) A
Var69 : Con69 β Ty69 β Set; Var69
= Ξ» Ξ A β
(Var69 : Con69 β Ty69 β Set)
(vz : β Ξ A β Var69 (snoc69 Ξ A) A)
(vs : β Ξ B A β Var69 Ξ A β Var69 (snoc69 Ξ B) A)
β Var69 Ξ A
vz69 : β{Ξ A} β Var69 (snoc69 Ξ A) A; vz69
= Ξ» Var69 vz69 vs β vz69 _ _
vs69 : β{Ξ B A} β Var69 Ξ A β Var69 (snoc69 Ξ B) A; vs69
= Ξ» x Var69 vz69 vs69 β vs69 _ _ _ (x Var69 vz69 vs69)
Tm69 : Con69 β Ty69 β Set; Tm69
= Ξ» Ξ A β
(Tm69 : Con69 β Ty69 β Set)
(var : β Ξ A β Var69 Ξ A β Tm69 Ξ A)
(lam : β Ξ A B β Tm69 (snoc69 Ξ A) B β Tm69 Ξ (arr69 A B))
(app : β Ξ A B β Tm69 Ξ (arr69 A B) β Tm69 Ξ A β Tm69 Ξ B)
(tt : β Ξ β Tm69 Ξ top69)
(pair : β Ξ A B β Tm69 Ξ A β Tm69 Ξ B β Tm69 Ξ (prod69 A B))
(fst : β Ξ A B β Tm69 Ξ (prod69 A B) β Tm69 Ξ A)
(snd : β Ξ A B β Tm69 Ξ (prod69 A B) β Tm69 Ξ B)
(left : β Ξ A B β Tm69 Ξ A β Tm69 Ξ (sum69 A B))
(right : β Ξ A B β Tm69 Ξ B β Tm69 Ξ (sum69 A B))
(case : β Ξ A B C β Tm69 Ξ (sum69 A B) β Tm69 Ξ (arr69 A C) β Tm69 Ξ (arr69 B C) β Tm69 Ξ C)
(zero : β Ξ β Tm69 Ξ nat69)
(suc : β Ξ β Tm69 Ξ nat69 β Tm69 Ξ nat69)
(rec : β Ξ A β Tm69 Ξ nat69 β Tm69 Ξ (arr69 nat69 (arr69 A A)) β Tm69 Ξ A β Tm69 Ξ A)
β Tm69 Ξ A
var69 : β{Ξ A} β Var69 Ξ A β Tm69 Ξ A; var69
= Ξ» x Tm69 var69 lam app tt pair fst snd left right case zero suc rec β
var69 _ _ x
lam69 : β{Ξ A B} β Tm69 (snoc69 Ξ A) B β Tm69 Ξ (arr69 A B); lam69
= Ξ» t Tm69 var69 lam69 app tt pair fst snd left right case zero suc rec β
lam69 _ _ _ (t Tm69 var69 lam69 app tt pair fst snd left right case zero suc rec)
app69 : β{Ξ A B} β Tm69 Ξ (arr69 A B) β Tm69 Ξ A β Tm69 Ξ B; app69
= Ξ» t u Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec β
app69 _ _ _ (t Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec)
(u Tm69 var69 lam69 app69 tt pair fst snd left right case zero suc rec)
tt69 : β{Ξ} β Tm69 Ξ top69; tt69
= Ξ» Tm69 var69 lam69 app69 tt69 pair fst snd left right case zero suc rec β tt69 _
pair69 : β{Ξ A B} β Tm69 Ξ A β Tm69 Ξ B β Tm69 Ξ (prod69 A B); pair69
= Ξ» t u Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec β
pair69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec)
(u Tm69 var69 lam69 app69 tt69 pair69 fst snd left right case zero suc rec)
fst69 : β{Ξ A B} β Tm69 Ξ (prod69 A B) β Tm69 Ξ A; fst69
= Ξ» t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd left right case zero suc rec β
fst69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd left right case zero suc rec)
snd69 : β{Ξ A B} β Tm69 Ξ (prod69 A B) β Tm69 Ξ B; snd69
= Ξ» t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left right case zero suc rec β
snd69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left right case zero suc rec)
left69 : β{Ξ A B} β Tm69 Ξ A β Tm69 Ξ (sum69 A B); left69
= Ξ» t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right case zero suc rec β
left69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right case zero suc rec)
right69 : β{Ξ A B} β Tm69 Ξ B β Tm69 Ξ (sum69 A B); right69
= Ξ» t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case zero suc rec β
right69 _ _ _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case zero suc rec)
case69 : β{Ξ A B C} β Tm69 Ξ (sum69 A B) β Tm69 Ξ (arr69 A C) β Tm69 Ξ (arr69 B C) β Tm69 Ξ C; case69
= Ξ» t u v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec β
case69 _ _ _ _
(t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
(u Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
(v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero suc rec)
zero69 : β{Ξ} β Tm69 Ξ nat69; zero69
= Ξ» Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc rec β zero69 _
suc69 : β{Ξ} β Tm69 Ξ nat69 β Tm69 Ξ nat69; suc69
= Ξ» t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec β
suc69 _ (t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec)
rec69 : β{Ξ A} β Tm69 Ξ nat69 β Tm69 Ξ (arr69 nat69 (arr69 A A)) β Tm69 Ξ A β Tm69 Ξ A; rec69
= Ξ» t u v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69 β
rec69 _ _
(t Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
(u Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
(v Tm69 var69 lam69 app69 tt69 pair69 fst69 snd69 left69 right69 case69 zero69 suc69 rec69)
v069 : β{Ξ A} β Tm69 (snoc69 Ξ A) A; v069
= var69 vz69
v169 : β{Ξ A B} β Tm69 (snoc69 (snoc69 Ξ A) B) A; v169
= var69 (vs69 vz69)
v269 : β{Ξ A B C} β Tm69 (snoc69 (snoc69 (snoc69 Ξ A) B) C) A; v269
= var69 (vs69 (vs69 vz69))
v369 : β{Ξ A B C D} β Tm69 (snoc69 (snoc69 (snoc69 (snoc69 Ξ A) B) C) D) A; v369
= var69 (vs69 (vs69 (vs69 vz69)))
tbool69 : Ty69; tbool69
= sum69 top69 top69
true69 : β{Ξ} β Tm69 Ξ tbool69; true69
= left69 tt69
tfalse69 : β{Ξ} β Tm69 Ξ tbool69; tfalse69
= right69 tt69
ifthenelse69 : β{Ξ A} β Tm69 Ξ (arr69 tbool69 (arr69 A (arr69 A A))); ifthenelse69
= lam69 (lam69 (lam69 (case69 v269 (lam69 v269) (lam69 v169))))
times469 : β{Ξ A} β Tm69 Ξ (arr69 (arr69 A A) (arr69 A A)); times469
= lam69 (lam69 (app69 v169 (app69 v169 (app69 v169 (app69 v169 v069)))))
add69 : β{Ξ} β Tm69 Ξ (arr69 nat69 (arr69 nat69 nat69)); add69
= lam69 (rec69 v069
(lam69 (lam69 (lam69 (suc69 (app69 v169 v069)))))
(lam69 v069))
mul69 : β{Ξ} β Tm69 Ξ (arr69 nat69 (arr69 nat69 nat69)); mul69
= lam69 (rec69 v069
(lam69 (lam69 (lam69 (app69 (app69 add69 (app69 v169 v069)) v069))))
(lam69 zero69))
fact69 : β{Ξ} β Tm69 Ξ (arr69 nat69 nat69); fact69
= lam69 (rec69 v069 (lam69 (lam69 (app69 (app69 mul69 (suc69 v169)) v069)))
(suc69 zero69))
{-# OPTIONS --type-in-type #-}
Ty70 : Set
Ty70 =
(Ty70 : Set)
(nat top bot : Ty70)
(arr prod sum : Ty70 β Ty70 β Ty70)
β Ty70
nat70 : Ty70; nat70 = Ξ» _ nat70 _ _ _ _ _ β nat70
top70 : Ty70; top70 = Ξ» _ _ top70 _ _ _ _ β top70
bot70 : Ty70; bot70 = Ξ» _ _ _ bot70 _ _ _ β bot70
arr70 : Ty70 β Ty70 β Ty70; arr70
= Ξ» A B Ty70 nat70 top70 bot70 arr70 prod sum β
arr70 (A Ty70 nat70 top70 bot70 arr70 prod sum) (B Ty70 nat70 top70 bot70 arr70 prod sum)
prod70 : Ty70 β Ty70 β Ty70; prod70
= Ξ» A B Ty70 nat70 top70 bot70 arr70 prod70 sum β
prod70 (A Ty70 nat70 top70 bot70 arr70 prod70 sum) (B Ty70 nat70 top70 bot70 arr70 prod70 sum)
sum70 : Ty70 β Ty70 β Ty70; sum70
= Ξ» A B Ty70 nat70 top70 bot70 arr70 prod70 sum70 β
sum70 (A Ty70 nat70 top70 bot70 arr70 prod70 sum70) (B Ty70 nat70 top70 bot70 arr70 prod70 sum70)
Con70 : Set; Con70
= (Con70 : Set)
(nil : Con70)
(snoc : Con70 β Ty70 β Con70)
β Con70
nil70 : Con70; nil70
= Ξ» Con70 nil70 snoc β nil70
snoc70 : Con70 β Ty70 β Con70; snoc70
= Ξ» Ξ A Con70 nil70 snoc70 β snoc70 (Ξ Con70 nil70 snoc70) A
Var70 : Con70 β Ty70 β Set; Var70
= Ξ» Ξ A β
(Var70 : Con70 β Ty70 β Set)
(vz : β Ξ A β Var70 (snoc70 Ξ A) A)
(vs : β Ξ B A β Var70 Ξ A β Var70 (snoc70 Ξ B) A)
β Var70 Ξ A
vz70 : β{Ξ A} β Var70 (snoc70 Ξ A) A; vz70
= Ξ» Var70 vz70 vs β vz70 _ _
vs70 : β{Ξ B A} β Var70 Ξ A β Var70 (snoc70 Ξ B) A; vs70
= Ξ» x Var70 vz70 vs70 β vs70 _ _ _ (x Var70 vz70 vs70)
Tm70 : Con70 β Ty70 β Set; Tm70
= Ξ» Ξ A β
(Tm70 : Con70 β Ty70 β Set)
(var : β Ξ A β Var70 Ξ A β Tm70 Ξ A)
(lam : β Ξ A B β Tm70 (snoc70 Ξ A) B β Tm70 Ξ (arr70 A B))
(app : β Ξ A B β Tm70 Ξ (arr70 A B) β Tm70 Ξ A β Tm70 Ξ B)
(tt : β Ξ β Tm70 Ξ top70)
(pair : β Ξ A B β Tm70 Ξ A β Tm70 Ξ B β Tm70 Ξ (prod70 A B))
(fst : β Ξ A B β Tm70 Ξ (prod70 A B) β Tm70 Ξ A)
(snd : β Ξ A B β Tm70 Ξ (prod70 A B) β Tm70 Ξ B)
(left : β Ξ A B β Tm70 Ξ A β Tm70 Ξ (sum70 A B))
(right : β Ξ A B β Tm70 Ξ B β Tm70 Ξ (sum70 A B))
(case : β Ξ A B C β Tm70 Ξ (sum70 A B) β Tm70 Ξ (arr70 A C) β Tm70 Ξ (arr70 B C) β Tm70 Ξ C)
(zero : β Ξ β Tm70 Ξ nat70)
(suc : β Ξ β Tm70 Ξ nat70 β Tm70 Ξ nat70)
(rec : β Ξ A β Tm70 Ξ nat70 β Tm70 Ξ (arr70 nat70 (arr70 A A)) β Tm70 Ξ A β Tm70 Ξ A)
β Tm70 Ξ A
var70 : β{Ξ A} β Var70 Ξ A β Tm70 Ξ A; var70
= Ξ» x Tm70 var70 lam app tt pair fst snd left right case zero suc rec β
var70 _ _ x
lam70 : β{Ξ A B} β Tm70 (snoc70 Ξ A) B β Tm70 Ξ (arr70 A B); lam70
= Ξ» t Tm70 var70 lam70 app tt pair fst snd left right case zero suc rec β
lam70 _ _ _ (t Tm70 var70 lam70 app tt pair fst snd left right case zero suc rec)
app70 : β{Ξ A B} β Tm70 Ξ (arr70 A B) β Tm70 Ξ A β Tm70 Ξ B; app70
= Ξ» t u Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec β
app70 _ _ _ (t Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec)
(u Tm70 var70 lam70 app70 tt pair fst snd left right case zero suc rec)
tt70 : β{Ξ} β Tm70 Ξ top70; tt70
= Ξ» Tm70 var70 lam70 app70 tt70 pair fst snd left right case zero suc rec β tt70 _
pair70 : β{Ξ A B} β Tm70 Ξ A β Tm70 Ξ B β Tm70 Ξ (prod70 A B); pair70
= Ξ» t u Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec β
pair70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec)
(u Tm70 var70 lam70 app70 tt70 pair70 fst snd left right case zero suc rec)
fst70 : β{Ξ A B} β Tm70 Ξ (prod70 A B) β Tm70 Ξ A; fst70
= Ξ» t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd left right case zero suc rec β
fst70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd left right case zero suc rec)
snd70 : β{Ξ A B} β Tm70 Ξ (prod70 A B) β Tm70 Ξ B; snd70
= Ξ» t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left right case zero suc rec β
snd70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left right case zero suc rec)
left70 : β{Ξ A B} β Tm70 Ξ A β Tm70 Ξ (sum70 A B); left70
= Ξ» t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right case zero suc rec β
left70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right case zero suc rec)
right70 : β{Ξ A B} β Tm70 Ξ B β Tm70 Ξ (sum70 A B); right70
= Ξ» t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case zero suc rec β
right70 _ _ _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case zero suc rec)
case70 : β{Ξ A B C} β Tm70 Ξ (sum70 A B) β Tm70 Ξ (arr70 A C) β Tm70 Ξ (arr70 B C) β Tm70 Ξ C; case70
= Ξ» t u v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec β
case70 _ _ _ _
(t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
(u Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
(v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero suc rec)
zero70 : β{Ξ} β Tm70 Ξ nat70; zero70
= Ξ» Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc rec β zero70 _
suc70 : β{Ξ} β Tm70 Ξ nat70 β Tm70 Ξ nat70; suc70
= Ξ» t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec β
suc70 _ (t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec)
rec70 : β{Ξ A} β Tm70 Ξ nat70 β Tm70 Ξ (arr70 nat70 (arr70 A A)) β Tm70 Ξ A β Tm70 Ξ A; rec70
= Ξ» t u v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70 β
rec70 _ _
(t Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
(u Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
(v Tm70 var70 lam70 app70 tt70 pair70 fst70 snd70 left70 right70 case70 zero70 suc70 rec70)
v070 : β{Ξ A} β Tm70 (snoc70 Ξ A) A; v070
= var70 vz70
v170 : β{Ξ A B} β Tm70 (snoc70 (snoc70 Ξ A) B) A; v170
= var70 (vs70 vz70)
v270 : β{Ξ A B C} β Tm70 (snoc70 (snoc70 (snoc70 Ξ A) B) C) A; v270
= var70 (vs70 (vs70 vz70))
v370 : β{Ξ A B C D} β Tm70 (snoc70 (snoc70 (snoc70 (snoc70 Ξ A) B) C) D) A; v370
= var70 (vs70 (vs70 (vs70 vz70)))
tbool70 : Ty70; tbool70
= sum70 top70 top70
true70 : β{Ξ} β Tm70 Ξ tbool70; true70
= left70 tt70
tfalse70 : β{Ξ} β Tm70 Ξ tbool70; tfalse70
= right70 tt70
ifthenelse70 : β{Ξ A} β Tm70 Ξ (arr70 tbool70 (arr70 A (arr70 A A))); ifthenelse70
= lam70 (lam70 (lam70 (case70 v270 (lam70 v270) (lam70 v170))))
times470 : β{Ξ A} β Tm70 Ξ (arr70 (arr70 A A) (arr70 A A)); times470
= lam70 (lam70 (app70 v170 (app70 v170 (app70 v170 (app70 v170 v070)))))
add70 : β{Ξ} β Tm70 Ξ (arr70 nat70 (arr70 nat70 nat70)); add70
= lam70 (rec70 v070
(lam70 (lam70 (lam70 (suc70 (app70 v170 v070)))))
(lam70 v070))
mul70 : β{Ξ} β Tm70 Ξ (arr70 nat70 (arr70 nat70 nat70)); mul70
= lam70 (rec70 v070
(lam70 (lam70 (lam70 (app70 (app70 add70 (app70 v170 v070)) v070))))
(lam70 zero70))
fact70 : β{Ξ} β Tm70 Ξ (arr70 nat70 nat70); fact70
= lam70 (rec70 v070 (lam70 (lam70 (app70 (app70 mul70 (suc70 v170)) v070)))
(suc70 zero70))
{-# OPTIONS --type-in-type #-}
Ty71 : Set
Ty71 =
(Ty71 : Set)
(nat top bot : Ty71)
(arr prod sum : Ty71 β Ty71 β Ty71)
β Ty71
nat71 : Ty71; nat71 = Ξ» _ nat71 _ _ _ _ _ β nat71
top71 : Ty71; top71 = Ξ» _ _ top71 _ _ _ _ β top71
bot71 : Ty71; bot71 = Ξ» _ _ _ bot71 _ _ _ β bot71
arr71 : Ty71 β Ty71 β Ty71; arr71
= Ξ» A B Ty71 nat71 top71 bot71 arr71 prod sum β
arr71 (A Ty71 nat71 top71 bot71 arr71 prod sum) (B Ty71 nat71 top71 bot71 arr71 prod sum)
prod71 : Ty71 β Ty71 β Ty71; prod71
= Ξ» A B Ty71 nat71 top71 bot71 arr71 prod71 sum β
prod71 (A Ty71 nat71 top71 bot71 arr71 prod71 sum) (B Ty71 nat71 top71 bot71 arr71 prod71 sum)
sum71 : Ty71 β Ty71 β Ty71; sum71
= Ξ» A B Ty71 nat71 top71 bot71 arr71 prod71 sum71 β
sum71 (A Ty71 nat71 top71 bot71 arr71 prod71 sum71) (B Ty71 nat71 top71 bot71 arr71 prod71 sum71)
Con71 : Set; Con71
= (Con71 : Set)
(nil : Con71)
(snoc : Con71 β Ty71 β Con71)
β Con71
nil71 : Con71; nil71
= Ξ» Con71 nil71 snoc β nil71
snoc71 : Con71 β Ty71 β Con71; snoc71
= Ξ» Ξ A Con71 nil71 snoc71 β snoc71 (Ξ Con71 nil71 snoc71) A
Var71 : Con71 β Ty71 β Set; Var71
= Ξ» Ξ A β
(Var71 : Con71 β Ty71 β Set)
(vz : β Ξ A β Var71 (snoc71 Ξ A) A)
(vs : β Ξ B A β Var71 Ξ A β Var71 (snoc71 Ξ B) A)
β Var71 Ξ A
vz71 : β{Ξ A} β Var71 (snoc71 Ξ A) A; vz71
= Ξ» Var71 vz71 vs β vz71 _ _
vs71 : β{Ξ B A} β Var71 Ξ A β Var71 (snoc71 Ξ B) A; vs71
= Ξ» x Var71 vz71 vs71 β vs71 _ _ _ (x Var71 vz71 vs71)
Tm71 : Con71 β Ty71 β Set; Tm71
= Ξ» Ξ A β
(Tm71 : Con71 β Ty71 β Set)
(var : β Ξ A β Var71 Ξ A β Tm71 Ξ A)
(lam : β Ξ A B β Tm71 (snoc71 Ξ A) B β Tm71 Ξ (arr71 A B))
(app : β Ξ A B β Tm71 Ξ (arr71 A B) β Tm71 Ξ A β Tm71 Ξ B)
(tt : β Ξ β Tm71 Ξ top71)
(pair : β Ξ A B β Tm71 Ξ A β Tm71 Ξ B β Tm71 Ξ (prod71 A B))
(fst : β Ξ A B β Tm71 Ξ (prod71 A B) β Tm71 Ξ A)
(snd : β Ξ A B β Tm71 Ξ (prod71 A B) β Tm71 Ξ B)
(left : β Ξ A B β Tm71 Ξ A β Tm71 Ξ (sum71 A B))
(right : β Ξ A B β Tm71 Ξ B β Tm71 Ξ (sum71 A B))
(case : β Ξ A B C β Tm71 Ξ (sum71 A B) β Tm71 Ξ (arr71 A C) β Tm71 Ξ (arr71 B C) β Tm71 Ξ C)
(zero : β Ξ β Tm71 Ξ nat71)
(suc : β Ξ β Tm71 Ξ nat71 β Tm71 Ξ nat71)
(rec : β Ξ A β Tm71 Ξ nat71 β Tm71 Ξ (arr71 nat71 (arr71 A A)) β Tm71 Ξ A β Tm71 Ξ A)
β Tm71 Ξ A
var71 : β{Ξ A} β Var71 Ξ A β Tm71 Ξ A; var71
= Ξ» x Tm71 var71 lam app tt pair fst snd left right case zero suc rec β
var71 _ _ x
lam71 : β{Ξ A B} β Tm71 (snoc71 Ξ A) B β Tm71 Ξ (arr71 A B); lam71
= Ξ» t Tm71 var71 lam71 app tt pair fst snd left right case zero suc rec β
lam71 _ _ _ (t Tm71 var71 lam71 app tt pair fst snd left right case zero suc rec)
app71 : β{Ξ A B} β Tm71 Ξ (arr71 A B) β Tm71 Ξ A β Tm71 Ξ B; app71
= Ξ» t u Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec β
app71 _ _ _ (t Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec)
(u Tm71 var71 lam71 app71 tt pair fst snd left right case zero suc rec)
tt71 : β{Ξ} β Tm71 Ξ top71; tt71
= Ξ» Tm71 var71 lam71 app71 tt71 pair fst snd left right case zero suc rec β tt71 _
pair71 : β{Ξ A B} β Tm71 Ξ A β Tm71 Ξ B β Tm71 Ξ (prod71 A B); pair71
= Ξ» t u Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec β
pair71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec)
(u Tm71 var71 lam71 app71 tt71 pair71 fst snd left right case zero suc rec)
fst71 : β{Ξ A B} β Tm71 Ξ (prod71 A B) β Tm71 Ξ A; fst71
= Ξ» t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd left right case zero suc rec β
fst71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd left right case zero suc rec)
snd71 : β{Ξ A B} β Tm71 Ξ (prod71 A B) β Tm71 Ξ B; snd71
= Ξ» t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left right case zero suc rec β
snd71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left right case zero suc rec)
left71 : β{Ξ A B} β Tm71 Ξ A β Tm71 Ξ (sum71 A B); left71
= Ξ» t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right case zero suc rec β
left71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right case zero suc rec)
right71 : β{Ξ A B} β Tm71 Ξ B β Tm71 Ξ (sum71 A B); right71
= Ξ» t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case zero suc rec β
right71 _ _ _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case zero suc rec)
case71 : β{Ξ A B C} β Tm71 Ξ (sum71 A B) β Tm71 Ξ (arr71 A C) β Tm71 Ξ (arr71 B C) β Tm71 Ξ C; case71
= Ξ» t u v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec β
case71 _ _ _ _
(t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
(u Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
(v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero suc rec)
zero71 : β{Ξ} β Tm71 Ξ nat71; zero71
= Ξ» Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc rec β zero71 _
suc71 : β{Ξ} β Tm71 Ξ nat71 β Tm71 Ξ nat71; suc71
= Ξ» t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec β
suc71 _ (t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec)
rec71 : β{Ξ A} β Tm71 Ξ nat71 β Tm71 Ξ (arr71 nat71 (arr71 A A)) β Tm71 Ξ A β Tm71 Ξ A; rec71
= Ξ» t u v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71 β
rec71 _ _
(t Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
(u Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
(v Tm71 var71 lam71 app71 tt71 pair71 fst71 snd71 left71 right71 case71 zero71 suc71 rec71)
v071 : β{Ξ A} β Tm71 (snoc71 Ξ A) A; v071
= var71 vz71
v171 : β{Ξ A B} β Tm71 (snoc71 (snoc71 Ξ A) B) A; v171
= var71 (vs71 vz71)
v271 : β{Ξ A B C} β Tm71 (snoc71 (snoc71 (snoc71 Ξ A) B) C) A; v271
= var71 (vs71 (vs71 vz71))
v371 : β{Ξ A B C D} β Tm71 (snoc71 (snoc71 (snoc71 (snoc71 Ξ A) B) C) D) A; v371
= var71 (vs71 (vs71 (vs71 vz71)))
tbool71 : Ty71; tbool71
= sum71 top71 top71
true71 : β{Ξ} β Tm71 Ξ tbool71; true71
= left71 tt71
tfalse71 : β{Ξ} β Tm71 Ξ tbool71; tfalse71
= right71 tt71
ifthenelse71 : β{Ξ A} β Tm71 Ξ (arr71 tbool71 (arr71 A (arr71 A A))); ifthenelse71
= lam71 (lam71 (lam71 (case71 v271 (lam71 v271) (lam71 v171))))
times471 : β{Ξ A} β Tm71 Ξ (arr71 (arr71 A A) (arr71 A A)); times471
= lam71 (lam71 (app71 v171 (app71 v171 (app71 v171 (app71 v171 v071)))))
add71 : β{Ξ} β Tm71 Ξ (arr71 nat71 (arr71 nat71 nat71)); add71
= lam71 (rec71 v071
(lam71 (lam71 (lam71 (suc71 (app71 v171 v071)))))
(lam71 v071))
mul71 : β{Ξ} β Tm71 Ξ (arr71 nat71 (arr71 nat71 nat71)); mul71
= lam71 (rec71 v071
(lam71 (lam71 (lam71 (app71 (app71 add71 (app71 v171 v071)) v071))))
(lam71 zero71))
fact71 : β{Ξ} β Tm71 Ξ (arr71 nat71 nat71); fact71
= lam71 (rec71 v071 (lam71 (lam71 (app71 (app71 mul71 (suc71 v171)) v071)))
(suc71 zero71))
{-# OPTIONS --type-in-type #-}
Ty72 : Set
Ty72 =
(Ty72 : Set)
(nat top bot : Ty72)
(arr prod sum : Ty72 β Ty72 β Ty72)
β Ty72
nat72 : Ty72; nat72 = Ξ» _ nat72 _ _ _ _ _ β nat72
top72 : Ty72; top72 = Ξ» _ _ top72 _ _ _ _ β top72
bot72 : Ty72; bot72 = Ξ» _ _ _ bot72 _ _ _ β bot72
arr72 : Ty72 β Ty72 β Ty72; arr72
= Ξ» A B Ty72 nat72 top72 bot72 arr72 prod sum β
arr72 (A Ty72 nat72 top72 bot72 arr72 prod sum) (B Ty72 nat72 top72 bot72 arr72 prod sum)
prod72 : Ty72 β Ty72 β Ty72; prod72
= Ξ» A B Ty72 nat72 top72 bot72 arr72 prod72 sum β
prod72 (A Ty72 nat72 top72 bot72 arr72 prod72 sum) (B Ty72 nat72 top72 bot72 arr72 prod72 sum)
sum72 : Ty72 β Ty72 β Ty72; sum72
= Ξ» A B Ty72 nat72 top72 bot72 arr72 prod72 sum72 β
sum72 (A Ty72 nat72 top72 bot72 arr72 prod72 sum72) (B Ty72 nat72 top72 bot72 arr72 prod72 sum72)
Con72 : Set; Con72
= (Con72 : Set)
(nil : Con72)
(snoc : Con72 β Ty72 β Con72)
β Con72
nil72 : Con72; nil72
= Ξ» Con72 nil72 snoc β nil72
snoc72 : Con72 β Ty72 β Con72; snoc72
= Ξ» Ξ A Con72 nil72 snoc72 β snoc72 (Ξ Con72 nil72 snoc72) A
Var72 : Con72 β Ty72 β Set; Var72
= Ξ» Ξ A β
(Var72 : Con72 β Ty72 β Set)
(vz : β Ξ A β Var72 (snoc72 Ξ A) A)
(vs : β Ξ B A β Var72 Ξ A β Var72 (snoc72 Ξ B) A)
β Var72 Ξ A
vz72 : β{Ξ A} β Var72 (snoc72 Ξ A) A; vz72
= Ξ» Var72 vz72 vs β vz72 _ _
vs72 : β{Ξ B A} β Var72 Ξ A β Var72 (snoc72 Ξ B) A; vs72
= Ξ» x Var72 vz72 vs72 β vs72 _ _ _ (x Var72 vz72 vs72)
Tm72 : Con72 β Ty72 β Set; Tm72
= Ξ» Ξ A β
(Tm72 : Con72 β Ty72 β Set)
(var : β Ξ A β Var72 Ξ A β Tm72 Ξ A)
(lam : β Ξ A B β Tm72 (snoc72 Ξ A) B β Tm72 Ξ (arr72 A B))
(app : β Ξ A B β Tm72 Ξ (arr72 A B) β Tm72 Ξ A β Tm72 Ξ B)
(tt : β Ξ β Tm72 Ξ top72)
(pair : β Ξ A B β Tm72 Ξ A β Tm72 Ξ B β Tm72 Ξ (prod72 A B))
(fst : β Ξ A B β Tm72 Ξ (prod72 A B) β Tm72 Ξ A)
(snd : β Ξ A B β Tm72 Ξ (prod72 A B) β Tm72 Ξ B)
(left : β Ξ A B β Tm72 Ξ A β Tm72 Ξ (sum72 A B))
(right : β Ξ A B β Tm72 Ξ B β Tm72 Ξ (sum72 A B))
(case : β Ξ A B C β Tm72 Ξ (sum72 A B) β Tm72 Ξ (arr72 A C) β Tm72 Ξ (arr72 B C) β Tm72 Ξ C)
(zero : β Ξ β Tm72 Ξ nat72)
(suc : β Ξ β Tm72 Ξ nat72 β Tm72 Ξ nat72)
(rec : β Ξ A β Tm72 Ξ nat72 β Tm72 Ξ (arr72 nat72 (arr72 A A)) β Tm72 Ξ A β Tm72 Ξ A)
β Tm72 Ξ A
var72 : β{Ξ A} β Var72 Ξ A β Tm72 Ξ A; var72
= Ξ» x Tm72 var72 lam app tt pair fst snd left right case zero suc rec β
var72 _ _ x
lam72 : β{Ξ A B} β Tm72 (snoc72 Ξ A) B β Tm72 Ξ (arr72 A B); lam72
= Ξ» t Tm72 var72 lam72 app tt pair fst snd left right case zero suc rec β
lam72 _ _ _ (t Tm72 var72 lam72 app tt pair fst snd left right case zero suc rec)
app72 : β{Ξ A B} β Tm72 Ξ (arr72 A B) β Tm72 Ξ A β Tm72 Ξ B; app72
= Ξ» t u Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec β
app72 _ _ _ (t Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec)
(u Tm72 var72 lam72 app72 tt pair fst snd left right case zero suc rec)
tt72 : β{Ξ} β Tm72 Ξ top72; tt72
= Ξ» Tm72 var72 lam72 app72 tt72 pair fst snd left right case zero suc rec β tt72 _
pair72 : β{Ξ A B} β Tm72 Ξ A β Tm72 Ξ B β Tm72 Ξ (prod72 A B); pair72
= Ξ» t u Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec β
pair72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec)
(u Tm72 var72 lam72 app72 tt72 pair72 fst snd left right case zero suc rec)
fst72 : β{Ξ A B} β Tm72 Ξ (prod72 A B) β Tm72 Ξ A; fst72
= Ξ» t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd left right case zero suc rec β
fst72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd left right case zero suc rec)
snd72 : β{Ξ A B} β Tm72 Ξ (prod72 A B) β Tm72 Ξ B; snd72
= Ξ» t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left right case zero suc rec β
snd72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left right case zero suc rec)
left72 : β{Ξ A B} β Tm72 Ξ A β Tm72 Ξ (sum72 A B); left72
= Ξ» t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right case zero suc rec β
left72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right case zero suc rec)
right72 : β{Ξ A B} β Tm72 Ξ B β Tm72 Ξ (sum72 A B); right72
= Ξ» t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case zero suc rec β
right72 _ _ _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case zero suc rec)
case72 : β{Ξ A B C} β Tm72 Ξ (sum72 A B) β Tm72 Ξ (arr72 A C) β Tm72 Ξ (arr72 B C) β Tm72 Ξ C; case72
= Ξ» t u v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec β
case72 _ _ _ _
(t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
(u Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
(v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero suc rec)
zero72 : β{Ξ} β Tm72 Ξ nat72; zero72
= Ξ» Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc rec β zero72 _
suc72 : β{Ξ} β Tm72 Ξ nat72 β Tm72 Ξ nat72; suc72
= Ξ» t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec β
suc72 _ (t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec)
rec72 : β{Ξ A} β Tm72 Ξ nat72 β Tm72 Ξ (arr72 nat72 (arr72 A A)) β Tm72 Ξ A β Tm72 Ξ A; rec72
= Ξ» t u v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72 β
rec72 _ _
(t Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
(u Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
(v Tm72 var72 lam72 app72 tt72 pair72 fst72 snd72 left72 right72 case72 zero72 suc72 rec72)
v072 : β{Ξ A} β Tm72 (snoc72 Ξ A) A; v072
= var72 vz72
v172 : β{Ξ A B} β Tm72 (snoc72 (snoc72 Ξ A) B) A; v172
= var72 (vs72 vz72)
v272 : β{Ξ A B C} β Tm72 (snoc72 (snoc72 (snoc72 Ξ A) B) C) A; v272
= var72 (vs72 (vs72 vz72))
v372 : β{Ξ A B C D} β Tm72 (snoc72 (snoc72 (snoc72 (snoc72 Ξ A) B) C) D) A; v372
= var72 (vs72 (vs72 (vs72 vz72)))
tbool72 : Ty72; tbool72
= sum72 top72 top72
true72 : β{Ξ} β Tm72 Ξ tbool72; true72
= left72 tt72
tfalse72 : β{Ξ} β Tm72 Ξ tbool72; tfalse72
= right72 tt72
ifthenelse72 : β{Ξ A} β Tm72 Ξ (arr72 tbool72 (arr72 A (arr72 A A))); ifthenelse72
= lam72 (lam72 (lam72 (case72 v272 (lam72 v272) (lam72 v172))))
times472 : β{Ξ A} β Tm72 Ξ (arr72 (arr72 A A) (arr72 A A)); times472
= lam72 (lam72 (app72 v172 (app72 v172 (app72 v172 (app72 v172 v072)))))
add72 : β{Ξ} β Tm72 Ξ (arr72 nat72 (arr72 nat72 nat72)); add72
= lam72 (rec72 v072
(lam72 (lam72 (lam72 (suc72 (app72 v172 v072)))))
(lam72 v072))
mul72 : β{Ξ} β Tm72 Ξ (arr72 nat72 (arr72 nat72 nat72)); mul72
= lam72 (rec72 v072
(lam72 (lam72 (lam72 (app72 (app72 add72 (app72 v172 v072)) v072))))
(lam72 zero72))
fact72 : β{Ξ} β Tm72 Ξ (arr72 nat72 nat72); fact72
= lam72 (rec72 v072 (lam72 (lam72 (app72 (app72 mul72 (suc72 v172)) v072)))
(suc72 zero72))
{-# OPTIONS --type-in-type #-}
Ty73 : Set
Ty73 =
(Ty73 : Set)
(nat top bot : Ty73)
(arr prod sum : Ty73 β Ty73 β Ty73)
β Ty73
nat73 : Ty73; nat73 = Ξ» _ nat73 _ _ _ _ _ β nat73
top73 : Ty73; top73 = Ξ» _ _ top73 _ _ _ _ β top73
bot73 : Ty73; bot73 = Ξ» _ _ _ bot73 _ _ _ β bot73
arr73 : Ty73 β Ty73 β Ty73; arr73
= Ξ» A B Ty73 nat73 top73 bot73 arr73 prod sum β
arr73 (A Ty73 nat73 top73 bot73 arr73 prod sum) (B Ty73 nat73 top73 bot73 arr73 prod sum)
prod73 : Ty73 β Ty73 β Ty73; prod73
= Ξ» A B Ty73 nat73 top73 bot73 arr73 prod73 sum β
prod73 (A Ty73 nat73 top73 bot73 arr73 prod73 sum) (B Ty73 nat73 top73 bot73 arr73 prod73 sum)
sum73 : Ty73 β Ty73 β Ty73; sum73
= Ξ» A B Ty73 nat73 top73 bot73 arr73 prod73 sum73 β
sum73 (A Ty73 nat73 top73 bot73 arr73 prod73 sum73) (B Ty73 nat73 top73 bot73 arr73 prod73 sum73)
Con73 : Set; Con73
= (Con73 : Set)
(nil : Con73)
(snoc : Con73 β Ty73 β Con73)
β Con73
nil73 : Con73; nil73
= Ξ» Con73 nil73 snoc β nil73
snoc73 : Con73 β Ty73 β Con73; snoc73
= Ξ» Ξ A Con73 nil73 snoc73 β snoc73 (Ξ Con73 nil73 snoc73) A
Var73 : Con73 β Ty73 β Set; Var73
= Ξ» Ξ A β
(Var73 : Con73 β Ty73 β Set)
(vz : β Ξ A β Var73 (snoc73 Ξ A) A)
(vs : β Ξ B A β Var73 Ξ A β Var73 (snoc73 Ξ B) A)
β Var73 Ξ A
vz73 : β{Ξ A} β Var73 (snoc73 Ξ A) A; vz73
= Ξ» Var73 vz73 vs β vz73 _ _
vs73 : β{Ξ B A} β Var73 Ξ A β Var73 (snoc73 Ξ B) A; vs73
= Ξ» x Var73 vz73 vs73 β vs73 _ _ _ (x Var73 vz73 vs73)
Tm73 : Con73 β Ty73 β Set; Tm73
= Ξ» Ξ A β
(Tm73 : Con73 β Ty73 β Set)
(var : β Ξ A β Var73 Ξ A β Tm73 Ξ A)
(lam : β Ξ A B β Tm73 (snoc73 Ξ A) B β Tm73 Ξ (arr73 A B))
(app : β Ξ A B β Tm73 Ξ (arr73 A B) β Tm73 Ξ A β Tm73 Ξ B)
(tt : β Ξ β Tm73 Ξ top73)
(pair : β Ξ A B β Tm73 Ξ A β Tm73 Ξ B β Tm73 Ξ (prod73 A B))
(fst : β Ξ A B β Tm73 Ξ (prod73 A B) β Tm73 Ξ A)
(snd : β Ξ A B β Tm73 Ξ (prod73 A B) β Tm73 Ξ B)
(left : β Ξ A B β Tm73 Ξ A β Tm73 Ξ (sum73 A B))
(right : β Ξ A B β Tm73 Ξ B β Tm73 Ξ (sum73 A B))
(case : β Ξ A B C β Tm73 Ξ (sum73 A B) β Tm73 Ξ (arr73 A C) β Tm73 Ξ (arr73 B C) β Tm73 Ξ C)
(zero : β Ξ β Tm73 Ξ nat73)
(suc : β Ξ β Tm73 Ξ nat73 β Tm73 Ξ nat73)
(rec : β Ξ A β Tm73 Ξ nat73 β Tm73 Ξ (arr73 nat73 (arr73 A A)) β Tm73 Ξ A β Tm73 Ξ A)
β Tm73 Ξ A
var73 : β{Ξ A} β Var73 Ξ A β Tm73 Ξ A; var73
= Ξ» x Tm73 var73 lam app tt pair fst snd left right case zero suc rec β
var73 _ _ x
lam73 : β{Ξ A B} β Tm73 (snoc73 Ξ A) B β Tm73 Ξ (arr73 A B); lam73
= Ξ» t Tm73 var73 lam73 app tt pair fst snd left right case zero suc rec β
lam73 _ _ _ (t Tm73 var73 lam73 app tt pair fst snd left right case zero suc rec)
app73 : β{Ξ A B} β Tm73 Ξ (arr73 A B) β Tm73 Ξ A β Tm73 Ξ B; app73
= Ξ» t u Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec β
app73 _ _ _ (t Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec)
(u Tm73 var73 lam73 app73 tt pair fst snd left right case zero suc rec)
tt73 : β{Ξ} β Tm73 Ξ top73; tt73
= Ξ» Tm73 var73 lam73 app73 tt73 pair fst snd left right case zero suc rec β tt73 _
pair73 : β{Ξ A B} β Tm73 Ξ A β Tm73 Ξ B β Tm73 Ξ (prod73 A B); pair73
= Ξ» t u Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec β
pair73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec)
(u Tm73 var73 lam73 app73 tt73 pair73 fst snd left right case zero suc rec)
fst73 : β{Ξ A B} β Tm73 Ξ (prod73 A B) β Tm73 Ξ A; fst73
= Ξ» t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd left right case zero suc rec β
fst73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd left right case zero suc rec)
snd73 : β{Ξ A B} β Tm73 Ξ (prod73 A B) β Tm73 Ξ B; snd73
= Ξ» t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left right case zero suc rec β
snd73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left right case zero suc rec)
left73 : β{Ξ A B} β Tm73 Ξ A β Tm73 Ξ (sum73 A B); left73
= Ξ» t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right case zero suc rec β
left73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right case zero suc rec)
right73 : β{Ξ A B} β Tm73 Ξ B β Tm73 Ξ (sum73 A B); right73
= Ξ» t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case zero suc rec β
right73 _ _ _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case zero suc rec)
case73 : β{Ξ A B C} β Tm73 Ξ (sum73 A B) β Tm73 Ξ (arr73 A C) β Tm73 Ξ (arr73 B C) β Tm73 Ξ C; case73
= Ξ» t u v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec β
case73 _ _ _ _
(t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
(u Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
(v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero suc rec)
zero73 : β{Ξ} β Tm73 Ξ nat73; zero73
= Ξ» Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc rec β zero73 _
suc73 : β{Ξ} β Tm73 Ξ nat73 β Tm73 Ξ nat73; suc73
= Ξ» t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec β
suc73 _ (t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec)
rec73 : β{Ξ A} β Tm73 Ξ nat73 β Tm73 Ξ (arr73 nat73 (arr73 A A)) β Tm73 Ξ A β Tm73 Ξ A; rec73
= Ξ» t u v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73 β
rec73 _ _
(t Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
(u Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
(v Tm73 var73 lam73 app73 tt73 pair73 fst73 snd73 left73 right73 case73 zero73 suc73 rec73)
v073 : β{Ξ A} β Tm73 (snoc73 Ξ A) A; v073
= var73 vz73
v173 : β{Ξ A B} β Tm73 (snoc73 (snoc73 Ξ A) B) A; v173
= var73 (vs73 vz73)
v273 : β{Ξ A B C} β Tm73 (snoc73 (snoc73 (snoc73 Ξ A) B) C) A; v273
= var73 (vs73 (vs73 vz73))
v373 : β{Ξ A B C D} β Tm73 (snoc73 (snoc73 (snoc73 (snoc73 Ξ A) B) C) D) A; v373
= var73 (vs73 (vs73 (vs73 vz73)))
tbool73 : Ty73; tbool73
= sum73 top73 top73
true73 : β{Ξ} β Tm73 Ξ tbool73; true73
= left73 tt73
tfalse73 : β{Ξ} β Tm73 Ξ tbool73; tfalse73
= right73 tt73
ifthenelse73 : β{Ξ A} β Tm73 Ξ (arr73 tbool73 (arr73 A (arr73 A A))); ifthenelse73
= lam73 (lam73 (lam73 (case73 v273 (lam73 v273) (lam73 v173))))
times473 : β{Ξ A} β Tm73 Ξ (arr73 (arr73 A A) (arr73 A A)); times473
= lam73 (lam73 (app73 v173 (app73 v173 (app73 v173 (app73 v173 v073)))))
add73 : β{Ξ} β Tm73 Ξ (arr73 nat73 (arr73 nat73 nat73)); add73
= lam73 (rec73 v073
(lam73 (lam73 (lam73 (suc73 (app73 v173 v073)))))
(lam73 v073))
mul73 : β{Ξ} β Tm73 Ξ (arr73 nat73 (arr73 nat73 nat73)); mul73
= lam73 (rec73 v073
(lam73 (lam73 (lam73 (app73 (app73 add73 (app73 v173 v073)) v073))))
(lam73 zero73))
fact73 : β{Ξ} β Tm73 Ξ (arr73 nat73 nat73); fact73
= lam73 (rec73 v073 (lam73 (lam73 (app73 (app73 mul73 (suc73 v173)) v073)))
(suc73 zero73))
{-# OPTIONS --type-in-type #-}
Ty74 : Set
Ty74 =
(Ty74 : Set)
(nat top bot : Ty74)
(arr prod sum : Ty74 β Ty74 β Ty74)
β Ty74
nat74 : Ty74; nat74 = Ξ» _ nat74 _ _ _ _ _ β nat74
top74 : Ty74; top74 = Ξ» _ _ top74 _ _ _ _ β top74
bot74 : Ty74; bot74 = Ξ» _ _ _ bot74 _ _ _ β bot74
arr74 : Ty74 β Ty74 β Ty74; arr74
= Ξ» A B Ty74 nat74 top74 bot74 arr74 prod sum β
arr74 (A Ty74 nat74 top74 bot74 arr74 prod sum) (B Ty74 nat74 top74 bot74 arr74 prod sum)
prod74 : Ty74 β Ty74 β Ty74; prod74
= Ξ» A B Ty74 nat74 top74 bot74 arr74 prod74 sum β
prod74 (A Ty74 nat74 top74 bot74 arr74 prod74 sum) (B Ty74 nat74 top74 bot74 arr74 prod74 sum)
sum74 : Ty74 β Ty74 β Ty74; sum74
= Ξ» A B Ty74 nat74 top74 bot74 arr74 prod74 sum74 β
sum74 (A Ty74 nat74 top74 bot74 arr74 prod74 sum74) (B Ty74 nat74 top74 bot74 arr74 prod74 sum74)
Con74 : Set; Con74
= (Con74 : Set)
(nil : Con74)
(snoc : Con74 β Ty74 β Con74)
β Con74
nil74 : Con74; nil74
= Ξ» Con74 nil74 snoc β nil74
snoc74 : Con74 β Ty74 β Con74; snoc74
= Ξ» Ξ A Con74 nil74 snoc74 β snoc74 (Ξ Con74 nil74 snoc74) A
Var74 : Con74 β Ty74 β Set; Var74
= Ξ» Ξ A β
(Var74 : Con74 β Ty74 β Set)
(vz : β Ξ A β Var74 (snoc74 Ξ A) A)
(vs : β Ξ B A β Var74 Ξ A β Var74 (snoc74 Ξ B) A)
β Var74 Ξ A
vz74 : β{Ξ A} β Var74 (snoc74 Ξ A) A; vz74
= Ξ» Var74 vz74 vs β vz74 _ _
vs74 : β{Ξ B A} β Var74 Ξ A β Var74 (snoc74 Ξ B) A; vs74
= Ξ» x Var74 vz74 vs74 β vs74 _ _ _ (x Var74 vz74 vs74)
Tm74 : Con74 β Ty74 β Set; Tm74
= Ξ» Ξ A β
(Tm74 : Con74 β Ty74 β Set)
(var : β Ξ A β Var74 Ξ A β Tm74 Ξ A)
(lam : β Ξ A B β Tm74 (snoc74 Ξ A) B β Tm74 Ξ (arr74 A B))
(app : β Ξ A B β Tm74 Ξ (arr74 A B) β Tm74 Ξ A β Tm74 Ξ B)
(tt : β Ξ β Tm74 Ξ top74)
(pair : β Ξ A B β Tm74 Ξ A β Tm74 Ξ B β Tm74 Ξ (prod74 A B))
(fst : β Ξ A B β Tm74 Ξ (prod74 A B) β Tm74 Ξ A)
(snd : β Ξ A B β Tm74 Ξ (prod74 A B) β Tm74 Ξ B)
(left : β Ξ A B β Tm74 Ξ A β Tm74 Ξ (sum74 A B))
(right : β Ξ A B β Tm74 Ξ B β Tm74 Ξ (sum74 A B))
(case : β Ξ A B C β Tm74 Ξ (sum74 A B) β Tm74 Ξ (arr74 A C) β Tm74 Ξ (arr74 B C) β Tm74 Ξ C)
(zero : β Ξ β Tm74 Ξ nat74)
(suc : β Ξ β Tm74 Ξ nat74 β Tm74 Ξ nat74)
(rec : β Ξ A β Tm74 Ξ nat74 β Tm74 Ξ (arr74 nat74 (arr74 A A)) β Tm74 Ξ A β Tm74 Ξ A)
β Tm74 Ξ A
var74 : β{Ξ A} β Var74 Ξ A β Tm74 Ξ A; var74
= Ξ» x Tm74 var74 lam app tt pair fst snd left right case zero suc rec β
var74 _ _ x
lam74 : β{Ξ A B} β Tm74 (snoc74 Ξ A) B β Tm74 Ξ (arr74 A B); lam74
= Ξ» t Tm74 var74 lam74 app tt pair fst snd left right case zero suc rec β
lam74 _ _ _ (t Tm74 var74 lam74 app tt pair fst snd left right case zero suc rec)
app74 : β{Ξ A B} β Tm74 Ξ (arr74 A B) β Tm74 Ξ A β Tm74 Ξ B; app74
= Ξ» t u Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec β
app74 _ _ _ (t Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec)
(u Tm74 var74 lam74 app74 tt pair fst snd left right case zero suc rec)
tt74 : β{Ξ} β Tm74 Ξ top74; tt74
= Ξ» Tm74 var74 lam74 app74 tt74 pair fst snd left right case zero suc rec β tt74 _
pair74 : β{Ξ A B} β Tm74 Ξ A β Tm74 Ξ B β Tm74 Ξ (prod74 A B); pair74
= Ξ» t u Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec β
pair74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec)
(u Tm74 var74 lam74 app74 tt74 pair74 fst snd left right case zero suc rec)
fst74 : β{Ξ A B} β Tm74 Ξ (prod74 A B) β Tm74 Ξ A; fst74
= Ξ» t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd left right case zero suc rec β
fst74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd left right case zero suc rec)
snd74 : β{Ξ A B} β Tm74 Ξ (prod74 A B) β Tm74 Ξ B; snd74
= Ξ» t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left right case zero suc rec β
snd74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left right case zero suc rec)
left74 : β{Ξ A B} β Tm74 Ξ A β Tm74 Ξ (sum74 A B); left74
= Ξ» t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right case zero suc rec β
left74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right case zero suc rec)
right74 : β{Ξ A B} β Tm74 Ξ B β Tm74 Ξ (sum74 A B); right74
= Ξ» t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case zero suc rec β
right74 _ _ _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case zero suc rec)
case74 : β{Ξ A B C} β Tm74 Ξ (sum74 A B) β Tm74 Ξ (arr74 A C) β Tm74 Ξ (arr74 B C) β Tm74 Ξ C; case74
= Ξ» t u v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec β
case74 _ _ _ _
(t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
(u Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
(v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero suc rec)
zero74 : β{Ξ} β Tm74 Ξ nat74; zero74
= Ξ» Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc rec β zero74 _
suc74 : β{Ξ} β Tm74 Ξ nat74 β Tm74 Ξ nat74; suc74
= Ξ» t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec β
suc74 _ (t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec)
rec74 : β{Ξ A} β Tm74 Ξ nat74 β Tm74 Ξ (arr74 nat74 (arr74 A A)) β Tm74 Ξ A β Tm74 Ξ A; rec74
= Ξ» t u v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74 β
rec74 _ _
(t Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
(u Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
(v Tm74 var74 lam74 app74 tt74 pair74 fst74 snd74 left74 right74 case74 zero74 suc74 rec74)
v074 : β{Ξ A} β Tm74 (snoc74 Ξ A) A; v074
= var74 vz74
v174 : β{Ξ A B} β Tm74 (snoc74 (snoc74 Ξ A) B) A; v174
= var74 (vs74 vz74)
v274 : β{Ξ A B C} β Tm74 (snoc74 (snoc74 (snoc74 Ξ A) B) C) A; v274
= var74 (vs74 (vs74 vz74))
v374 : β{Ξ A B C D} β Tm74 (snoc74 (snoc74 (snoc74 (snoc74 Ξ A) B) C) D) A; v374
= var74 (vs74 (vs74 (vs74 vz74)))
tbool74 : Ty74; tbool74
= sum74 top74 top74
true74 : β{Ξ} β Tm74 Ξ tbool74; true74
= left74 tt74
tfalse74 : β{Ξ} β Tm74 Ξ tbool74; tfalse74
= right74 tt74
ifthenelse74 : β{Ξ A} β Tm74 Ξ (arr74 tbool74 (arr74 A (arr74 A A))); ifthenelse74
= lam74 (lam74 (lam74 (case74 v274 (lam74 v274) (lam74 v174))))
times474 : β{Ξ A} β Tm74 Ξ (arr74 (arr74 A A) (arr74 A A)); times474
= lam74 (lam74 (app74 v174 (app74 v174 (app74 v174 (app74 v174 v074)))))
add74 : β{Ξ} β Tm74 Ξ (arr74 nat74 (arr74 nat74 nat74)); add74
= lam74 (rec74 v074
(lam74 (lam74 (lam74 (suc74 (app74 v174 v074)))))
(lam74 v074))
mul74 : β{Ξ} β Tm74 Ξ (arr74 nat74 (arr74 nat74 nat74)); mul74
= lam74 (rec74 v074
(lam74 (lam74 (lam74 (app74 (app74 add74 (app74 v174 v074)) v074))))
(lam74 zero74))
fact74 : β{Ξ} β Tm74 Ξ (arr74 nat74 nat74); fact74
= lam74 (rec74 v074 (lam74 (lam74 (app74 (app74 mul74 (suc74 v174)) v074)))
(suc74 zero74))
{-# OPTIONS --type-in-type #-}
Ty75 : Set
Ty75 =
(Ty75 : Set)
(nat top bot : Ty75)
(arr prod sum : Ty75 β Ty75 β Ty75)
β Ty75
nat75 : Ty75; nat75 = Ξ» _ nat75 _ _ _ _ _ β nat75
top75 : Ty75; top75 = Ξ» _ _ top75 _ _ _ _ β top75
bot75 : Ty75; bot75 = Ξ» _ _ _ bot75 _ _ _ β bot75
arr75 : Ty75 β Ty75 β Ty75; arr75
= Ξ» A B Ty75 nat75 top75 bot75 arr75 prod sum β
arr75 (A Ty75 nat75 top75 bot75 arr75 prod sum) (B Ty75 nat75 top75 bot75 arr75 prod sum)
prod75 : Ty75 β Ty75 β Ty75; prod75
= Ξ» A B Ty75 nat75 top75 bot75 arr75 prod75 sum β
prod75 (A Ty75 nat75 top75 bot75 arr75 prod75 sum) (B Ty75 nat75 top75 bot75 arr75 prod75 sum)
sum75 : Ty75 β Ty75 β Ty75; sum75
= Ξ» A B Ty75 nat75 top75 bot75 arr75 prod75 sum75 β
sum75 (A Ty75 nat75 top75 bot75 arr75 prod75 sum75) (B Ty75 nat75 top75 bot75 arr75 prod75 sum75)
Con75 : Set; Con75
= (Con75 : Set)
(nil : Con75)
(snoc : Con75 β Ty75 β Con75)
β Con75
nil75 : Con75; nil75
= Ξ» Con75 nil75 snoc β nil75
snoc75 : Con75 β Ty75 β Con75; snoc75
= Ξ» Ξ A Con75 nil75 snoc75 β snoc75 (Ξ Con75 nil75 snoc75) A
Var75 : Con75 β Ty75 β Set; Var75
= Ξ» Ξ A β
(Var75 : Con75 β Ty75 β Set)
(vz : β Ξ A β Var75 (snoc75 Ξ A) A)
(vs : β Ξ B A β Var75 Ξ A β Var75 (snoc75 Ξ B) A)
β Var75 Ξ A
vz75 : β{Ξ A} β Var75 (snoc75 Ξ A) A; vz75
= Ξ» Var75 vz75 vs β vz75 _ _
vs75 : β{Ξ B A} β Var75 Ξ A β Var75 (snoc75 Ξ B) A; vs75
= Ξ» x Var75 vz75 vs75 β vs75 _ _ _ (x Var75 vz75 vs75)
Tm75 : Con75 β Ty75 β Set; Tm75
= Ξ» Ξ A β
(Tm75 : Con75 β Ty75 β Set)
(var : β Ξ A β Var75 Ξ A β Tm75 Ξ A)
(lam : β Ξ A B β Tm75 (snoc75 Ξ A) B β Tm75 Ξ (arr75 A B))
(app : β Ξ A B β Tm75 Ξ (arr75 A B) β Tm75 Ξ A β Tm75 Ξ B)
(tt : β Ξ β Tm75 Ξ top75)
(pair : β Ξ A B β Tm75 Ξ A β Tm75 Ξ B β Tm75 Ξ (prod75 A B))
(fst : β Ξ A B β Tm75 Ξ (prod75 A B) β Tm75 Ξ A)
(snd : β Ξ A B β Tm75 Ξ (prod75 A B) β Tm75 Ξ B)
(left : β Ξ A B β Tm75 Ξ A β Tm75 Ξ (sum75 A B))
(right : β Ξ A B β Tm75 Ξ B β Tm75 Ξ (sum75 A B))
(case : β Ξ A B C β Tm75 Ξ (sum75 A B) β Tm75 Ξ (arr75 A C) β Tm75 Ξ (arr75 B C) β Tm75 Ξ C)
(zero : β Ξ β Tm75 Ξ nat75)
(suc : β Ξ β Tm75 Ξ nat75 β Tm75 Ξ nat75)
(rec : β Ξ A β Tm75 Ξ nat75 β Tm75 Ξ (arr75 nat75 (arr75 A A)) β Tm75 Ξ A β Tm75 Ξ A)
β Tm75 Ξ A
var75 : β{Ξ A} β Var75 Ξ A β Tm75 Ξ A; var75
= Ξ» x Tm75 var75 lam app tt pair fst snd left right case zero suc rec β
var75 _ _ x
lam75 : β{Ξ A B} β Tm75 (snoc75 Ξ A) B β Tm75 Ξ (arr75 A B); lam75
= Ξ» t Tm75 var75 lam75 app tt pair fst snd left right case zero suc rec β
lam75 _ _ _ (t Tm75 var75 lam75 app tt pair fst snd left right case zero suc rec)
app75 : β{Ξ A B} β Tm75 Ξ (arr75 A B) β Tm75 Ξ A β Tm75 Ξ B; app75
= Ξ» t u Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec β
app75 _ _ _ (t Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec)
(u Tm75 var75 lam75 app75 tt pair fst snd left right case zero suc rec)
tt75 : β{Ξ} β Tm75 Ξ top75; tt75
= Ξ» Tm75 var75 lam75 app75 tt75 pair fst snd left right case zero suc rec β tt75 _
pair75 : β{Ξ A B} β Tm75 Ξ A β Tm75 Ξ B β Tm75 Ξ (prod75 A B); pair75
= Ξ» t u Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec β
pair75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec)
(u Tm75 var75 lam75 app75 tt75 pair75 fst snd left right case zero suc rec)
fst75 : β{Ξ A B} β Tm75 Ξ (prod75 A B) β Tm75 Ξ A; fst75
= Ξ» t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd left right case zero suc rec β
fst75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd left right case zero suc rec)
snd75 : β{Ξ A B} β Tm75 Ξ (prod75 A B) β Tm75 Ξ B; snd75
= Ξ» t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left right case zero suc rec β
snd75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left right case zero suc rec)
left75 : β{Ξ A B} β Tm75 Ξ A β Tm75 Ξ (sum75 A B); left75
= Ξ» t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right case zero suc rec β
left75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right case zero suc rec)
right75 : β{Ξ A B} β Tm75 Ξ B β Tm75 Ξ (sum75 A B); right75
= Ξ» t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case zero suc rec β
right75 _ _ _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case zero suc rec)
case75 : β{Ξ A B C} β Tm75 Ξ (sum75 A B) β Tm75 Ξ (arr75 A C) β Tm75 Ξ (arr75 B C) β Tm75 Ξ C; case75
= Ξ» t u v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec β
case75 _ _ _ _
(t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
(u Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
(v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero suc rec)
zero75 : β{Ξ} β Tm75 Ξ nat75; zero75
= Ξ» Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc rec β zero75 _
suc75 : β{Ξ} β Tm75 Ξ nat75 β Tm75 Ξ nat75; suc75
= Ξ» t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec β
suc75 _ (t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec)
rec75 : β{Ξ A} β Tm75 Ξ nat75 β Tm75 Ξ (arr75 nat75 (arr75 A A)) β Tm75 Ξ A β Tm75 Ξ A; rec75
= Ξ» t u v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75 β
rec75 _ _
(t Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
(u Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
(v Tm75 var75 lam75 app75 tt75 pair75 fst75 snd75 left75 right75 case75 zero75 suc75 rec75)
v075 : β{Ξ A} β Tm75 (snoc75 Ξ A) A; v075
= var75 vz75
v175 : β{Ξ A B} β Tm75 (snoc75 (snoc75 Ξ A) B) A; v175
= var75 (vs75 vz75)
v275 : β{Ξ A B C} β Tm75 (snoc75 (snoc75 (snoc75 Ξ A) B) C) A; v275
= var75 (vs75 (vs75 vz75))
v375 : β{Ξ A B C D} β Tm75 (snoc75 (snoc75 (snoc75 (snoc75 Ξ A) B) C) D) A; v375
= var75 (vs75 (vs75 (vs75 vz75)))
tbool75 : Ty75; tbool75
= sum75 top75 top75
true75 : β{Ξ} β Tm75 Ξ tbool75; true75
= left75 tt75
tfalse75 : β{Ξ} β Tm75 Ξ tbool75; tfalse75
= right75 tt75
ifthenelse75 : β{Ξ A} β Tm75 Ξ (arr75 tbool75 (arr75 A (arr75 A A))); ifthenelse75
= lam75 (lam75 (lam75 (case75 v275 (lam75 v275) (lam75 v175))))
times475 : β{Ξ A} β Tm75 Ξ (arr75 (arr75 A A) (arr75 A A)); times475
= lam75 (lam75 (app75 v175 (app75 v175 (app75 v175 (app75 v175 v075)))))
add75 : β{Ξ} β Tm75 Ξ (arr75 nat75 (arr75 nat75 nat75)); add75
= lam75 (rec75 v075
(lam75 (lam75 (lam75 (suc75 (app75 v175 v075)))))
(lam75 v075))
mul75 : β{Ξ} β Tm75 Ξ (arr75 nat75 (arr75 nat75 nat75)); mul75
= lam75 (rec75 v075
(lam75 (lam75 (lam75 (app75 (app75 add75 (app75 v175 v075)) v075))))
(lam75 zero75))
fact75 : β{Ξ} β Tm75 Ξ (arr75 nat75 nat75); fact75
= lam75 (rec75 v075 (lam75 (lam75 (app75 (app75 mul75 (suc75 v175)) v075)))
(suc75 zero75))
{-# OPTIONS --type-in-type #-}
Ty76 : Set
Ty76 =
(Ty76 : Set)
(nat top bot : Ty76)
(arr prod sum : Ty76 β Ty76 β Ty76)
β Ty76
nat76 : Ty76; nat76 = Ξ» _ nat76 _ _ _ _ _ β nat76
top76 : Ty76; top76 = Ξ» _ _ top76 _ _ _ _ β top76
bot76 : Ty76; bot76 = Ξ» _ _ _ bot76 _ _ _ β bot76
arr76 : Ty76 β Ty76 β Ty76; arr76
= Ξ» A B Ty76 nat76 top76 bot76 arr76 prod sum β
arr76 (A Ty76 nat76 top76 bot76 arr76 prod sum) (B Ty76 nat76 top76 bot76 arr76 prod sum)
prod76 : Ty76 β Ty76 β Ty76; prod76
= Ξ» A B Ty76 nat76 top76 bot76 arr76 prod76 sum β
prod76 (A Ty76 nat76 top76 bot76 arr76 prod76 sum) (B Ty76 nat76 top76 bot76 arr76 prod76 sum)
sum76 : Ty76 β Ty76 β Ty76; sum76
= Ξ» A B Ty76 nat76 top76 bot76 arr76 prod76 sum76 β
sum76 (A Ty76 nat76 top76 bot76 arr76 prod76 sum76) (B Ty76 nat76 top76 bot76 arr76 prod76 sum76)
Con76 : Set; Con76
= (Con76 : Set)
(nil : Con76)
(snoc : Con76 β Ty76 β Con76)
β Con76
nil76 : Con76; nil76
= Ξ» Con76 nil76 snoc β nil76
snoc76 : Con76 β Ty76 β Con76; snoc76
= Ξ» Ξ A Con76 nil76 snoc76 β snoc76 (Ξ Con76 nil76 snoc76) A
Var76 : Con76 β Ty76 β Set; Var76
= Ξ» Ξ A β
(Var76 : Con76 β Ty76 β Set)
(vz : β Ξ A β Var76 (snoc76 Ξ A) A)
(vs : β Ξ B A β Var76 Ξ A β Var76 (snoc76 Ξ B) A)
β Var76 Ξ A
vz76 : β{Ξ A} β Var76 (snoc76 Ξ A) A; vz76
= Ξ» Var76 vz76 vs β vz76 _ _
vs76 : β{Ξ B A} β Var76 Ξ A β Var76 (snoc76 Ξ B) A; vs76
= Ξ» x Var76 vz76 vs76 β vs76 _ _ _ (x Var76 vz76 vs76)
Tm76 : Con76 β Ty76 β Set; Tm76
= Ξ» Ξ A β
(Tm76 : Con76 β Ty76 β Set)
(var : β Ξ A β Var76 Ξ A β Tm76 Ξ A)
(lam : β Ξ A B β Tm76 (snoc76 Ξ A) B β Tm76 Ξ (arr76 A B))
(app : β Ξ A B β Tm76 Ξ (arr76 A B) β Tm76 Ξ A β Tm76 Ξ B)
(tt : β Ξ β Tm76 Ξ top76)
(pair : β Ξ A B β Tm76 Ξ A β Tm76 Ξ B β Tm76 Ξ (prod76 A B))
(fst : β Ξ A B β Tm76 Ξ (prod76 A B) β Tm76 Ξ A)
(snd : β Ξ A B β Tm76 Ξ (prod76 A B) β Tm76 Ξ B)
(left : β Ξ A B β Tm76 Ξ A β Tm76 Ξ (sum76 A B))
(right : β Ξ A B β Tm76 Ξ B β Tm76 Ξ (sum76 A B))
(case : β Ξ A B C β Tm76 Ξ (sum76 A B) β Tm76 Ξ (arr76 A C) β Tm76 Ξ (arr76 B C) β Tm76 Ξ C)
(zero : β Ξ β Tm76 Ξ nat76)
(suc : β Ξ β Tm76 Ξ nat76 β Tm76 Ξ nat76)
(rec : β Ξ A β Tm76 Ξ nat76 β Tm76 Ξ (arr76 nat76 (arr76 A A)) β Tm76 Ξ A β Tm76 Ξ A)
β Tm76 Ξ A
var76 : β{Ξ A} β Var76 Ξ A β Tm76 Ξ A; var76
= Ξ» x Tm76 var76 lam app tt pair fst snd left right case zero suc rec β
var76 _ _ x
lam76 : β{Ξ A B} β Tm76 (snoc76 Ξ A) B β Tm76 Ξ (arr76 A B); lam76
= Ξ» t Tm76 var76 lam76 app tt pair fst snd left right case zero suc rec β
lam76 _ _ _ (t Tm76 var76 lam76 app tt pair fst snd left right case zero suc rec)
app76 : β{Ξ A B} β Tm76 Ξ (arr76 A B) β Tm76 Ξ A β Tm76 Ξ B; app76
= Ξ» t u Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec β
app76 _ _ _ (t Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec)
(u Tm76 var76 lam76 app76 tt pair fst snd left right case zero suc rec)
tt76 : β{Ξ} β Tm76 Ξ top76; tt76
= Ξ» Tm76 var76 lam76 app76 tt76 pair fst snd left right case zero suc rec β tt76 _
pair76 : β{Ξ A B} β Tm76 Ξ A β Tm76 Ξ B β Tm76 Ξ (prod76 A B); pair76
= Ξ» t u Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec β
pair76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec)
(u Tm76 var76 lam76 app76 tt76 pair76 fst snd left right case zero suc rec)
fst76 : β{Ξ A B} β Tm76 Ξ (prod76 A B) β Tm76 Ξ A; fst76
= Ξ» t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd left right case zero suc rec β
fst76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd left right case zero suc rec)
snd76 : β{Ξ A B} β Tm76 Ξ (prod76 A B) β Tm76 Ξ B; snd76
= Ξ» t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left right case zero suc rec β
snd76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left right case zero suc rec)
left76 : β{Ξ A B} β Tm76 Ξ A β Tm76 Ξ (sum76 A B); left76
= Ξ» t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right case zero suc rec β
left76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right case zero suc rec)
right76 : β{Ξ A B} β Tm76 Ξ B β Tm76 Ξ (sum76 A B); right76
= Ξ» t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case zero suc rec β
right76 _ _ _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case zero suc rec)
case76 : β{Ξ A B C} β Tm76 Ξ (sum76 A B) β Tm76 Ξ (arr76 A C) β Tm76 Ξ (arr76 B C) β Tm76 Ξ C; case76
= Ξ» t u v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec β
case76 _ _ _ _
(t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
(u Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
(v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero suc rec)
zero76 : β{Ξ} β Tm76 Ξ nat76; zero76
= Ξ» Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc rec β zero76 _
suc76 : β{Ξ} β Tm76 Ξ nat76 β Tm76 Ξ nat76; suc76
= Ξ» t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec β
suc76 _ (t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec)
rec76 : β{Ξ A} β Tm76 Ξ nat76 β Tm76 Ξ (arr76 nat76 (arr76 A A)) β Tm76 Ξ A β Tm76 Ξ A; rec76
= Ξ» t u v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76 β
rec76 _ _
(t Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
(u Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
(v Tm76 var76 lam76 app76 tt76 pair76 fst76 snd76 left76 right76 case76 zero76 suc76 rec76)
v076 : β{Ξ A} β Tm76 (snoc76 Ξ A) A; v076
= var76 vz76
v176 : β{Ξ A B} β Tm76 (snoc76 (snoc76 Ξ A) B) A; v176
= var76 (vs76 vz76)
v276 : β{Ξ A B C} β Tm76 (snoc76 (snoc76 (snoc76 Ξ A) B) C) A; v276
= var76 (vs76 (vs76 vz76))
v376 : β{Ξ A B C D} β Tm76 (snoc76 (snoc76 (snoc76 (snoc76 Ξ A) B) C) D) A; v376
= var76 (vs76 (vs76 (vs76 vz76)))
tbool76 : Ty76; tbool76
= sum76 top76 top76
true76 : β{Ξ} β Tm76 Ξ tbool76; true76
= left76 tt76
tfalse76 : β{Ξ} β Tm76 Ξ tbool76; tfalse76
= right76 tt76
ifthenelse76 : β{Ξ A} β Tm76 Ξ (arr76 tbool76 (arr76 A (arr76 A A))); ifthenelse76
= lam76 (lam76 (lam76 (case76 v276 (lam76 v276) (lam76 v176))))
times476 : β{Ξ A} β Tm76 Ξ (arr76 (arr76 A A) (arr76 A A)); times476
= lam76 (lam76 (app76 v176 (app76 v176 (app76 v176 (app76 v176 v076)))))
add76 : β{Ξ} β Tm76 Ξ (arr76 nat76 (arr76 nat76 nat76)); add76
= lam76 (rec76 v076
(lam76 (lam76 (lam76 (suc76 (app76 v176 v076)))))
(lam76 v076))
mul76 : β{Ξ} β Tm76 Ξ (arr76 nat76 (arr76 nat76 nat76)); mul76
= lam76 (rec76 v076
(lam76 (lam76 (lam76 (app76 (app76 add76 (app76 v176 v076)) v076))))
(lam76 zero76))
fact76 : β{Ξ} β Tm76 Ξ (arr76 nat76 nat76); fact76
= lam76 (rec76 v076 (lam76 (lam76 (app76 (app76 mul76 (suc76 v176)) v076)))
(suc76 zero76))
{-# OPTIONS --type-in-type #-}
Ty77 : Set
Ty77 =
(Ty77 : Set)
(nat top bot : Ty77)
(arr prod sum : Ty77 β Ty77 β Ty77)
β Ty77
nat77 : Ty77; nat77 = Ξ» _ nat77 _ _ _ _ _ β nat77
top77 : Ty77; top77 = Ξ» _ _ top77 _ _ _ _ β top77
bot77 : Ty77; bot77 = Ξ» _ _ _ bot77 _ _ _ β bot77
arr77 : Ty77 β Ty77 β Ty77; arr77
= Ξ» A B Ty77 nat77 top77 bot77 arr77 prod sum β
arr77 (A Ty77 nat77 top77 bot77 arr77 prod sum) (B Ty77 nat77 top77 bot77 arr77 prod sum)
prod77 : Ty77 β Ty77 β Ty77; prod77
= Ξ» A B Ty77 nat77 top77 bot77 arr77 prod77 sum β
prod77 (A Ty77 nat77 top77 bot77 arr77 prod77 sum) (B Ty77 nat77 top77 bot77 arr77 prod77 sum)
sum77 : Ty77 β Ty77 β Ty77; sum77
= Ξ» A B Ty77 nat77 top77 bot77 arr77 prod77 sum77 β
sum77 (A Ty77 nat77 top77 bot77 arr77 prod77 sum77) (B Ty77 nat77 top77 bot77 arr77 prod77 sum77)
Con77 : Set; Con77
= (Con77 : Set)
(nil : Con77)
(snoc : Con77 β Ty77 β Con77)
β Con77
nil77 : Con77; nil77
= Ξ» Con77 nil77 snoc β nil77
snoc77 : Con77 β Ty77 β Con77; snoc77
= Ξ» Ξ A Con77 nil77 snoc77 β snoc77 (Ξ Con77 nil77 snoc77) A
Var77 : Con77 β Ty77 β Set; Var77
= Ξ» Ξ A β
(Var77 : Con77 β Ty77 β Set)
(vz : β Ξ A β Var77 (snoc77 Ξ A) A)
(vs : β Ξ B A β Var77 Ξ A β Var77 (snoc77 Ξ B) A)
β Var77 Ξ A
vz77 : β{Ξ A} β Var77 (snoc77 Ξ A) A; vz77
= Ξ» Var77 vz77 vs β vz77 _ _
vs77 : β{Ξ B A} β Var77 Ξ A β Var77 (snoc77 Ξ B) A; vs77
= Ξ» x Var77 vz77 vs77 β vs77 _ _ _ (x Var77 vz77 vs77)
Tm77 : Con77 β Ty77 β Set; Tm77
= Ξ» Ξ A β
(Tm77 : Con77 β Ty77 β Set)
(var : β Ξ A β Var77 Ξ A β Tm77 Ξ A)
(lam : β Ξ A B β Tm77 (snoc77 Ξ A) B β Tm77 Ξ (arr77 A B))
(app : β Ξ A B β Tm77 Ξ (arr77 A B) β Tm77 Ξ A β Tm77 Ξ B)
(tt : β Ξ β Tm77 Ξ top77)
(pair : β Ξ A B β Tm77 Ξ A β Tm77 Ξ B β Tm77 Ξ (prod77 A B))
(fst : β Ξ A B β Tm77 Ξ (prod77 A B) β Tm77 Ξ A)
(snd : β Ξ A B β Tm77 Ξ (prod77 A B) β Tm77 Ξ B)
(left : β Ξ A B β Tm77 Ξ A β Tm77 Ξ (sum77 A B))
(right : β Ξ A B β Tm77 Ξ B β Tm77 Ξ (sum77 A B))
(case : β Ξ A B C β Tm77 Ξ (sum77 A B) β Tm77 Ξ (arr77 A C) β Tm77 Ξ (arr77 B C) β Tm77 Ξ C)
(zero : β Ξ β Tm77 Ξ nat77)
(suc : β Ξ β Tm77 Ξ nat77 β Tm77 Ξ nat77)
(rec : β Ξ A β Tm77 Ξ nat77 β Tm77 Ξ (arr77 nat77 (arr77 A A)) β Tm77 Ξ A β Tm77 Ξ A)
β Tm77 Ξ A
var77 : β{Ξ A} β Var77 Ξ A β Tm77 Ξ A; var77
= Ξ» x Tm77 var77 lam app tt pair fst snd left right case zero suc rec β
var77 _ _ x
lam77 : β{Ξ A B} β Tm77 (snoc77 Ξ A) B β Tm77 Ξ (arr77 A B); lam77
= Ξ» t Tm77 var77 lam77 app tt pair fst snd left right case zero suc rec β
lam77 _ _ _ (t Tm77 var77 lam77 app tt pair fst snd left right case zero suc rec)
app77 : β{Ξ A B} β Tm77 Ξ (arr77 A B) β Tm77 Ξ A β Tm77 Ξ B; app77
= Ξ» t u Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec β
app77 _ _ _ (t Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec)
(u Tm77 var77 lam77 app77 tt pair fst snd left right case zero suc rec)
tt77 : β{Ξ} β Tm77 Ξ top77; tt77
= Ξ» Tm77 var77 lam77 app77 tt77 pair fst snd left right case zero suc rec β tt77 _
pair77 : β{Ξ A B} β Tm77 Ξ A β Tm77 Ξ B β Tm77 Ξ (prod77 A B); pair77
= Ξ» t u Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec β
pair77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec)
(u Tm77 var77 lam77 app77 tt77 pair77 fst snd left right case zero suc rec)
fst77 : β{Ξ A B} β Tm77 Ξ (prod77 A B) β Tm77 Ξ A; fst77
= Ξ» t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd left right case zero suc rec β
fst77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd left right case zero suc rec)
snd77 : β{Ξ A B} β Tm77 Ξ (prod77 A B) β Tm77 Ξ B; snd77
= Ξ» t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left right case zero suc rec β
snd77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left right case zero suc rec)
left77 : β{Ξ A B} β Tm77 Ξ A β Tm77 Ξ (sum77 A B); left77
= Ξ» t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right case zero suc rec β
left77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right case zero suc rec)
right77 : β{Ξ A B} β Tm77 Ξ B β Tm77 Ξ (sum77 A B); right77
= Ξ» t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case zero suc rec β
right77 _ _ _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case zero suc rec)
case77 : β{Ξ A B C} β Tm77 Ξ (sum77 A B) β Tm77 Ξ (arr77 A C) β Tm77 Ξ (arr77 B C) β Tm77 Ξ C; case77
= Ξ» t u v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec β
case77 _ _ _ _
(t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
(u Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
(v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero suc rec)
zero77 : β{Ξ} β Tm77 Ξ nat77; zero77
= Ξ» Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc rec β zero77 _
suc77 : β{Ξ} β Tm77 Ξ nat77 β Tm77 Ξ nat77; suc77
= Ξ» t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec β
suc77 _ (t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec)
rec77 : β{Ξ A} β Tm77 Ξ nat77 β Tm77 Ξ (arr77 nat77 (arr77 A A)) β Tm77 Ξ A β Tm77 Ξ A; rec77
= Ξ» t u v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77 β
rec77 _ _
(t Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
(u Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
(v Tm77 var77 lam77 app77 tt77 pair77 fst77 snd77 left77 right77 case77 zero77 suc77 rec77)
v077 : β{Ξ A} β Tm77 (snoc77 Ξ A) A; v077
= var77 vz77
v177 : β{Ξ A B} β Tm77 (snoc77 (snoc77 Ξ A) B) A; v177
= var77 (vs77 vz77)
v277 : β{Ξ A B C} β Tm77 (snoc77 (snoc77 (snoc77 Ξ A) B) C) A; v277
= var77 (vs77 (vs77 vz77))
v377 : β{Ξ A B C D} β Tm77 (snoc77 (snoc77 (snoc77 (snoc77 Ξ A) B) C) D) A; v377
= var77 (vs77 (vs77 (vs77 vz77)))
tbool77 : Ty77; tbool77
= sum77 top77 top77
true77 : β{Ξ} β Tm77 Ξ tbool77; true77
= left77 tt77
tfalse77 : β{Ξ} β Tm77 Ξ tbool77; tfalse77
= right77 tt77
ifthenelse77 : β{Ξ A} β Tm77 Ξ (arr77 tbool77 (arr77 A (arr77 A A))); ifthenelse77
= lam77 (lam77 (lam77 (case77 v277 (lam77 v277) (lam77 v177))))
times477 : β{Ξ A} β Tm77 Ξ (arr77 (arr77 A A) (arr77 A A)); times477
= lam77 (lam77 (app77 v177 (app77 v177 (app77 v177 (app77 v177 v077)))))
add77 : β{Ξ} β Tm77 Ξ (arr77 nat77 (arr77 nat77 nat77)); add77
= lam77 (rec77 v077
(lam77 (lam77 (lam77 (suc77 (app77 v177 v077)))))
(lam77 v077))
mul77 : β{Ξ} β Tm77 Ξ (arr77 nat77 (arr77 nat77 nat77)); mul77
= lam77 (rec77 v077
(lam77 (lam77 (lam77 (app77 (app77 add77 (app77 v177 v077)) v077))))
(lam77 zero77))
fact77 : β{Ξ} β Tm77 Ξ (arr77 nat77 nat77); fact77
= lam77 (rec77 v077 (lam77 (lam77 (app77 (app77 mul77 (suc77 v177)) v077)))
(suc77 zero77))
{-# OPTIONS --type-in-type #-}
Ty78 : Set
Ty78 =
(Ty78 : Set)
(nat top bot : Ty78)
(arr prod sum : Ty78 β Ty78 β Ty78)
β Ty78
nat78 : Ty78; nat78 = Ξ» _ nat78 _ _ _ _ _ β nat78
top78 : Ty78; top78 = Ξ» _ _ top78 _ _ _ _ β top78
bot78 : Ty78; bot78 = Ξ» _ _ _ bot78 _ _ _ β bot78
arr78 : Ty78 β Ty78 β Ty78; arr78
= Ξ» A B Ty78 nat78 top78 bot78 arr78 prod sum β
arr78 (A Ty78 nat78 top78 bot78 arr78 prod sum) (B Ty78 nat78 top78 bot78 arr78 prod sum)
prod78 : Ty78 β Ty78 β Ty78; prod78
= Ξ» A B Ty78 nat78 top78 bot78 arr78 prod78 sum β
prod78 (A Ty78 nat78 top78 bot78 arr78 prod78 sum) (B Ty78 nat78 top78 bot78 arr78 prod78 sum)
sum78 : Ty78 β Ty78 β Ty78; sum78
= Ξ» A B Ty78 nat78 top78 bot78 arr78 prod78 sum78 β
sum78 (A Ty78 nat78 top78 bot78 arr78 prod78 sum78) (B Ty78 nat78 top78 bot78 arr78 prod78 sum78)
Con78 : Set; Con78
= (Con78 : Set)
(nil : Con78)
(snoc : Con78 β Ty78 β Con78)
β Con78
nil78 : Con78; nil78
= Ξ» Con78 nil78 snoc β nil78
snoc78 : Con78 β Ty78 β Con78; snoc78
= Ξ» Ξ A Con78 nil78 snoc78 β snoc78 (Ξ Con78 nil78 snoc78) A
Var78 : Con78 β Ty78 β Set; Var78
= Ξ» Ξ A β
(Var78 : Con78 β Ty78 β Set)
(vz : β Ξ A β Var78 (snoc78 Ξ A) A)
(vs : β Ξ B A β Var78 Ξ A β Var78 (snoc78 Ξ B) A)
β Var78 Ξ A
vz78 : β{Ξ A} β Var78 (snoc78 Ξ A) A; vz78
= Ξ» Var78 vz78 vs β vz78 _ _
vs78 : β{Ξ B A} β Var78 Ξ A β Var78 (snoc78 Ξ B) A; vs78
= Ξ» x Var78 vz78 vs78 β vs78 _ _ _ (x Var78 vz78 vs78)
Tm78 : Con78 β Ty78 β Set; Tm78
= Ξ» Ξ A β
(Tm78 : Con78 β Ty78 β Set)
(var : β Ξ A β Var78 Ξ A β Tm78 Ξ A)
(lam : β Ξ A B β Tm78 (snoc78 Ξ A) B β Tm78 Ξ (arr78 A B))
(app : β Ξ A B β Tm78 Ξ (arr78 A B) β Tm78 Ξ A β Tm78 Ξ B)
(tt : β Ξ β Tm78 Ξ top78)
(pair : β Ξ A B β Tm78 Ξ A β Tm78 Ξ B β Tm78 Ξ (prod78 A B))
(fst : β Ξ A B β Tm78 Ξ (prod78 A B) β Tm78 Ξ A)
(snd : β Ξ A B β Tm78 Ξ (prod78 A B) β Tm78 Ξ B)
(left : β Ξ A B β Tm78 Ξ A β Tm78 Ξ (sum78 A B))
(right : β Ξ A B β Tm78 Ξ B β Tm78 Ξ (sum78 A B))
(case : β Ξ A B C β Tm78 Ξ (sum78 A B) β Tm78 Ξ (arr78 A C) β Tm78 Ξ (arr78 B C) β Tm78 Ξ C)
(zero : β Ξ β Tm78 Ξ nat78)
(suc : β Ξ β Tm78 Ξ nat78 β Tm78 Ξ nat78)
(rec : β Ξ A β Tm78 Ξ nat78 β Tm78 Ξ (arr78 nat78 (arr78 A A)) β Tm78 Ξ A β Tm78 Ξ A)
β Tm78 Ξ A
var78 : β{Ξ A} β Var78 Ξ A β Tm78 Ξ A; var78
= Ξ» x Tm78 var78 lam app tt pair fst snd left right case zero suc rec β
var78 _ _ x
lam78 : β{Ξ A B} β Tm78 (snoc78 Ξ A) B β Tm78 Ξ (arr78 A B); lam78
= Ξ» t Tm78 var78 lam78 app tt pair fst snd left right case zero suc rec β
lam78 _ _ _ (t Tm78 var78 lam78 app tt pair fst snd left right case zero suc rec)
app78 : β{Ξ A B} β Tm78 Ξ (arr78 A B) β Tm78 Ξ A β Tm78 Ξ B; app78
= Ξ» t u Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec β
app78 _ _ _ (t Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec)
(u Tm78 var78 lam78 app78 tt pair fst snd left right case zero suc rec)
tt78 : β{Ξ} β Tm78 Ξ top78; tt78
= Ξ» Tm78 var78 lam78 app78 tt78 pair fst snd left right case zero suc rec β tt78 _
pair78 : β{Ξ A B} β Tm78 Ξ A β Tm78 Ξ B β Tm78 Ξ (prod78 A B); pair78
= Ξ» t u Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec β
pair78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec)
(u Tm78 var78 lam78 app78 tt78 pair78 fst snd left right case zero suc rec)
fst78 : β{Ξ A B} β Tm78 Ξ (prod78 A B) β Tm78 Ξ A; fst78
= Ξ» t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd left right case zero suc rec β
fst78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd left right case zero suc rec)
snd78 : β{Ξ A B} β Tm78 Ξ (prod78 A B) β Tm78 Ξ B; snd78
= Ξ» t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left right case zero suc rec β
snd78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left right case zero suc rec)
left78 : β{Ξ A B} β Tm78 Ξ A β Tm78 Ξ (sum78 A B); left78
= Ξ» t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right case zero suc rec β
left78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right case zero suc rec)
right78 : β{Ξ A B} β Tm78 Ξ B β Tm78 Ξ (sum78 A B); right78
= Ξ» t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case zero suc rec β
right78 _ _ _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case zero suc rec)
case78 : β{Ξ A B C} β Tm78 Ξ (sum78 A B) β Tm78 Ξ (arr78 A C) β Tm78 Ξ (arr78 B C) β Tm78 Ξ C; case78
= Ξ» t u v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec β
case78 _ _ _ _
(t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
(u Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
(v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero suc rec)
zero78 : β{Ξ} β Tm78 Ξ nat78; zero78
= Ξ» Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc rec β zero78 _
suc78 : β{Ξ} β Tm78 Ξ nat78 β Tm78 Ξ nat78; suc78
= Ξ» t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec β
suc78 _ (t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec)
rec78 : β{Ξ A} β Tm78 Ξ nat78 β Tm78 Ξ (arr78 nat78 (arr78 A A)) β Tm78 Ξ A β Tm78 Ξ A; rec78
= Ξ» t u v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78 β
rec78 _ _
(t Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
(u Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
(v Tm78 var78 lam78 app78 tt78 pair78 fst78 snd78 left78 right78 case78 zero78 suc78 rec78)
v078 : β{Ξ A} β Tm78 (snoc78 Ξ A) A; v078
= var78 vz78
v178 : β{Ξ A B} β Tm78 (snoc78 (snoc78 Ξ A) B) A; v178
= var78 (vs78 vz78)
v278 : β{Ξ A B C} β Tm78 (snoc78 (snoc78 (snoc78 Ξ A) B) C) A; v278
= var78 (vs78 (vs78 vz78))
v378 : β{Ξ A B C D} β Tm78 (snoc78 (snoc78 (snoc78 (snoc78 Ξ A) B) C) D) A; v378
= var78 (vs78 (vs78 (vs78 vz78)))
tbool78 : Ty78; tbool78
= sum78 top78 top78
true78 : β{Ξ} β Tm78 Ξ tbool78; true78
= left78 tt78
tfalse78 : β{Ξ} β Tm78 Ξ tbool78; tfalse78
= right78 tt78
ifthenelse78 : β{Ξ A} β Tm78 Ξ (arr78 tbool78 (arr78 A (arr78 A A))); ifthenelse78
= lam78 (lam78 (lam78 (case78 v278 (lam78 v278) (lam78 v178))))
times478 : β{Ξ A} β Tm78 Ξ (arr78 (arr78 A A) (arr78 A A)); times478
= lam78 (lam78 (app78 v178 (app78 v178 (app78 v178 (app78 v178 v078)))))
add78 : β{Ξ} β Tm78 Ξ (arr78 nat78 (arr78 nat78 nat78)); add78
= lam78 (rec78 v078
(lam78 (lam78 (lam78 (suc78 (app78 v178 v078)))))
(lam78 v078))
mul78 : β{Ξ} β Tm78 Ξ (arr78 nat78 (arr78 nat78 nat78)); mul78
= lam78 (rec78 v078
(lam78 (lam78 (lam78 (app78 (app78 add78 (app78 v178 v078)) v078))))
(lam78 zero78))
fact78 : β{Ξ} β Tm78 Ξ (arr78 nat78 nat78); fact78
= lam78 (rec78 v078 (lam78 (lam78 (app78 (app78 mul78 (suc78 v178)) v078)))
(suc78 zero78))
{-# OPTIONS --type-in-type #-}
Ty79 : Set
Ty79 =
(Ty79 : Set)
(nat top bot : Ty79)
(arr prod sum : Ty79 β Ty79 β Ty79)
β Ty79
nat79 : Ty79; nat79 = Ξ» _ nat79 _ _ _ _ _ β nat79
top79 : Ty79; top79 = Ξ» _ _ top79 _ _ _ _ β top79
bot79 : Ty79; bot79 = Ξ» _ _ _ bot79 _ _ _ β bot79
arr79 : Ty79 β Ty79 β Ty79; arr79
= Ξ» A B Ty79 nat79 top79 bot79 arr79 prod sum β
arr79 (A Ty79 nat79 top79 bot79 arr79 prod sum) (B Ty79 nat79 top79 bot79 arr79 prod sum)
prod79 : Ty79 β Ty79 β Ty79; prod79
= Ξ» A B Ty79 nat79 top79 bot79 arr79 prod79 sum β
prod79 (A Ty79 nat79 top79 bot79 arr79 prod79 sum) (B Ty79 nat79 top79 bot79 arr79 prod79 sum)
sum79 : Ty79 β Ty79 β Ty79; sum79
= Ξ» A B Ty79 nat79 top79 bot79 arr79 prod79 sum79 β
sum79 (A Ty79 nat79 top79 bot79 arr79 prod79 sum79) (B Ty79 nat79 top79 bot79 arr79 prod79 sum79)
Con79 : Set; Con79
= (Con79 : Set)
(nil : Con79)
(snoc : Con79 β Ty79 β Con79)
β Con79
nil79 : Con79; nil79
= Ξ» Con79 nil79 snoc β nil79
snoc79 : Con79 β Ty79 β Con79; snoc79
= Ξ» Ξ A Con79 nil79 snoc79 β snoc79 (Ξ Con79 nil79 snoc79) A
Var79 : Con79 β Ty79 β Set; Var79
= Ξ» Ξ A β
(Var79 : Con79 β Ty79 β Set)
(vz : β Ξ A β Var79 (snoc79 Ξ A) A)
(vs : β Ξ B A β Var79 Ξ A β Var79 (snoc79 Ξ B) A)
β Var79 Ξ A
vz79 : β{Ξ A} β Var79 (snoc79 Ξ A) A; vz79
= Ξ» Var79 vz79 vs β vz79 _ _
vs79 : β{Ξ B A} β Var79 Ξ A β Var79 (snoc79 Ξ B) A; vs79
= Ξ» x Var79 vz79 vs79 β vs79 _ _ _ (x Var79 vz79 vs79)
Tm79 : Con79 β Ty79 β Set; Tm79
= Ξ» Ξ A β
(Tm79 : Con79 β Ty79 β Set)
(var : β Ξ A β Var79 Ξ A β Tm79 Ξ A)
(lam : β Ξ A B β Tm79 (snoc79 Ξ A) B β Tm79 Ξ (arr79 A B))
(app : β Ξ A B β Tm79 Ξ (arr79 A B) β Tm79 Ξ A β Tm79 Ξ B)
(tt : β Ξ β Tm79 Ξ top79)
(pair : β Ξ A B β Tm79 Ξ A β Tm79 Ξ B β Tm79 Ξ (prod79 A B))
(fst : β Ξ A B β Tm79 Ξ (prod79 A B) β Tm79 Ξ A)
(snd : β Ξ A B β Tm79 Ξ (prod79 A B) β Tm79 Ξ B)
(left : β Ξ A B β Tm79 Ξ A β Tm79 Ξ (sum79 A B))
(right : β Ξ A B β Tm79 Ξ B β Tm79 Ξ (sum79 A B))
(case : β Ξ A B C β Tm79 Ξ (sum79 A B) β Tm79 Ξ (arr79 A C) β Tm79 Ξ (arr79 B C) β Tm79 Ξ C)
(zero : β Ξ β Tm79 Ξ nat79)
(suc : β Ξ β Tm79 Ξ nat79 β Tm79 Ξ nat79)
(rec : β Ξ A β Tm79 Ξ nat79 β Tm79 Ξ (arr79 nat79 (arr79 A A)) β Tm79 Ξ A β Tm79 Ξ A)
β Tm79 Ξ A
var79 : β{Ξ A} β Var79 Ξ A β Tm79 Ξ A; var79
= Ξ» x Tm79 var79 lam app tt pair fst snd left right case zero suc rec β
var79 _ _ x
lam79 : β{Ξ A B} β Tm79 (snoc79 Ξ A) B β Tm79 Ξ (arr79 A B); lam79
= Ξ» t Tm79 var79 lam79 app tt pair fst snd left right case zero suc rec β
lam79 _ _ _ (t Tm79 var79 lam79 app tt pair fst snd left right case zero suc rec)
app79 : β{Ξ A B} β Tm79 Ξ (arr79 A B) β Tm79 Ξ A β Tm79 Ξ B; app79
= Ξ» t u Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec β
app79 _ _ _ (t Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec)
(u Tm79 var79 lam79 app79 tt pair fst snd left right case zero suc rec)
tt79 : β{Ξ} β Tm79 Ξ top79; tt79
= Ξ» Tm79 var79 lam79 app79 tt79 pair fst snd left right case zero suc rec β tt79 _
pair79 : β{Ξ A B} β Tm79 Ξ A β Tm79 Ξ B β Tm79 Ξ (prod79 A B); pair79
= Ξ» t u Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec β
pair79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec)
(u Tm79 var79 lam79 app79 tt79 pair79 fst snd left right case zero suc rec)
fst79 : β{Ξ A B} β Tm79 Ξ (prod79 A B) β Tm79 Ξ A; fst79
= Ξ» t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd left right case zero suc rec β
fst79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd left right case zero suc rec)
snd79 : β{Ξ A B} β Tm79 Ξ (prod79 A B) β Tm79 Ξ B; snd79
= Ξ» t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left right case zero suc rec β
snd79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left right case zero suc rec)
left79 : β{Ξ A B} β Tm79 Ξ A β Tm79 Ξ (sum79 A B); left79
= Ξ» t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right case zero suc rec β
left79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right case zero suc rec)
right79 : β{Ξ A B} β Tm79 Ξ B β Tm79 Ξ (sum79 A B); right79
= Ξ» t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case zero suc rec β
right79 _ _ _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case zero suc rec)
case79 : β{Ξ A B C} β Tm79 Ξ (sum79 A B) β Tm79 Ξ (arr79 A C) β Tm79 Ξ (arr79 B C) β Tm79 Ξ C; case79
= Ξ» t u v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec β
case79 _ _ _ _
(t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
(u Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
(v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero suc rec)
zero79 : β{Ξ} β Tm79 Ξ nat79; zero79
= Ξ» Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc rec β zero79 _
suc79 : β{Ξ} β Tm79 Ξ nat79 β Tm79 Ξ nat79; suc79
= Ξ» t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec β
suc79 _ (t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec)
rec79 : β{Ξ A} β Tm79 Ξ nat79 β Tm79 Ξ (arr79 nat79 (arr79 A A)) β Tm79 Ξ A β Tm79 Ξ A; rec79
= Ξ» t u v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79 β
rec79 _ _
(t Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
(u Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
(v Tm79 var79 lam79 app79 tt79 pair79 fst79 snd79 left79 right79 case79 zero79 suc79 rec79)
v079 : β{Ξ A} β Tm79 (snoc79 Ξ A) A; v079
= var79 vz79
v179 : β{Ξ A B} β Tm79 (snoc79 (snoc79 Ξ A) B) A; v179
= var79 (vs79 vz79)
v279 : β{Ξ A B C} β Tm79 (snoc79 (snoc79 (snoc79 Ξ A) B) C) A; v279
= var79 (vs79 (vs79 vz79))
v379 : β{Ξ A B C D} β Tm79 (snoc79 (snoc79 (snoc79 (snoc79 Ξ A) B) C) D) A; v379
= var79 (vs79 (vs79 (vs79 vz79)))
tbool79 : Ty79; tbool79
= sum79 top79 top79
true79 : β{Ξ} β Tm79 Ξ tbool79; true79
= left79 tt79
tfalse79 : β{Ξ} β Tm79 Ξ tbool79; tfalse79
= right79 tt79
ifthenelse79 : β{Ξ A} β Tm79 Ξ (arr79 tbool79 (arr79 A (arr79 A A))); ifthenelse79
= lam79 (lam79 (lam79 (case79 v279 (lam79 v279) (lam79 v179))))
times479 : β{Ξ A} β Tm79 Ξ (arr79 (arr79 A A) (arr79 A A)); times479
= lam79 (lam79 (app79 v179 (app79 v179 (app79 v179 (app79 v179 v079)))))
add79 : β{Ξ} β Tm79 Ξ (arr79 nat79 (arr79 nat79 nat79)); add79
= lam79 (rec79 v079
(lam79 (lam79 (lam79 (suc79 (app79 v179 v079)))))
(lam79 v079))
mul79 : β{Ξ} β Tm79 Ξ (arr79 nat79 (arr79 nat79 nat79)); mul79
= lam79 (rec79 v079
(lam79 (lam79 (lam79 (app79 (app79 add79 (app79 v179 v079)) v079))))
(lam79 zero79))
fact79 : β{Ξ} β Tm79 Ξ (arr79 nat79 nat79); fact79
= lam79 (rec79 v079 (lam79 (lam79 (app79 (app79 mul79 (suc79 v179)) v079)))
(suc79 zero79))
|
Are you seeking volunteer opportunities for teens?
Internships for high school students?
Global leadership adventures for university students?
Our planet will face huge challenges during your lifetime. Some of these challenges have already begun to surface.
Plastic in the ocean, for example. Every day approximately 8 million pieces of plastic pollution find their way into our oceans. Itβs estimated that there may now be around 5.25 trillion macro and microplastic pieces floating in the open ocean, with a combined weight of 269,000 tonnes. Plastic pollution is responsible for the deaths of 100,000 marine mammals and turtles and one million seabirds every year.
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Poverty. Over three billion people β thatβs almost half the world β live on less than $2.50 a day, and at least 80% of humanity lives on less than $10 a day. Statistics from Unicef show that 22,000 children die each day due to poverty, in some of the poorest villages on earth.
Gender equality. For the past decade, the World Economic Forum has been measuring the pace of change through the Global Gender Gap Report. While weβre getting closer to achieving gender equality, at current rates, it would take the world another 115 years β or until 2133 β to close the economic gap entirely.
Thatβs just four of the biggest problems the world is facing right now β and thereβs a whole lot more where that came from. But the good news is, weβre on a mission to reverse these problems β and make the world a better place in the process.
We believe that global problems demand new and powerful forms of leadership. Therefore, we want to unite young people around the worldβs most pressing problems and empower them to become driven and visionary leaders through extraordinary experiences.
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Plastic in the ocean, for example. Like the idea of volunteering with animals and being part of the transition to a plastic-free world? Youβll participate in beach clean-ups and contribute to the conservation and protection of marine species on our sea turtle internship in Costa Rica.
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And thereβs more where that came from. Species extinction? Weβre on it (check out our sloth conservation program.) Combating loneliness? We place volunteers in Elderly Care Centers across the globe. Preventing the spread of HIV? We have HIV awareness programs that are helping to manage the virus in local communities. And other 245 social and conservation programs.
We want to help you understand the world and its problems. Our goal is to give you the confidence and the tools to be a leader and generate a ripple of change in the world, by taking you to global communities where you can see the effects first-hand, and work alongside locals to create positive change.
We work closely with schools to design a bespoke itinerary and trip curriculum that meets each establishmentβs individual needs and objectives.
We require that at least one faculty member or parent participates in the trip alongside the students (one per 10 students). In fact, one teacher gets a discount for every group of 10 paying students! Why? In addition to ensuring students stay safe on our trips, we also want educators as involved as possible in order to impact the larger school community and ensure the learning continues after the trip has ended.
Click here to see what our services include on this trip, and for lots more useful information.
As well as tackling global issues head on, outdoor adventure and cultural explorations also form an important part of these volunteer opportunities. Options include: trekking in the Andean highlands of Peru or Nepal, snorkeling in Honduras or Chile, visiting Machu Picchu in Peru or going on safari in Tanzania or Uganda.
Our service project ideas extend past high school and college. We have also partnered with universities not just across the US, but across the world, to create volunteer leadership programs across sectors including health, HIV awareness, teaching abroad, orphanage support, building and restoration, vocational training, animal welfare and ecology, sea turtle conservation and so much more.
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As a 5***** trusted volunteer organization thatβs been up and running for over 11 years, weβre super familiar with hosting group tours. Weβve hosted student-led alternative spring trips, faculty-led service learning trips, medical and dental brigades, corporate retreat programs, family volunteer vacations, and high school language & cultural immersion tours.
With this in mind, please rest assured that safety is our number one priority. For all the parents and teachers reading this: we have a 100% safety record.Weβve build safety into every level of our trips, including highly experienced in-country directors and the communities themselves, which are small and close-knit. Our host families take the safety of our students very seriously, and we can also offer hostels or hotels as alternative accommodation.
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Ready to sign up? You can either register as a volunteer or contact usfor more information. |
[STATEMENT]
lemma fresh_fun_simp_AndL2:
assumes a: "z'\<sharp>P" "z'\<sharp>M" "z'\<sharp>x"
shows "fresh_fun (\<lambda>z'. Cut <c>.P (z').AndL2 (x).M z') = Cut <c>.P (z').AndL2 (x).M z'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. fresh_fun (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z') = Cut <c>.P z'.AndL2 x.M z'
[PROOF STEP]
using a
[PROOF STATE]
proof (prove)
using this:
z' \<sharp> P
z' \<sharp> M
z' \<sharp> x
goal (1 subgoal):
1. fresh_fun (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z') = Cut <c>.P z'.AndL2 x.M z'
[PROOF STEP]
apply -
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> fresh_fun (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z') = Cut <c>.P z'.AndL2 x.M z'
[PROOF STEP]
apply(rule fresh_fun_app)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> pt TYPE(trm) TYPE(name)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> at TYPE(name)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> finite (supp (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z'))
4. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
5. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(rule pt_name_inst)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> at TYPE(name)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> finite (supp (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z'))
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
4. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(rule at_name_inst)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> finite (supp (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z'))
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(finite_guess)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(subgoal_tac "\<exists>n::name. n\<sharp>(c,P,x,M)")
[PROOF STATE]
proof (prove)
goal (3 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x; \<exists>n. n \<sharp> (c, P, x, M)\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>n. n \<sharp> (c, P, x, M)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(erule exE)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
1. \<And>n. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x; n \<sharp> (c, P, x, M)\<rbrakk> \<Longrightarrow> \<exists>a. a \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P a.AndL2 x.M a)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>n. n \<sharp> (c, P, x, M)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(rule_tac x="n" in exI)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
1. \<And>n. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x; n \<sharp> (c, P, x, M)\<rbrakk> \<Longrightarrow> n \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z', Cut <c>.P n.AndL2 x.M n)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>n. n \<sharp> (c, P, x, M)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(simp add: fresh_prod abs_fresh)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
1. \<And>n. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x; n \<sharp> c \<and> n \<sharp> P \<and> n \<sharp> x \<and> n \<sharp> M\<rbrakk> \<Longrightarrow> n \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>n. n \<sharp> (c, P, x, M)
3. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(fresh_guess)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> \<exists>n. n \<sharp> (c, P, x, M)
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(rule exists_fresh')
[PROOF STATE]
proof (prove)
goal (2 subgoals):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> finite (supp (c, P, x, M))
2. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(simp add: fin_supp)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>z' \<sharp> P; z' \<sharp> M; z' \<sharp> x\<rbrakk> \<Longrightarrow> z' \<sharp> (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')
[PROOF STEP]
apply(fresh_guess)
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done |
Following his retirement , Zhou serves for a time as an advisor to General Liu <unk> ( <unk> ) , whose troops are garrisoned in Henan Province . But Zhou later becomes an outlaw himself after he aids the heroes of the Water Margin and is forced to flee from government forces . Meanwhile , he learns his elderly classmate Jin Tai is close to death and hurries to Shaolin ( where the general had become a Buddhist monk after the murder of his family ) to pay his last respects . As the oldest of Tan 's pupils , Jin orders Zhou to find a talented youth to pass on all of his martial arts knowledge to . However , this reunion is cut short when the troops track him to Shaolin . He flees to Wine Spring mountain and lives in hiding for sometime before being invited by his old friend Wang Ming ( <unk> ) to become the precept of the Wang family in Unicorn Village .
|
Not to be confused with Subsidiary or subsidy.
This article is about the general principle of subsidiarity, with particular reference to European Union law. For the Catholic social teaching, see Subsidiarity (Catholicism).
Subsidiarity is perhaps presently best known as a general principle of European Union law.
The Oxford English Dictionary defines subsidiarity as "the principle that a central authority should have a subsidiary function, performing only those tasks which cannot be performed at a more local level". The concept is applicable in the fields of government, political science, neuropsychology, cybernetics, management and in military command (mission command). Subsidiarity is a general principle of European Union law. In the United States of America, the principle of States' Rights is enshrined in the constitution. Although the principle is older, its expression in the term "subsidiarity" was first coined in 1891 by the Roman Catholic Church for its social teaching.
The OED adds that the term "subsidiarity" in English follows the early German usage of "SubsidiaritΓ€t". More distantly, it is derived from the Latin verb subsidio (to aid or help), and the related noun subsidium (aid or assistance). The concept as discussed here was first described formally in Catholic social teaching. Coupled with another Christian democratic principle, sphere sovereignty, subsidiarity is said to have led to the creation of corporatist welfare states throughout the world.
Alexis de Tocqueville's classic study, Democracy in America, may be viewed as an examination of the operation of the principle of subsidiarity in early 19th century America. De Tocqueville noted that the French Revolution began with "a push towards decentralization ... in the end, an extension of centralization". He wrote that "Decentralization has, not only an administrative value, but also a civic dimension, since it increases the opportunities for citizens to take interest in public affairs; it makes them get accustomed to using freedom. And from the accumulation of these local, active, persnickety freedoms, is born the most efficient counterweight against the claims of the central government, even if it were supported by an impersonal, collective will."
As Christian Democratic political parties were formed, they adopted the Catholic social teaching of subsidiarity, as well as the neo-Calvinist theological teaching of sphere sovereignty, with both Protestants and Roman Catholics agreeing "that the principles of sphere sovereignty and subsidiarity boiled down to the same thing".
Will the American people never learn that, as a principle, to expect swift response and efficiency from government is fatuous? Will we never heed the principle of subsidiarity (in which our fathers were bred), namely that no public agency should do what a private agency can do better, and that no higher-level public agency should attempt to do what a lower-level agency can do better β that to the degree the principle of subsidiarity is violated, first local government, the state government, and then federal government wax in inefficiency? Moreover, the more powers that are invested in government, and the more powers that are wielded by government, the less well does government discharge its primary responsibilities, which are (1) defence of the commonwealth, (2) protection of the rights of citizens, and (3) support of just order.
Decentralization, or decentralising governance, refers to the restructuring or reorganisation of authority so that there is a system of co-responsibility between institutions of governance at the central, regional and local levels according to the principle of subsidiarity, thus increasing the overall quality and effectiveness of the system of governance, while increasing the authority and capacities of sub-national levels.
Systemic failures of the type seen in the crash of 2007/08 can largely be avoided, since diverse solutions to common problems avoid common mode failure.
Individual and group initiative is given maximum scope to solve problems.
The systemic problem of moral hazard is largely avoided. In particular, the vexing problem of atrophied local initiative/responsibility is avoided.
When a genuine principle of liberty is recognised by a higher political entity but not all subsidiary entities, implementation of that principle can be delayed at the more local level.
When a genuinely efficacious economic principle is recognised by a higher political entity, but not all subsidiary entities, implementation of that principle can be delayed at the more local level.
In areas where the local use of common resources has a broad regional, or even global, impact (such as in the generation of pollutants), higher levels of authority may have a natural mandate to supersede local authority.
Subsidiarity is perhaps presently best known as a general principle of European Union law. According to this principle, the EU may only act (i.e. make laws) where action of individual countries is insufficient. The principle was established in the 1992 Treaty of Maastricht. However, at the local level it was already a key element of the European Charter of Local Self-Government, an instrument of the Council of Europe promulgated in 1985 (see Article 4, Paragraph 3 of the Charter) (which states that the exercise of public responsibilities should be decentralised). Subsidiarity is related in essence to, but should not be confused with, the concept of a margin of appreciation.
A more descriptive analysis of the principle can be found in Protocol 2 to the European Treaties.
The Court of Justice of the European Union in Luxembourg is the authority that has to decide whether a regulation falls within the exclusive competence[a] of the Union, as defined by the Treaty of European Union and its predecessors. As the concept of subsidiarity has a political as well as a legal dimension, the Court of Justice has a reserved attitude toward judging whether EU legislation is consistent with the concept. The Court will examine only marginally whether the principle is fulfilled. A detailed explanation of the legislation is not required; it is enough that the EU institutions explain why national legislation seems inadequate and that Union law has an added value.
Furthermore, in the fifth recital the Parliament and the Council stated that the action taken by the Member States in response to the Commission's Recommendation has not fully achieved the desired result. The Community legislature therefore found that the objective of its action could not be achieved sufficiently by the Member States.
Consequently, it is apparent that, on any view, the Parliament and the Council did explain why they considered that their action was in conformity with the principle of subsidiarity and, accordingly, that they complied with the obligation to give reasons as required under Article 190 of the Treaty. An express reference to that principle cannot be required.
^ Oxford English Dictionary. https://en.oxforddictionaries.com/definition/subsidiarity Definition: "[mass noun] (in politics) the principle that a central authority should have a subsidiary function, performing only those tasks which cannot be performed at a more local level:"
^ Early German usage: SubsidiaritΓ€t(1809 or earlier in legal use; 1931 in the context of Catholic social doctrine, in Β§80 of Rundschreiben ΓΌber die gesellschaftliche Ordnung ("Encyclical concerning the societal order"), the German version of Pope Pius XI's encyclical Quadragesimo anno (1931))".
^ "Das SubsidiaritΓ€tsprinzip als wirtschaftliches Ordnungsprinzip", Wirtschaftliche Entwicklung und soziale Ordnung. Degenfeld-Festschrift, Vienna: von Lagler and J. Messner, 1952, pp. 81β92 , cited in Helmut Zenz, DE .
^ Bak, Hans; Holthoon, F. L. van; Krabbendam, Hans; Edward L. Ayers (1 January 1996). Social and Secure?: Politics and Culture of the Welfare State: a Comparative Inquiry. VU University Press. ISBN 9789053834589. The Christian democrats promoted a corporatist welfare state, based on the principles of the so-called "sphere sovereignty" and "subsidiarity" in social policy.
^ Schmidt, Vivien A, Democratizing France: The Political and Administrative History of Decentralization, p. 10 .
^ Segell, Glen (2000). Is There a Third Way?. Glen Segell Publishers. p. 80. ISBN 9781901414189. When the Dutch Protestant and Catholic parties combined, to form the Christian Democrats, the two parties agreed that the principles of sphere sovereignty and subsidiarity boiled down to the same thing.
^ Reid Buckley, An American Family β The Buckleys, Simon & Schuster, 2008, p. 177.
^ Decentralization: A Sampling of Definitions, Joint UNDP (United Nations Development Programme)-Government of Germany evaluation of the UNDP role in decentralization and local governance, at the United Nations Development Programme website, October 1999, pp. 2, 16, 26.
^ a b Macrory, Richard, 2008, Regulation, Enforcement and Governance in Environmental Law, Cameron May, London, p. 657.
^ Shelton, Dinah (2003). "The Boundaries of Human Rights Jurisdiction in Europe". Duke Journal of Comparative and International Law. Duke University School of Law. 13 (1): 95β154. Retrieved 17 April 2017.
^ Protocol 2 to the European Treaties.
^ Exclusive competencies are those matters that the member states have agreed with each other by treaty are those that they should achieve jointly (typically through the European Commission). All other matters remain as "national competences" (each member decides its own policy independently). International trade agreements are an example of the former, taxation is an example of the latter.
Look up subsidiarity in Wiktionary, the free dictionary.
This page was last edited on 7 September 2018, at 21:45 (UTC). |
\documentclass{report}
\title{\textsc{A Digital Signal Processing Report on \\ \Huge Image Steganography using LSB Matching}}
\author{\\\\ {\bf \underline{Member Details}} \\\\{\bf MEMBER-1: Himanshu Sharma} \\ {\bf Roll Number: $1610110149$} \\ {\bf Dept. of Electrical Engineering (ECE)} \\\\{\bf MEMBER-2: Mridul Agarwal} \\ {\bf Roll Number: $1610110199$} \\ {\bf Dept. of Electrical Engineering (EEE)} \\\\ {\bf MEMBER-3: Vedansh Gupta} \\ {\bf Roll Number: $1610110429$} \\ {\bf Dept. of Electrical Engineering (ECE)}\\\\\\\\ {\bf Shiv Nadar University,} \\ {\bf Gautam Buddh Nagar, Greater Noida, Uttar Pradesh 201314} \\\\\\ \textsc{ Under the guidance of Professor Vijay K. Chakka}}
\date{}
\usepackage[margin=0.8in]{geometry}
\usepackage{graphicx}
\usepackage{float}
\usepackage{amsmath}
\usepackage{algorithm}
\usepackage[noend]{algpseudocode}
\usepackage{fourier-orns}
\usepackage{csquotes}
\newenvironment{ppl}{\fontfamily{pcr}\selectfont}{\par}
\begin{document}
\maketitle
\pagenumbering{gobble}
\renewcommand{\thesection}{\arabic{section}}
% include the paper here.
\section{Paper Comprehension by Member 1 - Himanshu Sharma}
{\it Image Steganography} is the art of hiding information inside a digital image. Steagnography does not restrict to only plain text but it includes text, video, audio and image hiding also. In this report, the author has restricted himself to text data type only. Even when doing text embedding, there are immense possible algorithms to choose from. The most popular among them is the LSB replacement or Least Significant Bit replacement. With this type of algorithm, the least significant bit of each pixel (only one channel out of the RGB pallete) is changed by a bit decided according to the message bit. Therefore, we should not expect much change in the image after data hiding because the least significant bit is changed. For example, lets take a blue channel with value of 145. In binary, it is equivalent of 10010001. If by some technique, the last bit is changed to 0, then the decimal equivalent would become 144, which does not change the image much. This is the power of LSB replacement. It basically relies on the fact that hiding message bits in the LSB of each pixel does not affect the image much, in fact, the image practically remains as it is. There are, however, techniques now in modern signal processing science which can detect that some data is hidden in the image or not, a field called {\it steganalysis}. A successful hiding algorithm would be that, that would allow high payload and still look similar to the original image.
\subsection{LSB Matching}
In LSB matching, if the message bit does not match with the pixel's LSB, then $\pm 1$ is randomly added to that pixel value. Unlike LSB replacement, where the pixel's LSB is just replaced by the message bit, here, a set of conditions are used to modify the pixel. LSB Matching could not be detected using the techniques used to detect LSB replacement. However, now it has been proved that LSB matching acts like a low pass filter on the digital images and therefore, this fact is utilized to detect whether LSB matching is applied on an image or not. \par Usually, two consecutive pixels are taken alongwith two consecutive message bits. The consecutive pairs are chosen randomly based on the {\it pseudo-random number generator} (PRNG). If the LSB matching is used as it is, then any pixel could be chosen with every pixel pair having equal chances of being selected by the algorithm. This has a flaw. This type of approach makes it difficult to disguise a changed pixel to the pixel surrounding it. For example, if lot of pixels are changed in a close vicinity, then they could be easily identified if the region is light in color, like sky which is light blue in color. The paper chosen here tries to rectify this problem by chosing those pixels on the image which lie on the edges. Edges are usually sharper than the surrounding regions and therefore its not easy to identify the change in pixel color on the edges. Similar papers have already been published. All of them suggest to use what is called the {\it pixel-value difference} (PVD). \par In PVD, what we do is that when we try to hide a message bit in a pixel's LSB, the pixel value is compared with its neighbouring pixels. If the difference is large, then more bits can be accomodated in that region without them being easily identified. Why? Because if there is a large difference in the pixel values then on changing the pixel value will not generate any significant difference. For example, if a pixel that is to be modified has a value of 45 and it's neighbouring pixel has a value of 145, then the difference $\displaystyle \Delta = 145-45=100$. Now, if by some technique if this pixel value if changed to 46, then the difference would become $\displaystyle \Delta' = 145-46 = 99$. To a human eye, this difference is not accountable. PVD is a good approach, in fact, far more better that PRNG because it utilizes the fact that sharp changes can be used to hide the information. \par In both LSB replacement and LSB matching, a travelling order is generated using PRNG which also acts as a key for decoding the stego-image. In both of these algorithms, the LSB of the selected pixel becomes equal to the message bit. According to the algorithm, if the two consecutive pixels are $x_{i}$ and $x_{i+1}$ and the consecutive message bits are $m_{i}$ and $m_{i+1}$, then the pixels are modified such that $x_{i}$ becomes $x_{i}^{'}$ and $x_{i+1}$ becomes $x_{i+1}^{'}$ and the following relation holds.
\begin{center}
$ \displaystyle LSB(x_{i}^{'})=m_{i} \textrm{ and } LSB \Big( \Big\lfloor{\frac{x_{i}^{'}}{2}}\Big \rfloor + x_{i+1}^{'} \Big) = m_{i+1}$
\end{center}
From experiments done by the authors of the paper it has been shown that even if the cover image has rough textures, it will still have smooth regions in every $5 \times 5$ non-overlapping blocks. So, if by any chance, a pixel is selected in that region for message hiding then it could be easily identified. This is what the paper aims to solve.
\subsection{Bit Planes}
We now come to a brief discussion on what is called the bit planes. Bit planes help us visualize the importance of the significant bits used to represent the images. They clearly display the fact that altering the LSB of pixels is less damaging to the original image than altering the MSB of the same original image. Before we discuss them in great detail, let me show an example. Suppose we have a cover image which is shown below.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{images/desktop.jpg}
\caption{Cover Image}
\end{figure}
Its bit planes are shown below. The LSB plane and the MSB planes are what we display first.
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/plane1.png}
\caption{MSB Plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/plane8.png}
\caption{LSB Plane}
\end{minipage}
\end{figure}
Changing a pixel value in the MSB plane is more prone to human eye detection as compared to the LSB plane. This is because a MSB plane has more structured black and white regions, and so, changing any pixel value, for example say, that a pixel is changed from white to black in the white region, then it would be easily detected. Whereas, on a LSB plane, the black and white regions are more uniformly aligned, just like {\it white noise} on a television screen. Inverting any color on the LSB plane won't change the visual effect. Hence, it is always advised to change the LSB plane in stegnography.
\par The idea is to generate these images using single values. Since we have 3 values associated with a single pixel {\it (R, G, B)}, it would be a great idea to convert the image to a grayscale image, so that each pixel is represented by a single value. Now each pixel value, which initially was a three dimensional array, is converted to a one dimensional array having a single value. This matrix of single values (grayscale image) is passed to a function which converts these decimal pixel values to 8 bit binary number. As per the request of the user, the $i^{th}$ bit from each binary pixel is accessed. So, for example, if a pixel has 8-bit binary representation as 11110101 and the user asks for fifth bit plane, then the fifth bit from starting would be accessed and that is 0 for this example. These values are stored in another matrix. So now we have a matrix with only 1 and 0s. We now multiply each and every value of the matrix by 255. The matrix becomes full of 255 and 0s. Zero represent black color and 255 represent white color. When saved, this image is a black and white $i^{th}$ bit plane.
\par Below, I have shown all the bit planes of the above cover image.
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/1.png}
\caption{$1^{st}$ bit plane or the MSB Plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/2.png}
\caption{$2^{nd}$ bit plane}
\end{minipage}
\vfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/3.png}
\caption{$3^{rd}$ bit plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/4.png}
\caption{$4^{th}$ bit plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/5.png}
\caption{$5^{th}$ bit plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/6.png}
\caption{$6^{th}$ bit plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/7.png}
\caption{$7^{th}$ bit plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/8.png}
\caption{$8^{th}$ bit plane or the LSB plane}
\end{minipage}
\end{figure}
Clearly, as we increase the bit plane number, we are bound to get more better results. Increasing the bit plane number implies less human eye detection in case of a pixel change. Now let us see the mathematical reasoning behind the bit planes. \newpage Let us define a closed form expression which converts a binary number to its decimal equivalent.
\begin{equation}
\displaystyle d(n) =\sum_{i=1}^{n}g(n+1-i,\textrm{ } b)2^{n-i}
\end{equation}
where $g(k, b)$ returns the $k^{th}$ bit from left of the binary number $b$ and $n$ denotes the length of the binary number. Here we are dealing with $n=8$. Let us say that the $j^{th}$ bit is changed while doing some steganography. The $j^{th}$ bit could be LSB or MSB or any other bit in between, we are not concerned about that for now. So, we know that $ 1 \leq j \leq n$. Let the original binary number be $\alpha$ and the binary number after the changed bit be denoted by $\beta$. Hence we get,
\begin{center}
$\displaystyle d_{1}(n) =\sum_{i=1}^{n}g(n+1-i,\textrm{ } \alpha)2^{n-i}$ and
$\displaystyle d_{2}(n) =\sum_{i=1}^{n}g(n+1-i,\textrm{ } \beta)2^{n-i}$
\end{center}
If we take the difference $\Delta d(n) = (d_{1} - d_{2})(n)$, the we get,
\begin{equation}
\displaystyle \Delta d(n) =\sum_{i=1}^{n}g(n+1-i,\textrm{ } \alpha)2^{n-i} - \sum_{i=1}^{n}g(n+1-i,\textrm{ } \beta)2^{n-i}
\end{equation}
Assuming all other bits of $\alpha$ and $\beta$ to be same except for the $j^{th}$ bit, we get,
\begin{center}
$\displaystyle \Delta d(n) = g(n+1-j, \alpha)2^{n-j} - g(n+1-j, \beta)2^{n-j} = 2^{n-j} \Delta g(n, j)$
\end{center}
where, $\Delta g(n, j) = g(n+1-j, \alpha) - g(n+1-j, \beta)$ and thus we get,
\begin{equation}
\displaystyle \delta (n) = |\Delta d(n)| = 2^{n-j}|g(n, j)|
\end{equation}
Note that $\Delta d(n)$ is the difference in the decimal values of the binary numbers which can be positive, negative or 0, $\delta (n)$ is always non-negative. Now, $|\Delta g(n, j)|$ can be defined as follows;
\[
|\Delta g(n, j)| =
\begin{cases}
1, & j^{th} \text{ bit of $\alpha$ and $\beta$ are different} \\
0, & \text{otherwise}
\end{cases}
\]
The {\it otherwise} case is trivial because if there is no difference in even the $j^{th}$ bit of these binary numbers, then both of them are identical. This means that the number was not changed. But if the number has change, then $|\Delta g(n, j)|$, for sure, is 1. Now, it is in this part we are concerned with the difference. We need to minimize this difference $\delta(n)$ because then only our stego-image will be statistically close to the original image. The value of $\delta(n)$ could be minimum only if $n=j$ (see equation 3), i.e., $\delta(n) = |\Delta g(n, j)|$ which is 1 if we assume that the pixel bit was changed. That is, the difference in decimal value has the minimum magnitude of 1, 0 being the trivial case that the binary numbers were never changed. Since the decimal difference value will decide the grayscale of the stego-image, we want to minimize it. This proves that if we want statistically similar features in the stego-image, we should take $j=n$, i.e., we should consider the LSB only, because that would affect the orignal decimal value by $\pm 1$ only. \par If we do consider $j=1$, then $\delta(n) = 2^{n-1} |\Delta g(n, 1)|$. Again, $\Delta g(n, 1) = \pm 1$ if we consider that the pixel value was changed. However, it is clear that $2^{n-1}$ will yield the maximum difference, and therefore, $j=1$ should never be taken, or, MSB should never be changed.
\subsection{The Algorithm}
The paper provides a scheme to optimize the traditional LSB algorithm. It covers an extra mile by embedding the key information in the stego-image itself. We can also say that first the pixels are changed according to the algorithm and then the key information is embedded into the image. Finally, the image thus obtained is the stego-image. \par Let me now explain how the data embedding is done in the image.
\newpage
\underline{\large Data Embedding} \\
\par Data embedding in the image starts by a new process which is generally not found in traditional steganography algorithms. First the image is divided into non-overlapping blocks of size $B \times B$ (as per the paper). Then we rotate each and every individual block by some random angle $\theta$, where $\theta \in \{0, 90, 180, 270\}$ in degrees. \par Dividing the image into non-overlapping blocks and then rotating each block by a random angle increases the security. That is because if we rotate the image before we embedd message bits in it and then later on rotate back those blocks to form the cover-like image, we have actually gained a great deal of security as compared to the image on which the embedding was done directly. Think about it, since the image is rotated back after embedding, all the embedded bits have been shuffled. Now, if some attacker tries to bruteforce, then even if he guesses the correct key, he won't be able to get the correct message. \par Now, the image is raster scanned. Raster scan is like converting an image into a row vector. In language like Python which I am using, this transformation can be achieved by using the statement \texttt{img.flatten()}, where \texttt{img} is the image as a numpy array. To read about numpy, please visit the docs. After this step, consecutive pixels are selected in pairs and following set is defined.
\begin{equation}
S(t) = \{(x_{i}, x_{i+1})| |x_{i} - x_{i+1}| \geq t, \forall (x_{i}, x_{i+1}) \in V \}
\end{equation}
where $V$ is the raster scanned image and $t$ is some value in the set $\{0, 1, 2, ..., 31 \}$. In other words, $S(t)$ is the set of all those consecutive pixels in the raster scanned image which have a differece greater than or equal to the parameter $t$. After this, the threshold value $T$ is calculated by the following method.
\begin{equation}
T = \textrm{argmax}_{t}\{2 \times n(S(t)) \geq n(M)\}
\end{equation}
where, $n(S)$ denotes the number of elements in $S$ and $M$ is the message. This equation, that is, equation 5 checks whether the message bits can be embedded in the image or not. Now let us understand what this equation actually means. Suppose a hypothetical case of $S(t) = \{ (x_{1}, x_{2}), (x_{3}, x_{4}), (x_{5}, x_{6}) \}$ for some $t=t_{1}$ and let $M$ be a 5 bit stream. Then, we find that $2 \times 3 \geq 5$ where $n(S(t)) = 3$. We chose some other $t$ now which is $t=t_{2} > t_{1}$. For this, let us say that we get $n(S(t)) = 2$, impling $2 \times 2 \geq 5$ which is false and therefore $T=t_{1}$. This is what equation 5 tries to say. It gives us that maximum $t$ for which the condition in the equaiton holds true. It is clear why we are multiplying by $2$. Thats because each pixel can hold one bit from the message stream and $S$ contains a tuple of such pixels. In total, we have twice the length pixels. A good observation is that if $T=0 \textrm{ }\exists \textrm{ }t$, then that would mean that our equation 5 is satisfied only for max $t=0$. Hence, our equation 4 would become,
\begin{center}
$ S(0) = \{(x_{i}, x_{i+1})| |x_{i} - x_{i+1}| \geq 0, \forall (x_{i}, x_{i+1}) \in V \} $
\end{center}
This is what a conventional LSB steganography technique does. It just embedds the message stream without thinking about the difference between the consecutive pixels. \par We now come to the discusssion of the main pseudocode which actually embedds the data in the image. Below, I have shown the algorithm used for LSB matching.
\begin{algorithm}
\caption{LSB Matching}\label{euclid}
\begin{algorithmic}[1]
\Procedure{hide}{}
\If {$LSB(x_{i}) = m_{i}$}
\If {$f(x_{i}, x_{i+1})=m_{i+1}$}
\State $(x_{i}^{'}, x_{i+1}^{'}) = (x_{i}, x_{i+1})$
\Else
\State $(x_{i}^{'}, x_{i+1}^{'}) = (x_{i}, x_{i+1}+r)$
\EndIf
\EndIf
\If {$LSB(x_{i})\neq m_{i}$}
\If {$f(x_{i}-1, x_{i+1})=m_{i+1}$}
\State $(x_{i}^{'}, x_{i+1}^{'}) = (x_{i}-1, x_{i+1})$
\Else
\State $(x_{i}^{'}, x_{i+1}^{'}) = (x_{i}+1, x_{i+1})$
\EndIf
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
\\ This algorithm is used for hiding the text data inside the image. $r$ is any random value in $\{ 1, -1\}$. Where, $(x_{i}^{'}, x_{i+1}^{'})$ are the new pixel values after data hiding. After this, the image blocks are rotated back to get the orignal image like image back and the angle values, the threshold $T$ and the block size are bundled into a binary file and returned as the key, back to the user. \newpage
\underline{\large Data Extraction} \\
\par Now, to get back the message stream, the image is first divided into blocks of size $B \times B$ again and then each of these blocks is rotated by the angles provided in the key file. Then those pixels are found in which data hiding was actually done using the inequality $|x_{i+1}-x_{i}|\geq T$. Now, to get back two units of the stream, the formula given in section 1.1 is used (refer section 1.1).
\subsection{Conclusions}
In the case of data embedding, the lower the value of $T$, more number of bits from the message can be embedded in the cover image. The following reasoning proves this; if we have two thresholds $T_{1}$ and $T_{2}$ with $T_{1} < T_{2}$, then this would mean
\begin{center}
$\textrm{argmax}_{t}\{2 \times n(S_{1}(t)) \geq n(M)\} < \textrm{argmax}_{t}\{2 \times n(S_{2}(t)) \geq n(M)\}$
\end{center}
where,
\begin{center}
$ S_{1}(t) = \{ (x_{i}, x_{i+1})||x_{i} - x_{i+1}| > t_{1}, \forall (x_{i}, x_{i+1}) \in V\} $ \\
$ S_{2}(t) = \{ (x_{i}, x_{i+1})||x_{i} - x_{i+1}| > t_{2}, \forall (x_{i}, x_{i+1}) \in V\} $
\end{center}
Because $T_{1} < T_{2} \implies t_{1} < t_{2}$ and that would mean that $S_{1}$ has lesser threshold value when compared to $S_{2}$. This would mean that more pixels would be available which satisfies $|x_{i} - x_{i+1}| > t_{1}$ than the pixels which satisfy $|x_{i} - x_{i+1}| > t_{2}$. Hence, $S_{1}$ would contain more elements or tuples than $S_{2}$. This is what the very first statement is saying, that if, threshold is small, more pixels could be accomodated. \par With steganography in mind, in general, we always need to compare the images for steganalysis. For this we first calculate the {\it mean square error} (MSE) between the cover $(A)$ and the stego-image $(B)$ by the following relation,
\begin{equation}
MSE(A, B) = \frac{1}{N} \sum_{i=0}^{N} (A_{i} - B_{i})^{2}
\end{equation}
Where, the subscript $i$ denotes the $i^{th}$ pixel of the image.
Here, actually, the MSE is the noise introduced in the cover image after steganography has been performed on it. So, in decibels (dB), the Peak Signal to Noise Ratio (PSNR) is given by,
\begin{equation}
PSNR_{dB} = 10 \log_{10}\Bigg|\frac{max^{2}(A)}{MSE}\Bigg|
\end{equation}
Where $max(A)$ is the maximum pixel value in the image or the peak signal available in the image. \par The techinque used in this paper produces exactly identical image after doing steganography as was the original cover image. Below, I have shown the eighth bit plane of a cover image and its stego-image as an example.
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/konsole.png}
\caption{Cover Image LSB plane}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\includegraphics[width=\textwidth]{images/konsole-stego.png}
\caption{Stego Image LSB plane}
\end{minipage}
\end{figure}
Clearly, we see that both the bit planes are identical looking, although they are not in exact sense. If the planes are zoomed and then viewed, then it is easy to spot differences. These differences occur because of resizing done by the code and some due to the embedded text also. Since there is such an uniformity in the LSB planes of both cover and stego-image, its practically impossible for someone to spot the difference in the cover and stego-image. I thus conclude by saying that image steganography is successfully implemented by me. \underline{Full code base is written in Python and is available on my GitHub repository at} \begin{center} {\bf https://github.com/hmnhGeek/DSP-Project-Report-LaTeX} \end{center}
This report is also available with it's complete \LaTeX code in the same repository under the \texttt{Report} folder. I would like the reader to surely go through the repository once.
\\
\par {\bf My contributions include the following,}
\begin{enumerate}
\item Full code base in Python dedicated to this project.
\item Report writing done with \LaTeX.
\end{enumerate}
\section{Paper Comprehension by Member 2 - Mridul Agarwal}
The science which deals with the hidden communication is called steganography whereras steganlaysis aims to detect the presence of hidden secret message in the stego media. There are different kinds of steganographic techniques which are complex and which have strong and weak points in hiding the information in various file formats. The file format used in this paper is images. The images which are used to hide the message $(M)$ are called cover images whereas the altered (steganographed) images are called stegos. The message hidden here is a binary stream. \par A steganography method is secure only when the statistics of the cover image and the stego are similar to each other. Image steganography is of two types-lossy compression and lossless compression. Lossy compression may not preserve the original image where as lossless compression preserves the original image. Examples of lossless compression formats are GIF,BMP,PNG etc. and that of lossy compression are JPEG. Also embedding capacity is also an important factor in designing a steganographic algorithm. \par Most of the existing steganographic techniques, the choice of embedding positions within a cover image mainly depends on a pseudorandom number generator (PRNG) without considering the relationship between the image content itself and the size of the secret message. Here the researchers have extended the LSB matching revisited (LSBMR) image steganography and propose an edge adaptive scheme which can select the embedding regions according to the size of secret message and the difference between two consecutive pixels in the cover image.
\subsection{LSB Replacement}
In this embedding technique, only the LSB of the cover image pixel is overwritten with the secret bit stream according to a pseudorandom number generator (PRNG). As a result, some structural asymmetry is introduced, and thus it is very easy to detect the existence of hidden message even at a low embedding rate.
\subsection{LSB Matching (LSBM)}
LSB matching (LSBM) employs a minor modification to LSB replacement. If the secret bit does not match the LSB of the cover image, then $\pm 1$ is randomly added to the corresponding pixel value. Statistically, the probability of increasing or decreasing for each modified pixel value is the same and so the obvious asymmetry introduced by LSB replacement can be easily avoided. Therefore, the common approaches used to detect LSB replacement are totally ineffective at detecting the LSBM.
\subsection{LSB Matching Revisited}
Unlike LSB replacement and LSBM, which deal with the pixel values independently, LSB matching revisited (LSBMR) uses a pair of pixels as an embedding unit, in which the LSB of the first pixel carries one bit of secret message, and the relationship (oddβeven combination) of the two pixel values carries another bit of secret message. \par The regions located at the sharper edges present more complicated statistical features and are highly dependent on the image contents. It is more difficult to observe changes at the sharper edges than those in smooth regions.
\newpage
LSBMR applies a pixel pair $(x_{i}, x_{i+1})$ in the cover image as an embedding unit. After message embedding, the unit is modified as $(x_{i}^{'}, x_{i+1}^{'})$ in stego image which satisfies,
\begin{center}
$ \displaystyle LSB(x_{i}^{'})=m_{i} \textrm{ and } LSB \Big( \Big\lfloor{\frac{x_{i}^{'}}{2}}\Big \rfloor + x_{i+1}^{'} \Big) = m_{i+1}$
\end{center}
By using the relationship (oddβeven combination) of adjacent pixels, the modification rate of pixels in LSBMR would decrease compared with LSB replacement and LSBM at the same embedding rate. It also does not introduce the LSB replacement style asymmetry. Similarly, in data extraction, it first generates a traveling order by a PRNG with a shared key. And then for each embedding unit along the order, two bits can be extracted. The first secret bit is the LSB of the first pixel value, and the second bit can be obtained by calculating the relationship between the two pixels given by above formula. Our human vision is sensitive to slight changes in the smooth regions, while it can tolerate more severe changes in the edge regions.
This proposed scheme will first embed the secret bits into edge regions as far as possible while keeping other smooth regions as they are.
\subsection{Proposed Idea}
The flow diagram of proposed scheme is illustrated in figure below. In the data embedding stage, the scheme first initializes some parameters, which are used for data pre-processing and region selection, and then estimatesthe capacity of those selected regions. If the regions are large enough for hiding the given secret message , then data hiding is performed on the selected regions. Finally, it does some postprocessing to obtain the stego image. Otherwise the scheme needs to redo the parameters, and then repeats region selection and capacity estimation until can be embedded completely.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{images/proposed_scheme.png}
\caption{Proposed Scheme (as given in the paper)}
\end{figure}
They use the absolute difference between two adjacent pixels as the criterion for region selection, and use LSBMR as the data hiding algorithm.
\par The {\bf step 1} of the algorithm mainly first divides the whole image into nonoverlapping blocks of $B_{z}\times B_{z}$ pixels and then rotates each block by $\{0,90,180,270\}$ as determined by a secret key.The resultant image is reshaped into an row vector which is then divided into non overlapping embedding units (two consecutive pixels).
\par {\bf Step 2} calculates the threshold $T$ for region selection using given formula.
\par {\bf Step 3} Implements data hiding according to four discussed cases.If the modifications are out of constraints then they are readjusted.
\par {\bf Step 4} is similar to step 1 except that the blocks are rotated by same degrees but in opposite direction.
The parameters for $(T,B_{z})$ are hidden in a preset area.
\par {\bf Analysis} - One of the important properties of this steganographic method is that it can first choose the sharper edge regions for data hiding according to the size of the secret message by adjusting a threshold $T$.
The experimental results are summarized in table-3 for different embedding rates in the paper itself, show that it has the minimum accuracy of being detected in most of the cases amongst the seven steganographic algorithms.
\newpage
\section{Paper Comprehension by Member 3 - Vedansh Gupta}
In this
paper, we expand the LSB matching revisited image steganography propose an edge adaptive scheme which can select the
embedding regions according to the size of secret message and the difference between two consecutive pixels in the cover image.For lower embedding rates, only sharper edge regions are used while keeping the other smoother regions as they are.
LSB replacement is a well-known steganographic method. In
this embedding scheme, only the LSB plane of the cover image
is overwritten with the secret bit stream according to a pseudorandom
number generator (PRNG). \par LSB matching (LSBM) employs a minor modification to LSB
replacement. If the secret bit does not match the LSB of the
cover image, then $\pm 1$ is randomly added to the corresponding
pixel value. The experimental results demonstrated that
the method was more effective on uncompressed grayscale im
ages. \par LSB matching revisited (LSBMR) uses a pair of pixels as an embedding unit, in which the LSB
of the first pixel carries one bit of secret message, and the relationship
(oddβeven combination) of the two pixel values carries
another bit of secret message. In such a way, the modification
rate of pixels can decrease from 0.5 to 0.375 bits/pixel
(bpp) in the case of a maximum embedding rate, meaning fewer
changes to the cover image at the same payload compared to
LSB replacement and LSBM. The typical LSB-based approaches, including LSB replacement,
LSBM, and LSBMR, deal with each given pixel/pixelpair
without considering the difference between the pixel and
its neighbors. \par The pixel-value differencing (PVD)-based scheme
is another kind of edge adaptive scheme, in
which the number of embedded bits is determined by the difference
between a pixel and its neighbor. The larger the difference,
the larger the number of secret bits that can be embedded.
Usually, PVD-based approaches can provide a larger
embedding capacity. Generally, the regions
located at the sharper edges present more complicated statistical
features and are highly dependent on the image contents.
Moreover, it is more difficult to observe changes at the sharper
edges than those in smooth regions. \par For LSB replacement, the secret bit
simply overwrites the LSB of the pixel, i.e., the first bit plane,
while the higher bit planes
are preserved. For the LSBM
scheme, if the secret bit is not equal to the LSB of the given
pixel, then
1 is added randomly to the pixel while keeping the
altered pixel in the range of $[0, 255]$.
\par
Our human vision is sensitive to slight changes in the smooth
regions, while it can tolerate more severe changes in the edge
regions.The
basic idea of PVD-based approaches is to first divide the cover
image into many nonoverlapping units with two consecutive
pixels and then deal with the embedding unit along a pseudorandom
order which is also determined by a PRNG.
The larger the difference between the two pixels, the larger the number of
secret bits that can be embedded into the unit.
\par We find that uncompressed natural images usually
contain some flat regions (it may be as small as 5X5 and it
is hard to notice), If we embed data into these regions, the LSB of stego images would become more and more random.
Our proposed scheme will first embed the secret bits into edge regions as far
as possible while keeping other smooth regions as they are.
We have not much focused on the security and safety of our data and image.
In this paper, we apply such a region adaptive scheme to the
spatial LSB domain. We use the absolute difference between
two adjacent pixels as the criterion for region selection, and use
LSBMR as the data hiding algorithm. We have taken two parameters for data hiding approach. The first one is the block size for block dividing in data preprocessing;
another is the threshold for embedding region selection.
The larger the
number of secret bits to be embedded, the smaller the threshold $T$
becomes, which means that more embedding units with
lower gradients in the cover image can be released.
When $T$ is 0, all the embedding units within the cover become
available. In such a case, our method can achieve the maximum
embedding capacity of 100 \%.
\par Our technique of implementation takes LSB of pixels in a ordered manner i.e. divide the image into blocks, convert them into a row of block vector (just for the ease of computation), rotate each block by a certain random angle, store data using LSBMR technique and bring it back to as the blocks were placed in the previous image.
Rotating images and then storing data could make the data scrambled and somebody who tries to debug could not be able to get the data in a particular manner.
What more could we do was when the blocks were made, we couldβve traversed the last two or three blocks from all the edge (depending on the bits of the message) to a certain place keeping a record of it and then rotating them and storing the data. Then moving them back to their original place.
This could make debugging even more difficult.
\newpage
\section{Conclusions and Results}
We conclude the report by saying that what we aimed at while implemenenting the paper has been achieved. We ought to target image steganography and for that we tried using edge adaptive techniques. Although, the author of the code did not implemented the paper in its full essence but the code is working as expected by the author. Few points that must be noted about what implementations have been achieved and what has been not before we conclude with the results. Note: The code has been authored by Himanshu Sharma (Member - 1). \\
\underline{\large What has been achieved in the Python code base}
\begin{enumerate}
\item Graphical User Interface has been developed so that anyone can use this software piece.
\item Division of image into blocks of dimension $B \times B$ and then their random rotations has been implemented so as to increase the security as given in the paper.
\item Bit Planes were implemented as they were given in the paper. A special Python script is dedicated for this only.
\item LSB Matching algorithm as given in the paper has been implemented successfully.
\item Image steganography has been done successfully.
\end{enumerate}
\par
\underline{\large What the author was unable to implement in the code}
\begin{enumerate}
\item Because the ``argmin" and ``argmax" operations seemed difficult and therefore steps 2 and 3 of the algorithm (refer the paper) suggested in the paper were not implemented precisely. Instead, a linear form of steganography was done as a substitute to these steps.
\end{enumerate}
If we have to summarize, we summarize the results by operating on the popular test image used in the field of image processing, i.e., the popular photograph of model Lenna S\"{o}derberg.
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/lenna.png}
\caption{Cover Image}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/stego-lenna.png}
\caption{Stego Image}
\end{minipage}
\end{figure}
Clearly, there is no difference in the cover and the stego image. Just to convey the information, I am also showing the MSB and LSB planes of the cover and the stego-image. Note how easy would it have been for an attacker if we would store our data inside the MSB plane (see the figures below). The following text is hidden in the stego image.
\begin{center}
\begin{ppl}
Steganography includes the concealment of information within computer files. In digital steganography, electronic communications may include steganographic coding inside of a transport layer, such as a document file, image file, program or protocol. Media files are ideal for steganographic transmission because of their large size. For example, a sender might start with an innocuous image file and adjust the color of every hundredth pixel to correspond to a letter in the alphabet. The change is so subtle that someone who is not specifically looking for it is unlikely to notice the change.
\end{ppl}
\end{center}
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/covermsb.png}
\caption{MSB plane of the Cover}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/coverlsb.png}
\caption{LSB plane of the Cover}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/stegomsb.png}
\caption{MSB plane of the Stego}
\end{minipage}
\hfill
\begin{minipage}{0.46\linewidth}
\centering
\includegraphics[width=0.7\textwidth]{images/stegolsb.png}
\caption{LSB plane of the Stego}
\end{minipage}
\end{figure}
We conclude by saying that image steganography has been implemented successfully and we got the results as expected.
\\
\par \begin{center} \textbf{IMPORTANT NOTE} \end{center}
\par Full code base is available at Member 1's Github repository - {\bf https://github.com/hmnhGeek/DSP-Project-Report-LaTeX}. We request the reader to please visit and see the code base for atleast once.
\begin{center}
\decosix\decosix\decosix
\end{center}
\end{document} |
//==================================================================================================
/**
Copyright 2016 NumScale SAS
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
**/
//==================================================================================================
#ifndef BOOST_SIMD_ARCH_COMMON_SIMD_FUNCTION_GENMASKC_HPP_INCLUDED
#define BOOST_SIMD_ARCH_COMMON_SIMD_FUNCTION_GENMASKC_HPP_INCLUDED
#include <boost/simd/detail/overload.hpp>
#include <boost/simd/constant/allbits.hpp>
#include <boost/simd/function/bitwise_cast.hpp>
#include <boost/simd/function/complement.hpp>
#include <boost/simd/function/genmask.hpp>
#include <boost/simd/function/if_zero_else.hpp>
#include <boost/simd/function/is_eqz.hpp>
namespace boost { namespace simd { namespace ext
{
namespace bd = boost::dispatch;
namespace bs = boost::simd;
BOOST_DISPATCH_OVERLOAD_IF( genmaskc_
, (typename A0, typename X)
, (detail::is_native<X>)
, bd::cpu_
, bs::pack_<bd::fundamental_<A0>, X>
)
{
BOOST_FORCEINLINE A0 operator()( const A0& a0) const BOOST_NOEXCEPT // TODO bool
{
return if_zero_else(a0, Allbits<A0>());
}
};
BOOST_DISPATCH_OVERLOAD_IF( genmaskc_
, (typename A0, typename X)
, (detail::is_native<X>)
, bd::cpu_
, bs::pack_<bd::arithmetic_<A0>, X>
)
{
BOOST_FORCEINLINE A0 operator()( const A0& a0) const BOOST_NOEXCEPT
{
return genmask(is_eqz(a0));
}
};
BOOST_DISPATCH_OVERLOAD_IF( genmaskc_
, (typename A0, typename X)
, (detail::is_native<X>)
, bd::cpu_
, bs::pack_<bs::logical_<A0>, X>
)
{
using result_t = as_arithmetic_t<A0>;
BOOST_FORCEINLINE result_t operator()( A0 const& a0 ) const BOOST_NOEXCEPT
{
return do_(a0, typename is_bitwise_logical<A0>::type{});
}
BOOST_FORCEINLINE result_t do_( A0 const& a0, tt::true_type const& ) const BOOST_NOEXCEPT
{
return complement(bitwise_cast<result_t>(a0));
}
BOOST_FORCEINLINE result_t do_( A0 const& a0, tt::false_type const& ) const BOOST_NOEXCEPT
{
return if_else(a0, result_t(0), Allbits<result_t>());
}
};
} } }
#endif
|
[STATEMENT]
lemma symmetry_preserves_per:
assumes "Per B P A" and
"B Midpoint A A'" and
"B Midpoint P P'"
shows "Per B P' A'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
obtain C where P1: "P Midpoint A C"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>C. P Midpoint A C \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using symmetric_point_construction
[PROOF STATE]
proof (prove)
using this:
\<exists>P'. ?A Midpoint ?P P'
goal (1 subgoal):
1. (\<And>C. P Midpoint A C \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
P Midpoint A C
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
obtain C' where P2: "B Midpoint C C'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>C'. B Midpoint C C' \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using symmetric_point_construction
[PROOF STATE]
proof (prove)
using this:
\<exists>P'. ?A Midpoint ?P P'
goal (1 subgoal):
1. (\<And>C'. B Midpoint C C' \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
B Midpoint C C'
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
have P3: "P' Midpoint A' C'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. P' Midpoint A' C'
[PROOF STEP]
using P1 P2 assms(2) assms(3) symmetry_preserves_midpoint
[PROOF STATE]
proof (prove)
using this:
P Midpoint A C
B Midpoint C C'
B Midpoint A A'
B Midpoint P P'
\<lbrakk>?Z Midpoint ?A ?D; ?Z Midpoint ?B ?E; ?Z Midpoint ?C ?F; ?B Midpoint ?A ?C\<rbrakk> \<Longrightarrow> ?E Midpoint ?D ?F
goal (1 subgoal):
1. P' Midpoint A' C'
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
P' Midpoint A' C'
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
have "Cong B A' B C'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. Cong B A' B C'
[PROOF STEP]
by (meson P1 P2 assms(1) assms(2) l7_16 l7_3_2 per_double_cong)
[PROOF STATE]
proof (state)
this:
Cong B A' B C'
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
Cong B A' B C'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
Cong B A' B C'
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
using P3 Per_def
[PROOF STATE]
proof (prove)
using this:
Cong B A' B C'
P' Midpoint A' C'
Per ?A ?B ?C \<equiv> \<exists>C'. ?B Midpoint ?C C' \<and> Cong ?A ?C ?A C'
goal (1 subgoal):
1. Per B P' A'
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
Per B P' A'
goal:
No subgoals!
[PROOF STEP]
qed |
{-# OPTIONS --type-in-type #-}
Ty : Set
Ty =
(Ty : Set)
(nat top bot : Ty)
(arr prod sum : Ty β Ty β Ty)
β Ty
nat : Ty; nat = Ξ» _ nat _ _ _ _ _ β nat
top : Ty; top = Ξ» _ _ top _ _ _ _ β top
bot : Ty; bot = Ξ» _ _ _ bot _ _ _ β bot
arr : Ty β Ty β Ty; arr
= Ξ» A B Ty nat top bot arr prod sum β
arr (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
prod : Ty β Ty β Ty; prod
= Ξ» A B Ty nat top bot arr prod sum β
prod (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
sum : Ty β Ty β Ty; sum
= Ξ» A B Ty nat top bot arr prod sum β
sum (A Ty nat top bot arr prod sum) (B Ty nat top bot arr prod sum)
Con : Set; Con
= (Con : Set)
(nil : Con)
(snoc : Con β Ty β Con)
β Con
nil : Con; nil
= Ξ» Con nil snoc β nil
snoc : Con β Ty β Con; snoc
= Ξ» Ξ A Con nil snoc β snoc (Ξ Con nil snoc) A
Var : Con β Ty β Set; Var
= Ξ» Ξ A β
(Var : Con β Ty β Set)
(vz : β Ξ A β Var (snoc Ξ A) A)
(vs : β Ξ B A β Var Ξ A β Var (snoc Ξ B) A)
β Var Ξ A
vz : β{Ξ A} β Var (snoc Ξ A) A; vz
= Ξ» Var vz vs β vz _ _
vs : β{Ξ B A} β Var Ξ A β Var (snoc Ξ B) A; vs
= Ξ» x Var vz vs β vs _ _ _ (x Var vz vs)
Tm : Con β Ty β Set; Tm
= Ξ» Ξ A β
(Tm : Con β Ty β Set)
(var : β Ξ A β Var Ξ A β Tm Ξ A)
(lam : β Ξ A B β Tm (snoc Ξ A) B β Tm Ξ (arr A B))
(app : β Ξ A B β Tm Ξ (arr A B) β Tm Ξ A β Tm Ξ B)
(tt : β Ξ β Tm Ξ top)
(pair : β Ξ A B β Tm Ξ A β Tm Ξ B β Tm Ξ (prod A B))
(fst : β Ξ A B β Tm Ξ (prod A B) β Tm Ξ A)
(snd : β Ξ A B β Tm Ξ (prod A B) β Tm Ξ B)
(left : β Ξ A B β Tm Ξ A β Tm Ξ (sum A B))
(right : β Ξ A B β Tm Ξ B β Tm Ξ (sum A B))
(case : β Ξ A B C β Tm Ξ (sum A B) β Tm Ξ (arr A C) β Tm Ξ (arr B C) β Tm Ξ C)
(zero : β Ξ β Tm Ξ nat)
(suc : β Ξ β Tm Ξ nat β Tm Ξ nat)
(rec : β Ξ A β Tm Ξ nat β Tm Ξ (arr nat (arr A A)) β Tm Ξ A β Tm Ξ A)
β Tm Ξ A
var : β{Ξ A} β Var Ξ A β Tm Ξ A; var
= Ξ» x Tm var lam app tt pair fst snd left right case zero suc rec β
var _ _ x
lam : β{Ξ A B} β Tm (snoc Ξ A) B β Tm Ξ (arr A B); lam
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
lam _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
app : β{Ξ A B} β Tm Ξ (arr A B) β Tm Ξ A β Tm Ξ B; app
= Ξ» t u Tm var lam app tt pair fst snd left right case zero suc rec β
app _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
tt : β{Ξ} β Tm Ξ top; tt
= Ξ» Tm var lam app tt pair fst snd left right case zero suc rec β tt _
pair : β{Ξ A B} β Tm Ξ A β Tm Ξ B β Tm Ξ (prod A B); pair
= Ξ» t u Tm var lam app tt pair fst snd left right case zero suc rec β
pair _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
fst : β{Ξ A B} β Tm Ξ (prod A B) β Tm Ξ A; fst
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
fst _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
snd : β{Ξ A B} β Tm Ξ (prod A B) β Tm Ξ B; snd
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
snd _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
left : β{Ξ A B} β Tm Ξ A β Tm Ξ (sum A B); left
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
left _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
right : β{Ξ A B} β Tm Ξ B β Tm Ξ (sum A B); right
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
right _ _ _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
case : β{Ξ A B C} β Tm Ξ (sum A B) β Tm Ξ (arr A C) β Tm Ξ (arr B C) β Tm Ξ C; case
= Ξ» t u v Tm var lam app tt pair fst snd left right case zero suc rec β
case _ _ _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
zero : β{Ξ} β Tm Ξ nat; zero
= Ξ» Tm var lam app tt pair fst snd left right case zero suc rec β zero _
suc : β{Ξ} β Tm Ξ nat β Tm Ξ nat; suc
= Ξ» t Tm var lam app tt pair fst snd left right case zero suc rec β
suc _ (t Tm var lam app tt pair fst snd left right case zero suc rec)
rec : β{Ξ A} β Tm Ξ nat β Tm Ξ (arr nat (arr A A)) β Tm Ξ A β Tm Ξ A; rec
= Ξ» t u v Tm var lam app tt pair fst snd left right case zero suc rec β
rec _ _
(t Tm var lam app tt pair fst snd left right case zero suc rec)
(u Tm var lam app tt pair fst snd left right case zero suc rec)
(v Tm var lam app tt pair fst snd left right case zero suc rec)
v0 : β{Ξ A} β Tm (snoc Ξ A) A; v0
= var vz
v1 : β{Ξ A B} β Tm (snoc (snoc Ξ A) B) A; v1
= var (vs vz)
v2 : β{Ξ A B C} β Tm (snoc (snoc (snoc Ξ A) B) C) A; v2
= var (vs (vs vz))
v3 : β{Ξ A B C D} β Tm (snoc (snoc (snoc (snoc Ξ A) B) C) D) A; v3
= var (vs (vs (vs vz)))
tbool : Ty; tbool
= sum top top
true : β{Ξ} β Tm Ξ tbool; true
= left tt
tfalse : β{Ξ} β Tm Ξ tbool; tfalse
= right tt
ifthenelse : β{Ξ A} β Tm Ξ (arr tbool (arr A (arr A A))); ifthenelse
= lam (lam (lam (case v2 (lam v2) (lam v1))))
times4 : β{Ξ A} β Tm Ξ (arr (arr A A) (arr A A)); times4
= lam (lam (app v1 (app v1 (app v1 (app v1 v0)))))
add : β{Ξ} β Tm Ξ (arr nat (arr nat nat)); add
= lam (rec v0
(lam (lam (lam (suc (app v1 v0)))))
(lam v0))
mul : β{Ξ} β Tm Ξ (arr nat (arr nat nat)); mul
= lam (rec v0
(lam (lam (lam (app (app add (app v1 v0)) v0))))
(lam zero))
fact : β{Ξ} β Tm Ξ (arr nat nat); fact
= lam (rec v0 (lam (lam (app (app mul (suc v1)) v0)))
(suc zero))
{-# OPTIONS --type-in-type #-}
Ty1 : Set
Ty1 =
(Ty1 : Set)
(nat top bot : Ty1)
(arr prod sum : Ty1 β Ty1 β Ty1)
β Ty1
nat1 : Ty1; nat1 = Ξ» _ nat1 _ _ _ _ _ β nat1
top1 : Ty1; top1 = Ξ» _ _ top1 _ _ _ _ β top1
bot1 : Ty1; bot1 = Ξ» _ _ _ bot1 _ _ _ β bot1
arr1 : Ty1 β Ty1 β Ty1; arr1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod sum β
arr1 (A Ty1 nat1 top1 bot1 arr1 prod sum) (B Ty1 nat1 top1 bot1 arr1 prod sum)
prod1 : Ty1 β Ty1 β Ty1; prod1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod1 sum β
prod1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum) (B Ty1 nat1 top1 bot1 arr1 prod1 sum)
sum1 : Ty1 β Ty1 β Ty1; sum1
= Ξ» A B Ty1 nat1 top1 bot1 arr1 prod1 sum1 β
sum1 (A Ty1 nat1 top1 bot1 arr1 prod1 sum1) (B Ty1 nat1 top1 bot1 arr1 prod1 sum1)
Con1 : Set; Con1
= (Con1 : Set)
(nil : Con1)
(snoc : Con1 β Ty1 β Con1)
β Con1
nil1 : Con1; nil1
= Ξ» Con1 nil1 snoc β nil1
snoc1 : Con1 β Ty1 β Con1; snoc1
= Ξ» Ξ A Con1 nil1 snoc1 β snoc1 (Ξ Con1 nil1 snoc1) A
Var1 : Con1 β Ty1 β Set; Var1
= Ξ» Ξ A β
(Var1 : Con1 β Ty1 β Set)
(vz : β Ξ A β Var1 (snoc1 Ξ A) A)
(vs : β Ξ B A β Var1 Ξ A β Var1 (snoc1 Ξ B) A)
β Var1 Ξ A
vz1 : β{Ξ A} β Var1 (snoc1 Ξ A) A; vz1
= Ξ» Var1 vz1 vs β vz1 _ _
vs1 : β{Ξ B A} β Var1 Ξ A β Var1 (snoc1 Ξ B) A; vs1
= Ξ» x Var1 vz1 vs1 β vs1 _ _ _ (x Var1 vz1 vs1)
Tm1 : Con1 β Ty1 β Set; Tm1
= Ξ» Ξ A β
(Tm1 : Con1 β Ty1 β Set)
(var : β Ξ A β Var1 Ξ A β Tm1 Ξ A)
(lam : β Ξ A B β Tm1 (snoc1 Ξ A) B β Tm1 Ξ (arr1 A B))
(app : β Ξ A B β Tm1 Ξ (arr1 A B) β Tm1 Ξ A β Tm1 Ξ B)
(tt : β Ξ β Tm1 Ξ top1)
(pair : β Ξ A B β Tm1 Ξ A β Tm1 Ξ B β Tm1 Ξ (prod1 A B))
(fst : β Ξ A B β Tm1 Ξ (prod1 A B) β Tm1 Ξ A)
(snd : β Ξ A B β Tm1 Ξ (prod1 A B) β Tm1 Ξ B)
(left : β Ξ A B β Tm1 Ξ A β Tm1 Ξ (sum1 A B))
(right : β Ξ A B β Tm1 Ξ B β Tm1 Ξ (sum1 A B))
(case : β Ξ A B C β Tm1 Ξ (sum1 A B) β Tm1 Ξ (arr1 A C) β Tm1 Ξ (arr1 B C) β Tm1 Ξ C)
(zero : β Ξ β Tm1 Ξ nat1)
(suc : β Ξ β Tm1 Ξ nat1 β Tm1 Ξ nat1)
(rec : β Ξ A β Tm1 Ξ nat1 β Tm1 Ξ (arr1 nat1 (arr1 A A)) β Tm1 Ξ A β Tm1 Ξ A)
β Tm1 Ξ A
var1 : β{Ξ A} β Var1 Ξ A β Tm1 Ξ A; var1
= Ξ» x Tm1 var1 lam app tt pair fst snd left right case zero suc rec β
var1 _ _ x
lam1 : β{Ξ A B} β Tm1 (snoc1 Ξ A) B β Tm1 Ξ (arr1 A B); lam1
= Ξ» t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec β
lam1 _ _ _ (t Tm1 var1 lam1 app tt pair fst snd left right case zero suc rec)
app1 : β{Ξ A B} β Tm1 Ξ (arr1 A B) β Tm1 Ξ A β Tm1 Ξ B; app1
= Ξ» t u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec β
app1 _ _ _ (t Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt pair fst snd left right case zero suc rec)
tt1 : β{Ξ} β Tm1 Ξ top1; tt1
= Ξ» Tm1 var1 lam1 app1 tt1 pair fst snd left right case zero suc rec β tt1 _
pair1 : β{Ξ A B} β Tm1 Ξ A β Tm1 Ξ B β Tm1 Ξ (prod1 A B); pair1
= Ξ» t u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec β
pair1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst snd left right case zero suc rec)
fst1 : β{Ξ A B} β Tm1 Ξ (prod1 A B) β Tm1 Ξ A; fst1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec β
fst1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd left right case zero suc rec)
snd1 : β{Ξ A B} β Tm1 Ξ (prod1 A B) β Tm1 Ξ B; snd1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec β
snd1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left right case zero suc rec)
left1 : β{Ξ A B} β Tm1 Ξ A β Tm1 Ξ (sum1 A B); left1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec β
left1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right case zero suc rec)
right1 : β{Ξ A B} β Tm1 Ξ B β Tm1 Ξ (sum1 A B); right1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec β
right1 _ _ _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case zero suc rec)
case1 : β{Ξ A B C} β Tm1 Ξ (sum1 A B) β Tm1 Ξ (arr1 A C) β Tm1 Ξ (arr1 B C) β Tm1 Ξ C; case1
= Ξ» t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec β
case1 _ _ _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero suc rec)
zero1 : β{Ξ} β Tm1 Ξ nat1; zero1
= Ξ» Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc rec β zero1 _
suc1 : β{Ξ} β Tm1 Ξ nat1 β Tm1 Ξ nat1; suc1
= Ξ» t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec β
suc1 _ (t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec)
rec1 : β{Ξ A} β Tm1 Ξ nat1 β Tm1 Ξ (arr1 nat1 (arr1 A A)) β Tm1 Ξ A β Tm1 Ξ A; rec1
= Ξ» t u v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1 β
rec1 _ _
(t Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(u Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
(v Tm1 var1 lam1 app1 tt1 pair1 fst1 snd1 left1 right1 case1 zero1 suc1 rec1)
v01 : β{Ξ A} β Tm1 (snoc1 Ξ A) A; v01
= var1 vz1
v11 : β{Ξ A B} β Tm1 (snoc1 (snoc1 Ξ A) B) A; v11
= var1 (vs1 vz1)
v21 : β{Ξ A B C} β Tm1 (snoc1 (snoc1 (snoc1 Ξ A) B) C) A; v21
= var1 (vs1 (vs1 vz1))
v31 : β{Ξ A B C D} β Tm1 (snoc1 (snoc1 (snoc1 (snoc1 Ξ A) B) C) D) A; v31
= var1 (vs1 (vs1 (vs1 vz1)))
tbool1 : Ty1; tbool1
= sum1 top1 top1
true1 : β{Ξ} β Tm1 Ξ tbool1; true1
= left1 tt1
tfalse1 : β{Ξ} β Tm1 Ξ tbool1; tfalse1
= right1 tt1
ifthenelse1 : β{Ξ A} β Tm1 Ξ (arr1 tbool1 (arr1 A (arr1 A A))); ifthenelse1
= lam1 (lam1 (lam1 (case1 v21 (lam1 v21) (lam1 v11))))
times41 : β{Ξ A} β Tm1 Ξ (arr1 (arr1 A A) (arr1 A A)); times41
= lam1 (lam1 (app1 v11 (app1 v11 (app1 v11 (app1 v11 v01)))))
add1 : β{Ξ} β Tm1 Ξ (arr1 nat1 (arr1 nat1 nat1)); add1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (suc1 (app1 v11 v01)))))
(lam1 v01))
mul1 : β{Ξ} β Tm1 Ξ (arr1 nat1 (arr1 nat1 nat1)); mul1
= lam1 (rec1 v01
(lam1 (lam1 (lam1 (app1 (app1 add1 (app1 v11 v01)) v01))))
(lam1 zero1))
fact1 : β{Ξ} β Tm1 Ξ (arr1 nat1 nat1); fact1
= lam1 (rec1 v01 (lam1 (lam1 (app1 (app1 mul1 (suc1 v11)) v01)))
(suc1 zero1))
{-# OPTIONS --type-in-type #-}
Ty2 : Set
Ty2 =
(Ty2 : Set)
(nat top bot : Ty2)
(arr prod sum : Ty2 β Ty2 β Ty2)
β Ty2
nat2 : Ty2; nat2 = Ξ» _ nat2 _ _ _ _ _ β nat2
top2 : Ty2; top2 = Ξ» _ _ top2 _ _ _ _ β top2
bot2 : Ty2; bot2 = Ξ» _ _ _ bot2 _ _ _ β bot2
arr2 : Ty2 β Ty2 β Ty2; arr2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod sum β
arr2 (A Ty2 nat2 top2 bot2 arr2 prod sum) (B Ty2 nat2 top2 bot2 arr2 prod sum)
prod2 : Ty2 β Ty2 β Ty2; prod2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod2 sum β
prod2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum) (B Ty2 nat2 top2 bot2 arr2 prod2 sum)
sum2 : Ty2 β Ty2 β Ty2; sum2
= Ξ» A B Ty2 nat2 top2 bot2 arr2 prod2 sum2 β
sum2 (A Ty2 nat2 top2 bot2 arr2 prod2 sum2) (B Ty2 nat2 top2 bot2 arr2 prod2 sum2)
Con2 : Set; Con2
= (Con2 : Set)
(nil : Con2)
(snoc : Con2 β Ty2 β Con2)
β Con2
nil2 : Con2; nil2
= Ξ» Con2 nil2 snoc β nil2
snoc2 : Con2 β Ty2 β Con2; snoc2
= Ξ» Ξ A Con2 nil2 snoc2 β snoc2 (Ξ Con2 nil2 snoc2) A
Var2 : Con2 β Ty2 β Set; Var2
= Ξ» Ξ A β
(Var2 : Con2 β Ty2 β Set)
(vz : β Ξ A β Var2 (snoc2 Ξ A) A)
(vs : β Ξ B A β Var2 Ξ A β Var2 (snoc2 Ξ B) A)
β Var2 Ξ A
vz2 : β{Ξ A} β Var2 (snoc2 Ξ A) A; vz2
= Ξ» Var2 vz2 vs β vz2 _ _
vs2 : β{Ξ B A} β Var2 Ξ A β Var2 (snoc2 Ξ B) A; vs2
= Ξ» x Var2 vz2 vs2 β vs2 _ _ _ (x Var2 vz2 vs2)
Tm2 : Con2 β Ty2 β Set; Tm2
= Ξ» Ξ A β
(Tm2 : Con2 β Ty2 β Set)
(var : β Ξ A β Var2 Ξ A β Tm2 Ξ A)
(lam : β Ξ A B β Tm2 (snoc2 Ξ A) B β Tm2 Ξ (arr2 A B))
(app : β Ξ A B β Tm2 Ξ (arr2 A B) β Tm2 Ξ A β Tm2 Ξ B)
(tt : β Ξ β Tm2 Ξ top2)
(pair : β Ξ A B β Tm2 Ξ A β Tm2 Ξ B β Tm2 Ξ (prod2 A B))
(fst : β Ξ A B β Tm2 Ξ (prod2 A B) β Tm2 Ξ A)
(snd : β Ξ A B β Tm2 Ξ (prod2 A B) β Tm2 Ξ B)
(left : β Ξ A B β Tm2 Ξ A β Tm2 Ξ (sum2 A B))
(right : β Ξ A B β Tm2 Ξ B β Tm2 Ξ (sum2 A B))
(case : β Ξ A B C β Tm2 Ξ (sum2 A B) β Tm2 Ξ (arr2 A C) β Tm2 Ξ (arr2 B C) β Tm2 Ξ C)
(zero : β Ξ β Tm2 Ξ nat2)
(suc : β Ξ β Tm2 Ξ nat2 β Tm2 Ξ nat2)
(rec : β Ξ A β Tm2 Ξ nat2 β Tm2 Ξ (arr2 nat2 (arr2 A A)) β Tm2 Ξ A β Tm2 Ξ A)
β Tm2 Ξ A
var2 : β{Ξ A} β Var2 Ξ A β Tm2 Ξ A; var2
= Ξ» x Tm2 var2 lam app tt pair fst snd left right case zero suc rec β
var2 _ _ x
lam2 : β{Ξ A B} β Tm2 (snoc2 Ξ A) B β Tm2 Ξ (arr2 A B); lam2
= Ξ» t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec β
lam2 _ _ _ (t Tm2 var2 lam2 app tt pair fst snd left right case zero suc rec)
app2 : β{Ξ A B} β Tm2 Ξ (arr2 A B) β Tm2 Ξ A β Tm2 Ξ B; app2
= Ξ» t u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec β
app2 _ _ _ (t Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt pair fst snd left right case zero suc rec)
tt2 : β{Ξ} β Tm2 Ξ top2; tt2
= Ξ» Tm2 var2 lam2 app2 tt2 pair fst snd left right case zero suc rec β tt2 _
pair2 : β{Ξ A B} β Tm2 Ξ A β Tm2 Ξ B β Tm2 Ξ (prod2 A B); pair2
= Ξ» t u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec β
pair2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst snd left right case zero suc rec)
fst2 : β{Ξ A B} β Tm2 Ξ (prod2 A B) β Tm2 Ξ A; fst2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec β
fst2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd left right case zero suc rec)
snd2 : β{Ξ A B} β Tm2 Ξ (prod2 A B) β Tm2 Ξ B; snd2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec β
snd2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left right case zero suc rec)
left2 : β{Ξ A B} β Tm2 Ξ A β Tm2 Ξ (sum2 A B); left2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec β
left2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right case zero suc rec)
right2 : β{Ξ A B} β Tm2 Ξ B β Tm2 Ξ (sum2 A B); right2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec β
right2 _ _ _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case zero suc rec)
case2 : β{Ξ A B C} β Tm2 Ξ (sum2 A B) β Tm2 Ξ (arr2 A C) β Tm2 Ξ (arr2 B C) β Tm2 Ξ C; case2
= Ξ» t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec β
case2 _ _ _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero suc rec)
zero2 : β{Ξ} β Tm2 Ξ nat2; zero2
= Ξ» Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc rec β zero2 _
suc2 : β{Ξ} β Tm2 Ξ nat2 β Tm2 Ξ nat2; suc2
= Ξ» t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec β
suc2 _ (t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec)
rec2 : β{Ξ A} β Tm2 Ξ nat2 β Tm2 Ξ (arr2 nat2 (arr2 A A)) β Tm2 Ξ A β Tm2 Ξ A; rec2
= Ξ» t u v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2 β
rec2 _ _
(t Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(u Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
(v Tm2 var2 lam2 app2 tt2 pair2 fst2 snd2 left2 right2 case2 zero2 suc2 rec2)
v02 : β{Ξ A} β Tm2 (snoc2 Ξ A) A; v02
= var2 vz2
v12 : β{Ξ A B} β Tm2 (snoc2 (snoc2 Ξ A) B) A; v12
= var2 (vs2 vz2)
v22 : β{Ξ A B C} β Tm2 (snoc2 (snoc2 (snoc2 Ξ A) B) C) A; v22
= var2 (vs2 (vs2 vz2))
v32 : β{Ξ A B C D} β Tm2 (snoc2 (snoc2 (snoc2 (snoc2 Ξ A) B) C) D) A; v32
= var2 (vs2 (vs2 (vs2 vz2)))
tbool2 : Ty2; tbool2
= sum2 top2 top2
true2 : β{Ξ} β Tm2 Ξ tbool2; true2
= left2 tt2
tfalse2 : β{Ξ} β Tm2 Ξ tbool2; tfalse2
= right2 tt2
ifthenelse2 : β{Ξ A} β Tm2 Ξ (arr2 tbool2 (arr2 A (arr2 A A))); ifthenelse2
= lam2 (lam2 (lam2 (case2 v22 (lam2 v22) (lam2 v12))))
times42 : β{Ξ A} β Tm2 Ξ (arr2 (arr2 A A) (arr2 A A)); times42
= lam2 (lam2 (app2 v12 (app2 v12 (app2 v12 (app2 v12 v02)))))
add2 : β{Ξ} β Tm2 Ξ (arr2 nat2 (arr2 nat2 nat2)); add2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (suc2 (app2 v12 v02)))))
(lam2 v02))
mul2 : β{Ξ} β Tm2 Ξ (arr2 nat2 (arr2 nat2 nat2)); mul2
= lam2 (rec2 v02
(lam2 (lam2 (lam2 (app2 (app2 add2 (app2 v12 v02)) v02))))
(lam2 zero2))
fact2 : β{Ξ} β Tm2 Ξ (arr2 nat2 nat2); fact2
= lam2 (rec2 v02 (lam2 (lam2 (app2 (app2 mul2 (suc2 v12)) v02)))
(suc2 zero2))
{-# OPTIONS --type-in-type #-}
Ty3 : Set
Ty3 =
(Ty3 : Set)
(nat top bot : Ty3)
(arr prod sum : Ty3 β Ty3 β Ty3)
β Ty3
nat3 : Ty3; nat3 = Ξ» _ nat3 _ _ _ _ _ β nat3
top3 : Ty3; top3 = Ξ» _ _ top3 _ _ _ _ β top3
bot3 : Ty3; bot3 = Ξ» _ _ _ bot3 _ _ _ β bot3
arr3 : Ty3 β Ty3 β Ty3; arr3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod sum β
arr3 (A Ty3 nat3 top3 bot3 arr3 prod sum) (B Ty3 nat3 top3 bot3 arr3 prod sum)
prod3 : Ty3 β Ty3 β Ty3; prod3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod3 sum β
prod3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum) (B Ty3 nat3 top3 bot3 arr3 prod3 sum)
sum3 : Ty3 β Ty3 β Ty3; sum3
= Ξ» A B Ty3 nat3 top3 bot3 arr3 prod3 sum3 β
sum3 (A Ty3 nat3 top3 bot3 arr3 prod3 sum3) (B Ty3 nat3 top3 bot3 arr3 prod3 sum3)
Con3 : Set; Con3
= (Con3 : Set)
(nil : Con3)
(snoc : Con3 β Ty3 β Con3)
β Con3
nil3 : Con3; nil3
= Ξ» Con3 nil3 snoc β nil3
snoc3 : Con3 β Ty3 β Con3; snoc3
= Ξ» Ξ A Con3 nil3 snoc3 β snoc3 (Ξ Con3 nil3 snoc3) A
Var3 : Con3 β Ty3 β Set; Var3
= Ξ» Ξ A β
(Var3 : Con3 β Ty3 β Set)
(vz : β Ξ A β Var3 (snoc3 Ξ A) A)
(vs : β Ξ B A β Var3 Ξ A β Var3 (snoc3 Ξ B) A)
β Var3 Ξ A
vz3 : β{Ξ A} β Var3 (snoc3 Ξ A) A; vz3
= Ξ» Var3 vz3 vs β vz3 _ _
vs3 : β{Ξ B A} β Var3 Ξ A β Var3 (snoc3 Ξ B) A; vs3
= Ξ» x Var3 vz3 vs3 β vs3 _ _ _ (x Var3 vz3 vs3)
Tm3 : Con3 β Ty3 β Set; Tm3
= Ξ» Ξ A β
(Tm3 : Con3 β Ty3 β Set)
(var : β Ξ A β Var3 Ξ A β Tm3 Ξ A)
(lam : β Ξ A B β Tm3 (snoc3 Ξ A) B β Tm3 Ξ (arr3 A B))
(app : β Ξ A B β Tm3 Ξ (arr3 A B) β Tm3 Ξ A β Tm3 Ξ B)
(tt : β Ξ β Tm3 Ξ top3)
(pair : β Ξ A B β Tm3 Ξ A β Tm3 Ξ B β Tm3 Ξ (prod3 A B))
(fst : β Ξ A B β Tm3 Ξ (prod3 A B) β Tm3 Ξ A)
(snd : β Ξ A B β Tm3 Ξ (prod3 A B) β Tm3 Ξ B)
(left : β Ξ A B β Tm3 Ξ A β Tm3 Ξ (sum3 A B))
(right : β Ξ A B β Tm3 Ξ B β Tm3 Ξ (sum3 A B))
(case : β Ξ A B C β Tm3 Ξ (sum3 A B) β Tm3 Ξ (arr3 A C) β Tm3 Ξ (arr3 B C) β Tm3 Ξ C)
(zero : β Ξ β Tm3 Ξ nat3)
(suc : β Ξ β Tm3 Ξ nat3 β Tm3 Ξ nat3)
(rec : β Ξ A β Tm3 Ξ nat3 β Tm3 Ξ (arr3 nat3 (arr3 A A)) β Tm3 Ξ A β Tm3 Ξ A)
β Tm3 Ξ A
var3 : β{Ξ A} β Var3 Ξ A β Tm3 Ξ A; var3
= Ξ» x Tm3 var3 lam app tt pair fst snd left right case zero suc rec β
var3 _ _ x
lam3 : β{Ξ A B} β Tm3 (snoc3 Ξ A) B β Tm3 Ξ (arr3 A B); lam3
= Ξ» t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec β
lam3 _ _ _ (t Tm3 var3 lam3 app tt pair fst snd left right case zero suc rec)
app3 : β{Ξ A B} β Tm3 Ξ (arr3 A B) β Tm3 Ξ A β Tm3 Ξ B; app3
= Ξ» t u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec β
app3 _ _ _ (t Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt pair fst snd left right case zero suc rec)
tt3 : β{Ξ} β Tm3 Ξ top3; tt3
= Ξ» Tm3 var3 lam3 app3 tt3 pair fst snd left right case zero suc rec β tt3 _
pair3 : β{Ξ A B} β Tm3 Ξ A β Tm3 Ξ B β Tm3 Ξ (prod3 A B); pair3
= Ξ» t u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec β
pair3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst snd left right case zero suc rec)
fst3 : β{Ξ A B} β Tm3 Ξ (prod3 A B) β Tm3 Ξ A; fst3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec β
fst3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd left right case zero suc rec)
snd3 : β{Ξ A B} β Tm3 Ξ (prod3 A B) β Tm3 Ξ B; snd3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec β
snd3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left right case zero suc rec)
left3 : β{Ξ A B} β Tm3 Ξ A β Tm3 Ξ (sum3 A B); left3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec β
left3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right case zero suc rec)
right3 : β{Ξ A B} β Tm3 Ξ B β Tm3 Ξ (sum3 A B); right3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec β
right3 _ _ _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case zero suc rec)
case3 : β{Ξ A B C} β Tm3 Ξ (sum3 A B) β Tm3 Ξ (arr3 A C) β Tm3 Ξ (arr3 B C) β Tm3 Ξ C; case3
= Ξ» t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec β
case3 _ _ _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero suc rec)
zero3 : β{Ξ} β Tm3 Ξ nat3; zero3
= Ξ» Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc rec β zero3 _
suc3 : β{Ξ} β Tm3 Ξ nat3 β Tm3 Ξ nat3; suc3
= Ξ» t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec β
suc3 _ (t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec)
rec3 : β{Ξ A} β Tm3 Ξ nat3 β Tm3 Ξ (arr3 nat3 (arr3 A A)) β Tm3 Ξ A β Tm3 Ξ A; rec3
= Ξ» t u v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3 β
rec3 _ _
(t Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(u Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
(v Tm3 var3 lam3 app3 tt3 pair3 fst3 snd3 left3 right3 case3 zero3 suc3 rec3)
v03 : β{Ξ A} β Tm3 (snoc3 Ξ A) A; v03
= var3 vz3
v13 : β{Ξ A B} β Tm3 (snoc3 (snoc3 Ξ A) B) A; v13
= var3 (vs3 vz3)
v23 : β{Ξ A B C} β Tm3 (snoc3 (snoc3 (snoc3 Ξ A) B) C) A; v23
= var3 (vs3 (vs3 vz3))
v33 : β{Ξ A B C D} β Tm3 (snoc3 (snoc3 (snoc3 (snoc3 Ξ A) B) C) D) A; v33
= var3 (vs3 (vs3 (vs3 vz3)))
tbool3 : Ty3; tbool3
= sum3 top3 top3
true3 : β{Ξ} β Tm3 Ξ tbool3; true3
= left3 tt3
tfalse3 : β{Ξ} β Tm3 Ξ tbool3; tfalse3
= right3 tt3
ifthenelse3 : β{Ξ A} β Tm3 Ξ (arr3 tbool3 (arr3 A (arr3 A A))); ifthenelse3
= lam3 (lam3 (lam3 (case3 v23 (lam3 v23) (lam3 v13))))
times43 : β{Ξ A} β Tm3 Ξ (arr3 (arr3 A A) (arr3 A A)); times43
= lam3 (lam3 (app3 v13 (app3 v13 (app3 v13 (app3 v13 v03)))))
add3 : β{Ξ} β Tm3 Ξ (arr3 nat3 (arr3 nat3 nat3)); add3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (suc3 (app3 v13 v03)))))
(lam3 v03))
mul3 : β{Ξ} β Tm3 Ξ (arr3 nat3 (arr3 nat3 nat3)); mul3
= lam3 (rec3 v03
(lam3 (lam3 (lam3 (app3 (app3 add3 (app3 v13 v03)) v03))))
(lam3 zero3))
fact3 : β{Ξ} β Tm3 Ξ (arr3 nat3 nat3); fact3
= lam3 (rec3 v03 (lam3 (lam3 (app3 (app3 mul3 (suc3 v13)) v03)))
(suc3 zero3))
{-# OPTIONS --type-in-type #-}
Ty4 : Set
Ty4 =
(Ty4 : Set)
(nat top bot : Ty4)
(arr prod sum : Ty4 β Ty4 β Ty4)
β Ty4
nat4 : Ty4; nat4 = Ξ» _ nat4 _ _ _ _ _ β nat4
top4 : Ty4; top4 = Ξ» _ _ top4 _ _ _ _ β top4
bot4 : Ty4; bot4 = Ξ» _ _ _ bot4 _ _ _ β bot4
arr4 : Ty4 β Ty4 β Ty4; arr4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod sum β
arr4 (A Ty4 nat4 top4 bot4 arr4 prod sum) (B Ty4 nat4 top4 bot4 arr4 prod sum)
prod4 : Ty4 β Ty4 β Ty4; prod4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod4 sum β
prod4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum) (B Ty4 nat4 top4 bot4 arr4 prod4 sum)
sum4 : Ty4 β Ty4 β Ty4; sum4
= Ξ» A B Ty4 nat4 top4 bot4 arr4 prod4 sum4 β
sum4 (A Ty4 nat4 top4 bot4 arr4 prod4 sum4) (B Ty4 nat4 top4 bot4 arr4 prod4 sum4)
Con4 : Set; Con4
= (Con4 : Set)
(nil : Con4)
(snoc : Con4 β Ty4 β Con4)
β Con4
nil4 : Con4; nil4
= Ξ» Con4 nil4 snoc β nil4
snoc4 : Con4 β Ty4 β Con4; snoc4
= Ξ» Ξ A Con4 nil4 snoc4 β snoc4 (Ξ Con4 nil4 snoc4) A
Var4 : Con4 β Ty4 β Set; Var4
= Ξ» Ξ A β
(Var4 : Con4 β Ty4 β Set)
(vz : β Ξ A β Var4 (snoc4 Ξ A) A)
(vs : β Ξ B A β Var4 Ξ A β Var4 (snoc4 Ξ B) A)
β Var4 Ξ A
vz4 : β{Ξ A} β Var4 (snoc4 Ξ A) A; vz4
= Ξ» Var4 vz4 vs β vz4 _ _
vs4 : β{Ξ B A} β Var4 Ξ A β Var4 (snoc4 Ξ B) A; vs4
= Ξ» x Var4 vz4 vs4 β vs4 _ _ _ (x Var4 vz4 vs4)
Tm4 : Con4 β Ty4 β Set; Tm4
= Ξ» Ξ A β
(Tm4 : Con4 β Ty4 β Set)
(var : β Ξ A β Var4 Ξ A β Tm4 Ξ A)
(lam : β Ξ A B β Tm4 (snoc4 Ξ A) B β Tm4 Ξ (arr4 A B))
(app : β Ξ A B β Tm4 Ξ (arr4 A B) β Tm4 Ξ A β Tm4 Ξ B)
(tt : β Ξ β Tm4 Ξ top4)
(pair : β Ξ A B β Tm4 Ξ A β Tm4 Ξ B β Tm4 Ξ (prod4 A B))
(fst : β Ξ A B β Tm4 Ξ (prod4 A B) β Tm4 Ξ A)
(snd : β Ξ A B β Tm4 Ξ (prod4 A B) β Tm4 Ξ B)
(left : β Ξ A B β Tm4 Ξ A β Tm4 Ξ (sum4 A B))
(right : β Ξ A B β Tm4 Ξ B β Tm4 Ξ (sum4 A B))
(case : β Ξ A B C β Tm4 Ξ (sum4 A B) β Tm4 Ξ (arr4 A C) β Tm4 Ξ (arr4 B C) β Tm4 Ξ C)
(zero : β Ξ β Tm4 Ξ nat4)
(suc : β Ξ β Tm4 Ξ nat4 β Tm4 Ξ nat4)
(rec : β Ξ A β Tm4 Ξ nat4 β Tm4 Ξ (arr4 nat4 (arr4 A A)) β Tm4 Ξ A β Tm4 Ξ A)
β Tm4 Ξ A
var4 : β{Ξ A} β Var4 Ξ A β Tm4 Ξ A; var4
= Ξ» x Tm4 var4 lam app tt pair fst snd left right case zero suc rec β
var4 _ _ x
lam4 : β{Ξ A B} β Tm4 (snoc4 Ξ A) B β Tm4 Ξ (arr4 A B); lam4
= Ξ» t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec β
lam4 _ _ _ (t Tm4 var4 lam4 app tt pair fst snd left right case zero suc rec)
app4 : β{Ξ A B} β Tm4 Ξ (arr4 A B) β Tm4 Ξ A β Tm4 Ξ B; app4
= Ξ» t u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec β
app4 _ _ _ (t Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt pair fst snd left right case zero suc rec)
tt4 : β{Ξ} β Tm4 Ξ top4; tt4
= Ξ» Tm4 var4 lam4 app4 tt4 pair fst snd left right case zero suc rec β tt4 _
pair4 : β{Ξ A B} β Tm4 Ξ A β Tm4 Ξ B β Tm4 Ξ (prod4 A B); pair4
= Ξ» t u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec β
pair4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst snd left right case zero suc rec)
fst4 : β{Ξ A B} β Tm4 Ξ (prod4 A B) β Tm4 Ξ A; fst4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec β
fst4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd left right case zero suc rec)
snd4 : β{Ξ A B} β Tm4 Ξ (prod4 A B) β Tm4 Ξ B; snd4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec β
snd4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left right case zero suc rec)
left4 : β{Ξ A B} β Tm4 Ξ A β Tm4 Ξ (sum4 A B); left4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec β
left4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right case zero suc rec)
right4 : β{Ξ A B} β Tm4 Ξ B β Tm4 Ξ (sum4 A B); right4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec β
right4 _ _ _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case zero suc rec)
case4 : β{Ξ A B C} β Tm4 Ξ (sum4 A B) β Tm4 Ξ (arr4 A C) β Tm4 Ξ (arr4 B C) β Tm4 Ξ C; case4
= Ξ» t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec β
case4 _ _ _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero suc rec)
zero4 : β{Ξ} β Tm4 Ξ nat4; zero4
= Ξ» Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc rec β zero4 _
suc4 : β{Ξ} β Tm4 Ξ nat4 β Tm4 Ξ nat4; suc4
= Ξ» t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec β
suc4 _ (t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec)
rec4 : β{Ξ A} β Tm4 Ξ nat4 β Tm4 Ξ (arr4 nat4 (arr4 A A)) β Tm4 Ξ A β Tm4 Ξ A; rec4
= Ξ» t u v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4 β
rec4 _ _
(t Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(u Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
(v Tm4 var4 lam4 app4 tt4 pair4 fst4 snd4 left4 right4 case4 zero4 suc4 rec4)
v04 : β{Ξ A} β Tm4 (snoc4 Ξ A) A; v04
= var4 vz4
v14 : β{Ξ A B} β Tm4 (snoc4 (snoc4 Ξ A) B) A; v14
= var4 (vs4 vz4)
v24 : β{Ξ A B C} β Tm4 (snoc4 (snoc4 (snoc4 Ξ A) B) C) A; v24
= var4 (vs4 (vs4 vz4))
v34 : β{Ξ A B C D} β Tm4 (snoc4 (snoc4 (snoc4 (snoc4 Ξ A) B) C) D) A; v34
= var4 (vs4 (vs4 (vs4 vz4)))
tbool4 : Ty4; tbool4
= sum4 top4 top4
true4 : β{Ξ} β Tm4 Ξ tbool4; true4
= left4 tt4
tfalse4 : β{Ξ} β Tm4 Ξ tbool4; tfalse4
= right4 tt4
ifthenelse4 : β{Ξ A} β Tm4 Ξ (arr4 tbool4 (arr4 A (arr4 A A))); ifthenelse4
= lam4 (lam4 (lam4 (case4 v24 (lam4 v24) (lam4 v14))))
times44 : β{Ξ A} β Tm4 Ξ (arr4 (arr4 A A) (arr4 A A)); times44
= lam4 (lam4 (app4 v14 (app4 v14 (app4 v14 (app4 v14 v04)))))
add4 : β{Ξ} β Tm4 Ξ (arr4 nat4 (arr4 nat4 nat4)); add4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (suc4 (app4 v14 v04)))))
(lam4 v04))
mul4 : β{Ξ} β Tm4 Ξ (arr4 nat4 (arr4 nat4 nat4)); mul4
= lam4 (rec4 v04
(lam4 (lam4 (lam4 (app4 (app4 add4 (app4 v14 v04)) v04))))
(lam4 zero4))
fact4 : β{Ξ} β Tm4 Ξ (arr4 nat4 nat4); fact4
= lam4 (rec4 v04 (lam4 (lam4 (app4 (app4 mul4 (suc4 v14)) v04)))
(suc4 zero4))
{-# OPTIONS --type-in-type #-}
Ty5 : Set
Ty5 =
(Ty5 : Set)
(nat top bot : Ty5)
(arr prod sum : Ty5 β Ty5 β Ty5)
β Ty5
nat5 : Ty5; nat5 = Ξ» _ nat5 _ _ _ _ _ β nat5
top5 : Ty5; top5 = Ξ» _ _ top5 _ _ _ _ β top5
bot5 : Ty5; bot5 = Ξ» _ _ _ bot5 _ _ _ β bot5
arr5 : Ty5 β Ty5 β Ty5; arr5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod sum β
arr5 (A Ty5 nat5 top5 bot5 arr5 prod sum) (B Ty5 nat5 top5 bot5 arr5 prod sum)
prod5 : Ty5 β Ty5 β Ty5; prod5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod5 sum β
prod5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum) (B Ty5 nat5 top5 bot5 arr5 prod5 sum)
sum5 : Ty5 β Ty5 β Ty5; sum5
= Ξ» A B Ty5 nat5 top5 bot5 arr5 prod5 sum5 β
sum5 (A Ty5 nat5 top5 bot5 arr5 prod5 sum5) (B Ty5 nat5 top5 bot5 arr5 prod5 sum5)
Con5 : Set; Con5
= (Con5 : Set)
(nil : Con5)
(snoc : Con5 β Ty5 β Con5)
β Con5
nil5 : Con5; nil5
= Ξ» Con5 nil5 snoc β nil5
snoc5 : Con5 β Ty5 β Con5; snoc5
= Ξ» Ξ A Con5 nil5 snoc5 β snoc5 (Ξ Con5 nil5 snoc5) A
Var5 : Con5 β Ty5 β Set; Var5
= Ξ» Ξ A β
(Var5 : Con5 β Ty5 β Set)
(vz : β Ξ A β Var5 (snoc5 Ξ A) A)
(vs : β Ξ B A β Var5 Ξ A β Var5 (snoc5 Ξ B) A)
β Var5 Ξ A
vz5 : β{Ξ A} β Var5 (snoc5 Ξ A) A; vz5
= Ξ» Var5 vz5 vs β vz5 _ _
vs5 : β{Ξ B A} β Var5 Ξ A β Var5 (snoc5 Ξ B) A; vs5
= Ξ» x Var5 vz5 vs5 β vs5 _ _ _ (x Var5 vz5 vs5)
Tm5 : Con5 β Ty5 β Set; Tm5
= Ξ» Ξ A β
(Tm5 : Con5 β Ty5 β Set)
(var : β Ξ A β Var5 Ξ A β Tm5 Ξ A)
(lam : β Ξ A B β Tm5 (snoc5 Ξ A) B β Tm5 Ξ (arr5 A B))
(app : β Ξ A B β Tm5 Ξ (arr5 A B) β Tm5 Ξ A β Tm5 Ξ B)
(tt : β Ξ β Tm5 Ξ top5)
(pair : β Ξ A B β Tm5 Ξ A β Tm5 Ξ B β Tm5 Ξ (prod5 A B))
(fst : β Ξ A B β Tm5 Ξ (prod5 A B) β Tm5 Ξ A)
(snd : β Ξ A B β Tm5 Ξ (prod5 A B) β Tm5 Ξ B)
(left : β Ξ A B β Tm5 Ξ A β Tm5 Ξ (sum5 A B))
(right : β Ξ A B β Tm5 Ξ B β Tm5 Ξ (sum5 A B))
(case : β Ξ A B C β Tm5 Ξ (sum5 A B) β Tm5 Ξ (arr5 A C) β Tm5 Ξ (arr5 B C) β Tm5 Ξ C)
(zero : β Ξ β Tm5 Ξ nat5)
(suc : β Ξ β Tm5 Ξ nat5 β Tm5 Ξ nat5)
(rec : β Ξ A β Tm5 Ξ nat5 β Tm5 Ξ (arr5 nat5 (arr5 A A)) β Tm5 Ξ A β Tm5 Ξ A)
β Tm5 Ξ A
var5 : β{Ξ A} β Var5 Ξ A β Tm5 Ξ A; var5
= Ξ» x Tm5 var5 lam app tt pair fst snd left right case zero suc rec β
var5 _ _ x
lam5 : β{Ξ A B} β Tm5 (snoc5 Ξ A) B β Tm5 Ξ (arr5 A B); lam5
= Ξ» t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec β
lam5 _ _ _ (t Tm5 var5 lam5 app tt pair fst snd left right case zero suc rec)
app5 : β{Ξ A B} β Tm5 Ξ (arr5 A B) β Tm5 Ξ A β Tm5 Ξ B; app5
= Ξ» t u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec β
app5 _ _ _ (t Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt pair fst snd left right case zero suc rec)
tt5 : β{Ξ} β Tm5 Ξ top5; tt5
= Ξ» Tm5 var5 lam5 app5 tt5 pair fst snd left right case zero suc rec β tt5 _
pair5 : β{Ξ A B} β Tm5 Ξ A β Tm5 Ξ B β Tm5 Ξ (prod5 A B); pair5
= Ξ» t u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec β
pair5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst snd left right case zero suc rec)
fst5 : β{Ξ A B} β Tm5 Ξ (prod5 A B) β Tm5 Ξ A; fst5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec β
fst5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd left right case zero suc rec)
snd5 : β{Ξ A B} β Tm5 Ξ (prod5 A B) β Tm5 Ξ B; snd5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec β
snd5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left right case zero suc rec)
left5 : β{Ξ A B} β Tm5 Ξ A β Tm5 Ξ (sum5 A B); left5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec β
left5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right case zero suc rec)
right5 : β{Ξ A B} β Tm5 Ξ B β Tm5 Ξ (sum5 A B); right5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec β
right5 _ _ _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case zero suc rec)
case5 : β{Ξ A B C} β Tm5 Ξ (sum5 A B) β Tm5 Ξ (arr5 A C) β Tm5 Ξ (arr5 B C) β Tm5 Ξ C; case5
= Ξ» t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec β
case5 _ _ _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero suc rec)
zero5 : β{Ξ} β Tm5 Ξ nat5; zero5
= Ξ» Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc rec β zero5 _
suc5 : β{Ξ} β Tm5 Ξ nat5 β Tm5 Ξ nat5; suc5
= Ξ» t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec β
suc5 _ (t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec)
rec5 : β{Ξ A} β Tm5 Ξ nat5 β Tm5 Ξ (arr5 nat5 (arr5 A A)) β Tm5 Ξ A β Tm5 Ξ A; rec5
= Ξ» t u v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5 β
rec5 _ _
(t Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(u Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
(v Tm5 var5 lam5 app5 tt5 pair5 fst5 snd5 left5 right5 case5 zero5 suc5 rec5)
v05 : β{Ξ A} β Tm5 (snoc5 Ξ A) A; v05
= var5 vz5
v15 : β{Ξ A B} β Tm5 (snoc5 (snoc5 Ξ A) B) A; v15
= var5 (vs5 vz5)
v25 : β{Ξ A B C} β Tm5 (snoc5 (snoc5 (snoc5 Ξ A) B) C) A; v25
= var5 (vs5 (vs5 vz5))
v35 : β{Ξ A B C D} β Tm5 (snoc5 (snoc5 (snoc5 (snoc5 Ξ A) B) C) D) A; v35
= var5 (vs5 (vs5 (vs5 vz5)))
tbool5 : Ty5; tbool5
= sum5 top5 top5
true5 : β{Ξ} β Tm5 Ξ tbool5; true5
= left5 tt5
tfalse5 : β{Ξ} β Tm5 Ξ tbool5; tfalse5
= right5 tt5
ifthenelse5 : β{Ξ A} β Tm5 Ξ (arr5 tbool5 (arr5 A (arr5 A A))); ifthenelse5
= lam5 (lam5 (lam5 (case5 v25 (lam5 v25) (lam5 v15))))
times45 : β{Ξ A} β Tm5 Ξ (arr5 (arr5 A A) (arr5 A A)); times45
= lam5 (lam5 (app5 v15 (app5 v15 (app5 v15 (app5 v15 v05)))))
add5 : β{Ξ} β Tm5 Ξ (arr5 nat5 (arr5 nat5 nat5)); add5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (suc5 (app5 v15 v05)))))
(lam5 v05))
mul5 : β{Ξ} β Tm5 Ξ (arr5 nat5 (arr5 nat5 nat5)); mul5
= lam5 (rec5 v05
(lam5 (lam5 (lam5 (app5 (app5 add5 (app5 v15 v05)) v05))))
(lam5 zero5))
fact5 : β{Ξ} β Tm5 Ξ (arr5 nat5 nat5); fact5
= lam5 (rec5 v05 (lam5 (lam5 (app5 (app5 mul5 (suc5 v15)) v05)))
(suc5 zero5))
{-# OPTIONS --type-in-type #-}
Ty6 : Set
Ty6 =
(Ty6 : Set)
(nat top bot : Ty6)
(arr prod sum : Ty6 β Ty6 β Ty6)
β Ty6
nat6 : Ty6; nat6 = Ξ» _ nat6 _ _ _ _ _ β nat6
top6 : Ty6; top6 = Ξ» _ _ top6 _ _ _ _ β top6
bot6 : Ty6; bot6 = Ξ» _ _ _ bot6 _ _ _ β bot6
arr6 : Ty6 β Ty6 β Ty6; arr6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod sum β
arr6 (A Ty6 nat6 top6 bot6 arr6 prod sum) (B Ty6 nat6 top6 bot6 arr6 prod sum)
prod6 : Ty6 β Ty6 β Ty6; prod6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod6 sum β
prod6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum) (B Ty6 nat6 top6 bot6 arr6 prod6 sum)
sum6 : Ty6 β Ty6 β Ty6; sum6
= Ξ» A B Ty6 nat6 top6 bot6 arr6 prod6 sum6 β
sum6 (A Ty6 nat6 top6 bot6 arr6 prod6 sum6) (B Ty6 nat6 top6 bot6 arr6 prod6 sum6)
Con6 : Set; Con6
= (Con6 : Set)
(nil : Con6)
(snoc : Con6 β Ty6 β Con6)
β Con6
nil6 : Con6; nil6
= Ξ» Con6 nil6 snoc β nil6
snoc6 : Con6 β Ty6 β Con6; snoc6
= Ξ» Ξ A Con6 nil6 snoc6 β snoc6 (Ξ Con6 nil6 snoc6) A
Var6 : Con6 β Ty6 β Set; Var6
= Ξ» Ξ A β
(Var6 : Con6 β Ty6 β Set)
(vz : β Ξ A β Var6 (snoc6 Ξ A) A)
(vs : β Ξ B A β Var6 Ξ A β Var6 (snoc6 Ξ B) A)
β Var6 Ξ A
vz6 : β{Ξ A} β Var6 (snoc6 Ξ A) A; vz6
= Ξ» Var6 vz6 vs β vz6 _ _
vs6 : β{Ξ B A} β Var6 Ξ A β Var6 (snoc6 Ξ B) A; vs6
= Ξ» x Var6 vz6 vs6 β vs6 _ _ _ (x Var6 vz6 vs6)
Tm6 : Con6 β Ty6 β Set; Tm6
= Ξ» Ξ A β
(Tm6 : Con6 β Ty6 β Set)
(var : β Ξ A β Var6 Ξ A β Tm6 Ξ A)
(lam : β Ξ A B β Tm6 (snoc6 Ξ A) B β Tm6 Ξ (arr6 A B))
(app : β Ξ A B β Tm6 Ξ (arr6 A B) β Tm6 Ξ A β Tm6 Ξ B)
(tt : β Ξ β Tm6 Ξ top6)
(pair : β Ξ A B β Tm6 Ξ A β Tm6 Ξ B β Tm6 Ξ (prod6 A B))
(fst : β Ξ A B β Tm6 Ξ (prod6 A B) β Tm6 Ξ A)
(snd : β Ξ A B β Tm6 Ξ (prod6 A B) β Tm6 Ξ B)
(left : β Ξ A B β Tm6 Ξ A β Tm6 Ξ (sum6 A B))
(right : β Ξ A B β Tm6 Ξ B β Tm6 Ξ (sum6 A B))
(case : β Ξ A B C β Tm6 Ξ (sum6 A B) β Tm6 Ξ (arr6 A C) β Tm6 Ξ (arr6 B C) β Tm6 Ξ C)
(zero : β Ξ β Tm6 Ξ nat6)
(suc : β Ξ β Tm6 Ξ nat6 β Tm6 Ξ nat6)
(rec : β Ξ A β Tm6 Ξ nat6 β Tm6 Ξ (arr6 nat6 (arr6 A A)) β Tm6 Ξ A β Tm6 Ξ A)
β Tm6 Ξ A
var6 : β{Ξ A} β Var6 Ξ A β Tm6 Ξ A; var6
= Ξ» x Tm6 var6 lam app tt pair fst snd left right case zero suc rec β
var6 _ _ x
lam6 : β{Ξ A B} β Tm6 (snoc6 Ξ A) B β Tm6 Ξ (arr6 A B); lam6
= Ξ» t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec β
lam6 _ _ _ (t Tm6 var6 lam6 app tt pair fst snd left right case zero suc rec)
app6 : β{Ξ A B} β Tm6 Ξ (arr6 A B) β Tm6 Ξ A β Tm6 Ξ B; app6
= Ξ» t u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec β
app6 _ _ _ (t Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt pair fst snd left right case zero suc rec)
tt6 : β{Ξ} β Tm6 Ξ top6; tt6
= Ξ» Tm6 var6 lam6 app6 tt6 pair fst snd left right case zero suc rec β tt6 _
pair6 : β{Ξ A B} β Tm6 Ξ A β Tm6 Ξ B β Tm6 Ξ (prod6 A B); pair6
= Ξ» t u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec β
pair6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst snd left right case zero suc rec)
fst6 : β{Ξ A B} β Tm6 Ξ (prod6 A B) β Tm6 Ξ A; fst6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec β
fst6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd left right case zero suc rec)
snd6 : β{Ξ A B} β Tm6 Ξ (prod6 A B) β Tm6 Ξ B; snd6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec β
snd6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left right case zero suc rec)
left6 : β{Ξ A B} β Tm6 Ξ A β Tm6 Ξ (sum6 A B); left6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec β
left6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right case zero suc rec)
right6 : β{Ξ A B} β Tm6 Ξ B β Tm6 Ξ (sum6 A B); right6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec β
right6 _ _ _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case zero suc rec)
case6 : β{Ξ A B C} β Tm6 Ξ (sum6 A B) β Tm6 Ξ (arr6 A C) β Tm6 Ξ (arr6 B C) β Tm6 Ξ C; case6
= Ξ» t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec β
case6 _ _ _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero suc rec)
zero6 : β{Ξ} β Tm6 Ξ nat6; zero6
= Ξ» Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc rec β zero6 _
suc6 : β{Ξ} β Tm6 Ξ nat6 β Tm6 Ξ nat6; suc6
= Ξ» t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec β
suc6 _ (t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec)
rec6 : β{Ξ A} β Tm6 Ξ nat6 β Tm6 Ξ (arr6 nat6 (arr6 A A)) β Tm6 Ξ A β Tm6 Ξ A; rec6
= Ξ» t u v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6 β
rec6 _ _
(t Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(u Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
(v Tm6 var6 lam6 app6 tt6 pair6 fst6 snd6 left6 right6 case6 zero6 suc6 rec6)
v06 : β{Ξ A} β Tm6 (snoc6 Ξ A) A; v06
= var6 vz6
v16 : β{Ξ A B} β Tm6 (snoc6 (snoc6 Ξ A) B) A; v16
= var6 (vs6 vz6)
v26 : β{Ξ A B C} β Tm6 (snoc6 (snoc6 (snoc6 Ξ A) B) C) A; v26
= var6 (vs6 (vs6 vz6))
v36 : β{Ξ A B C D} β Tm6 (snoc6 (snoc6 (snoc6 (snoc6 Ξ A) B) C) D) A; v36
= var6 (vs6 (vs6 (vs6 vz6)))
tbool6 : Ty6; tbool6
= sum6 top6 top6
true6 : β{Ξ} β Tm6 Ξ tbool6; true6
= left6 tt6
tfalse6 : β{Ξ} β Tm6 Ξ tbool6; tfalse6
= right6 tt6
ifthenelse6 : β{Ξ A} β Tm6 Ξ (arr6 tbool6 (arr6 A (arr6 A A))); ifthenelse6
= lam6 (lam6 (lam6 (case6 v26 (lam6 v26) (lam6 v16))))
times46 : β{Ξ A} β Tm6 Ξ (arr6 (arr6 A A) (arr6 A A)); times46
= lam6 (lam6 (app6 v16 (app6 v16 (app6 v16 (app6 v16 v06)))))
add6 : β{Ξ} β Tm6 Ξ (arr6 nat6 (arr6 nat6 nat6)); add6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (suc6 (app6 v16 v06)))))
(lam6 v06))
mul6 : β{Ξ} β Tm6 Ξ (arr6 nat6 (arr6 nat6 nat6)); mul6
= lam6 (rec6 v06
(lam6 (lam6 (lam6 (app6 (app6 add6 (app6 v16 v06)) v06))))
(lam6 zero6))
fact6 : β{Ξ} β Tm6 Ξ (arr6 nat6 nat6); fact6
= lam6 (rec6 v06 (lam6 (lam6 (app6 (app6 mul6 (suc6 v16)) v06)))
(suc6 zero6))
{-# OPTIONS --type-in-type #-}
Ty7 : Set
Ty7 =
(Ty7 : Set)
(nat top bot : Ty7)
(arr prod sum : Ty7 β Ty7 β Ty7)
β Ty7
nat7 : Ty7; nat7 = Ξ» _ nat7 _ _ _ _ _ β nat7
top7 : Ty7; top7 = Ξ» _ _ top7 _ _ _ _ β top7
bot7 : Ty7; bot7 = Ξ» _ _ _ bot7 _ _ _ β bot7
arr7 : Ty7 β Ty7 β Ty7; arr7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod sum β
arr7 (A Ty7 nat7 top7 bot7 arr7 prod sum) (B Ty7 nat7 top7 bot7 arr7 prod sum)
prod7 : Ty7 β Ty7 β Ty7; prod7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod7 sum β
prod7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum) (B Ty7 nat7 top7 bot7 arr7 prod7 sum)
sum7 : Ty7 β Ty7 β Ty7; sum7
= Ξ» A B Ty7 nat7 top7 bot7 arr7 prod7 sum7 β
sum7 (A Ty7 nat7 top7 bot7 arr7 prod7 sum7) (B Ty7 nat7 top7 bot7 arr7 prod7 sum7)
Con7 : Set; Con7
= (Con7 : Set)
(nil : Con7)
(snoc : Con7 β Ty7 β Con7)
β Con7
nil7 : Con7; nil7
= Ξ» Con7 nil7 snoc β nil7
snoc7 : Con7 β Ty7 β Con7; snoc7
= Ξ» Ξ A Con7 nil7 snoc7 β snoc7 (Ξ Con7 nil7 snoc7) A
Var7 : Con7 β Ty7 β Set; Var7
= Ξ» Ξ A β
(Var7 : Con7 β Ty7 β Set)
(vz : β Ξ A β Var7 (snoc7 Ξ A) A)
(vs : β Ξ B A β Var7 Ξ A β Var7 (snoc7 Ξ B) A)
β Var7 Ξ A
vz7 : β{Ξ A} β Var7 (snoc7 Ξ A) A; vz7
= Ξ» Var7 vz7 vs β vz7 _ _
vs7 : β{Ξ B A} β Var7 Ξ A β Var7 (snoc7 Ξ B) A; vs7
= Ξ» x Var7 vz7 vs7 β vs7 _ _ _ (x Var7 vz7 vs7)
Tm7 : Con7 β Ty7 β Set; Tm7
= Ξ» Ξ A β
(Tm7 : Con7 β Ty7 β Set)
(var : β Ξ A β Var7 Ξ A β Tm7 Ξ A)
(lam : β Ξ A B β Tm7 (snoc7 Ξ A) B β Tm7 Ξ (arr7 A B))
(app : β Ξ A B β Tm7 Ξ (arr7 A B) β Tm7 Ξ A β Tm7 Ξ B)
(tt : β Ξ β Tm7 Ξ top7)
(pair : β Ξ A B β Tm7 Ξ A β Tm7 Ξ B β Tm7 Ξ (prod7 A B))
(fst : β Ξ A B β Tm7 Ξ (prod7 A B) β Tm7 Ξ A)
(snd : β Ξ A B β Tm7 Ξ (prod7 A B) β Tm7 Ξ B)
(left : β Ξ A B β Tm7 Ξ A β Tm7 Ξ (sum7 A B))
(right : β Ξ A B β Tm7 Ξ B β Tm7 Ξ (sum7 A B))
(case : β Ξ A B C β Tm7 Ξ (sum7 A B) β Tm7 Ξ (arr7 A C) β Tm7 Ξ (arr7 B C) β Tm7 Ξ C)
(zero : β Ξ β Tm7 Ξ nat7)
(suc : β Ξ β Tm7 Ξ nat7 β Tm7 Ξ nat7)
(rec : β Ξ A β Tm7 Ξ nat7 β Tm7 Ξ (arr7 nat7 (arr7 A A)) β Tm7 Ξ A β Tm7 Ξ A)
β Tm7 Ξ A
var7 : β{Ξ A} β Var7 Ξ A β Tm7 Ξ A; var7
= Ξ» x Tm7 var7 lam app tt pair fst snd left right case zero suc rec β
var7 _ _ x
lam7 : β{Ξ A B} β Tm7 (snoc7 Ξ A) B β Tm7 Ξ (arr7 A B); lam7
= Ξ» t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec β
lam7 _ _ _ (t Tm7 var7 lam7 app tt pair fst snd left right case zero suc rec)
app7 : β{Ξ A B} β Tm7 Ξ (arr7 A B) β Tm7 Ξ A β Tm7 Ξ B; app7
= Ξ» t u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec β
app7 _ _ _ (t Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt pair fst snd left right case zero suc rec)
tt7 : β{Ξ} β Tm7 Ξ top7; tt7
= Ξ» Tm7 var7 lam7 app7 tt7 pair fst snd left right case zero suc rec β tt7 _
pair7 : β{Ξ A B} β Tm7 Ξ A β Tm7 Ξ B β Tm7 Ξ (prod7 A B); pair7
= Ξ» t u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec β
pair7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst snd left right case zero suc rec)
fst7 : β{Ξ A B} β Tm7 Ξ (prod7 A B) β Tm7 Ξ A; fst7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec β
fst7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd left right case zero suc rec)
snd7 : β{Ξ A B} β Tm7 Ξ (prod7 A B) β Tm7 Ξ B; snd7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec β
snd7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left right case zero suc rec)
left7 : β{Ξ A B} β Tm7 Ξ A β Tm7 Ξ (sum7 A B); left7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec β
left7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right case zero suc rec)
right7 : β{Ξ A B} β Tm7 Ξ B β Tm7 Ξ (sum7 A B); right7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec β
right7 _ _ _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case zero suc rec)
case7 : β{Ξ A B C} β Tm7 Ξ (sum7 A B) β Tm7 Ξ (arr7 A C) β Tm7 Ξ (arr7 B C) β Tm7 Ξ C; case7
= Ξ» t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec β
case7 _ _ _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero suc rec)
zero7 : β{Ξ} β Tm7 Ξ nat7; zero7
= Ξ» Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc rec β zero7 _
suc7 : β{Ξ} β Tm7 Ξ nat7 β Tm7 Ξ nat7; suc7
= Ξ» t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec β
suc7 _ (t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec)
rec7 : β{Ξ A} β Tm7 Ξ nat7 β Tm7 Ξ (arr7 nat7 (arr7 A A)) β Tm7 Ξ A β Tm7 Ξ A; rec7
= Ξ» t u v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7 β
rec7 _ _
(t Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(u Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
(v Tm7 var7 lam7 app7 tt7 pair7 fst7 snd7 left7 right7 case7 zero7 suc7 rec7)
v07 : β{Ξ A} β Tm7 (snoc7 Ξ A) A; v07
= var7 vz7
v17 : β{Ξ A B} β Tm7 (snoc7 (snoc7 Ξ A) B) A; v17
= var7 (vs7 vz7)
v27 : β{Ξ A B C} β Tm7 (snoc7 (snoc7 (snoc7 Ξ A) B) C) A; v27
= var7 (vs7 (vs7 vz7))
v37 : β{Ξ A B C D} β Tm7 (snoc7 (snoc7 (snoc7 (snoc7 Ξ A) B) C) D) A; v37
= var7 (vs7 (vs7 (vs7 vz7)))
tbool7 : Ty7; tbool7
= sum7 top7 top7
true7 : β{Ξ} β Tm7 Ξ tbool7; true7
= left7 tt7
tfalse7 : β{Ξ} β Tm7 Ξ tbool7; tfalse7
= right7 tt7
ifthenelse7 : β{Ξ A} β Tm7 Ξ (arr7 tbool7 (arr7 A (arr7 A A))); ifthenelse7
= lam7 (lam7 (lam7 (case7 v27 (lam7 v27) (lam7 v17))))
times47 : β{Ξ A} β Tm7 Ξ (arr7 (arr7 A A) (arr7 A A)); times47
= lam7 (lam7 (app7 v17 (app7 v17 (app7 v17 (app7 v17 v07)))))
add7 : β{Ξ} β Tm7 Ξ (arr7 nat7 (arr7 nat7 nat7)); add7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (suc7 (app7 v17 v07)))))
(lam7 v07))
mul7 : β{Ξ} β Tm7 Ξ (arr7 nat7 (arr7 nat7 nat7)); mul7
= lam7 (rec7 v07
(lam7 (lam7 (lam7 (app7 (app7 add7 (app7 v17 v07)) v07))))
(lam7 zero7))
fact7 : β{Ξ} β Tm7 Ξ (arr7 nat7 nat7); fact7
= lam7 (rec7 v07 (lam7 (lam7 (app7 (app7 mul7 (suc7 v17)) v07)))
(suc7 zero7))
{-# OPTIONS --type-in-type #-}
Ty8 : Set
Ty8 =
(Ty8 : Set)
(nat top bot : Ty8)
(arr prod sum : Ty8 β Ty8 β Ty8)
β Ty8
nat8 : Ty8; nat8 = Ξ» _ nat8 _ _ _ _ _ β nat8
top8 : Ty8; top8 = Ξ» _ _ top8 _ _ _ _ β top8
bot8 : Ty8; bot8 = Ξ» _ _ _ bot8 _ _ _ β bot8
arr8 : Ty8 β Ty8 β Ty8; arr8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod sum β
arr8 (A Ty8 nat8 top8 bot8 arr8 prod sum) (B Ty8 nat8 top8 bot8 arr8 prod sum)
prod8 : Ty8 β Ty8 β Ty8; prod8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod8 sum β
prod8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum) (B Ty8 nat8 top8 bot8 arr8 prod8 sum)
sum8 : Ty8 β Ty8 β Ty8; sum8
= Ξ» A B Ty8 nat8 top8 bot8 arr8 prod8 sum8 β
sum8 (A Ty8 nat8 top8 bot8 arr8 prod8 sum8) (B Ty8 nat8 top8 bot8 arr8 prod8 sum8)
Con8 : Set; Con8
= (Con8 : Set)
(nil : Con8)
(snoc : Con8 β Ty8 β Con8)
β Con8
nil8 : Con8; nil8
= Ξ» Con8 nil8 snoc β nil8
snoc8 : Con8 β Ty8 β Con8; snoc8
= Ξ» Ξ A Con8 nil8 snoc8 β snoc8 (Ξ Con8 nil8 snoc8) A
Var8 : Con8 β Ty8 β Set; Var8
= Ξ» Ξ A β
(Var8 : Con8 β Ty8 β Set)
(vz : β Ξ A β Var8 (snoc8 Ξ A) A)
(vs : β Ξ B A β Var8 Ξ A β Var8 (snoc8 Ξ B) A)
β Var8 Ξ A
vz8 : β{Ξ A} β Var8 (snoc8 Ξ A) A; vz8
= Ξ» Var8 vz8 vs β vz8 _ _
vs8 : β{Ξ B A} β Var8 Ξ A β Var8 (snoc8 Ξ B) A; vs8
= Ξ» x Var8 vz8 vs8 β vs8 _ _ _ (x Var8 vz8 vs8)
Tm8 : Con8 β Ty8 β Set; Tm8
= Ξ» Ξ A β
(Tm8 : Con8 β Ty8 β Set)
(var : β Ξ A β Var8 Ξ A β Tm8 Ξ A)
(lam : β Ξ A B β Tm8 (snoc8 Ξ A) B β Tm8 Ξ (arr8 A B))
(app : β Ξ A B β Tm8 Ξ (arr8 A B) β Tm8 Ξ A β Tm8 Ξ B)
(tt : β Ξ β Tm8 Ξ top8)
(pair : β Ξ A B β Tm8 Ξ A β Tm8 Ξ B β Tm8 Ξ (prod8 A B))
(fst : β Ξ A B β Tm8 Ξ (prod8 A B) β Tm8 Ξ A)
(snd : β Ξ A B β Tm8 Ξ (prod8 A B) β Tm8 Ξ B)
(left : β Ξ A B β Tm8 Ξ A β Tm8 Ξ (sum8 A B))
(right : β Ξ A B β Tm8 Ξ B β Tm8 Ξ (sum8 A B))
(case : β Ξ A B C β Tm8 Ξ (sum8 A B) β Tm8 Ξ (arr8 A C) β Tm8 Ξ (arr8 B C) β Tm8 Ξ C)
(zero : β Ξ β Tm8 Ξ nat8)
(suc : β Ξ β Tm8 Ξ nat8 β Tm8 Ξ nat8)
(rec : β Ξ A β Tm8 Ξ nat8 β Tm8 Ξ (arr8 nat8 (arr8 A A)) β Tm8 Ξ A β Tm8 Ξ A)
β Tm8 Ξ A
var8 : β{Ξ A} β Var8 Ξ A β Tm8 Ξ A; var8
= Ξ» x Tm8 var8 lam app tt pair fst snd left right case zero suc rec β
var8 _ _ x
lam8 : β{Ξ A B} β Tm8 (snoc8 Ξ A) B β Tm8 Ξ (arr8 A B); lam8
= Ξ» t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec β
lam8 _ _ _ (t Tm8 var8 lam8 app tt pair fst snd left right case zero suc rec)
app8 : β{Ξ A B} β Tm8 Ξ (arr8 A B) β Tm8 Ξ A β Tm8 Ξ B; app8
= Ξ» t u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec β
app8 _ _ _ (t Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt pair fst snd left right case zero suc rec)
tt8 : β{Ξ} β Tm8 Ξ top8; tt8
= Ξ» Tm8 var8 lam8 app8 tt8 pair fst snd left right case zero suc rec β tt8 _
pair8 : β{Ξ A B} β Tm8 Ξ A β Tm8 Ξ B β Tm8 Ξ (prod8 A B); pair8
= Ξ» t u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec β
pair8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst snd left right case zero suc rec)
fst8 : β{Ξ A B} β Tm8 Ξ (prod8 A B) β Tm8 Ξ A; fst8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec β
fst8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd left right case zero suc rec)
snd8 : β{Ξ A B} β Tm8 Ξ (prod8 A B) β Tm8 Ξ B; snd8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec β
snd8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left right case zero suc rec)
left8 : β{Ξ A B} β Tm8 Ξ A β Tm8 Ξ (sum8 A B); left8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec β
left8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right case zero suc rec)
right8 : β{Ξ A B} β Tm8 Ξ B β Tm8 Ξ (sum8 A B); right8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec β
right8 _ _ _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case zero suc rec)
case8 : β{Ξ A B C} β Tm8 Ξ (sum8 A B) β Tm8 Ξ (arr8 A C) β Tm8 Ξ (arr8 B C) β Tm8 Ξ C; case8
= Ξ» t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec β
case8 _ _ _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero suc rec)
zero8 : β{Ξ} β Tm8 Ξ nat8; zero8
= Ξ» Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc rec β zero8 _
suc8 : β{Ξ} β Tm8 Ξ nat8 β Tm8 Ξ nat8; suc8
= Ξ» t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec β
suc8 _ (t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec)
rec8 : β{Ξ A} β Tm8 Ξ nat8 β Tm8 Ξ (arr8 nat8 (arr8 A A)) β Tm8 Ξ A β Tm8 Ξ A; rec8
= Ξ» t u v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8 β
rec8 _ _
(t Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(u Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
(v Tm8 var8 lam8 app8 tt8 pair8 fst8 snd8 left8 right8 case8 zero8 suc8 rec8)
v08 : β{Ξ A} β Tm8 (snoc8 Ξ A) A; v08
= var8 vz8
v18 : β{Ξ A B} β Tm8 (snoc8 (snoc8 Ξ A) B) A; v18
= var8 (vs8 vz8)
v28 : β{Ξ A B C} β Tm8 (snoc8 (snoc8 (snoc8 Ξ A) B) C) A; v28
= var8 (vs8 (vs8 vz8))
v38 : β{Ξ A B C D} β Tm8 (snoc8 (snoc8 (snoc8 (snoc8 Ξ A) B) C) D) A; v38
= var8 (vs8 (vs8 (vs8 vz8)))
tbool8 : Ty8; tbool8
= sum8 top8 top8
true8 : β{Ξ} β Tm8 Ξ tbool8; true8
= left8 tt8
tfalse8 : β{Ξ} β Tm8 Ξ tbool8; tfalse8
= right8 tt8
ifthenelse8 : β{Ξ A} β Tm8 Ξ (arr8 tbool8 (arr8 A (arr8 A A))); ifthenelse8
= lam8 (lam8 (lam8 (case8 v28 (lam8 v28) (lam8 v18))))
times48 : β{Ξ A} β Tm8 Ξ (arr8 (arr8 A A) (arr8 A A)); times48
= lam8 (lam8 (app8 v18 (app8 v18 (app8 v18 (app8 v18 v08)))))
add8 : β{Ξ} β Tm8 Ξ (arr8 nat8 (arr8 nat8 nat8)); add8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (suc8 (app8 v18 v08)))))
(lam8 v08))
mul8 : β{Ξ} β Tm8 Ξ (arr8 nat8 (arr8 nat8 nat8)); mul8
= lam8 (rec8 v08
(lam8 (lam8 (lam8 (app8 (app8 add8 (app8 v18 v08)) v08))))
(lam8 zero8))
fact8 : β{Ξ} β Tm8 Ξ (arr8 nat8 nat8); fact8
= lam8 (rec8 v08 (lam8 (lam8 (app8 (app8 mul8 (suc8 v18)) v08)))
(suc8 zero8))
{-# OPTIONS --type-in-type #-}
Ty9 : Set
Ty9 =
(Ty9 : Set)
(nat top bot : Ty9)
(arr prod sum : Ty9 β Ty9 β Ty9)
β Ty9
nat9 : Ty9; nat9 = Ξ» _ nat9 _ _ _ _ _ β nat9
top9 : Ty9; top9 = Ξ» _ _ top9 _ _ _ _ β top9
bot9 : Ty9; bot9 = Ξ» _ _ _ bot9 _ _ _ β bot9
arr9 : Ty9 β Ty9 β Ty9; arr9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod sum β
arr9 (A Ty9 nat9 top9 bot9 arr9 prod sum) (B Ty9 nat9 top9 bot9 arr9 prod sum)
prod9 : Ty9 β Ty9 β Ty9; prod9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod9 sum β
prod9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum) (B Ty9 nat9 top9 bot9 arr9 prod9 sum)
sum9 : Ty9 β Ty9 β Ty9; sum9
= Ξ» A B Ty9 nat9 top9 bot9 arr9 prod9 sum9 β
sum9 (A Ty9 nat9 top9 bot9 arr9 prod9 sum9) (B Ty9 nat9 top9 bot9 arr9 prod9 sum9)
Con9 : Set; Con9
= (Con9 : Set)
(nil : Con9)
(snoc : Con9 β Ty9 β Con9)
β Con9
nil9 : Con9; nil9
= Ξ» Con9 nil9 snoc β nil9
snoc9 : Con9 β Ty9 β Con9; snoc9
= Ξ» Ξ A Con9 nil9 snoc9 β snoc9 (Ξ Con9 nil9 snoc9) A
Var9 : Con9 β Ty9 β Set; Var9
= Ξ» Ξ A β
(Var9 : Con9 β Ty9 β Set)
(vz : β Ξ A β Var9 (snoc9 Ξ A) A)
(vs : β Ξ B A β Var9 Ξ A β Var9 (snoc9 Ξ B) A)
β Var9 Ξ A
vz9 : β{Ξ A} β Var9 (snoc9 Ξ A) A; vz9
= Ξ» Var9 vz9 vs β vz9 _ _
vs9 : β{Ξ B A} β Var9 Ξ A β Var9 (snoc9 Ξ B) A; vs9
= Ξ» x Var9 vz9 vs9 β vs9 _ _ _ (x Var9 vz9 vs9)
Tm9 : Con9 β Ty9 β Set; Tm9
= Ξ» Ξ A β
(Tm9 : Con9 β Ty9 β Set)
(var : β Ξ A β Var9 Ξ A β Tm9 Ξ A)
(lam : β Ξ A B β Tm9 (snoc9 Ξ A) B β Tm9 Ξ (arr9 A B))
(app : β Ξ A B β Tm9 Ξ (arr9 A B) β Tm9 Ξ A β Tm9 Ξ B)
(tt : β Ξ β Tm9 Ξ top9)
(pair : β Ξ A B β Tm9 Ξ A β Tm9 Ξ B β Tm9 Ξ (prod9 A B))
(fst : β Ξ A B β Tm9 Ξ (prod9 A B) β Tm9 Ξ A)
(snd : β Ξ A B β Tm9 Ξ (prod9 A B) β Tm9 Ξ B)
(left : β Ξ A B β Tm9 Ξ A β Tm9 Ξ (sum9 A B))
(right : β Ξ A B β Tm9 Ξ B β Tm9 Ξ (sum9 A B))
(case : β Ξ A B C β Tm9 Ξ (sum9 A B) β Tm9 Ξ (arr9 A C) β Tm9 Ξ (arr9 B C) β Tm9 Ξ C)
(zero : β Ξ β Tm9 Ξ nat9)
(suc : β Ξ β Tm9 Ξ nat9 β Tm9 Ξ nat9)
(rec : β Ξ A β Tm9 Ξ nat9 β Tm9 Ξ (arr9 nat9 (arr9 A A)) β Tm9 Ξ A β Tm9 Ξ A)
β Tm9 Ξ A
var9 : β{Ξ A} β Var9 Ξ A β Tm9 Ξ A; var9
= Ξ» x Tm9 var9 lam app tt pair fst snd left right case zero suc rec β
var9 _ _ x
lam9 : β{Ξ A B} β Tm9 (snoc9 Ξ A) B β Tm9 Ξ (arr9 A B); lam9
= Ξ» t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec β
lam9 _ _ _ (t Tm9 var9 lam9 app tt pair fst snd left right case zero suc rec)
app9 : β{Ξ A B} β Tm9 Ξ (arr9 A B) β Tm9 Ξ A β Tm9 Ξ B; app9
= Ξ» t u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec β
app9 _ _ _ (t Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt pair fst snd left right case zero suc rec)
tt9 : β{Ξ} β Tm9 Ξ top9; tt9
= Ξ» Tm9 var9 lam9 app9 tt9 pair fst snd left right case zero suc rec β tt9 _
pair9 : β{Ξ A B} β Tm9 Ξ A β Tm9 Ξ B β Tm9 Ξ (prod9 A B); pair9
= Ξ» t u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec β
pair9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst snd left right case zero suc rec)
fst9 : β{Ξ A B} β Tm9 Ξ (prod9 A B) β Tm9 Ξ A; fst9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec β
fst9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd left right case zero suc rec)
snd9 : β{Ξ A B} β Tm9 Ξ (prod9 A B) β Tm9 Ξ B; snd9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec β
snd9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left right case zero suc rec)
left9 : β{Ξ A B} β Tm9 Ξ A β Tm9 Ξ (sum9 A B); left9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec β
left9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right case zero suc rec)
right9 : β{Ξ A B} β Tm9 Ξ B β Tm9 Ξ (sum9 A B); right9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec β
right9 _ _ _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case zero suc rec)
case9 : β{Ξ A B C} β Tm9 Ξ (sum9 A B) β Tm9 Ξ (arr9 A C) β Tm9 Ξ (arr9 B C) β Tm9 Ξ C; case9
= Ξ» t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec β
case9 _ _ _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero suc rec)
zero9 : β{Ξ} β Tm9 Ξ nat9; zero9
= Ξ» Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc rec β zero9 _
suc9 : β{Ξ} β Tm9 Ξ nat9 β Tm9 Ξ nat9; suc9
= Ξ» t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec β
suc9 _ (t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec)
rec9 : β{Ξ A} β Tm9 Ξ nat9 β Tm9 Ξ (arr9 nat9 (arr9 A A)) β Tm9 Ξ A β Tm9 Ξ A; rec9
= Ξ» t u v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9 β
rec9 _ _
(t Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(u Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
(v Tm9 var9 lam9 app9 tt9 pair9 fst9 snd9 left9 right9 case9 zero9 suc9 rec9)
v09 : β{Ξ A} β Tm9 (snoc9 Ξ A) A; v09
= var9 vz9
v19 : β{Ξ A B} β Tm9 (snoc9 (snoc9 Ξ A) B) A; v19
= var9 (vs9 vz9)
v29 : β{Ξ A B C} β Tm9 (snoc9 (snoc9 (snoc9 Ξ A) B) C) A; v29
= var9 (vs9 (vs9 vz9))
v39 : β{Ξ A B C D} β Tm9 (snoc9 (snoc9 (snoc9 (snoc9 Ξ A) B) C) D) A; v39
= var9 (vs9 (vs9 (vs9 vz9)))
tbool9 : Ty9; tbool9
= sum9 top9 top9
true9 : β{Ξ} β Tm9 Ξ tbool9; true9
= left9 tt9
tfalse9 : β{Ξ} β Tm9 Ξ tbool9; tfalse9
= right9 tt9
ifthenelse9 : β{Ξ A} β Tm9 Ξ (arr9 tbool9 (arr9 A (arr9 A A))); ifthenelse9
= lam9 (lam9 (lam9 (case9 v29 (lam9 v29) (lam9 v19))))
times49 : β{Ξ A} β Tm9 Ξ (arr9 (arr9 A A) (arr9 A A)); times49
= lam9 (lam9 (app9 v19 (app9 v19 (app9 v19 (app9 v19 v09)))))
add9 : β{Ξ} β Tm9 Ξ (arr9 nat9 (arr9 nat9 nat9)); add9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (suc9 (app9 v19 v09)))))
(lam9 v09))
mul9 : β{Ξ} β Tm9 Ξ (arr9 nat9 (arr9 nat9 nat9)); mul9
= lam9 (rec9 v09
(lam9 (lam9 (lam9 (app9 (app9 add9 (app9 v19 v09)) v09))))
(lam9 zero9))
fact9 : β{Ξ} β Tm9 Ξ (arr9 nat9 nat9); fact9
= lam9 (rec9 v09 (lam9 (lam9 (app9 (app9 mul9 (suc9 v19)) v09)))
(suc9 zero9))
{-# OPTIONS --type-in-type #-}
Ty10 : Set
Ty10 =
(Ty10 : Set)
(nat top bot : Ty10)
(arr prod sum : Ty10 β Ty10 β Ty10)
β Ty10
nat10 : Ty10; nat10 = Ξ» _ nat10 _ _ _ _ _ β nat10
top10 : Ty10; top10 = Ξ» _ _ top10 _ _ _ _ β top10
bot10 : Ty10; bot10 = Ξ» _ _ _ bot10 _ _ _ β bot10
arr10 : Ty10 β Ty10 β Ty10; arr10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod sum β
arr10 (A Ty10 nat10 top10 bot10 arr10 prod sum) (B Ty10 nat10 top10 bot10 arr10 prod sum)
prod10 : Ty10 β Ty10 β Ty10; prod10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod10 sum β
prod10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum) (B Ty10 nat10 top10 bot10 arr10 prod10 sum)
sum10 : Ty10 β Ty10 β Ty10; sum10
= Ξ» A B Ty10 nat10 top10 bot10 arr10 prod10 sum10 β
sum10 (A Ty10 nat10 top10 bot10 arr10 prod10 sum10) (B Ty10 nat10 top10 bot10 arr10 prod10 sum10)
Con10 : Set; Con10
= (Con10 : Set)
(nil : Con10)
(snoc : Con10 β Ty10 β Con10)
β Con10
nil10 : Con10; nil10
= Ξ» Con10 nil10 snoc β nil10
snoc10 : Con10 β Ty10 β Con10; snoc10
= Ξ» Ξ A Con10 nil10 snoc10 β snoc10 (Ξ Con10 nil10 snoc10) A
Var10 : Con10 β Ty10 β Set; Var10
= Ξ» Ξ A β
(Var10 : Con10 β Ty10 β Set)
(vz : β Ξ A β Var10 (snoc10 Ξ A) A)
(vs : β Ξ B A β Var10 Ξ A β Var10 (snoc10 Ξ B) A)
β Var10 Ξ A
vz10 : β{Ξ A} β Var10 (snoc10 Ξ A) A; vz10
= Ξ» Var10 vz10 vs β vz10 _ _
vs10 : β{Ξ B A} β Var10 Ξ A β Var10 (snoc10 Ξ B) A; vs10
= Ξ» x Var10 vz10 vs10 β vs10 _ _ _ (x Var10 vz10 vs10)
Tm10 : Con10 β Ty10 β Set; Tm10
= Ξ» Ξ A β
(Tm10 : Con10 β Ty10 β Set)
(var : β Ξ A β Var10 Ξ A β Tm10 Ξ A)
(lam : β Ξ A B β Tm10 (snoc10 Ξ A) B β Tm10 Ξ (arr10 A B))
(app : β Ξ A B β Tm10 Ξ (arr10 A B) β Tm10 Ξ A β Tm10 Ξ B)
(tt : β Ξ β Tm10 Ξ top10)
(pair : β Ξ A B β Tm10 Ξ A β Tm10 Ξ B β Tm10 Ξ (prod10 A B))
(fst : β Ξ A B β Tm10 Ξ (prod10 A B) β Tm10 Ξ A)
(snd : β Ξ A B β Tm10 Ξ (prod10 A B) β Tm10 Ξ B)
(left : β Ξ A B β Tm10 Ξ A β Tm10 Ξ (sum10 A B))
(right : β Ξ A B β Tm10 Ξ B β Tm10 Ξ (sum10 A B))
(case : β Ξ A B C β Tm10 Ξ (sum10 A B) β Tm10 Ξ (arr10 A C) β Tm10 Ξ (arr10 B C) β Tm10 Ξ C)
(zero : β Ξ β Tm10 Ξ nat10)
(suc : β Ξ β Tm10 Ξ nat10 β Tm10 Ξ nat10)
(rec : β Ξ A β Tm10 Ξ nat10 β Tm10 Ξ (arr10 nat10 (arr10 A A)) β Tm10 Ξ A β Tm10 Ξ A)
β Tm10 Ξ A
var10 : β{Ξ A} β Var10 Ξ A β Tm10 Ξ A; var10
= Ξ» x Tm10 var10 lam app tt pair fst snd left right case zero suc rec β
var10 _ _ x
lam10 : β{Ξ A B} β Tm10 (snoc10 Ξ A) B β Tm10 Ξ (arr10 A B); lam10
= Ξ» t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec β
lam10 _ _ _ (t Tm10 var10 lam10 app tt pair fst snd left right case zero suc rec)
app10 : β{Ξ A B} β Tm10 Ξ (arr10 A B) β Tm10 Ξ A β Tm10 Ξ B; app10
= Ξ» t u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec β
app10 _ _ _ (t Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt pair fst snd left right case zero suc rec)
tt10 : β{Ξ} β Tm10 Ξ top10; tt10
= Ξ» Tm10 var10 lam10 app10 tt10 pair fst snd left right case zero suc rec β tt10 _
pair10 : β{Ξ A B} β Tm10 Ξ A β Tm10 Ξ B β Tm10 Ξ (prod10 A B); pair10
= Ξ» t u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec β
pair10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst snd left right case zero suc rec)
fst10 : β{Ξ A B} β Tm10 Ξ (prod10 A B) β Tm10 Ξ A; fst10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec β
fst10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd left right case zero suc rec)
snd10 : β{Ξ A B} β Tm10 Ξ (prod10 A B) β Tm10 Ξ B; snd10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec β
snd10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left right case zero suc rec)
left10 : β{Ξ A B} β Tm10 Ξ A β Tm10 Ξ (sum10 A B); left10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec β
left10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right case zero suc rec)
right10 : β{Ξ A B} β Tm10 Ξ B β Tm10 Ξ (sum10 A B); right10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec β
right10 _ _ _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case zero suc rec)
case10 : β{Ξ A B C} β Tm10 Ξ (sum10 A B) β Tm10 Ξ (arr10 A C) β Tm10 Ξ (arr10 B C) β Tm10 Ξ C; case10
= Ξ» t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec β
case10 _ _ _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero suc rec)
zero10 : β{Ξ} β Tm10 Ξ nat10; zero10
= Ξ» Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc rec β zero10 _
suc10 : β{Ξ} β Tm10 Ξ nat10 β Tm10 Ξ nat10; suc10
= Ξ» t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec β
suc10 _ (t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec)
rec10 : β{Ξ A} β Tm10 Ξ nat10 β Tm10 Ξ (arr10 nat10 (arr10 A A)) β Tm10 Ξ A β Tm10 Ξ A; rec10
= Ξ» t u v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10 β
rec10 _ _
(t Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(u Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
(v Tm10 var10 lam10 app10 tt10 pair10 fst10 snd10 left10 right10 case10 zero10 suc10 rec10)
v010 : β{Ξ A} β Tm10 (snoc10 Ξ A) A; v010
= var10 vz10
v110 : β{Ξ A B} β Tm10 (snoc10 (snoc10 Ξ A) B) A; v110
= var10 (vs10 vz10)
v210 : β{Ξ A B C} β Tm10 (snoc10 (snoc10 (snoc10 Ξ A) B) C) A; v210
= var10 (vs10 (vs10 vz10))
v310 : β{Ξ A B C D} β Tm10 (snoc10 (snoc10 (snoc10 (snoc10 Ξ A) B) C) D) A; v310
= var10 (vs10 (vs10 (vs10 vz10)))
tbool10 : Ty10; tbool10
= sum10 top10 top10
true10 : β{Ξ} β Tm10 Ξ tbool10; true10
= left10 tt10
tfalse10 : β{Ξ} β Tm10 Ξ tbool10; tfalse10
= right10 tt10
ifthenelse10 : β{Ξ A} β Tm10 Ξ (arr10 tbool10 (arr10 A (arr10 A A))); ifthenelse10
= lam10 (lam10 (lam10 (case10 v210 (lam10 v210) (lam10 v110))))
times410 : β{Ξ A} β Tm10 Ξ (arr10 (arr10 A A) (arr10 A A)); times410
= lam10 (lam10 (app10 v110 (app10 v110 (app10 v110 (app10 v110 v010)))))
add10 : β{Ξ} β Tm10 Ξ (arr10 nat10 (arr10 nat10 nat10)); add10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (suc10 (app10 v110 v010)))))
(lam10 v010))
mul10 : β{Ξ} β Tm10 Ξ (arr10 nat10 (arr10 nat10 nat10)); mul10
= lam10 (rec10 v010
(lam10 (lam10 (lam10 (app10 (app10 add10 (app10 v110 v010)) v010))))
(lam10 zero10))
fact10 : β{Ξ} β Tm10 Ξ (arr10 nat10 nat10); fact10
= lam10 (rec10 v010 (lam10 (lam10 (app10 (app10 mul10 (suc10 v110)) v010)))
(suc10 zero10))
{-# OPTIONS --type-in-type #-}
Ty11 : Set
Ty11 =
(Ty11 : Set)
(nat top bot : Ty11)
(arr prod sum : Ty11 β Ty11 β Ty11)
β Ty11
nat11 : Ty11; nat11 = Ξ» _ nat11 _ _ _ _ _ β nat11
top11 : Ty11; top11 = Ξ» _ _ top11 _ _ _ _ β top11
bot11 : Ty11; bot11 = Ξ» _ _ _ bot11 _ _ _ β bot11
arr11 : Ty11 β Ty11 β Ty11; arr11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod sum β
arr11 (A Ty11 nat11 top11 bot11 arr11 prod sum) (B Ty11 nat11 top11 bot11 arr11 prod sum)
prod11 : Ty11 β Ty11 β Ty11; prod11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod11 sum β
prod11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum) (B Ty11 nat11 top11 bot11 arr11 prod11 sum)
sum11 : Ty11 β Ty11 β Ty11; sum11
= Ξ» A B Ty11 nat11 top11 bot11 arr11 prod11 sum11 β
sum11 (A Ty11 nat11 top11 bot11 arr11 prod11 sum11) (B Ty11 nat11 top11 bot11 arr11 prod11 sum11)
Con11 : Set; Con11
= (Con11 : Set)
(nil : Con11)
(snoc : Con11 β Ty11 β Con11)
β Con11
nil11 : Con11; nil11
= Ξ» Con11 nil11 snoc β nil11
snoc11 : Con11 β Ty11 β Con11; snoc11
= Ξ» Ξ A Con11 nil11 snoc11 β snoc11 (Ξ Con11 nil11 snoc11) A
Var11 : Con11 β Ty11 β Set; Var11
= Ξ» Ξ A β
(Var11 : Con11 β Ty11 β Set)
(vz : β Ξ A β Var11 (snoc11 Ξ A) A)
(vs : β Ξ B A β Var11 Ξ A β Var11 (snoc11 Ξ B) A)
β Var11 Ξ A
vz11 : β{Ξ A} β Var11 (snoc11 Ξ A) A; vz11
= Ξ» Var11 vz11 vs β vz11 _ _
vs11 : β{Ξ B A} β Var11 Ξ A β Var11 (snoc11 Ξ B) A; vs11
= Ξ» x Var11 vz11 vs11 β vs11 _ _ _ (x Var11 vz11 vs11)
Tm11 : Con11 β Ty11 β Set; Tm11
= Ξ» Ξ A β
(Tm11 : Con11 β Ty11 β Set)
(var : β Ξ A β Var11 Ξ A β Tm11 Ξ A)
(lam : β Ξ A B β Tm11 (snoc11 Ξ A) B β Tm11 Ξ (arr11 A B))
(app : β Ξ A B β Tm11 Ξ (arr11 A B) β Tm11 Ξ A β Tm11 Ξ B)
(tt : β Ξ β Tm11 Ξ top11)
(pair : β Ξ A B β Tm11 Ξ A β Tm11 Ξ B β Tm11 Ξ (prod11 A B))
(fst : β Ξ A B β Tm11 Ξ (prod11 A B) β Tm11 Ξ A)
(snd : β Ξ A B β Tm11 Ξ (prod11 A B) β Tm11 Ξ B)
(left : β Ξ A B β Tm11 Ξ A β Tm11 Ξ (sum11 A B))
(right : β Ξ A B β Tm11 Ξ B β Tm11 Ξ (sum11 A B))
(case : β Ξ A B C β Tm11 Ξ (sum11 A B) β Tm11 Ξ (arr11 A C) β Tm11 Ξ (arr11 B C) β Tm11 Ξ C)
(zero : β Ξ β Tm11 Ξ nat11)
(suc : β Ξ β Tm11 Ξ nat11 β Tm11 Ξ nat11)
(rec : β Ξ A β Tm11 Ξ nat11 β Tm11 Ξ (arr11 nat11 (arr11 A A)) β Tm11 Ξ A β Tm11 Ξ A)
β Tm11 Ξ A
var11 : β{Ξ A} β Var11 Ξ A β Tm11 Ξ A; var11
= Ξ» x Tm11 var11 lam app tt pair fst snd left right case zero suc rec β
var11 _ _ x
lam11 : β{Ξ A B} β Tm11 (snoc11 Ξ A) B β Tm11 Ξ (arr11 A B); lam11
= Ξ» t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec β
lam11 _ _ _ (t Tm11 var11 lam11 app tt pair fst snd left right case zero suc rec)
app11 : β{Ξ A B} β Tm11 Ξ (arr11 A B) β Tm11 Ξ A β Tm11 Ξ B; app11
= Ξ» t u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec β
app11 _ _ _ (t Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt pair fst snd left right case zero suc rec)
tt11 : β{Ξ} β Tm11 Ξ top11; tt11
= Ξ» Tm11 var11 lam11 app11 tt11 pair fst snd left right case zero suc rec β tt11 _
pair11 : β{Ξ A B} β Tm11 Ξ A β Tm11 Ξ B β Tm11 Ξ (prod11 A B); pair11
= Ξ» t u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec β
pair11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst snd left right case zero suc rec)
fst11 : β{Ξ A B} β Tm11 Ξ (prod11 A B) β Tm11 Ξ A; fst11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec β
fst11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd left right case zero suc rec)
snd11 : β{Ξ A B} β Tm11 Ξ (prod11 A B) β Tm11 Ξ B; snd11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec β
snd11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left right case zero suc rec)
left11 : β{Ξ A B} β Tm11 Ξ A β Tm11 Ξ (sum11 A B); left11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec β
left11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right case zero suc rec)
right11 : β{Ξ A B} β Tm11 Ξ B β Tm11 Ξ (sum11 A B); right11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec β
right11 _ _ _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case zero suc rec)
case11 : β{Ξ A B C} β Tm11 Ξ (sum11 A B) β Tm11 Ξ (arr11 A C) β Tm11 Ξ (arr11 B C) β Tm11 Ξ C; case11
= Ξ» t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec β
case11 _ _ _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero suc rec)
zero11 : β{Ξ} β Tm11 Ξ nat11; zero11
= Ξ» Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc rec β zero11 _
suc11 : β{Ξ} β Tm11 Ξ nat11 β Tm11 Ξ nat11; suc11
= Ξ» t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec β
suc11 _ (t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec)
rec11 : β{Ξ A} β Tm11 Ξ nat11 β Tm11 Ξ (arr11 nat11 (arr11 A A)) β Tm11 Ξ A β Tm11 Ξ A; rec11
= Ξ» t u v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11 β
rec11 _ _
(t Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(u Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
(v Tm11 var11 lam11 app11 tt11 pair11 fst11 snd11 left11 right11 case11 zero11 suc11 rec11)
v011 : β{Ξ A} β Tm11 (snoc11 Ξ A) A; v011
= var11 vz11
v111 : β{Ξ A B} β Tm11 (snoc11 (snoc11 Ξ A) B) A; v111
= var11 (vs11 vz11)
v211 : β{Ξ A B C} β Tm11 (snoc11 (snoc11 (snoc11 Ξ A) B) C) A; v211
= var11 (vs11 (vs11 vz11))
v311 : β{Ξ A B C D} β Tm11 (snoc11 (snoc11 (snoc11 (snoc11 Ξ A) B) C) D) A; v311
= var11 (vs11 (vs11 (vs11 vz11)))
tbool11 : Ty11; tbool11
= sum11 top11 top11
true11 : β{Ξ} β Tm11 Ξ tbool11; true11
= left11 tt11
tfalse11 : β{Ξ} β Tm11 Ξ tbool11; tfalse11
= right11 tt11
ifthenelse11 : β{Ξ A} β Tm11 Ξ (arr11 tbool11 (arr11 A (arr11 A A))); ifthenelse11
= lam11 (lam11 (lam11 (case11 v211 (lam11 v211) (lam11 v111))))
times411 : β{Ξ A} β Tm11 Ξ (arr11 (arr11 A A) (arr11 A A)); times411
= lam11 (lam11 (app11 v111 (app11 v111 (app11 v111 (app11 v111 v011)))))
add11 : β{Ξ} β Tm11 Ξ (arr11 nat11 (arr11 nat11 nat11)); add11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (suc11 (app11 v111 v011)))))
(lam11 v011))
mul11 : β{Ξ} β Tm11 Ξ (arr11 nat11 (arr11 nat11 nat11)); mul11
= lam11 (rec11 v011
(lam11 (lam11 (lam11 (app11 (app11 add11 (app11 v111 v011)) v011))))
(lam11 zero11))
fact11 : β{Ξ} β Tm11 Ξ (arr11 nat11 nat11); fact11
= lam11 (rec11 v011 (lam11 (lam11 (app11 (app11 mul11 (suc11 v111)) v011)))
(suc11 zero11))
{-# OPTIONS --type-in-type #-}
Ty12 : Set
Ty12 =
(Ty12 : Set)
(nat top bot : Ty12)
(arr prod sum : Ty12 β Ty12 β Ty12)
β Ty12
nat12 : Ty12; nat12 = Ξ» _ nat12 _ _ _ _ _ β nat12
top12 : Ty12; top12 = Ξ» _ _ top12 _ _ _ _ β top12
bot12 : Ty12; bot12 = Ξ» _ _ _ bot12 _ _ _ β bot12
arr12 : Ty12 β Ty12 β Ty12; arr12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod sum β
arr12 (A Ty12 nat12 top12 bot12 arr12 prod sum) (B Ty12 nat12 top12 bot12 arr12 prod sum)
prod12 : Ty12 β Ty12 β Ty12; prod12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod12 sum β
prod12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum) (B Ty12 nat12 top12 bot12 arr12 prod12 sum)
sum12 : Ty12 β Ty12 β Ty12; sum12
= Ξ» A B Ty12 nat12 top12 bot12 arr12 prod12 sum12 β
sum12 (A Ty12 nat12 top12 bot12 arr12 prod12 sum12) (B Ty12 nat12 top12 bot12 arr12 prod12 sum12)
Con12 : Set; Con12
= (Con12 : Set)
(nil : Con12)
(snoc : Con12 β Ty12 β Con12)
β Con12
nil12 : Con12; nil12
= Ξ» Con12 nil12 snoc β nil12
snoc12 : Con12 β Ty12 β Con12; snoc12
= Ξ» Ξ A Con12 nil12 snoc12 β snoc12 (Ξ Con12 nil12 snoc12) A
Var12 : Con12 β Ty12 β Set; Var12
= Ξ» Ξ A β
(Var12 : Con12 β Ty12 β Set)
(vz : β Ξ A β Var12 (snoc12 Ξ A) A)
(vs : β Ξ B A β Var12 Ξ A β Var12 (snoc12 Ξ B) A)
β Var12 Ξ A
vz12 : β{Ξ A} β Var12 (snoc12 Ξ A) A; vz12
= Ξ» Var12 vz12 vs β vz12 _ _
vs12 : β{Ξ B A} β Var12 Ξ A β Var12 (snoc12 Ξ B) A; vs12
= Ξ» x Var12 vz12 vs12 β vs12 _ _ _ (x Var12 vz12 vs12)
Tm12 : Con12 β Ty12 β Set; Tm12
= Ξ» Ξ A β
(Tm12 : Con12 β Ty12 β Set)
(var : β Ξ A β Var12 Ξ A β Tm12 Ξ A)
(lam : β Ξ A B β Tm12 (snoc12 Ξ A) B β Tm12 Ξ (arr12 A B))
(app : β Ξ A B β Tm12 Ξ (arr12 A B) β Tm12 Ξ A β Tm12 Ξ B)
(tt : β Ξ β Tm12 Ξ top12)
(pair : β Ξ A B β Tm12 Ξ A β Tm12 Ξ B β Tm12 Ξ (prod12 A B))
(fst : β Ξ A B β Tm12 Ξ (prod12 A B) β Tm12 Ξ A)
(snd : β Ξ A B β Tm12 Ξ (prod12 A B) β Tm12 Ξ B)
(left : β Ξ A B β Tm12 Ξ A β Tm12 Ξ (sum12 A B))
(right : β Ξ A B β Tm12 Ξ B β Tm12 Ξ (sum12 A B))
(case : β Ξ A B C β Tm12 Ξ (sum12 A B) β Tm12 Ξ (arr12 A C) β Tm12 Ξ (arr12 B C) β Tm12 Ξ C)
(zero : β Ξ β Tm12 Ξ nat12)
(suc : β Ξ β Tm12 Ξ nat12 β Tm12 Ξ nat12)
(rec : β Ξ A β Tm12 Ξ nat12 β Tm12 Ξ (arr12 nat12 (arr12 A A)) β Tm12 Ξ A β Tm12 Ξ A)
β Tm12 Ξ A
var12 : β{Ξ A} β Var12 Ξ A β Tm12 Ξ A; var12
= Ξ» x Tm12 var12 lam app tt pair fst snd left right case zero suc rec β
var12 _ _ x
lam12 : β{Ξ A B} β Tm12 (snoc12 Ξ A) B β Tm12 Ξ (arr12 A B); lam12
= Ξ» t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec β
lam12 _ _ _ (t Tm12 var12 lam12 app tt pair fst snd left right case zero suc rec)
app12 : β{Ξ A B} β Tm12 Ξ (arr12 A B) β Tm12 Ξ A β Tm12 Ξ B; app12
= Ξ» t u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec β
app12 _ _ _ (t Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt pair fst snd left right case zero suc rec)
tt12 : β{Ξ} β Tm12 Ξ top12; tt12
= Ξ» Tm12 var12 lam12 app12 tt12 pair fst snd left right case zero suc rec β tt12 _
pair12 : β{Ξ A B} β Tm12 Ξ A β Tm12 Ξ B β Tm12 Ξ (prod12 A B); pair12
= Ξ» t u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec β
pair12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst snd left right case zero suc rec)
fst12 : β{Ξ A B} β Tm12 Ξ (prod12 A B) β Tm12 Ξ A; fst12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec β
fst12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd left right case zero suc rec)
snd12 : β{Ξ A B} β Tm12 Ξ (prod12 A B) β Tm12 Ξ B; snd12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec β
snd12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left right case zero suc rec)
left12 : β{Ξ A B} β Tm12 Ξ A β Tm12 Ξ (sum12 A B); left12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec β
left12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right case zero suc rec)
right12 : β{Ξ A B} β Tm12 Ξ B β Tm12 Ξ (sum12 A B); right12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec β
right12 _ _ _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case zero suc rec)
case12 : β{Ξ A B C} β Tm12 Ξ (sum12 A B) β Tm12 Ξ (arr12 A C) β Tm12 Ξ (arr12 B C) β Tm12 Ξ C; case12
= Ξ» t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec β
case12 _ _ _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero suc rec)
zero12 : β{Ξ} β Tm12 Ξ nat12; zero12
= Ξ» Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc rec β zero12 _
suc12 : β{Ξ} β Tm12 Ξ nat12 β Tm12 Ξ nat12; suc12
= Ξ» t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec β
suc12 _ (t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec)
rec12 : β{Ξ A} β Tm12 Ξ nat12 β Tm12 Ξ (arr12 nat12 (arr12 A A)) β Tm12 Ξ A β Tm12 Ξ A; rec12
= Ξ» t u v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12 β
rec12 _ _
(t Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(u Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
(v Tm12 var12 lam12 app12 tt12 pair12 fst12 snd12 left12 right12 case12 zero12 suc12 rec12)
v012 : β{Ξ A} β Tm12 (snoc12 Ξ A) A; v012
= var12 vz12
v112 : β{Ξ A B} β Tm12 (snoc12 (snoc12 Ξ A) B) A; v112
= var12 (vs12 vz12)
v212 : β{Ξ A B C} β Tm12 (snoc12 (snoc12 (snoc12 Ξ A) B) C) A; v212
= var12 (vs12 (vs12 vz12))
v312 : β{Ξ A B C D} β Tm12 (snoc12 (snoc12 (snoc12 (snoc12 Ξ A) B) C) D) A; v312
= var12 (vs12 (vs12 (vs12 vz12)))
tbool12 : Ty12; tbool12
= sum12 top12 top12
true12 : β{Ξ} β Tm12 Ξ tbool12; true12
= left12 tt12
tfalse12 : β{Ξ} β Tm12 Ξ tbool12; tfalse12
= right12 tt12
ifthenelse12 : β{Ξ A} β Tm12 Ξ (arr12 tbool12 (arr12 A (arr12 A A))); ifthenelse12
= lam12 (lam12 (lam12 (case12 v212 (lam12 v212) (lam12 v112))))
times412 : β{Ξ A} β Tm12 Ξ (arr12 (arr12 A A) (arr12 A A)); times412
= lam12 (lam12 (app12 v112 (app12 v112 (app12 v112 (app12 v112 v012)))))
add12 : β{Ξ} β Tm12 Ξ (arr12 nat12 (arr12 nat12 nat12)); add12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (suc12 (app12 v112 v012)))))
(lam12 v012))
mul12 : β{Ξ} β Tm12 Ξ (arr12 nat12 (arr12 nat12 nat12)); mul12
= lam12 (rec12 v012
(lam12 (lam12 (lam12 (app12 (app12 add12 (app12 v112 v012)) v012))))
(lam12 zero12))
fact12 : β{Ξ} β Tm12 Ξ (arr12 nat12 nat12); fact12
= lam12 (rec12 v012 (lam12 (lam12 (app12 (app12 mul12 (suc12 v112)) v012)))
(suc12 zero12))
{-# OPTIONS --type-in-type #-}
Ty13 : Set
Ty13 =
(Ty13 : Set)
(nat top bot : Ty13)
(arr prod sum : Ty13 β Ty13 β Ty13)
β Ty13
nat13 : Ty13; nat13 = Ξ» _ nat13 _ _ _ _ _ β nat13
top13 : Ty13; top13 = Ξ» _ _ top13 _ _ _ _ β top13
bot13 : Ty13; bot13 = Ξ» _ _ _ bot13 _ _ _ β bot13
arr13 : Ty13 β Ty13 β Ty13; arr13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod sum β
arr13 (A Ty13 nat13 top13 bot13 arr13 prod sum) (B Ty13 nat13 top13 bot13 arr13 prod sum)
prod13 : Ty13 β Ty13 β Ty13; prod13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod13 sum β
prod13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum) (B Ty13 nat13 top13 bot13 arr13 prod13 sum)
sum13 : Ty13 β Ty13 β Ty13; sum13
= Ξ» A B Ty13 nat13 top13 bot13 arr13 prod13 sum13 β
sum13 (A Ty13 nat13 top13 bot13 arr13 prod13 sum13) (B Ty13 nat13 top13 bot13 arr13 prod13 sum13)
Con13 : Set; Con13
= (Con13 : Set)
(nil : Con13)
(snoc : Con13 β Ty13 β Con13)
β Con13
nil13 : Con13; nil13
= Ξ» Con13 nil13 snoc β nil13
snoc13 : Con13 β Ty13 β Con13; snoc13
= Ξ» Ξ A Con13 nil13 snoc13 β snoc13 (Ξ Con13 nil13 snoc13) A
Var13 : Con13 β Ty13 β Set; Var13
= Ξ» Ξ A β
(Var13 : Con13 β Ty13 β Set)
(vz : β Ξ A β Var13 (snoc13 Ξ A) A)
(vs : β Ξ B A β Var13 Ξ A β Var13 (snoc13 Ξ B) A)
β Var13 Ξ A
vz13 : β{Ξ A} β Var13 (snoc13 Ξ A) A; vz13
= Ξ» Var13 vz13 vs β vz13 _ _
vs13 : β{Ξ B A} β Var13 Ξ A β Var13 (snoc13 Ξ B) A; vs13
= Ξ» x Var13 vz13 vs13 β vs13 _ _ _ (x Var13 vz13 vs13)
Tm13 : Con13 β Ty13 β Set; Tm13
= Ξ» Ξ A β
(Tm13 : Con13 β Ty13 β Set)
(var : β Ξ A β Var13 Ξ A β Tm13 Ξ A)
(lam : β Ξ A B β Tm13 (snoc13 Ξ A) B β Tm13 Ξ (arr13 A B))
(app : β Ξ A B β Tm13 Ξ (arr13 A B) β Tm13 Ξ A β Tm13 Ξ B)
(tt : β Ξ β Tm13 Ξ top13)
(pair : β Ξ A B β Tm13 Ξ A β Tm13 Ξ B β Tm13 Ξ (prod13 A B))
(fst : β Ξ A B β Tm13 Ξ (prod13 A B) β Tm13 Ξ A)
(snd : β Ξ A B β Tm13 Ξ (prod13 A B) β Tm13 Ξ B)
(left : β Ξ A B β Tm13 Ξ A β Tm13 Ξ (sum13 A B))
(right : β Ξ A B β Tm13 Ξ B β Tm13 Ξ (sum13 A B))
(case : β Ξ A B C β Tm13 Ξ (sum13 A B) β Tm13 Ξ (arr13 A C) β Tm13 Ξ (arr13 B C) β Tm13 Ξ C)
(zero : β Ξ β Tm13 Ξ nat13)
(suc : β Ξ β Tm13 Ξ nat13 β Tm13 Ξ nat13)
(rec : β Ξ A β Tm13 Ξ nat13 β Tm13 Ξ (arr13 nat13 (arr13 A A)) β Tm13 Ξ A β Tm13 Ξ A)
β Tm13 Ξ A
var13 : β{Ξ A} β Var13 Ξ A β Tm13 Ξ A; var13
= Ξ» x Tm13 var13 lam app tt pair fst snd left right case zero suc rec β
var13 _ _ x
lam13 : β{Ξ A B} β Tm13 (snoc13 Ξ A) B β Tm13 Ξ (arr13 A B); lam13
= Ξ» t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec β
lam13 _ _ _ (t Tm13 var13 lam13 app tt pair fst snd left right case zero suc rec)
app13 : β{Ξ A B} β Tm13 Ξ (arr13 A B) β Tm13 Ξ A β Tm13 Ξ B; app13
= Ξ» t u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec β
app13 _ _ _ (t Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt pair fst snd left right case zero suc rec)
tt13 : β{Ξ} β Tm13 Ξ top13; tt13
= Ξ» Tm13 var13 lam13 app13 tt13 pair fst snd left right case zero suc rec β tt13 _
pair13 : β{Ξ A B} β Tm13 Ξ A β Tm13 Ξ B β Tm13 Ξ (prod13 A B); pair13
= Ξ» t u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec β
pair13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst snd left right case zero suc rec)
fst13 : β{Ξ A B} β Tm13 Ξ (prod13 A B) β Tm13 Ξ A; fst13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec β
fst13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd left right case zero suc rec)
snd13 : β{Ξ A B} β Tm13 Ξ (prod13 A B) β Tm13 Ξ B; snd13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec β
snd13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left right case zero suc rec)
left13 : β{Ξ A B} β Tm13 Ξ A β Tm13 Ξ (sum13 A B); left13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec β
left13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right case zero suc rec)
right13 : β{Ξ A B} β Tm13 Ξ B β Tm13 Ξ (sum13 A B); right13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec β
right13 _ _ _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case zero suc rec)
case13 : β{Ξ A B C} β Tm13 Ξ (sum13 A B) β Tm13 Ξ (arr13 A C) β Tm13 Ξ (arr13 B C) β Tm13 Ξ C; case13
= Ξ» t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec β
case13 _ _ _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero suc rec)
zero13 : β{Ξ} β Tm13 Ξ nat13; zero13
= Ξ» Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc rec β zero13 _
suc13 : β{Ξ} β Tm13 Ξ nat13 β Tm13 Ξ nat13; suc13
= Ξ» t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec β
suc13 _ (t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec)
rec13 : β{Ξ A} β Tm13 Ξ nat13 β Tm13 Ξ (arr13 nat13 (arr13 A A)) β Tm13 Ξ A β Tm13 Ξ A; rec13
= Ξ» t u v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13 β
rec13 _ _
(t Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(u Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
(v Tm13 var13 lam13 app13 tt13 pair13 fst13 snd13 left13 right13 case13 zero13 suc13 rec13)
v013 : β{Ξ A} β Tm13 (snoc13 Ξ A) A; v013
= var13 vz13
v113 : β{Ξ A B} β Tm13 (snoc13 (snoc13 Ξ A) B) A; v113
= var13 (vs13 vz13)
v213 : β{Ξ A B C} β Tm13 (snoc13 (snoc13 (snoc13 Ξ A) B) C) A; v213
= var13 (vs13 (vs13 vz13))
v313 : β{Ξ A B C D} β Tm13 (snoc13 (snoc13 (snoc13 (snoc13 Ξ A) B) C) D) A; v313
= var13 (vs13 (vs13 (vs13 vz13)))
tbool13 : Ty13; tbool13
= sum13 top13 top13
true13 : β{Ξ} β Tm13 Ξ tbool13; true13
= left13 tt13
tfalse13 : β{Ξ} β Tm13 Ξ tbool13; tfalse13
= right13 tt13
ifthenelse13 : β{Ξ A} β Tm13 Ξ (arr13 tbool13 (arr13 A (arr13 A A))); ifthenelse13
= lam13 (lam13 (lam13 (case13 v213 (lam13 v213) (lam13 v113))))
times413 : β{Ξ A} β Tm13 Ξ (arr13 (arr13 A A) (arr13 A A)); times413
= lam13 (lam13 (app13 v113 (app13 v113 (app13 v113 (app13 v113 v013)))))
add13 : β{Ξ} β Tm13 Ξ (arr13 nat13 (arr13 nat13 nat13)); add13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (suc13 (app13 v113 v013)))))
(lam13 v013))
mul13 : β{Ξ} β Tm13 Ξ (arr13 nat13 (arr13 nat13 nat13)); mul13
= lam13 (rec13 v013
(lam13 (lam13 (lam13 (app13 (app13 add13 (app13 v113 v013)) v013))))
(lam13 zero13))
fact13 : β{Ξ} β Tm13 Ξ (arr13 nat13 nat13); fact13
= lam13 (rec13 v013 (lam13 (lam13 (app13 (app13 mul13 (suc13 v113)) v013)))
(suc13 zero13))
{-# OPTIONS --type-in-type #-}
Ty14 : Set
Ty14 =
(Ty14 : Set)
(nat top bot : Ty14)
(arr prod sum : Ty14 β Ty14 β Ty14)
β Ty14
nat14 : Ty14; nat14 = Ξ» _ nat14 _ _ _ _ _ β nat14
top14 : Ty14; top14 = Ξ» _ _ top14 _ _ _ _ β top14
bot14 : Ty14; bot14 = Ξ» _ _ _ bot14 _ _ _ β bot14
arr14 : Ty14 β Ty14 β Ty14; arr14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod sum β
arr14 (A Ty14 nat14 top14 bot14 arr14 prod sum) (B Ty14 nat14 top14 bot14 arr14 prod sum)
prod14 : Ty14 β Ty14 β Ty14; prod14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod14 sum β
prod14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum) (B Ty14 nat14 top14 bot14 arr14 prod14 sum)
sum14 : Ty14 β Ty14 β Ty14; sum14
= Ξ» A B Ty14 nat14 top14 bot14 arr14 prod14 sum14 β
sum14 (A Ty14 nat14 top14 bot14 arr14 prod14 sum14) (B Ty14 nat14 top14 bot14 arr14 prod14 sum14)
Con14 : Set; Con14
= (Con14 : Set)
(nil : Con14)
(snoc : Con14 β Ty14 β Con14)
β Con14
nil14 : Con14; nil14
= Ξ» Con14 nil14 snoc β nil14
snoc14 : Con14 β Ty14 β Con14; snoc14
= Ξ» Ξ A Con14 nil14 snoc14 β snoc14 (Ξ Con14 nil14 snoc14) A
Var14 : Con14 β Ty14 β Set; Var14
= Ξ» Ξ A β
(Var14 : Con14 β Ty14 β Set)
(vz : β Ξ A β Var14 (snoc14 Ξ A) A)
(vs : β Ξ B A β Var14 Ξ A β Var14 (snoc14 Ξ B) A)
β Var14 Ξ A
vz14 : β{Ξ A} β Var14 (snoc14 Ξ A) A; vz14
= Ξ» Var14 vz14 vs β vz14 _ _
vs14 : β{Ξ B A} β Var14 Ξ A β Var14 (snoc14 Ξ B) A; vs14
= Ξ» x Var14 vz14 vs14 β vs14 _ _ _ (x Var14 vz14 vs14)
Tm14 : Con14 β Ty14 β Set; Tm14
= Ξ» Ξ A β
(Tm14 : Con14 β Ty14 β Set)
(var : β Ξ A β Var14 Ξ A β Tm14 Ξ A)
(lam : β Ξ A B β Tm14 (snoc14 Ξ A) B β Tm14 Ξ (arr14 A B))
(app : β Ξ A B β Tm14 Ξ (arr14 A B) β Tm14 Ξ A β Tm14 Ξ B)
(tt : β Ξ β Tm14 Ξ top14)
(pair : β Ξ A B β Tm14 Ξ A β Tm14 Ξ B β Tm14 Ξ (prod14 A B))
(fst : β Ξ A B β Tm14 Ξ (prod14 A B) β Tm14 Ξ A)
(snd : β Ξ A B β Tm14 Ξ (prod14 A B) β Tm14 Ξ B)
(left : β Ξ A B β Tm14 Ξ A β Tm14 Ξ (sum14 A B))
(right : β Ξ A B β Tm14 Ξ B β Tm14 Ξ (sum14 A B))
(case : β Ξ A B C β Tm14 Ξ (sum14 A B) β Tm14 Ξ (arr14 A C) β Tm14 Ξ (arr14 B C) β Tm14 Ξ C)
(zero : β Ξ β Tm14 Ξ nat14)
(suc : β Ξ β Tm14 Ξ nat14 β Tm14 Ξ nat14)
(rec : β Ξ A β Tm14 Ξ nat14 β Tm14 Ξ (arr14 nat14 (arr14 A A)) β Tm14 Ξ A β Tm14 Ξ A)
β Tm14 Ξ A
var14 : β{Ξ A} β Var14 Ξ A β Tm14 Ξ A; var14
= Ξ» x Tm14 var14 lam app tt pair fst snd left right case zero suc rec β
var14 _ _ x
lam14 : β{Ξ A B} β Tm14 (snoc14 Ξ A) B β Tm14 Ξ (arr14 A B); lam14
= Ξ» t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec β
lam14 _ _ _ (t Tm14 var14 lam14 app tt pair fst snd left right case zero suc rec)
app14 : β{Ξ A B} β Tm14 Ξ (arr14 A B) β Tm14 Ξ A β Tm14 Ξ B; app14
= Ξ» t u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec β
app14 _ _ _ (t Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt pair fst snd left right case zero suc rec)
tt14 : β{Ξ} β Tm14 Ξ top14; tt14
= Ξ» Tm14 var14 lam14 app14 tt14 pair fst snd left right case zero suc rec β tt14 _
pair14 : β{Ξ A B} β Tm14 Ξ A β Tm14 Ξ B β Tm14 Ξ (prod14 A B); pair14
= Ξ» t u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec β
pair14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst snd left right case zero suc rec)
fst14 : β{Ξ A B} β Tm14 Ξ (prod14 A B) β Tm14 Ξ A; fst14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec β
fst14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd left right case zero suc rec)
snd14 : β{Ξ A B} β Tm14 Ξ (prod14 A B) β Tm14 Ξ B; snd14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec β
snd14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left right case zero suc rec)
left14 : β{Ξ A B} β Tm14 Ξ A β Tm14 Ξ (sum14 A B); left14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec β
left14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right case zero suc rec)
right14 : β{Ξ A B} β Tm14 Ξ B β Tm14 Ξ (sum14 A B); right14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec β
right14 _ _ _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case zero suc rec)
case14 : β{Ξ A B C} β Tm14 Ξ (sum14 A B) β Tm14 Ξ (arr14 A C) β Tm14 Ξ (arr14 B C) β Tm14 Ξ C; case14
= Ξ» t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec β
case14 _ _ _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero suc rec)
zero14 : β{Ξ} β Tm14 Ξ nat14; zero14
= Ξ» Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc rec β zero14 _
suc14 : β{Ξ} β Tm14 Ξ nat14 β Tm14 Ξ nat14; suc14
= Ξ» t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec β
suc14 _ (t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec)
rec14 : β{Ξ A} β Tm14 Ξ nat14 β Tm14 Ξ (arr14 nat14 (arr14 A A)) β Tm14 Ξ A β Tm14 Ξ A; rec14
= Ξ» t u v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14 β
rec14 _ _
(t Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(u Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
(v Tm14 var14 lam14 app14 tt14 pair14 fst14 snd14 left14 right14 case14 zero14 suc14 rec14)
v014 : β{Ξ A} β Tm14 (snoc14 Ξ A) A; v014
= var14 vz14
v114 : β{Ξ A B} β Tm14 (snoc14 (snoc14 Ξ A) B) A; v114
= var14 (vs14 vz14)
v214 : β{Ξ A B C} β Tm14 (snoc14 (snoc14 (snoc14 Ξ A) B) C) A; v214
= var14 (vs14 (vs14 vz14))
v314 : β{Ξ A B C D} β Tm14 (snoc14 (snoc14 (snoc14 (snoc14 Ξ A) B) C) D) A; v314
= var14 (vs14 (vs14 (vs14 vz14)))
tbool14 : Ty14; tbool14
= sum14 top14 top14
true14 : β{Ξ} β Tm14 Ξ tbool14; true14
= left14 tt14
tfalse14 : β{Ξ} β Tm14 Ξ tbool14; tfalse14
= right14 tt14
ifthenelse14 : β{Ξ A} β Tm14 Ξ (arr14 tbool14 (arr14 A (arr14 A A))); ifthenelse14
= lam14 (lam14 (lam14 (case14 v214 (lam14 v214) (lam14 v114))))
times414 : β{Ξ A} β Tm14 Ξ (arr14 (arr14 A A) (arr14 A A)); times414
= lam14 (lam14 (app14 v114 (app14 v114 (app14 v114 (app14 v114 v014)))))
add14 : β{Ξ} β Tm14 Ξ (arr14 nat14 (arr14 nat14 nat14)); add14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (suc14 (app14 v114 v014)))))
(lam14 v014))
mul14 : β{Ξ} β Tm14 Ξ (arr14 nat14 (arr14 nat14 nat14)); mul14
= lam14 (rec14 v014
(lam14 (lam14 (lam14 (app14 (app14 add14 (app14 v114 v014)) v014))))
(lam14 zero14))
fact14 : β{Ξ} β Tm14 Ξ (arr14 nat14 nat14); fact14
= lam14 (rec14 v014 (lam14 (lam14 (app14 (app14 mul14 (suc14 v114)) v014)))
(suc14 zero14))
{-# OPTIONS --type-in-type #-}
Ty15 : Set
Ty15 =
(Ty15 : Set)
(nat top bot : Ty15)
(arr prod sum : Ty15 β Ty15 β Ty15)
β Ty15
nat15 : Ty15; nat15 = Ξ» _ nat15 _ _ _ _ _ β nat15
top15 : Ty15; top15 = Ξ» _ _ top15 _ _ _ _ β top15
bot15 : Ty15; bot15 = Ξ» _ _ _ bot15 _ _ _ β bot15
arr15 : Ty15 β Ty15 β Ty15; arr15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod sum β
arr15 (A Ty15 nat15 top15 bot15 arr15 prod sum) (B Ty15 nat15 top15 bot15 arr15 prod sum)
prod15 : Ty15 β Ty15 β Ty15; prod15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod15 sum β
prod15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum) (B Ty15 nat15 top15 bot15 arr15 prod15 sum)
sum15 : Ty15 β Ty15 β Ty15; sum15
= Ξ» A B Ty15 nat15 top15 bot15 arr15 prod15 sum15 β
sum15 (A Ty15 nat15 top15 bot15 arr15 prod15 sum15) (B Ty15 nat15 top15 bot15 arr15 prod15 sum15)
Con15 : Set; Con15
= (Con15 : Set)
(nil : Con15)
(snoc : Con15 β Ty15 β Con15)
β Con15
nil15 : Con15; nil15
= Ξ» Con15 nil15 snoc β nil15
snoc15 : Con15 β Ty15 β Con15; snoc15
= Ξ» Ξ A Con15 nil15 snoc15 β snoc15 (Ξ Con15 nil15 snoc15) A
Var15 : Con15 β Ty15 β Set; Var15
= Ξ» Ξ A β
(Var15 : Con15 β Ty15 β Set)
(vz : β Ξ A β Var15 (snoc15 Ξ A) A)
(vs : β Ξ B A β Var15 Ξ A β Var15 (snoc15 Ξ B) A)
β Var15 Ξ A
vz15 : β{Ξ A} β Var15 (snoc15 Ξ A) A; vz15
= Ξ» Var15 vz15 vs β vz15 _ _
vs15 : β{Ξ B A} β Var15 Ξ A β Var15 (snoc15 Ξ B) A; vs15
= Ξ» x Var15 vz15 vs15 β vs15 _ _ _ (x Var15 vz15 vs15)
Tm15 : Con15 β Ty15 β Set; Tm15
= Ξ» Ξ A β
(Tm15 : Con15 β Ty15 β Set)
(var : β Ξ A β Var15 Ξ A β Tm15 Ξ A)
(lam : β Ξ A B β Tm15 (snoc15 Ξ A) B β Tm15 Ξ (arr15 A B))
(app : β Ξ A B β Tm15 Ξ (arr15 A B) β Tm15 Ξ A β Tm15 Ξ B)
(tt : β Ξ β Tm15 Ξ top15)
(pair : β Ξ A B β Tm15 Ξ A β Tm15 Ξ B β Tm15 Ξ (prod15 A B))
(fst : β Ξ A B β Tm15 Ξ (prod15 A B) β Tm15 Ξ A)
(snd : β Ξ A B β Tm15 Ξ (prod15 A B) β Tm15 Ξ B)
(left : β Ξ A B β Tm15 Ξ A β Tm15 Ξ (sum15 A B))
(right : β Ξ A B β Tm15 Ξ B β Tm15 Ξ (sum15 A B))
(case : β Ξ A B C β Tm15 Ξ (sum15 A B) β Tm15 Ξ (arr15 A C) β Tm15 Ξ (arr15 B C) β Tm15 Ξ C)
(zero : β Ξ β Tm15 Ξ nat15)
(suc : β Ξ β Tm15 Ξ nat15 β Tm15 Ξ nat15)
(rec : β Ξ A β Tm15 Ξ nat15 β Tm15 Ξ (arr15 nat15 (arr15 A A)) β Tm15 Ξ A β Tm15 Ξ A)
β Tm15 Ξ A
var15 : β{Ξ A} β Var15 Ξ A β Tm15 Ξ A; var15
= Ξ» x Tm15 var15 lam app tt pair fst snd left right case zero suc rec β
var15 _ _ x
lam15 : β{Ξ A B} β Tm15 (snoc15 Ξ A) B β Tm15 Ξ (arr15 A B); lam15
= Ξ» t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec β
lam15 _ _ _ (t Tm15 var15 lam15 app tt pair fst snd left right case zero suc rec)
app15 : β{Ξ A B} β Tm15 Ξ (arr15 A B) β Tm15 Ξ A β Tm15 Ξ B; app15
= Ξ» t u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec β
app15 _ _ _ (t Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt pair fst snd left right case zero suc rec)
tt15 : β{Ξ} β Tm15 Ξ top15; tt15
= Ξ» Tm15 var15 lam15 app15 tt15 pair fst snd left right case zero suc rec β tt15 _
pair15 : β{Ξ A B} β Tm15 Ξ A β Tm15 Ξ B β Tm15 Ξ (prod15 A B); pair15
= Ξ» t u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec β
pair15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst snd left right case zero suc rec)
fst15 : β{Ξ A B} β Tm15 Ξ (prod15 A B) β Tm15 Ξ A; fst15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec β
fst15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd left right case zero suc rec)
snd15 : β{Ξ A B} β Tm15 Ξ (prod15 A B) β Tm15 Ξ B; snd15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec β
snd15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left right case zero suc rec)
left15 : β{Ξ A B} β Tm15 Ξ A β Tm15 Ξ (sum15 A B); left15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec β
left15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right case zero suc rec)
right15 : β{Ξ A B} β Tm15 Ξ B β Tm15 Ξ (sum15 A B); right15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec β
right15 _ _ _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case zero suc rec)
case15 : β{Ξ A B C} β Tm15 Ξ (sum15 A B) β Tm15 Ξ (arr15 A C) β Tm15 Ξ (arr15 B C) β Tm15 Ξ C; case15
= Ξ» t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec β
case15 _ _ _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero suc rec)
zero15 : β{Ξ} β Tm15 Ξ nat15; zero15
= Ξ» Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc rec β zero15 _
suc15 : β{Ξ} β Tm15 Ξ nat15 β Tm15 Ξ nat15; suc15
= Ξ» t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec β
suc15 _ (t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec)
rec15 : β{Ξ A} β Tm15 Ξ nat15 β Tm15 Ξ (arr15 nat15 (arr15 A A)) β Tm15 Ξ A β Tm15 Ξ A; rec15
= Ξ» t u v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15 β
rec15 _ _
(t Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(u Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
(v Tm15 var15 lam15 app15 tt15 pair15 fst15 snd15 left15 right15 case15 zero15 suc15 rec15)
v015 : β{Ξ A} β Tm15 (snoc15 Ξ A) A; v015
= var15 vz15
v115 : β{Ξ A B} β Tm15 (snoc15 (snoc15 Ξ A) B) A; v115
= var15 (vs15 vz15)
v215 : β{Ξ A B C} β Tm15 (snoc15 (snoc15 (snoc15 Ξ A) B) C) A; v215
= var15 (vs15 (vs15 vz15))
v315 : β{Ξ A B C D} β Tm15 (snoc15 (snoc15 (snoc15 (snoc15 Ξ A) B) C) D) A; v315
= var15 (vs15 (vs15 (vs15 vz15)))
tbool15 : Ty15; tbool15
= sum15 top15 top15
true15 : β{Ξ} β Tm15 Ξ tbool15; true15
= left15 tt15
tfalse15 : β{Ξ} β Tm15 Ξ tbool15; tfalse15
= right15 tt15
ifthenelse15 : β{Ξ A} β Tm15 Ξ (arr15 tbool15 (arr15 A (arr15 A A))); ifthenelse15
= lam15 (lam15 (lam15 (case15 v215 (lam15 v215) (lam15 v115))))
times415 : β{Ξ A} β Tm15 Ξ (arr15 (arr15 A A) (arr15 A A)); times415
= lam15 (lam15 (app15 v115 (app15 v115 (app15 v115 (app15 v115 v015)))))
add15 : β{Ξ} β Tm15 Ξ (arr15 nat15 (arr15 nat15 nat15)); add15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (suc15 (app15 v115 v015)))))
(lam15 v015))
mul15 : β{Ξ} β Tm15 Ξ (arr15 nat15 (arr15 nat15 nat15)); mul15
= lam15 (rec15 v015
(lam15 (lam15 (lam15 (app15 (app15 add15 (app15 v115 v015)) v015))))
(lam15 zero15))
fact15 : β{Ξ} β Tm15 Ξ (arr15 nat15 nat15); fact15
= lam15 (rec15 v015 (lam15 (lam15 (app15 (app15 mul15 (suc15 v115)) v015)))
(suc15 zero15))
{-# OPTIONS --type-in-type #-}
Ty16 : Set
Ty16 =
(Ty16 : Set)
(nat top bot : Ty16)
(arr prod sum : Ty16 β Ty16 β Ty16)
β Ty16
nat16 : Ty16; nat16 = Ξ» _ nat16 _ _ _ _ _ β nat16
top16 : Ty16; top16 = Ξ» _ _ top16 _ _ _ _ β top16
bot16 : Ty16; bot16 = Ξ» _ _ _ bot16 _ _ _ β bot16
arr16 : Ty16 β Ty16 β Ty16; arr16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod sum β
arr16 (A Ty16 nat16 top16 bot16 arr16 prod sum) (B Ty16 nat16 top16 bot16 arr16 prod sum)
prod16 : Ty16 β Ty16 β Ty16; prod16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod16 sum β
prod16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum) (B Ty16 nat16 top16 bot16 arr16 prod16 sum)
sum16 : Ty16 β Ty16 β Ty16; sum16
= Ξ» A B Ty16 nat16 top16 bot16 arr16 prod16 sum16 β
sum16 (A Ty16 nat16 top16 bot16 arr16 prod16 sum16) (B Ty16 nat16 top16 bot16 arr16 prod16 sum16)
Con16 : Set; Con16
= (Con16 : Set)
(nil : Con16)
(snoc : Con16 β Ty16 β Con16)
β Con16
nil16 : Con16; nil16
= Ξ» Con16 nil16 snoc β nil16
snoc16 : Con16 β Ty16 β Con16; snoc16
= Ξ» Ξ A Con16 nil16 snoc16 β snoc16 (Ξ Con16 nil16 snoc16) A
Var16 : Con16 β Ty16 β Set; Var16
= Ξ» Ξ A β
(Var16 : Con16 β Ty16 β Set)
(vz : β Ξ A β Var16 (snoc16 Ξ A) A)
(vs : β Ξ B A β Var16 Ξ A β Var16 (snoc16 Ξ B) A)
β Var16 Ξ A
vz16 : β{Ξ A} β Var16 (snoc16 Ξ A) A; vz16
= Ξ» Var16 vz16 vs β vz16 _ _
vs16 : β{Ξ B A} β Var16 Ξ A β Var16 (snoc16 Ξ B) A; vs16
= Ξ» x Var16 vz16 vs16 β vs16 _ _ _ (x Var16 vz16 vs16)
Tm16 : Con16 β Ty16 β Set; Tm16
= Ξ» Ξ A β
(Tm16 : Con16 β Ty16 β Set)
(var : β Ξ A β Var16 Ξ A β Tm16 Ξ A)
(lam : β Ξ A B β Tm16 (snoc16 Ξ A) B β Tm16 Ξ (arr16 A B))
(app : β Ξ A B β Tm16 Ξ (arr16 A B) β Tm16 Ξ A β Tm16 Ξ B)
(tt : β Ξ β Tm16 Ξ top16)
(pair : β Ξ A B β Tm16 Ξ A β Tm16 Ξ B β Tm16 Ξ (prod16 A B))
(fst : β Ξ A B β Tm16 Ξ (prod16 A B) β Tm16 Ξ A)
(snd : β Ξ A B β Tm16 Ξ (prod16 A B) β Tm16 Ξ B)
(left : β Ξ A B β Tm16 Ξ A β Tm16 Ξ (sum16 A B))
(right : β Ξ A B β Tm16 Ξ B β Tm16 Ξ (sum16 A B))
(case : β Ξ A B C β Tm16 Ξ (sum16 A B) β Tm16 Ξ (arr16 A C) β Tm16 Ξ (arr16 B C) β Tm16 Ξ C)
(zero : β Ξ β Tm16 Ξ nat16)
(suc : β Ξ β Tm16 Ξ nat16 β Tm16 Ξ nat16)
(rec : β Ξ A β Tm16 Ξ nat16 β Tm16 Ξ (arr16 nat16 (arr16 A A)) β Tm16 Ξ A β Tm16 Ξ A)
β Tm16 Ξ A
var16 : β{Ξ A} β Var16 Ξ A β Tm16 Ξ A; var16
= Ξ» x Tm16 var16 lam app tt pair fst snd left right case zero suc rec β
var16 _ _ x
lam16 : β{Ξ A B} β Tm16 (snoc16 Ξ A) B β Tm16 Ξ (arr16 A B); lam16
= Ξ» t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec β
lam16 _ _ _ (t Tm16 var16 lam16 app tt pair fst snd left right case zero suc rec)
app16 : β{Ξ A B} β Tm16 Ξ (arr16 A B) β Tm16 Ξ A β Tm16 Ξ B; app16
= Ξ» t u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec β
app16 _ _ _ (t Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt pair fst snd left right case zero suc rec)
tt16 : β{Ξ} β Tm16 Ξ top16; tt16
= Ξ» Tm16 var16 lam16 app16 tt16 pair fst snd left right case zero suc rec β tt16 _
pair16 : β{Ξ A B} β Tm16 Ξ A β Tm16 Ξ B β Tm16 Ξ (prod16 A B); pair16
= Ξ» t u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec β
pair16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst snd left right case zero suc rec)
fst16 : β{Ξ A B} β Tm16 Ξ (prod16 A B) β Tm16 Ξ A; fst16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec β
fst16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd left right case zero suc rec)
snd16 : β{Ξ A B} β Tm16 Ξ (prod16 A B) β Tm16 Ξ B; snd16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec β
snd16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left right case zero suc rec)
left16 : β{Ξ A B} β Tm16 Ξ A β Tm16 Ξ (sum16 A B); left16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec β
left16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right case zero suc rec)
right16 : β{Ξ A B} β Tm16 Ξ B β Tm16 Ξ (sum16 A B); right16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec β
right16 _ _ _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case zero suc rec)
case16 : β{Ξ A B C} β Tm16 Ξ (sum16 A B) β Tm16 Ξ (arr16 A C) β Tm16 Ξ (arr16 B C) β Tm16 Ξ C; case16
= Ξ» t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec β
case16 _ _ _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero suc rec)
zero16 : β{Ξ} β Tm16 Ξ nat16; zero16
= Ξ» Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc rec β zero16 _
suc16 : β{Ξ} β Tm16 Ξ nat16 β Tm16 Ξ nat16; suc16
= Ξ» t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec β
suc16 _ (t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec)
rec16 : β{Ξ A} β Tm16 Ξ nat16 β Tm16 Ξ (arr16 nat16 (arr16 A A)) β Tm16 Ξ A β Tm16 Ξ A; rec16
= Ξ» t u v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16 β
rec16 _ _
(t Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(u Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
(v Tm16 var16 lam16 app16 tt16 pair16 fst16 snd16 left16 right16 case16 zero16 suc16 rec16)
v016 : β{Ξ A} β Tm16 (snoc16 Ξ A) A; v016
= var16 vz16
v116 : β{Ξ A B} β Tm16 (snoc16 (snoc16 Ξ A) B) A; v116
= var16 (vs16 vz16)
v216 : β{Ξ A B C} β Tm16 (snoc16 (snoc16 (snoc16 Ξ A) B) C) A; v216
= var16 (vs16 (vs16 vz16))
v316 : β{Ξ A B C D} β Tm16 (snoc16 (snoc16 (snoc16 (snoc16 Ξ A) B) C) D) A; v316
= var16 (vs16 (vs16 (vs16 vz16)))
tbool16 : Ty16; tbool16
= sum16 top16 top16
true16 : β{Ξ} β Tm16 Ξ tbool16; true16
= left16 tt16
tfalse16 : β{Ξ} β Tm16 Ξ tbool16; tfalse16
= right16 tt16
ifthenelse16 : β{Ξ A} β Tm16 Ξ (arr16 tbool16 (arr16 A (arr16 A A))); ifthenelse16
= lam16 (lam16 (lam16 (case16 v216 (lam16 v216) (lam16 v116))))
times416 : β{Ξ A} β Tm16 Ξ (arr16 (arr16 A A) (arr16 A A)); times416
= lam16 (lam16 (app16 v116 (app16 v116 (app16 v116 (app16 v116 v016)))))
add16 : β{Ξ} β Tm16 Ξ (arr16 nat16 (arr16 nat16 nat16)); add16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (suc16 (app16 v116 v016)))))
(lam16 v016))
mul16 : β{Ξ} β Tm16 Ξ (arr16 nat16 (arr16 nat16 nat16)); mul16
= lam16 (rec16 v016
(lam16 (lam16 (lam16 (app16 (app16 add16 (app16 v116 v016)) v016))))
(lam16 zero16))
fact16 : β{Ξ} β Tm16 Ξ (arr16 nat16 nat16); fact16
= lam16 (rec16 v016 (lam16 (lam16 (app16 (app16 mul16 (suc16 v116)) v016)))
(suc16 zero16))
{-# OPTIONS --type-in-type #-}
Ty17 : Set
Ty17 =
(Ty17 : Set)
(nat top bot : Ty17)
(arr prod sum : Ty17 β Ty17 β Ty17)
β Ty17
nat17 : Ty17; nat17 = Ξ» _ nat17 _ _ _ _ _ β nat17
top17 : Ty17; top17 = Ξ» _ _ top17 _ _ _ _ β top17
bot17 : Ty17; bot17 = Ξ» _ _ _ bot17 _ _ _ β bot17
arr17 : Ty17 β Ty17 β Ty17; arr17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod sum β
arr17 (A Ty17 nat17 top17 bot17 arr17 prod sum) (B Ty17 nat17 top17 bot17 arr17 prod sum)
prod17 : Ty17 β Ty17 β Ty17; prod17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod17 sum β
prod17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum) (B Ty17 nat17 top17 bot17 arr17 prod17 sum)
sum17 : Ty17 β Ty17 β Ty17; sum17
= Ξ» A B Ty17 nat17 top17 bot17 arr17 prod17 sum17 β
sum17 (A Ty17 nat17 top17 bot17 arr17 prod17 sum17) (B Ty17 nat17 top17 bot17 arr17 prod17 sum17)
Con17 : Set; Con17
= (Con17 : Set)
(nil : Con17)
(snoc : Con17 β Ty17 β Con17)
β Con17
nil17 : Con17; nil17
= Ξ» Con17 nil17 snoc β nil17
snoc17 : Con17 β Ty17 β Con17; snoc17
= Ξ» Ξ A Con17 nil17 snoc17 β snoc17 (Ξ Con17 nil17 snoc17) A
Var17 : Con17 β Ty17 β Set; Var17
= Ξ» Ξ A β
(Var17 : Con17 β Ty17 β Set)
(vz : β Ξ A β Var17 (snoc17 Ξ A) A)
(vs : β Ξ B A β Var17 Ξ A β Var17 (snoc17 Ξ B) A)
β Var17 Ξ A
vz17 : β{Ξ A} β Var17 (snoc17 Ξ A) A; vz17
= Ξ» Var17 vz17 vs β vz17 _ _
vs17 : β{Ξ B A} β Var17 Ξ A β Var17 (snoc17 Ξ B) A; vs17
= Ξ» x Var17 vz17 vs17 β vs17 _ _ _ (x Var17 vz17 vs17)
Tm17 : Con17 β Ty17 β Set; Tm17
= Ξ» Ξ A β
(Tm17 : Con17 β Ty17 β Set)
(var : β Ξ A β Var17 Ξ A β Tm17 Ξ A)
(lam : β Ξ A B β Tm17 (snoc17 Ξ A) B β Tm17 Ξ (arr17 A B))
(app : β Ξ A B β Tm17 Ξ (arr17 A B) β Tm17 Ξ A β Tm17 Ξ B)
(tt : β Ξ β Tm17 Ξ top17)
(pair : β Ξ A B β Tm17 Ξ A β Tm17 Ξ B β Tm17 Ξ (prod17 A B))
(fst : β Ξ A B β Tm17 Ξ (prod17 A B) β Tm17 Ξ A)
(snd : β Ξ A B β Tm17 Ξ (prod17 A B) β Tm17 Ξ B)
(left : β Ξ A B β Tm17 Ξ A β Tm17 Ξ (sum17 A B))
(right : β Ξ A B β Tm17 Ξ B β Tm17 Ξ (sum17 A B))
(case : β Ξ A B C β Tm17 Ξ (sum17 A B) β Tm17 Ξ (arr17 A C) β Tm17 Ξ (arr17 B C) β Tm17 Ξ C)
(zero : β Ξ β Tm17 Ξ nat17)
(suc : β Ξ β Tm17 Ξ nat17 β Tm17 Ξ nat17)
(rec : β Ξ A β Tm17 Ξ nat17 β Tm17 Ξ (arr17 nat17 (arr17 A A)) β Tm17 Ξ A β Tm17 Ξ A)
β Tm17 Ξ A
var17 : β{Ξ A} β Var17 Ξ A β Tm17 Ξ A; var17
= Ξ» x Tm17 var17 lam app tt pair fst snd left right case zero suc rec β
var17 _ _ x
lam17 : β{Ξ A B} β Tm17 (snoc17 Ξ A) B β Tm17 Ξ (arr17 A B); lam17
= Ξ» t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec β
lam17 _ _ _ (t Tm17 var17 lam17 app tt pair fst snd left right case zero suc rec)
app17 : β{Ξ A B} β Tm17 Ξ (arr17 A B) β Tm17 Ξ A β Tm17 Ξ B; app17
= Ξ» t u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec β
app17 _ _ _ (t Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt pair fst snd left right case zero suc rec)
tt17 : β{Ξ} β Tm17 Ξ top17; tt17
= Ξ» Tm17 var17 lam17 app17 tt17 pair fst snd left right case zero suc rec β tt17 _
pair17 : β{Ξ A B} β Tm17 Ξ A β Tm17 Ξ B β Tm17 Ξ (prod17 A B); pair17
= Ξ» t u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec β
pair17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst snd left right case zero suc rec)
fst17 : β{Ξ A B} β Tm17 Ξ (prod17 A B) β Tm17 Ξ A; fst17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec β
fst17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd left right case zero suc rec)
snd17 : β{Ξ A B} β Tm17 Ξ (prod17 A B) β Tm17 Ξ B; snd17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec β
snd17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left right case zero suc rec)
left17 : β{Ξ A B} β Tm17 Ξ A β Tm17 Ξ (sum17 A B); left17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec β
left17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right case zero suc rec)
right17 : β{Ξ A B} β Tm17 Ξ B β Tm17 Ξ (sum17 A B); right17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec β
right17 _ _ _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case zero suc rec)
case17 : β{Ξ A B C} β Tm17 Ξ (sum17 A B) β Tm17 Ξ (arr17 A C) β Tm17 Ξ (arr17 B C) β Tm17 Ξ C; case17
= Ξ» t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec β
case17 _ _ _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero suc rec)
zero17 : β{Ξ} β Tm17 Ξ nat17; zero17
= Ξ» Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc rec β zero17 _
suc17 : β{Ξ} β Tm17 Ξ nat17 β Tm17 Ξ nat17; suc17
= Ξ» t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec β
suc17 _ (t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec)
rec17 : β{Ξ A} β Tm17 Ξ nat17 β Tm17 Ξ (arr17 nat17 (arr17 A A)) β Tm17 Ξ A β Tm17 Ξ A; rec17
= Ξ» t u v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17 β
rec17 _ _
(t Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(u Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
(v Tm17 var17 lam17 app17 tt17 pair17 fst17 snd17 left17 right17 case17 zero17 suc17 rec17)
v017 : β{Ξ A} β Tm17 (snoc17 Ξ A) A; v017
= var17 vz17
v117 : β{Ξ A B} β Tm17 (snoc17 (snoc17 Ξ A) B) A; v117
= var17 (vs17 vz17)
v217 : β{Ξ A B C} β Tm17 (snoc17 (snoc17 (snoc17 Ξ A) B) C) A; v217
= var17 (vs17 (vs17 vz17))
v317 : β{Ξ A B C D} β Tm17 (snoc17 (snoc17 (snoc17 (snoc17 Ξ A) B) C) D) A; v317
= var17 (vs17 (vs17 (vs17 vz17)))
tbool17 : Ty17; tbool17
= sum17 top17 top17
true17 : β{Ξ} β Tm17 Ξ tbool17; true17
= left17 tt17
tfalse17 : β{Ξ} β Tm17 Ξ tbool17; tfalse17
= right17 tt17
ifthenelse17 : β{Ξ A} β Tm17 Ξ (arr17 tbool17 (arr17 A (arr17 A A))); ifthenelse17
= lam17 (lam17 (lam17 (case17 v217 (lam17 v217) (lam17 v117))))
times417 : β{Ξ A} β Tm17 Ξ (arr17 (arr17 A A) (arr17 A A)); times417
= lam17 (lam17 (app17 v117 (app17 v117 (app17 v117 (app17 v117 v017)))))
add17 : β{Ξ} β Tm17 Ξ (arr17 nat17 (arr17 nat17 nat17)); add17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (suc17 (app17 v117 v017)))))
(lam17 v017))
mul17 : β{Ξ} β Tm17 Ξ (arr17 nat17 (arr17 nat17 nat17)); mul17
= lam17 (rec17 v017
(lam17 (lam17 (lam17 (app17 (app17 add17 (app17 v117 v017)) v017))))
(lam17 zero17))
fact17 : β{Ξ} β Tm17 Ξ (arr17 nat17 nat17); fact17
= lam17 (rec17 v017 (lam17 (lam17 (app17 (app17 mul17 (suc17 v117)) v017)))
(suc17 zero17))
{-# OPTIONS --type-in-type #-}
Ty18 : Set
Ty18 =
(Ty18 : Set)
(nat top bot : Ty18)
(arr prod sum : Ty18 β Ty18 β Ty18)
β Ty18
nat18 : Ty18; nat18 = Ξ» _ nat18 _ _ _ _ _ β nat18
top18 : Ty18; top18 = Ξ» _ _ top18 _ _ _ _ β top18
bot18 : Ty18; bot18 = Ξ» _ _ _ bot18 _ _ _ β bot18
arr18 : Ty18 β Ty18 β Ty18; arr18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod sum β
arr18 (A Ty18 nat18 top18 bot18 arr18 prod sum) (B Ty18 nat18 top18 bot18 arr18 prod sum)
prod18 : Ty18 β Ty18 β Ty18; prod18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod18 sum β
prod18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum) (B Ty18 nat18 top18 bot18 arr18 prod18 sum)
sum18 : Ty18 β Ty18 β Ty18; sum18
= Ξ» A B Ty18 nat18 top18 bot18 arr18 prod18 sum18 β
sum18 (A Ty18 nat18 top18 bot18 arr18 prod18 sum18) (B Ty18 nat18 top18 bot18 arr18 prod18 sum18)
Con18 : Set; Con18
= (Con18 : Set)
(nil : Con18)
(snoc : Con18 β Ty18 β Con18)
β Con18
nil18 : Con18; nil18
= Ξ» Con18 nil18 snoc β nil18
snoc18 : Con18 β Ty18 β Con18; snoc18
= Ξ» Ξ A Con18 nil18 snoc18 β snoc18 (Ξ Con18 nil18 snoc18) A
Var18 : Con18 β Ty18 β Set; Var18
= Ξ» Ξ A β
(Var18 : Con18 β Ty18 β Set)
(vz : β Ξ A β Var18 (snoc18 Ξ A) A)
(vs : β Ξ B A β Var18 Ξ A β Var18 (snoc18 Ξ B) A)
β Var18 Ξ A
vz18 : β{Ξ A} β Var18 (snoc18 Ξ A) A; vz18
= Ξ» Var18 vz18 vs β vz18 _ _
vs18 : β{Ξ B A} β Var18 Ξ A β Var18 (snoc18 Ξ B) A; vs18
= Ξ» x Var18 vz18 vs18 β vs18 _ _ _ (x Var18 vz18 vs18)
Tm18 : Con18 β Ty18 β Set; Tm18
= Ξ» Ξ A β
(Tm18 : Con18 β Ty18 β Set)
(var : β Ξ A β Var18 Ξ A β Tm18 Ξ A)
(lam : β Ξ A B β Tm18 (snoc18 Ξ A) B β Tm18 Ξ (arr18 A B))
(app : β Ξ A B β Tm18 Ξ (arr18 A B) β Tm18 Ξ A β Tm18 Ξ B)
(tt : β Ξ β Tm18 Ξ top18)
(pair : β Ξ A B β Tm18 Ξ A β Tm18 Ξ B β Tm18 Ξ (prod18 A B))
(fst : β Ξ A B β Tm18 Ξ (prod18 A B) β Tm18 Ξ A)
(snd : β Ξ A B β Tm18 Ξ (prod18 A B) β Tm18 Ξ B)
(left : β Ξ A B β Tm18 Ξ A β Tm18 Ξ (sum18 A B))
(right : β Ξ A B β Tm18 Ξ B β Tm18 Ξ (sum18 A B))
(case : β Ξ A B C β Tm18 Ξ (sum18 A B) β Tm18 Ξ (arr18 A C) β Tm18 Ξ (arr18 B C) β Tm18 Ξ C)
(zero : β Ξ β Tm18 Ξ nat18)
(suc : β Ξ β Tm18 Ξ nat18 β Tm18 Ξ nat18)
(rec : β Ξ A β Tm18 Ξ nat18 β Tm18 Ξ (arr18 nat18 (arr18 A A)) β Tm18 Ξ A β Tm18 Ξ A)
β Tm18 Ξ A
var18 : β{Ξ A} β Var18 Ξ A β Tm18 Ξ A; var18
= Ξ» x Tm18 var18 lam app tt pair fst snd left right case zero suc rec β
var18 _ _ x
lam18 : β{Ξ A B} β Tm18 (snoc18 Ξ A) B β Tm18 Ξ (arr18 A B); lam18
= Ξ» t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec β
lam18 _ _ _ (t Tm18 var18 lam18 app tt pair fst snd left right case zero suc rec)
app18 : β{Ξ A B} β Tm18 Ξ (arr18 A B) β Tm18 Ξ A β Tm18 Ξ B; app18
= Ξ» t u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec β
app18 _ _ _ (t Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt pair fst snd left right case zero suc rec)
tt18 : β{Ξ} β Tm18 Ξ top18; tt18
= Ξ» Tm18 var18 lam18 app18 tt18 pair fst snd left right case zero suc rec β tt18 _
pair18 : β{Ξ A B} β Tm18 Ξ A β Tm18 Ξ B β Tm18 Ξ (prod18 A B); pair18
= Ξ» t u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec β
pair18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst snd left right case zero suc rec)
fst18 : β{Ξ A B} β Tm18 Ξ (prod18 A B) β Tm18 Ξ A; fst18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec β
fst18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd left right case zero suc rec)
snd18 : β{Ξ A B} β Tm18 Ξ (prod18 A B) β Tm18 Ξ B; snd18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec β
snd18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left right case zero suc rec)
left18 : β{Ξ A B} β Tm18 Ξ A β Tm18 Ξ (sum18 A B); left18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec β
left18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right case zero suc rec)
right18 : β{Ξ A B} β Tm18 Ξ B β Tm18 Ξ (sum18 A B); right18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec β
right18 _ _ _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case zero suc rec)
case18 : β{Ξ A B C} β Tm18 Ξ (sum18 A B) β Tm18 Ξ (arr18 A C) β Tm18 Ξ (arr18 B C) β Tm18 Ξ C; case18
= Ξ» t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec β
case18 _ _ _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero suc rec)
zero18 : β{Ξ} β Tm18 Ξ nat18; zero18
= Ξ» Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc rec β zero18 _
suc18 : β{Ξ} β Tm18 Ξ nat18 β Tm18 Ξ nat18; suc18
= Ξ» t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec β
suc18 _ (t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec)
rec18 : β{Ξ A} β Tm18 Ξ nat18 β Tm18 Ξ (arr18 nat18 (arr18 A A)) β Tm18 Ξ A β Tm18 Ξ A; rec18
= Ξ» t u v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18 β
rec18 _ _
(t Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(u Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
(v Tm18 var18 lam18 app18 tt18 pair18 fst18 snd18 left18 right18 case18 zero18 suc18 rec18)
v018 : β{Ξ A} β Tm18 (snoc18 Ξ A) A; v018
= var18 vz18
v118 : β{Ξ A B} β Tm18 (snoc18 (snoc18 Ξ A) B) A; v118
= var18 (vs18 vz18)
v218 : β{Ξ A B C} β Tm18 (snoc18 (snoc18 (snoc18 Ξ A) B) C) A; v218
= var18 (vs18 (vs18 vz18))
v318 : β{Ξ A B C D} β Tm18 (snoc18 (snoc18 (snoc18 (snoc18 Ξ A) B) C) D) A; v318
= var18 (vs18 (vs18 (vs18 vz18)))
tbool18 : Ty18; tbool18
= sum18 top18 top18
true18 : β{Ξ} β Tm18 Ξ tbool18; true18
= left18 tt18
tfalse18 : β{Ξ} β Tm18 Ξ tbool18; tfalse18
= right18 tt18
ifthenelse18 : β{Ξ A} β Tm18 Ξ (arr18 tbool18 (arr18 A (arr18 A A))); ifthenelse18
= lam18 (lam18 (lam18 (case18 v218 (lam18 v218) (lam18 v118))))
times418 : β{Ξ A} β Tm18 Ξ (arr18 (arr18 A A) (arr18 A A)); times418
= lam18 (lam18 (app18 v118 (app18 v118 (app18 v118 (app18 v118 v018)))))
add18 : β{Ξ} β Tm18 Ξ (arr18 nat18 (arr18 nat18 nat18)); add18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (suc18 (app18 v118 v018)))))
(lam18 v018))
mul18 : β{Ξ} β Tm18 Ξ (arr18 nat18 (arr18 nat18 nat18)); mul18
= lam18 (rec18 v018
(lam18 (lam18 (lam18 (app18 (app18 add18 (app18 v118 v018)) v018))))
(lam18 zero18))
fact18 : β{Ξ} β Tm18 Ξ (arr18 nat18 nat18); fact18
= lam18 (rec18 v018 (lam18 (lam18 (app18 (app18 mul18 (suc18 v118)) v018)))
(suc18 zero18))
{-# OPTIONS --type-in-type #-}
Ty19 : Set
Ty19 =
(Ty19 : Set)
(nat top bot : Ty19)
(arr prod sum : Ty19 β Ty19 β Ty19)
β Ty19
nat19 : Ty19; nat19 = Ξ» _ nat19 _ _ _ _ _ β nat19
top19 : Ty19; top19 = Ξ» _ _ top19 _ _ _ _ β top19
bot19 : Ty19; bot19 = Ξ» _ _ _ bot19 _ _ _ β bot19
arr19 : Ty19 β Ty19 β Ty19; arr19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod sum β
arr19 (A Ty19 nat19 top19 bot19 arr19 prod sum) (B Ty19 nat19 top19 bot19 arr19 prod sum)
prod19 : Ty19 β Ty19 β Ty19; prod19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod19 sum β
prod19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum) (B Ty19 nat19 top19 bot19 arr19 prod19 sum)
sum19 : Ty19 β Ty19 β Ty19; sum19
= Ξ» A B Ty19 nat19 top19 bot19 arr19 prod19 sum19 β
sum19 (A Ty19 nat19 top19 bot19 arr19 prod19 sum19) (B Ty19 nat19 top19 bot19 arr19 prod19 sum19)
Con19 : Set; Con19
= (Con19 : Set)
(nil : Con19)
(snoc : Con19 β Ty19 β Con19)
β Con19
nil19 : Con19; nil19
= Ξ» Con19 nil19 snoc β nil19
snoc19 : Con19 β Ty19 β Con19; snoc19
= Ξ» Ξ A Con19 nil19 snoc19 β snoc19 (Ξ Con19 nil19 snoc19) A
Var19 : Con19 β Ty19 β Set; Var19
= Ξ» Ξ A β
(Var19 : Con19 β Ty19 β Set)
(vz : β Ξ A β Var19 (snoc19 Ξ A) A)
(vs : β Ξ B A β Var19 Ξ A β Var19 (snoc19 Ξ B) A)
β Var19 Ξ A
vz19 : β{Ξ A} β Var19 (snoc19 Ξ A) A; vz19
= Ξ» Var19 vz19 vs β vz19 _ _
vs19 : β{Ξ B A} β Var19 Ξ A β Var19 (snoc19 Ξ B) A; vs19
= Ξ» x Var19 vz19 vs19 β vs19 _ _ _ (x Var19 vz19 vs19)
Tm19 : Con19 β Ty19 β Set; Tm19
= Ξ» Ξ A β
(Tm19 : Con19 β Ty19 β Set)
(var : β Ξ A β Var19 Ξ A β Tm19 Ξ A)
(lam : β Ξ A B β Tm19 (snoc19 Ξ A) B β Tm19 Ξ (arr19 A B))
(app : β Ξ A B β Tm19 Ξ (arr19 A B) β Tm19 Ξ A β Tm19 Ξ B)
(tt : β Ξ β Tm19 Ξ top19)
(pair : β Ξ A B β Tm19 Ξ A β Tm19 Ξ B β Tm19 Ξ (prod19 A B))
(fst : β Ξ A B β Tm19 Ξ (prod19 A B) β Tm19 Ξ A)
(snd : β Ξ A B β Tm19 Ξ (prod19 A B) β Tm19 Ξ B)
(left : β Ξ A B β Tm19 Ξ A β Tm19 Ξ (sum19 A B))
(right : β Ξ A B β Tm19 Ξ B β Tm19 Ξ (sum19 A B))
(case : β Ξ A B C β Tm19 Ξ (sum19 A B) β Tm19 Ξ (arr19 A C) β Tm19 Ξ (arr19 B C) β Tm19 Ξ C)
(zero : β Ξ β Tm19 Ξ nat19)
(suc : β Ξ β Tm19 Ξ nat19 β Tm19 Ξ nat19)
(rec : β Ξ A β Tm19 Ξ nat19 β Tm19 Ξ (arr19 nat19 (arr19 A A)) β Tm19 Ξ A β Tm19 Ξ A)
β Tm19 Ξ A
var19 : β{Ξ A} β Var19 Ξ A β Tm19 Ξ A; var19
= Ξ» x Tm19 var19 lam app tt pair fst snd left right case zero suc rec β
var19 _ _ x
lam19 : β{Ξ A B} β Tm19 (snoc19 Ξ A) B β Tm19 Ξ (arr19 A B); lam19
= Ξ» t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec β
lam19 _ _ _ (t Tm19 var19 lam19 app tt pair fst snd left right case zero suc rec)
app19 : β{Ξ A B} β Tm19 Ξ (arr19 A B) β Tm19 Ξ A β Tm19 Ξ B; app19
= Ξ» t u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec β
app19 _ _ _ (t Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt pair fst snd left right case zero suc rec)
tt19 : β{Ξ} β Tm19 Ξ top19; tt19
= Ξ» Tm19 var19 lam19 app19 tt19 pair fst snd left right case zero suc rec β tt19 _
pair19 : β{Ξ A B} β Tm19 Ξ A β Tm19 Ξ B β Tm19 Ξ (prod19 A B); pair19
= Ξ» t u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec β
pair19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst snd left right case zero suc rec)
fst19 : β{Ξ A B} β Tm19 Ξ (prod19 A B) β Tm19 Ξ A; fst19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec β
fst19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd left right case zero suc rec)
snd19 : β{Ξ A B} β Tm19 Ξ (prod19 A B) β Tm19 Ξ B; snd19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec β
snd19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left right case zero suc rec)
left19 : β{Ξ A B} β Tm19 Ξ A β Tm19 Ξ (sum19 A B); left19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec β
left19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right case zero suc rec)
right19 : β{Ξ A B} β Tm19 Ξ B β Tm19 Ξ (sum19 A B); right19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec β
right19 _ _ _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case zero suc rec)
case19 : β{Ξ A B C} β Tm19 Ξ (sum19 A B) β Tm19 Ξ (arr19 A C) β Tm19 Ξ (arr19 B C) β Tm19 Ξ C; case19
= Ξ» t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec β
case19 _ _ _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero suc rec)
zero19 : β{Ξ} β Tm19 Ξ nat19; zero19
= Ξ» Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc rec β zero19 _
suc19 : β{Ξ} β Tm19 Ξ nat19 β Tm19 Ξ nat19; suc19
= Ξ» t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec β
suc19 _ (t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec)
rec19 : β{Ξ A} β Tm19 Ξ nat19 β Tm19 Ξ (arr19 nat19 (arr19 A A)) β Tm19 Ξ A β Tm19 Ξ A; rec19
= Ξ» t u v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19 β
rec19 _ _
(t Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(u Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
(v Tm19 var19 lam19 app19 tt19 pair19 fst19 snd19 left19 right19 case19 zero19 suc19 rec19)
v019 : β{Ξ A} β Tm19 (snoc19 Ξ A) A; v019
= var19 vz19
v119 : β{Ξ A B} β Tm19 (snoc19 (snoc19 Ξ A) B) A; v119
= var19 (vs19 vz19)
v219 : β{Ξ A B C} β Tm19 (snoc19 (snoc19 (snoc19 Ξ A) B) C) A; v219
= var19 (vs19 (vs19 vz19))
v319 : β{Ξ A B C D} β Tm19 (snoc19 (snoc19 (snoc19 (snoc19 Ξ A) B) C) D) A; v319
= var19 (vs19 (vs19 (vs19 vz19)))
tbool19 : Ty19; tbool19
= sum19 top19 top19
true19 : β{Ξ} β Tm19 Ξ tbool19; true19
= left19 tt19
tfalse19 : β{Ξ} β Tm19 Ξ tbool19; tfalse19
= right19 tt19
ifthenelse19 : β{Ξ A} β Tm19 Ξ (arr19 tbool19 (arr19 A (arr19 A A))); ifthenelse19
= lam19 (lam19 (lam19 (case19 v219 (lam19 v219) (lam19 v119))))
times419 : β{Ξ A} β Tm19 Ξ (arr19 (arr19 A A) (arr19 A A)); times419
= lam19 (lam19 (app19 v119 (app19 v119 (app19 v119 (app19 v119 v019)))))
add19 : β{Ξ} β Tm19 Ξ (arr19 nat19 (arr19 nat19 nat19)); add19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (suc19 (app19 v119 v019)))))
(lam19 v019))
mul19 : β{Ξ} β Tm19 Ξ (arr19 nat19 (arr19 nat19 nat19)); mul19
= lam19 (rec19 v019
(lam19 (lam19 (lam19 (app19 (app19 add19 (app19 v119 v019)) v019))))
(lam19 zero19))
fact19 : β{Ξ} β Tm19 Ξ (arr19 nat19 nat19); fact19
= lam19 (rec19 v019 (lam19 (lam19 (app19 (app19 mul19 (suc19 v119)) v019)))
(suc19 zero19))
{-# OPTIONS --type-in-type #-}
Ty20 : Set
Ty20 =
(Ty20 : Set)
(nat top bot : Ty20)
(arr prod sum : Ty20 β Ty20 β Ty20)
β Ty20
nat20 : Ty20; nat20 = Ξ» _ nat20 _ _ _ _ _ β nat20
top20 : Ty20; top20 = Ξ» _ _ top20 _ _ _ _ β top20
bot20 : Ty20; bot20 = Ξ» _ _ _ bot20 _ _ _ β bot20
arr20 : Ty20 β Ty20 β Ty20; arr20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod sum β
arr20 (A Ty20 nat20 top20 bot20 arr20 prod sum) (B Ty20 nat20 top20 bot20 arr20 prod sum)
prod20 : Ty20 β Ty20 β Ty20; prod20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod20 sum β
prod20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum) (B Ty20 nat20 top20 bot20 arr20 prod20 sum)
sum20 : Ty20 β Ty20 β Ty20; sum20
= Ξ» A B Ty20 nat20 top20 bot20 arr20 prod20 sum20 β
sum20 (A Ty20 nat20 top20 bot20 arr20 prod20 sum20) (B Ty20 nat20 top20 bot20 arr20 prod20 sum20)
Con20 : Set; Con20
= (Con20 : Set)
(nil : Con20)
(snoc : Con20 β Ty20 β Con20)
β Con20
nil20 : Con20; nil20
= Ξ» Con20 nil20 snoc β nil20
snoc20 : Con20 β Ty20 β Con20; snoc20
= Ξ» Ξ A Con20 nil20 snoc20 β snoc20 (Ξ Con20 nil20 snoc20) A
Var20 : Con20 β Ty20 β Set; Var20
= Ξ» Ξ A β
(Var20 : Con20 β Ty20 β Set)
(vz : β Ξ A β Var20 (snoc20 Ξ A) A)
(vs : β Ξ B A β Var20 Ξ A β Var20 (snoc20 Ξ B) A)
β Var20 Ξ A
vz20 : β{Ξ A} β Var20 (snoc20 Ξ A) A; vz20
= Ξ» Var20 vz20 vs β vz20 _ _
vs20 : β{Ξ B A} β Var20 Ξ A β Var20 (snoc20 Ξ B) A; vs20
= Ξ» x Var20 vz20 vs20 β vs20 _ _ _ (x Var20 vz20 vs20)
Tm20 : Con20 β Ty20 β Set; Tm20
= Ξ» Ξ A β
(Tm20 : Con20 β Ty20 β Set)
(var : β Ξ A β Var20 Ξ A β Tm20 Ξ A)
(lam : β Ξ A B β Tm20 (snoc20 Ξ A) B β Tm20 Ξ (arr20 A B))
(app : β Ξ A B β Tm20 Ξ (arr20 A B) β Tm20 Ξ A β Tm20 Ξ B)
(tt : β Ξ β Tm20 Ξ top20)
(pair : β Ξ A B β Tm20 Ξ A β Tm20 Ξ B β Tm20 Ξ (prod20 A B))
(fst : β Ξ A B β Tm20 Ξ (prod20 A B) β Tm20 Ξ A)
(snd : β Ξ A B β Tm20 Ξ (prod20 A B) β Tm20 Ξ B)
(left : β Ξ A B β Tm20 Ξ A β Tm20 Ξ (sum20 A B))
(right : β Ξ A B β Tm20 Ξ B β Tm20 Ξ (sum20 A B))
(case : β Ξ A B C β Tm20 Ξ (sum20 A B) β Tm20 Ξ (arr20 A C) β Tm20 Ξ (arr20 B C) β Tm20 Ξ C)
(zero : β Ξ β Tm20 Ξ nat20)
(suc : β Ξ β Tm20 Ξ nat20 β Tm20 Ξ nat20)
(rec : β Ξ A β Tm20 Ξ nat20 β Tm20 Ξ (arr20 nat20 (arr20 A A)) β Tm20 Ξ A β Tm20 Ξ A)
β Tm20 Ξ A
var20 : β{Ξ A} β Var20 Ξ A β Tm20 Ξ A; var20
= Ξ» x Tm20 var20 lam app tt pair fst snd left right case zero suc rec β
var20 _ _ x
lam20 : β{Ξ A B} β Tm20 (snoc20 Ξ A) B β Tm20 Ξ (arr20 A B); lam20
= Ξ» t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec β
lam20 _ _ _ (t Tm20 var20 lam20 app tt pair fst snd left right case zero suc rec)
app20 : β{Ξ A B} β Tm20 Ξ (arr20 A B) β Tm20 Ξ A β Tm20 Ξ B; app20
= Ξ» t u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec β
app20 _ _ _ (t Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt pair fst snd left right case zero suc rec)
tt20 : β{Ξ} β Tm20 Ξ top20; tt20
= Ξ» Tm20 var20 lam20 app20 tt20 pair fst snd left right case zero suc rec β tt20 _
pair20 : β{Ξ A B} β Tm20 Ξ A β Tm20 Ξ B β Tm20 Ξ (prod20 A B); pair20
= Ξ» t u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec β
pair20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst snd left right case zero suc rec)
fst20 : β{Ξ A B} β Tm20 Ξ (prod20 A B) β Tm20 Ξ A; fst20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec β
fst20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd left right case zero suc rec)
snd20 : β{Ξ A B} β Tm20 Ξ (prod20 A B) β Tm20 Ξ B; snd20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec β
snd20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left right case zero suc rec)
left20 : β{Ξ A B} β Tm20 Ξ A β Tm20 Ξ (sum20 A B); left20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec β
left20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right case zero suc rec)
right20 : β{Ξ A B} β Tm20 Ξ B β Tm20 Ξ (sum20 A B); right20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec β
right20 _ _ _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case zero suc rec)
case20 : β{Ξ A B C} β Tm20 Ξ (sum20 A B) β Tm20 Ξ (arr20 A C) β Tm20 Ξ (arr20 B C) β Tm20 Ξ C; case20
= Ξ» t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec β
case20 _ _ _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero suc rec)
zero20 : β{Ξ} β Tm20 Ξ nat20; zero20
= Ξ» Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc rec β zero20 _
suc20 : β{Ξ} β Tm20 Ξ nat20 β Tm20 Ξ nat20; suc20
= Ξ» t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec β
suc20 _ (t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec)
rec20 : β{Ξ A} β Tm20 Ξ nat20 β Tm20 Ξ (arr20 nat20 (arr20 A A)) β Tm20 Ξ A β Tm20 Ξ A; rec20
= Ξ» t u v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20 β
rec20 _ _
(t Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(u Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
(v Tm20 var20 lam20 app20 tt20 pair20 fst20 snd20 left20 right20 case20 zero20 suc20 rec20)
v020 : β{Ξ A} β Tm20 (snoc20 Ξ A) A; v020
= var20 vz20
v120 : β{Ξ A B} β Tm20 (snoc20 (snoc20 Ξ A) B) A; v120
= var20 (vs20 vz20)
v220 : β{Ξ A B C} β Tm20 (snoc20 (snoc20 (snoc20 Ξ A) B) C) A; v220
= var20 (vs20 (vs20 vz20))
v320 : β{Ξ A B C D} β Tm20 (snoc20 (snoc20 (snoc20 (snoc20 Ξ A) B) C) D) A; v320
= var20 (vs20 (vs20 (vs20 vz20)))
tbool20 : Ty20; tbool20
= sum20 top20 top20
true20 : β{Ξ} β Tm20 Ξ tbool20; true20
= left20 tt20
tfalse20 : β{Ξ} β Tm20 Ξ tbool20; tfalse20
= right20 tt20
ifthenelse20 : β{Ξ A} β Tm20 Ξ (arr20 tbool20 (arr20 A (arr20 A A))); ifthenelse20
= lam20 (lam20 (lam20 (case20 v220 (lam20 v220) (lam20 v120))))
times420 : β{Ξ A} β Tm20 Ξ (arr20 (arr20 A A) (arr20 A A)); times420
= lam20 (lam20 (app20 v120 (app20 v120 (app20 v120 (app20 v120 v020)))))
add20 : β{Ξ} β Tm20 Ξ (arr20 nat20 (arr20 nat20 nat20)); add20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (suc20 (app20 v120 v020)))))
(lam20 v020))
mul20 : β{Ξ} β Tm20 Ξ (arr20 nat20 (arr20 nat20 nat20)); mul20
= lam20 (rec20 v020
(lam20 (lam20 (lam20 (app20 (app20 add20 (app20 v120 v020)) v020))))
(lam20 zero20))
fact20 : β{Ξ} β Tm20 Ξ (arr20 nat20 nat20); fact20
= lam20 (rec20 v020 (lam20 (lam20 (app20 (app20 mul20 (suc20 v120)) v020)))
(suc20 zero20))
{-# OPTIONS --type-in-type #-}
Ty21 : Set
Ty21 =
(Ty21 : Set)
(nat top bot : Ty21)
(arr prod sum : Ty21 β Ty21 β Ty21)
β Ty21
nat21 : Ty21; nat21 = Ξ» _ nat21 _ _ _ _ _ β nat21
top21 : Ty21; top21 = Ξ» _ _ top21 _ _ _ _ β top21
bot21 : Ty21; bot21 = Ξ» _ _ _ bot21 _ _ _ β bot21
arr21 : Ty21 β Ty21 β Ty21; arr21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod sum β
arr21 (A Ty21 nat21 top21 bot21 arr21 prod sum) (B Ty21 nat21 top21 bot21 arr21 prod sum)
prod21 : Ty21 β Ty21 β Ty21; prod21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod21 sum β
prod21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum) (B Ty21 nat21 top21 bot21 arr21 prod21 sum)
sum21 : Ty21 β Ty21 β Ty21; sum21
= Ξ» A B Ty21 nat21 top21 bot21 arr21 prod21 sum21 β
sum21 (A Ty21 nat21 top21 bot21 arr21 prod21 sum21) (B Ty21 nat21 top21 bot21 arr21 prod21 sum21)
Con21 : Set; Con21
= (Con21 : Set)
(nil : Con21)
(snoc : Con21 β Ty21 β Con21)
β Con21
nil21 : Con21; nil21
= Ξ» Con21 nil21 snoc β nil21
snoc21 : Con21 β Ty21 β Con21; snoc21
= Ξ» Ξ A Con21 nil21 snoc21 β snoc21 (Ξ Con21 nil21 snoc21) A
Var21 : Con21 β Ty21 β Set; Var21
= Ξ» Ξ A β
(Var21 : Con21 β Ty21 β Set)
(vz : β Ξ A β Var21 (snoc21 Ξ A) A)
(vs : β Ξ B A β Var21 Ξ A β Var21 (snoc21 Ξ B) A)
β Var21 Ξ A
vz21 : β{Ξ A} β Var21 (snoc21 Ξ A) A; vz21
= Ξ» Var21 vz21 vs β vz21 _ _
vs21 : β{Ξ B A} β Var21 Ξ A β Var21 (snoc21 Ξ B) A; vs21
= Ξ» x Var21 vz21 vs21 β vs21 _ _ _ (x Var21 vz21 vs21)
Tm21 : Con21 β Ty21 β Set; Tm21
= Ξ» Ξ A β
(Tm21 : Con21 β Ty21 β Set)
(var : β Ξ A β Var21 Ξ A β Tm21 Ξ A)
(lam : β Ξ A B β Tm21 (snoc21 Ξ A) B β Tm21 Ξ (arr21 A B))
(app : β Ξ A B β Tm21 Ξ (arr21 A B) β Tm21 Ξ A β Tm21 Ξ B)
(tt : β Ξ β Tm21 Ξ top21)
(pair : β Ξ A B β Tm21 Ξ A β Tm21 Ξ B β Tm21 Ξ (prod21 A B))
(fst : β Ξ A B β Tm21 Ξ (prod21 A B) β Tm21 Ξ A)
(snd : β Ξ A B β Tm21 Ξ (prod21 A B) β Tm21 Ξ B)
(left : β Ξ A B β Tm21 Ξ A β Tm21 Ξ (sum21 A B))
(right : β Ξ A B β Tm21 Ξ B β Tm21 Ξ (sum21 A B))
(case : β Ξ A B C β Tm21 Ξ (sum21 A B) β Tm21 Ξ (arr21 A C) β Tm21 Ξ (arr21 B C) β Tm21 Ξ C)
(zero : β Ξ β Tm21 Ξ nat21)
(suc : β Ξ β Tm21 Ξ nat21 β Tm21 Ξ nat21)
(rec : β Ξ A β Tm21 Ξ nat21 β Tm21 Ξ (arr21 nat21 (arr21 A A)) β Tm21 Ξ A β Tm21 Ξ A)
β Tm21 Ξ A
var21 : β{Ξ A} β Var21 Ξ A β Tm21 Ξ A; var21
= Ξ» x Tm21 var21 lam app tt pair fst snd left right case zero suc rec β
var21 _ _ x
lam21 : β{Ξ A B} β Tm21 (snoc21 Ξ A) B β Tm21 Ξ (arr21 A B); lam21
= Ξ» t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec β
lam21 _ _ _ (t Tm21 var21 lam21 app tt pair fst snd left right case zero suc rec)
app21 : β{Ξ A B} β Tm21 Ξ (arr21 A B) β Tm21 Ξ A β Tm21 Ξ B; app21
= Ξ» t u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec β
app21 _ _ _ (t Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt pair fst snd left right case zero suc rec)
tt21 : β{Ξ} β Tm21 Ξ top21; tt21
= Ξ» Tm21 var21 lam21 app21 tt21 pair fst snd left right case zero suc rec β tt21 _
pair21 : β{Ξ A B} β Tm21 Ξ A β Tm21 Ξ B β Tm21 Ξ (prod21 A B); pair21
= Ξ» t u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec β
pair21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst snd left right case zero suc rec)
fst21 : β{Ξ A B} β Tm21 Ξ (prod21 A B) β Tm21 Ξ A; fst21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec β
fst21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd left right case zero suc rec)
snd21 : β{Ξ A B} β Tm21 Ξ (prod21 A B) β Tm21 Ξ B; snd21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec β
snd21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left right case zero suc rec)
left21 : β{Ξ A B} β Tm21 Ξ A β Tm21 Ξ (sum21 A B); left21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec β
left21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right case zero suc rec)
right21 : β{Ξ A B} β Tm21 Ξ B β Tm21 Ξ (sum21 A B); right21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec β
right21 _ _ _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case zero suc rec)
case21 : β{Ξ A B C} β Tm21 Ξ (sum21 A B) β Tm21 Ξ (arr21 A C) β Tm21 Ξ (arr21 B C) β Tm21 Ξ C; case21
= Ξ» t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec β
case21 _ _ _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero suc rec)
zero21 : β{Ξ} β Tm21 Ξ nat21; zero21
= Ξ» Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc rec β zero21 _
suc21 : β{Ξ} β Tm21 Ξ nat21 β Tm21 Ξ nat21; suc21
= Ξ» t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec β
suc21 _ (t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec)
rec21 : β{Ξ A} β Tm21 Ξ nat21 β Tm21 Ξ (arr21 nat21 (arr21 A A)) β Tm21 Ξ A β Tm21 Ξ A; rec21
= Ξ» t u v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21 β
rec21 _ _
(t Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(u Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
(v Tm21 var21 lam21 app21 tt21 pair21 fst21 snd21 left21 right21 case21 zero21 suc21 rec21)
v021 : β{Ξ A} β Tm21 (snoc21 Ξ A) A; v021
= var21 vz21
v121 : β{Ξ A B} β Tm21 (snoc21 (snoc21 Ξ A) B) A; v121
= var21 (vs21 vz21)
v221 : β{Ξ A B C} β Tm21 (snoc21 (snoc21 (snoc21 Ξ A) B) C) A; v221
= var21 (vs21 (vs21 vz21))
v321 : β{Ξ A B C D} β Tm21 (snoc21 (snoc21 (snoc21 (snoc21 Ξ A) B) C) D) A; v321
= var21 (vs21 (vs21 (vs21 vz21)))
tbool21 : Ty21; tbool21
= sum21 top21 top21
true21 : β{Ξ} β Tm21 Ξ tbool21; true21
= left21 tt21
tfalse21 : β{Ξ} β Tm21 Ξ tbool21; tfalse21
= right21 tt21
ifthenelse21 : β{Ξ A} β Tm21 Ξ (arr21 tbool21 (arr21 A (arr21 A A))); ifthenelse21
= lam21 (lam21 (lam21 (case21 v221 (lam21 v221) (lam21 v121))))
times421 : β{Ξ A} β Tm21 Ξ (arr21 (arr21 A A) (arr21 A A)); times421
= lam21 (lam21 (app21 v121 (app21 v121 (app21 v121 (app21 v121 v021)))))
add21 : β{Ξ} β Tm21 Ξ (arr21 nat21 (arr21 nat21 nat21)); add21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (suc21 (app21 v121 v021)))))
(lam21 v021))
mul21 : β{Ξ} β Tm21 Ξ (arr21 nat21 (arr21 nat21 nat21)); mul21
= lam21 (rec21 v021
(lam21 (lam21 (lam21 (app21 (app21 add21 (app21 v121 v021)) v021))))
(lam21 zero21))
fact21 : β{Ξ} β Tm21 Ξ (arr21 nat21 nat21); fact21
= lam21 (rec21 v021 (lam21 (lam21 (app21 (app21 mul21 (suc21 v121)) v021)))
(suc21 zero21))
{-# OPTIONS --type-in-type #-}
Ty22 : Set
Ty22 =
(Ty22 : Set)
(nat top bot : Ty22)
(arr prod sum : Ty22 β Ty22 β Ty22)
β Ty22
nat22 : Ty22; nat22 = Ξ» _ nat22 _ _ _ _ _ β nat22
top22 : Ty22; top22 = Ξ» _ _ top22 _ _ _ _ β top22
bot22 : Ty22; bot22 = Ξ» _ _ _ bot22 _ _ _ β bot22
arr22 : Ty22 β Ty22 β Ty22; arr22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod sum β
arr22 (A Ty22 nat22 top22 bot22 arr22 prod sum) (B Ty22 nat22 top22 bot22 arr22 prod sum)
prod22 : Ty22 β Ty22 β Ty22; prod22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod22 sum β
prod22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum) (B Ty22 nat22 top22 bot22 arr22 prod22 sum)
sum22 : Ty22 β Ty22 β Ty22; sum22
= Ξ» A B Ty22 nat22 top22 bot22 arr22 prod22 sum22 β
sum22 (A Ty22 nat22 top22 bot22 arr22 prod22 sum22) (B Ty22 nat22 top22 bot22 arr22 prod22 sum22)
Con22 : Set; Con22
= (Con22 : Set)
(nil : Con22)
(snoc : Con22 β Ty22 β Con22)
β Con22
nil22 : Con22; nil22
= Ξ» Con22 nil22 snoc β nil22
snoc22 : Con22 β Ty22 β Con22; snoc22
= Ξ» Ξ A Con22 nil22 snoc22 β snoc22 (Ξ Con22 nil22 snoc22) A
Var22 : Con22 β Ty22 β Set; Var22
= Ξ» Ξ A β
(Var22 : Con22 β Ty22 β Set)
(vz : β Ξ A β Var22 (snoc22 Ξ A) A)
(vs : β Ξ B A β Var22 Ξ A β Var22 (snoc22 Ξ B) A)
β Var22 Ξ A
vz22 : β{Ξ A} β Var22 (snoc22 Ξ A) A; vz22
= Ξ» Var22 vz22 vs β vz22 _ _
vs22 : β{Ξ B A} β Var22 Ξ A β Var22 (snoc22 Ξ B) A; vs22
= Ξ» x Var22 vz22 vs22 β vs22 _ _ _ (x Var22 vz22 vs22)
Tm22 : Con22 β Ty22 β Set; Tm22
= Ξ» Ξ A β
(Tm22 : Con22 β Ty22 β Set)
(var : β Ξ A β Var22 Ξ A β Tm22 Ξ A)
(lam : β Ξ A B β Tm22 (snoc22 Ξ A) B β Tm22 Ξ (arr22 A B))
(app : β Ξ A B β Tm22 Ξ (arr22 A B) β Tm22 Ξ A β Tm22 Ξ B)
(tt : β Ξ β Tm22 Ξ top22)
(pair : β Ξ A B β Tm22 Ξ A β Tm22 Ξ B β Tm22 Ξ (prod22 A B))
(fst : β Ξ A B β Tm22 Ξ (prod22 A B) β Tm22 Ξ A)
(snd : β Ξ A B β Tm22 Ξ (prod22 A B) β Tm22 Ξ B)
(left : β Ξ A B β Tm22 Ξ A β Tm22 Ξ (sum22 A B))
(right : β Ξ A B β Tm22 Ξ B β Tm22 Ξ (sum22 A B))
(case : β Ξ A B C β Tm22 Ξ (sum22 A B) β Tm22 Ξ (arr22 A C) β Tm22 Ξ (arr22 B C) β Tm22 Ξ C)
(zero : β Ξ β Tm22 Ξ nat22)
(suc : β Ξ β Tm22 Ξ nat22 β Tm22 Ξ nat22)
(rec : β Ξ A β Tm22 Ξ nat22 β Tm22 Ξ (arr22 nat22 (arr22 A A)) β Tm22 Ξ A β Tm22 Ξ A)
β Tm22 Ξ A
var22 : β{Ξ A} β Var22 Ξ A β Tm22 Ξ A; var22
= Ξ» x Tm22 var22 lam app tt pair fst snd left right case zero suc rec β
var22 _ _ x
lam22 : β{Ξ A B} β Tm22 (snoc22 Ξ A) B β Tm22 Ξ (arr22 A B); lam22
= Ξ» t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec β
lam22 _ _ _ (t Tm22 var22 lam22 app tt pair fst snd left right case zero suc rec)
app22 : β{Ξ A B} β Tm22 Ξ (arr22 A B) β Tm22 Ξ A β Tm22 Ξ B; app22
= Ξ» t u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec β
app22 _ _ _ (t Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt pair fst snd left right case zero suc rec)
tt22 : β{Ξ} β Tm22 Ξ top22; tt22
= Ξ» Tm22 var22 lam22 app22 tt22 pair fst snd left right case zero suc rec β tt22 _
pair22 : β{Ξ A B} β Tm22 Ξ A β Tm22 Ξ B β Tm22 Ξ (prod22 A B); pair22
= Ξ» t u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec β
pair22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst snd left right case zero suc rec)
fst22 : β{Ξ A B} β Tm22 Ξ (prod22 A B) β Tm22 Ξ A; fst22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec β
fst22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd left right case zero suc rec)
snd22 : β{Ξ A B} β Tm22 Ξ (prod22 A B) β Tm22 Ξ B; snd22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec β
snd22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left right case zero suc rec)
left22 : β{Ξ A B} β Tm22 Ξ A β Tm22 Ξ (sum22 A B); left22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec β
left22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right case zero suc rec)
right22 : β{Ξ A B} β Tm22 Ξ B β Tm22 Ξ (sum22 A B); right22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec β
right22 _ _ _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case zero suc rec)
case22 : β{Ξ A B C} β Tm22 Ξ (sum22 A B) β Tm22 Ξ (arr22 A C) β Tm22 Ξ (arr22 B C) β Tm22 Ξ C; case22
= Ξ» t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec β
case22 _ _ _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero suc rec)
zero22 : β{Ξ} β Tm22 Ξ nat22; zero22
= Ξ» Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc rec β zero22 _
suc22 : β{Ξ} β Tm22 Ξ nat22 β Tm22 Ξ nat22; suc22
= Ξ» t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec β
suc22 _ (t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec)
rec22 : β{Ξ A} β Tm22 Ξ nat22 β Tm22 Ξ (arr22 nat22 (arr22 A A)) β Tm22 Ξ A β Tm22 Ξ A; rec22
= Ξ» t u v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22 β
rec22 _ _
(t Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(u Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
(v Tm22 var22 lam22 app22 tt22 pair22 fst22 snd22 left22 right22 case22 zero22 suc22 rec22)
v022 : β{Ξ A} β Tm22 (snoc22 Ξ A) A; v022
= var22 vz22
v122 : β{Ξ A B} β Tm22 (snoc22 (snoc22 Ξ A) B) A; v122
= var22 (vs22 vz22)
v222 : β{Ξ A B C} β Tm22 (snoc22 (snoc22 (snoc22 Ξ A) B) C) A; v222
= var22 (vs22 (vs22 vz22))
v322 : β{Ξ A B C D} β Tm22 (snoc22 (snoc22 (snoc22 (snoc22 Ξ A) B) C) D) A; v322
= var22 (vs22 (vs22 (vs22 vz22)))
tbool22 : Ty22; tbool22
= sum22 top22 top22
true22 : β{Ξ} β Tm22 Ξ tbool22; true22
= left22 tt22
tfalse22 : β{Ξ} β Tm22 Ξ tbool22; tfalse22
= right22 tt22
ifthenelse22 : β{Ξ A} β Tm22 Ξ (arr22 tbool22 (arr22 A (arr22 A A))); ifthenelse22
= lam22 (lam22 (lam22 (case22 v222 (lam22 v222) (lam22 v122))))
times422 : β{Ξ A} β Tm22 Ξ (arr22 (arr22 A A) (arr22 A A)); times422
= lam22 (lam22 (app22 v122 (app22 v122 (app22 v122 (app22 v122 v022)))))
add22 : β{Ξ} β Tm22 Ξ (arr22 nat22 (arr22 nat22 nat22)); add22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (suc22 (app22 v122 v022)))))
(lam22 v022))
mul22 : β{Ξ} β Tm22 Ξ (arr22 nat22 (arr22 nat22 nat22)); mul22
= lam22 (rec22 v022
(lam22 (lam22 (lam22 (app22 (app22 add22 (app22 v122 v022)) v022))))
(lam22 zero22))
fact22 : β{Ξ} β Tm22 Ξ (arr22 nat22 nat22); fact22
= lam22 (rec22 v022 (lam22 (lam22 (app22 (app22 mul22 (suc22 v122)) v022)))
(suc22 zero22))
{-# OPTIONS --type-in-type #-}
Ty23 : Set
Ty23 =
(Ty23 : Set)
(nat top bot : Ty23)
(arr prod sum : Ty23 β Ty23 β Ty23)
β Ty23
nat23 : Ty23; nat23 = Ξ» _ nat23 _ _ _ _ _ β nat23
top23 : Ty23; top23 = Ξ» _ _ top23 _ _ _ _ β top23
bot23 : Ty23; bot23 = Ξ» _ _ _ bot23 _ _ _ β bot23
arr23 : Ty23 β Ty23 β Ty23; arr23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod sum β
arr23 (A Ty23 nat23 top23 bot23 arr23 prod sum) (B Ty23 nat23 top23 bot23 arr23 prod sum)
prod23 : Ty23 β Ty23 β Ty23; prod23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod23 sum β
prod23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum) (B Ty23 nat23 top23 bot23 arr23 prod23 sum)
sum23 : Ty23 β Ty23 β Ty23; sum23
= Ξ» A B Ty23 nat23 top23 bot23 arr23 prod23 sum23 β
sum23 (A Ty23 nat23 top23 bot23 arr23 prod23 sum23) (B Ty23 nat23 top23 bot23 arr23 prod23 sum23)
Con23 : Set; Con23
= (Con23 : Set)
(nil : Con23)
(snoc : Con23 β Ty23 β Con23)
β Con23
nil23 : Con23; nil23
= Ξ» Con23 nil23 snoc β nil23
snoc23 : Con23 β Ty23 β Con23; snoc23
= Ξ» Ξ A Con23 nil23 snoc23 β snoc23 (Ξ Con23 nil23 snoc23) A
Var23 : Con23 β Ty23 β Set; Var23
= Ξ» Ξ A β
(Var23 : Con23 β Ty23 β Set)
(vz : β Ξ A β Var23 (snoc23 Ξ A) A)
(vs : β Ξ B A β Var23 Ξ A β Var23 (snoc23 Ξ B) A)
β Var23 Ξ A
vz23 : β{Ξ A} β Var23 (snoc23 Ξ A) A; vz23
= Ξ» Var23 vz23 vs β vz23 _ _
vs23 : β{Ξ B A} β Var23 Ξ A β Var23 (snoc23 Ξ B) A; vs23
= Ξ» x Var23 vz23 vs23 β vs23 _ _ _ (x Var23 vz23 vs23)
Tm23 : Con23 β Ty23 β Set; Tm23
= Ξ» Ξ A β
(Tm23 : Con23 β Ty23 β Set)
(var : β Ξ A β Var23 Ξ A β Tm23 Ξ A)
(lam : β Ξ A B β Tm23 (snoc23 Ξ A) B β Tm23 Ξ (arr23 A B))
(app : β Ξ A B β Tm23 Ξ (arr23 A B) β Tm23 Ξ A β Tm23 Ξ B)
(tt : β Ξ β Tm23 Ξ top23)
(pair : β Ξ A B β Tm23 Ξ A β Tm23 Ξ B β Tm23 Ξ (prod23 A B))
(fst : β Ξ A B β Tm23 Ξ (prod23 A B) β Tm23 Ξ A)
(snd : β Ξ A B β Tm23 Ξ (prod23 A B) β Tm23 Ξ B)
(left : β Ξ A B β Tm23 Ξ A β Tm23 Ξ (sum23 A B))
(right : β Ξ A B β Tm23 Ξ B β Tm23 Ξ (sum23 A B))
(case : β Ξ A B C β Tm23 Ξ (sum23 A B) β Tm23 Ξ (arr23 A C) β Tm23 Ξ (arr23 B C) β Tm23 Ξ C)
(zero : β Ξ β Tm23 Ξ nat23)
(suc : β Ξ β Tm23 Ξ nat23 β Tm23 Ξ nat23)
(rec : β Ξ A β Tm23 Ξ nat23 β Tm23 Ξ (arr23 nat23 (arr23 A A)) β Tm23 Ξ A β Tm23 Ξ A)
β Tm23 Ξ A
var23 : β{Ξ A} β Var23 Ξ A β Tm23 Ξ A; var23
= Ξ» x Tm23 var23 lam app tt pair fst snd left right case zero suc rec β
var23 _ _ x
lam23 : β{Ξ A B} β Tm23 (snoc23 Ξ A) B β Tm23 Ξ (arr23 A B); lam23
= Ξ» t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec β
lam23 _ _ _ (t Tm23 var23 lam23 app tt pair fst snd left right case zero suc rec)
app23 : β{Ξ A B} β Tm23 Ξ (arr23 A B) β Tm23 Ξ A β Tm23 Ξ B; app23
= Ξ» t u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec β
app23 _ _ _ (t Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt pair fst snd left right case zero suc rec)
tt23 : β{Ξ} β Tm23 Ξ top23; tt23
= Ξ» Tm23 var23 lam23 app23 tt23 pair fst snd left right case zero suc rec β tt23 _
pair23 : β{Ξ A B} β Tm23 Ξ A β Tm23 Ξ B β Tm23 Ξ (prod23 A B); pair23
= Ξ» t u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec β
pair23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst snd left right case zero suc rec)
fst23 : β{Ξ A B} β Tm23 Ξ (prod23 A B) β Tm23 Ξ A; fst23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec β
fst23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd left right case zero suc rec)
snd23 : β{Ξ A B} β Tm23 Ξ (prod23 A B) β Tm23 Ξ B; snd23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec β
snd23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left right case zero suc rec)
left23 : β{Ξ A B} β Tm23 Ξ A β Tm23 Ξ (sum23 A B); left23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec β
left23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right case zero suc rec)
right23 : β{Ξ A B} β Tm23 Ξ B β Tm23 Ξ (sum23 A B); right23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec β
right23 _ _ _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case zero suc rec)
case23 : β{Ξ A B C} β Tm23 Ξ (sum23 A B) β Tm23 Ξ (arr23 A C) β Tm23 Ξ (arr23 B C) β Tm23 Ξ C; case23
= Ξ» t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec β
case23 _ _ _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero suc rec)
zero23 : β{Ξ} β Tm23 Ξ nat23; zero23
= Ξ» Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc rec β zero23 _
suc23 : β{Ξ} β Tm23 Ξ nat23 β Tm23 Ξ nat23; suc23
= Ξ» t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec β
suc23 _ (t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec)
rec23 : β{Ξ A} β Tm23 Ξ nat23 β Tm23 Ξ (arr23 nat23 (arr23 A A)) β Tm23 Ξ A β Tm23 Ξ A; rec23
= Ξ» t u v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23 β
rec23 _ _
(t Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(u Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
(v Tm23 var23 lam23 app23 tt23 pair23 fst23 snd23 left23 right23 case23 zero23 suc23 rec23)
v023 : β{Ξ A} β Tm23 (snoc23 Ξ A) A; v023
= var23 vz23
v123 : β{Ξ A B} β Tm23 (snoc23 (snoc23 Ξ A) B) A; v123
= var23 (vs23 vz23)
v223 : β{Ξ A B C} β Tm23 (snoc23 (snoc23 (snoc23 Ξ A) B) C) A; v223
= var23 (vs23 (vs23 vz23))
v323 : β{Ξ A B C D} β Tm23 (snoc23 (snoc23 (snoc23 (snoc23 Ξ A) B) C) D) A; v323
= var23 (vs23 (vs23 (vs23 vz23)))
tbool23 : Ty23; tbool23
= sum23 top23 top23
true23 : β{Ξ} β Tm23 Ξ tbool23; true23
= left23 tt23
tfalse23 : β{Ξ} β Tm23 Ξ tbool23; tfalse23
= right23 tt23
ifthenelse23 : β{Ξ A} β Tm23 Ξ (arr23 tbool23 (arr23 A (arr23 A A))); ifthenelse23
= lam23 (lam23 (lam23 (case23 v223 (lam23 v223) (lam23 v123))))
times423 : β{Ξ A} β Tm23 Ξ (arr23 (arr23 A A) (arr23 A A)); times423
= lam23 (lam23 (app23 v123 (app23 v123 (app23 v123 (app23 v123 v023)))))
add23 : β{Ξ} β Tm23 Ξ (arr23 nat23 (arr23 nat23 nat23)); add23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (suc23 (app23 v123 v023)))))
(lam23 v023))
mul23 : β{Ξ} β Tm23 Ξ (arr23 nat23 (arr23 nat23 nat23)); mul23
= lam23 (rec23 v023
(lam23 (lam23 (lam23 (app23 (app23 add23 (app23 v123 v023)) v023))))
(lam23 zero23))
fact23 : β{Ξ} β Tm23 Ξ (arr23 nat23 nat23); fact23
= lam23 (rec23 v023 (lam23 (lam23 (app23 (app23 mul23 (suc23 v123)) v023)))
(suc23 zero23))
{-# OPTIONS --type-in-type #-}
Ty24 : Set
Ty24 =
(Ty24 : Set)
(nat top bot : Ty24)
(arr prod sum : Ty24 β Ty24 β Ty24)
β Ty24
nat24 : Ty24; nat24 = Ξ» _ nat24 _ _ _ _ _ β nat24
top24 : Ty24; top24 = Ξ» _ _ top24 _ _ _ _ β top24
bot24 : Ty24; bot24 = Ξ» _ _ _ bot24 _ _ _ β bot24
arr24 : Ty24 β Ty24 β Ty24; arr24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod sum β
arr24 (A Ty24 nat24 top24 bot24 arr24 prod sum) (B Ty24 nat24 top24 bot24 arr24 prod sum)
prod24 : Ty24 β Ty24 β Ty24; prod24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod24 sum β
prod24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum) (B Ty24 nat24 top24 bot24 arr24 prod24 sum)
sum24 : Ty24 β Ty24 β Ty24; sum24
= Ξ» A B Ty24 nat24 top24 bot24 arr24 prod24 sum24 β
sum24 (A Ty24 nat24 top24 bot24 arr24 prod24 sum24) (B Ty24 nat24 top24 bot24 arr24 prod24 sum24)
Con24 : Set; Con24
= (Con24 : Set)
(nil : Con24)
(snoc : Con24 β Ty24 β Con24)
β Con24
nil24 : Con24; nil24
= Ξ» Con24 nil24 snoc β nil24
snoc24 : Con24 β Ty24 β Con24; snoc24
= Ξ» Ξ A Con24 nil24 snoc24 β snoc24 (Ξ Con24 nil24 snoc24) A
Var24 : Con24 β Ty24 β Set; Var24
= Ξ» Ξ A β
(Var24 : Con24 β Ty24 β Set)
(vz : β Ξ A β Var24 (snoc24 Ξ A) A)
(vs : β Ξ B A β Var24 Ξ A β Var24 (snoc24 Ξ B) A)
β Var24 Ξ A
vz24 : β{Ξ A} β Var24 (snoc24 Ξ A) A; vz24
= Ξ» Var24 vz24 vs β vz24 _ _
vs24 : β{Ξ B A} β Var24 Ξ A β Var24 (snoc24 Ξ B) A; vs24
= Ξ» x Var24 vz24 vs24 β vs24 _ _ _ (x Var24 vz24 vs24)
Tm24 : Con24 β Ty24 β Set; Tm24
= Ξ» Ξ A β
(Tm24 : Con24 β Ty24 β Set)
(var : β Ξ A β Var24 Ξ A β Tm24 Ξ A)
(lam : β Ξ A B β Tm24 (snoc24 Ξ A) B β Tm24 Ξ (arr24 A B))
(app : β Ξ A B β Tm24 Ξ (arr24 A B) β Tm24 Ξ A β Tm24 Ξ B)
(tt : β Ξ β Tm24 Ξ top24)
(pair : β Ξ A B β Tm24 Ξ A β Tm24 Ξ B β Tm24 Ξ (prod24 A B))
(fst : β Ξ A B β Tm24 Ξ (prod24 A B) β Tm24 Ξ A)
(snd : β Ξ A B β Tm24 Ξ (prod24 A B) β Tm24 Ξ B)
(left : β Ξ A B β Tm24 Ξ A β Tm24 Ξ (sum24 A B))
(right : β Ξ A B β Tm24 Ξ B β Tm24 Ξ (sum24 A B))
(case : β Ξ A B C β Tm24 Ξ (sum24 A B) β Tm24 Ξ (arr24 A C) β Tm24 Ξ (arr24 B C) β Tm24 Ξ C)
(zero : β Ξ β Tm24 Ξ nat24)
(suc : β Ξ β Tm24 Ξ nat24 β Tm24 Ξ nat24)
(rec : β Ξ A β Tm24 Ξ nat24 β Tm24 Ξ (arr24 nat24 (arr24 A A)) β Tm24 Ξ A β Tm24 Ξ A)
β Tm24 Ξ A
var24 : β{Ξ A} β Var24 Ξ A β Tm24 Ξ A; var24
= Ξ» x Tm24 var24 lam app tt pair fst snd left right case zero suc rec β
var24 _ _ x
lam24 : β{Ξ A B} β Tm24 (snoc24 Ξ A) B β Tm24 Ξ (arr24 A B); lam24
= Ξ» t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec β
lam24 _ _ _ (t Tm24 var24 lam24 app tt pair fst snd left right case zero suc rec)
app24 : β{Ξ A B} β Tm24 Ξ (arr24 A B) β Tm24 Ξ A β Tm24 Ξ B; app24
= Ξ» t u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec β
app24 _ _ _ (t Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt pair fst snd left right case zero suc rec)
tt24 : β{Ξ} β Tm24 Ξ top24; tt24
= Ξ» Tm24 var24 lam24 app24 tt24 pair fst snd left right case zero suc rec β tt24 _
pair24 : β{Ξ A B} β Tm24 Ξ A β Tm24 Ξ B β Tm24 Ξ (prod24 A B); pair24
= Ξ» t u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec β
pair24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst snd left right case zero suc rec)
fst24 : β{Ξ A B} β Tm24 Ξ (prod24 A B) β Tm24 Ξ A; fst24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec β
fst24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd left right case zero suc rec)
snd24 : β{Ξ A B} β Tm24 Ξ (prod24 A B) β Tm24 Ξ B; snd24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec β
snd24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left right case zero suc rec)
left24 : β{Ξ A B} β Tm24 Ξ A β Tm24 Ξ (sum24 A B); left24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec β
left24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right case zero suc rec)
right24 : β{Ξ A B} β Tm24 Ξ B β Tm24 Ξ (sum24 A B); right24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec β
right24 _ _ _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case zero suc rec)
case24 : β{Ξ A B C} β Tm24 Ξ (sum24 A B) β Tm24 Ξ (arr24 A C) β Tm24 Ξ (arr24 B C) β Tm24 Ξ C; case24
= Ξ» t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec β
case24 _ _ _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero suc rec)
zero24 : β{Ξ} β Tm24 Ξ nat24; zero24
= Ξ» Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc rec β zero24 _
suc24 : β{Ξ} β Tm24 Ξ nat24 β Tm24 Ξ nat24; suc24
= Ξ» t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec β
suc24 _ (t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec)
rec24 : β{Ξ A} β Tm24 Ξ nat24 β Tm24 Ξ (arr24 nat24 (arr24 A A)) β Tm24 Ξ A β Tm24 Ξ A; rec24
= Ξ» t u v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24 β
rec24 _ _
(t Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(u Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
(v Tm24 var24 lam24 app24 tt24 pair24 fst24 snd24 left24 right24 case24 zero24 suc24 rec24)
v024 : β{Ξ A} β Tm24 (snoc24 Ξ A) A; v024
= var24 vz24
v124 : β{Ξ A B} β Tm24 (snoc24 (snoc24 Ξ A) B) A; v124
= var24 (vs24 vz24)
v224 : β{Ξ A B C} β Tm24 (snoc24 (snoc24 (snoc24 Ξ A) B) C) A; v224
= var24 (vs24 (vs24 vz24))
v324 : β{Ξ A B C D} β Tm24 (snoc24 (snoc24 (snoc24 (snoc24 Ξ A) B) C) D) A; v324
= var24 (vs24 (vs24 (vs24 vz24)))
tbool24 : Ty24; tbool24
= sum24 top24 top24
true24 : β{Ξ} β Tm24 Ξ tbool24; true24
= left24 tt24
tfalse24 : β{Ξ} β Tm24 Ξ tbool24; tfalse24
= right24 tt24
ifthenelse24 : β{Ξ A} β Tm24 Ξ (arr24 tbool24 (arr24 A (arr24 A A))); ifthenelse24
= lam24 (lam24 (lam24 (case24 v224 (lam24 v224) (lam24 v124))))
times424 : β{Ξ A} β Tm24 Ξ (arr24 (arr24 A A) (arr24 A A)); times424
= lam24 (lam24 (app24 v124 (app24 v124 (app24 v124 (app24 v124 v024)))))
add24 : β{Ξ} β Tm24 Ξ (arr24 nat24 (arr24 nat24 nat24)); add24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (suc24 (app24 v124 v024)))))
(lam24 v024))
mul24 : β{Ξ} β Tm24 Ξ (arr24 nat24 (arr24 nat24 nat24)); mul24
= lam24 (rec24 v024
(lam24 (lam24 (lam24 (app24 (app24 add24 (app24 v124 v024)) v024))))
(lam24 zero24))
fact24 : β{Ξ} β Tm24 Ξ (arr24 nat24 nat24); fact24
= lam24 (rec24 v024 (lam24 (lam24 (app24 (app24 mul24 (suc24 v124)) v024)))
(suc24 zero24))
{-# OPTIONS --type-in-type #-}
Ty25 : Set
Ty25 =
(Ty25 : Set)
(nat top bot : Ty25)
(arr prod sum : Ty25 β Ty25 β Ty25)
β Ty25
nat25 : Ty25; nat25 = Ξ» _ nat25 _ _ _ _ _ β nat25
top25 : Ty25; top25 = Ξ» _ _ top25 _ _ _ _ β top25
bot25 : Ty25; bot25 = Ξ» _ _ _ bot25 _ _ _ β bot25
arr25 : Ty25 β Ty25 β Ty25; arr25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod sum β
arr25 (A Ty25 nat25 top25 bot25 arr25 prod sum) (B Ty25 nat25 top25 bot25 arr25 prod sum)
prod25 : Ty25 β Ty25 β Ty25; prod25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod25 sum β
prod25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum) (B Ty25 nat25 top25 bot25 arr25 prod25 sum)
sum25 : Ty25 β Ty25 β Ty25; sum25
= Ξ» A B Ty25 nat25 top25 bot25 arr25 prod25 sum25 β
sum25 (A Ty25 nat25 top25 bot25 arr25 prod25 sum25) (B Ty25 nat25 top25 bot25 arr25 prod25 sum25)
Con25 : Set; Con25
= (Con25 : Set)
(nil : Con25)
(snoc : Con25 β Ty25 β Con25)
β Con25
nil25 : Con25; nil25
= Ξ» Con25 nil25 snoc β nil25
snoc25 : Con25 β Ty25 β Con25; snoc25
= Ξ» Ξ A Con25 nil25 snoc25 β snoc25 (Ξ Con25 nil25 snoc25) A
Var25 : Con25 β Ty25 β Set; Var25
= Ξ» Ξ A β
(Var25 : Con25 β Ty25 β Set)
(vz : β Ξ A β Var25 (snoc25 Ξ A) A)
(vs : β Ξ B A β Var25 Ξ A β Var25 (snoc25 Ξ B) A)
β Var25 Ξ A
vz25 : β{Ξ A} β Var25 (snoc25 Ξ A) A; vz25
= Ξ» Var25 vz25 vs β vz25 _ _
vs25 : β{Ξ B A} β Var25 Ξ A β Var25 (snoc25 Ξ B) A; vs25
= Ξ» x Var25 vz25 vs25 β vs25 _ _ _ (x Var25 vz25 vs25)
Tm25 : Con25 β Ty25 β Set; Tm25
= Ξ» Ξ A β
(Tm25 : Con25 β Ty25 β Set)
(var : β Ξ A β Var25 Ξ A β Tm25 Ξ A)
(lam : β Ξ A B β Tm25 (snoc25 Ξ A) B β Tm25 Ξ (arr25 A B))
(app : β Ξ A B β Tm25 Ξ (arr25 A B) β Tm25 Ξ A β Tm25 Ξ B)
(tt : β Ξ β Tm25 Ξ top25)
(pair : β Ξ A B β Tm25 Ξ A β Tm25 Ξ B β Tm25 Ξ (prod25 A B))
(fst : β Ξ A B β Tm25 Ξ (prod25 A B) β Tm25 Ξ A)
(snd : β Ξ A B β Tm25 Ξ (prod25 A B) β Tm25 Ξ B)
(left : β Ξ A B β Tm25 Ξ A β Tm25 Ξ (sum25 A B))
(right : β Ξ A B β Tm25 Ξ B β Tm25 Ξ (sum25 A B))
(case : β Ξ A B C β Tm25 Ξ (sum25 A B) β Tm25 Ξ (arr25 A C) β Tm25 Ξ (arr25 B C) β Tm25 Ξ C)
(zero : β Ξ β Tm25 Ξ nat25)
(suc : β Ξ β Tm25 Ξ nat25 β Tm25 Ξ nat25)
(rec : β Ξ A β Tm25 Ξ nat25 β Tm25 Ξ (arr25 nat25 (arr25 A A)) β Tm25 Ξ A β Tm25 Ξ A)
β Tm25 Ξ A
var25 : β{Ξ A} β Var25 Ξ A β Tm25 Ξ A; var25
= Ξ» x Tm25 var25 lam app tt pair fst snd left right case zero suc rec β
var25 _ _ x
lam25 : β{Ξ A B} β Tm25 (snoc25 Ξ A) B β Tm25 Ξ (arr25 A B); lam25
= Ξ» t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec β
lam25 _ _ _ (t Tm25 var25 lam25 app tt pair fst snd left right case zero suc rec)
app25 : β{Ξ A B} β Tm25 Ξ (arr25 A B) β Tm25 Ξ A β Tm25 Ξ B; app25
= Ξ» t u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec β
app25 _ _ _ (t Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt pair fst snd left right case zero suc rec)
tt25 : β{Ξ} β Tm25 Ξ top25; tt25
= Ξ» Tm25 var25 lam25 app25 tt25 pair fst snd left right case zero suc rec β tt25 _
pair25 : β{Ξ A B} β Tm25 Ξ A β Tm25 Ξ B β Tm25 Ξ (prod25 A B); pair25
= Ξ» t u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec β
pair25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst snd left right case zero suc rec)
fst25 : β{Ξ A B} β Tm25 Ξ (prod25 A B) β Tm25 Ξ A; fst25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec β
fst25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd left right case zero suc rec)
snd25 : β{Ξ A B} β Tm25 Ξ (prod25 A B) β Tm25 Ξ B; snd25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec β
snd25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left right case zero suc rec)
left25 : β{Ξ A B} β Tm25 Ξ A β Tm25 Ξ (sum25 A B); left25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec β
left25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right case zero suc rec)
right25 : β{Ξ A B} β Tm25 Ξ B β Tm25 Ξ (sum25 A B); right25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec β
right25 _ _ _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case zero suc rec)
case25 : β{Ξ A B C} β Tm25 Ξ (sum25 A B) β Tm25 Ξ (arr25 A C) β Tm25 Ξ (arr25 B C) β Tm25 Ξ C; case25
= Ξ» t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec β
case25 _ _ _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero suc rec)
zero25 : β{Ξ} β Tm25 Ξ nat25; zero25
= Ξ» Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc rec β zero25 _
suc25 : β{Ξ} β Tm25 Ξ nat25 β Tm25 Ξ nat25; suc25
= Ξ» t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec β
suc25 _ (t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec)
rec25 : β{Ξ A} β Tm25 Ξ nat25 β Tm25 Ξ (arr25 nat25 (arr25 A A)) β Tm25 Ξ A β Tm25 Ξ A; rec25
= Ξ» t u v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25 β
rec25 _ _
(t Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(u Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
(v Tm25 var25 lam25 app25 tt25 pair25 fst25 snd25 left25 right25 case25 zero25 suc25 rec25)
v025 : β{Ξ A} β Tm25 (snoc25 Ξ A) A; v025
= var25 vz25
v125 : β{Ξ A B} β Tm25 (snoc25 (snoc25 Ξ A) B) A; v125
= var25 (vs25 vz25)
v225 : β{Ξ A B C} β Tm25 (snoc25 (snoc25 (snoc25 Ξ A) B) C) A; v225
= var25 (vs25 (vs25 vz25))
v325 : β{Ξ A B C D} β Tm25 (snoc25 (snoc25 (snoc25 (snoc25 Ξ A) B) C) D) A; v325
= var25 (vs25 (vs25 (vs25 vz25)))
tbool25 : Ty25; tbool25
= sum25 top25 top25
true25 : β{Ξ} β Tm25 Ξ tbool25; true25
= left25 tt25
tfalse25 : β{Ξ} β Tm25 Ξ tbool25; tfalse25
= right25 tt25
ifthenelse25 : β{Ξ A} β Tm25 Ξ (arr25 tbool25 (arr25 A (arr25 A A))); ifthenelse25
= lam25 (lam25 (lam25 (case25 v225 (lam25 v225) (lam25 v125))))
times425 : β{Ξ A} β Tm25 Ξ (arr25 (arr25 A A) (arr25 A A)); times425
= lam25 (lam25 (app25 v125 (app25 v125 (app25 v125 (app25 v125 v025)))))
add25 : β{Ξ} β Tm25 Ξ (arr25 nat25 (arr25 nat25 nat25)); add25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (suc25 (app25 v125 v025)))))
(lam25 v025))
mul25 : β{Ξ} β Tm25 Ξ (arr25 nat25 (arr25 nat25 nat25)); mul25
= lam25 (rec25 v025
(lam25 (lam25 (lam25 (app25 (app25 add25 (app25 v125 v025)) v025))))
(lam25 zero25))
fact25 : β{Ξ} β Tm25 Ξ (arr25 nat25 nat25); fact25
= lam25 (rec25 v025 (lam25 (lam25 (app25 (app25 mul25 (suc25 v125)) v025)))
(suc25 zero25))
{-# OPTIONS --type-in-type #-}
Ty26 : Set
Ty26 =
(Ty26 : Set)
(nat top bot : Ty26)
(arr prod sum : Ty26 β Ty26 β Ty26)
β Ty26
nat26 : Ty26; nat26 = Ξ» _ nat26 _ _ _ _ _ β nat26
top26 : Ty26; top26 = Ξ» _ _ top26 _ _ _ _ β top26
bot26 : Ty26; bot26 = Ξ» _ _ _ bot26 _ _ _ β bot26
arr26 : Ty26 β Ty26 β Ty26; arr26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod sum β
arr26 (A Ty26 nat26 top26 bot26 arr26 prod sum) (B Ty26 nat26 top26 bot26 arr26 prod sum)
prod26 : Ty26 β Ty26 β Ty26; prod26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod26 sum β
prod26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum) (B Ty26 nat26 top26 bot26 arr26 prod26 sum)
sum26 : Ty26 β Ty26 β Ty26; sum26
= Ξ» A B Ty26 nat26 top26 bot26 arr26 prod26 sum26 β
sum26 (A Ty26 nat26 top26 bot26 arr26 prod26 sum26) (B Ty26 nat26 top26 bot26 arr26 prod26 sum26)
Con26 : Set; Con26
= (Con26 : Set)
(nil : Con26)
(snoc : Con26 β Ty26 β Con26)
β Con26
nil26 : Con26; nil26
= Ξ» Con26 nil26 snoc β nil26
snoc26 : Con26 β Ty26 β Con26; snoc26
= Ξ» Ξ A Con26 nil26 snoc26 β snoc26 (Ξ Con26 nil26 snoc26) A
Var26 : Con26 β Ty26 β Set; Var26
= Ξ» Ξ A β
(Var26 : Con26 β Ty26 β Set)
(vz : β Ξ A β Var26 (snoc26 Ξ A) A)
(vs : β Ξ B A β Var26 Ξ A β Var26 (snoc26 Ξ B) A)
β Var26 Ξ A
vz26 : β{Ξ A} β Var26 (snoc26 Ξ A) A; vz26
= Ξ» Var26 vz26 vs β vz26 _ _
vs26 : β{Ξ B A} β Var26 Ξ A β Var26 (snoc26 Ξ B) A; vs26
= Ξ» x Var26 vz26 vs26 β vs26 _ _ _ (x Var26 vz26 vs26)
Tm26 : Con26 β Ty26 β Set; Tm26
= Ξ» Ξ A β
(Tm26 : Con26 β Ty26 β Set)
(var : β Ξ A β Var26 Ξ A β Tm26 Ξ A)
(lam : β Ξ A B β Tm26 (snoc26 Ξ A) B β Tm26 Ξ (arr26 A B))
(app : β Ξ A B β Tm26 Ξ (arr26 A B) β Tm26 Ξ A β Tm26 Ξ B)
(tt : β Ξ β Tm26 Ξ top26)
(pair : β Ξ A B β Tm26 Ξ A β Tm26 Ξ B β Tm26 Ξ (prod26 A B))
(fst : β Ξ A B β Tm26 Ξ (prod26 A B) β Tm26 Ξ A)
(snd : β Ξ A B β Tm26 Ξ (prod26 A B) β Tm26 Ξ B)
(left : β Ξ A B β Tm26 Ξ A β Tm26 Ξ (sum26 A B))
(right : β Ξ A B β Tm26 Ξ B β Tm26 Ξ (sum26 A B))
(case : β Ξ A B C β Tm26 Ξ (sum26 A B) β Tm26 Ξ (arr26 A C) β Tm26 Ξ (arr26 B C) β Tm26 Ξ C)
(zero : β Ξ β Tm26 Ξ nat26)
(suc : β Ξ β Tm26 Ξ nat26 β Tm26 Ξ nat26)
(rec : β Ξ A β Tm26 Ξ nat26 β Tm26 Ξ (arr26 nat26 (arr26 A A)) β Tm26 Ξ A β Tm26 Ξ A)
β Tm26 Ξ A
var26 : β{Ξ A} β Var26 Ξ A β Tm26 Ξ A; var26
= Ξ» x Tm26 var26 lam app tt pair fst snd left right case zero suc rec β
var26 _ _ x
lam26 : β{Ξ A B} β Tm26 (snoc26 Ξ A) B β Tm26 Ξ (arr26 A B); lam26
= Ξ» t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec β
lam26 _ _ _ (t Tm26 var26 lam26 app tt pair fst snd left right case zero suc rec)
app26 : β{Ξ A B} β Tm26 Ξ (arr26 A B) β Tm26 Ξ A β Tm26 Ξ B; app26
= Ξ» t u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec β
app26 _ _ _ (t Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt pair fst snd left right case zero suc rec)
tt26 : β{Ξ} β Tm26 Ξ top26; tt26
= Ξ» Tm26 var26 lam26 app26 tt26 pair fst snd left right case zero suc rec β tt26 _
pair26 : β{Ξ A B} β Tm26 Ξ A β Tm26 Ξ B β Tm26 Ξ (prod26 A B); pair26
= Ξ» t u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec β
pair26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst snd left right case zero suc rec)
fst26 : β{Ξ A B} β Tm26 Ξ (prod26 A B) β Tm26 Ξ A; fst26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec β
fst26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd left right case zero suc rec)
snd26 : β{Ξ A B} β Tm26 Ξ (prod26 A B) β Tm26 Ξ B; snd26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec β
snd26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left right case zero suc rec)
left26 : β{Ξ A B} β Tm26 Ξ A β Tm26 Ξ (sum26 A B); left26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec β
left26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right case zero suc rec)
right26 : β{Ξ A B} β Tm26 Ξ B β Tm26 Ξ (sum26 A B); right26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec β
right26 _ _ _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case zero suc rec)
case26 : β{Ξ A B C} β Tm26 Ξ (sum26 A B) β Tm26 Ξ (arr26 A C) β Tm26 Ξ (arr26 B C) β Tm26 Ξ C; case26
= Ξ» t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec β
case26 _ _ _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero suc rec)
zero26 : β{Ξ} β Tm26 Ξ nat26; zero26
= Ξ» Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc rec β zero26 _
suc26 : β{Ξ} β Tm26 Ξ nat26 β Tm26 Ξ nat26; suc26
= Ξ» t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec β
suc26 _ (t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec)
rec26 : β{Ξ A} β Tm26 Ξ nat26 β Tm26 Ξ (arr26 nat26 (arr26 A A)) β Tm26 Ξ A β Tm26 Ξ A; rec26
= Ξ» t u v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26 β
rec26 _ _
(t Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(u Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
(v Tm26 var26 lam26 app26 tt26 pair26 fst26 snd26 left26 right26 case26 zero26 suc26 rec26)
v026 : β{Ξ A} β Tm26 (snoc26 Ξ A) A; v026
= var26 vz26
v126 : β{Ξ A B} β Tm26 (snoc26 (snoc26 Ξ A) B) A; v126
= var26 (vs26 vz26)
v226 : β{Ξ A B C} β Tm26 (snoc26 (snoc26 (snoc26 Ξ A) B) C) A; v226
= var26 (vs26 (vs26 vz26))
v326 : β{Ξ A B C D} β Tm26 (snoc26 (snoc26 (snoc26 (snoc26 Ξ A) B) C) D) A; v326
= var26 (vs26 (vs26 (vs26 vz26)))
tbool26 : Ty26; tbool26
= sum26 top26 top26
true26 : β{Ξ} β Tm26 Ξ tbool26; true26
= left26 tt26
tfalse26 : β{Ξ} β Tm26 Ξ tbool26; tfalse26
= right26 tt26
ifthenelse26 : β{Ξ A} β Tm26 Ξ (arr26 tbool26 (arr26 A (arr26 A A))); ifthenelse26
= lam26 (lam26 (lam26 (case26 v226 (lam26 v226) (lam26 v126))))
times426 : β{Ξ A} β Tm26 Ξ (arr26 (arr26 A A) (arr26 A A)); times426
= lam26 (lam26 (app26 v126 (app26 v126 (app26 v126 (app26 v126 v026)))))
add26 : β{Ξ} β Tm26 Ξ (arr26 nat26 (arr26 nat26 nat26)); add26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (suc26 (app26 v126 v026)))))
(lam26 v026))
mul26 : β{Ξ} β Tm26 Ξ (arr26 nat26 (arr26 nat26 nat26)); mul26
= lam26 (rec26 v026
(lam26 (lam26 (lam26 (app26 (app26 add26 (app26 v126 v026)) v026))))
(lam26 zero26))
fact26 : β{Ξ} β Tm26 Ξ (arr26 nat26 nat26); fact26
= lam26 (rec26 v026 (lam26 (lam26 (app26 (app26 mul26 (suc26 v126)) v026)))
(suc26 zero26))
{-# OPTIONS --type-in-type #-}
Ty27 : Set
Ty27 =
(Ty27 : Set)
(nat top bot : Ty27)
(arr prod sum : Ty27 β Ty27 β Ty27)
β Ty27
nat27 : Ty27; nat27 = Ξ» _ nat27 _ _ _ _ _ β nat27
top27 : Ty27; top27 = Ξ» _ _ top27 _ _ _ _ β top27
bot27 : Ty27; bot27 = Ξ» _ _ _ bot27 _ _ _ β bot27
arr27 : Ty27 β Ty27 β Ty27; arr27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod sum β
arr27 (A Ty27 nat27 top27 bot27 arr27 prod sum) (B Ty27 nat27 top27 bot27 arr27 prod sum)
prod27 : Ty27 β Ty27 β Ty27; prod27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod27 sum β
prod27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum) (B Ty27 nat27 top27 bot27 arr27 prod27 sum)
sum27 : Ty27 β Ty27 β Ty27; sum27
= Ξ» A B Ty27 nat27 top27 bot27 arr27 prod27 sum27 β
sum27 (A Ty27 nat27 top27 bot27 arr27 prod27 sum27) (B Ty27 nat27 top27 bot27 arr27 prod27 sum27)
Con27 : Set; Con27
= (Con27 : Set)
(nil : Con27)
(snoc : Con27 β Ty27 β Con27)
β Con27
nil27 : Con27; nil27
= Ξ» Con27 nil27 snoc β nil27
snoc27 : Con27 β Ty27 β Con27; snoc27
= Ξ» Ξ A Con27 nil27 snoc27 β snoc27 (Ξ Con27 nil27 snoc27) A
Var27 : Con27 β Ty27 β Set; Var27
= Ξ» Ξ A β
(Var27 : Con27 β Ty27 β Set)
(vz : β Ξ A β Var27 (snoc27 Ξ A) A)
(vs : β Ξ B A β Var27 Ξ A β Var27 (snoc27 Ξ B) A)
β Var27 Ξ A
vz27 : β{Ξ A} β Var27 (snoc27 Ξ A) A; vz27
= Ξ» Var27 vz27 vs β vz27 _ _
vs27 : β{Ξ B A} β Var27 Ξ A β Var27 (snoc27 Ξ B) A; vs27
= Ξ» x Var27 vz27 vs27 β vs27 _ _ _ (x Var27 vz27 vs27)
Tm27 : Con27 β Ty27 β Set; Tm27
= Ξ» Ξ A β
(Tm27 : Con27 β Ty27 β Set)
(var : β Ξ A β Var27 Ξ A β Tm27 Ξ A)
(lam : β Ξ A B β Tm27 (snoc27 Ξ A) B β Tm27 Ξ (arr27 A B))
(app : β Ξ A B β Tm27 Ξ (arr27 A B) β Tm27 Ξ A β Tm27 Ξ B)
(tt : β Ξ β Tm27 Ξ top27)
(pair : β Ξ A B β Tm27 Ξ A β Tm27 Ξ B β Tm27 Ξ (prod27 A B))
(fst : β Ξ A B β Tm27 Ξ (prod27 A B) β Tm27 Ξ A)
(snd : β Ξ A B β Tm27 Ξ (prod27 A B) β Tm27 Ξ B)
(left : β Ξ A B β Tm27 Ξ A β Tm27 Ξ (sum27 A B))
(right : β Ξ A B β Tm27 Ξ B β Tm27 Ξ (sum27 A B))
(case : β Ξ A B C β Tm27 Ξ (sum27 A B) β Tm27 Ξ (arr27 A C) β Tm27 Ξ (arr27 B C) β Tm27 Ξ C)
(zero : β Ξ β Tm27 Ξ nat27)
(suc : β Ξ β Tm27 Ξ nat27 β Tm27 Ξ nat27)
(rec : β Ξ A β Tm27 Ξ nat27 β Tm27 Ξ (arr27 nat27 (arr27 A A)) β Tm27 Ξ A β Tm27 Ξ A)
β Tm27 Ξ A
var27 : β{Ξ A} β Var27 Ξ A β Tm27 Ξ A; var27
= Ξ» x Tm27 var27 lam app tt pair fst snd left right case zero suc rec β
var27 _ _ x
lam27 : β{Ξ A B} β Tm27 (snoc27 Ξ A) B β Tm27 Ξ (arr27 A B); lam27
= Ξ» t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec β
lam27 _ _ _ (t Tm27 var27 lam27 app tt pair fst snd left right case zero suc rec)
app27 : β{Ξ A B} β Tm27 Ξ (arr27 A B) β Tm27 Ξ A β Tm27 Ξ B; app27
= Ξ» t u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec β
app27 _ _ _ (t Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt pair fst snd left right case zero suc rec)
tt27 : β{Ξ} β Tm27 Ξ top27; tt27
= Ξ» Tm27 var27 lam27 app27 tt27 pair fst snd left right case zero suc rec β tt27 _
pair27 : β{Ξ A B} β Tm27 Ξ A β Tm27 Ξ B β Tm27 Ξ (prod27 A B); pair27
= Ξ» t u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec β
pair27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst snd left right case zero suc rec)
fst27 : β{Ξ A B} β Tm27 Ξ (prod27 A B) β Tm27 Ξ A; fst27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec β
fst27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd left right case zero suc rec)
snd27 : β{Ξ A B} β Tm27 Ξ (prod27 A B) β Tm27 Ξ B; snd27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec β
snd27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left right case zero suc rec)
left27 : β{Ξ A B} β Tm27 Ξ A β Tm27 Ξ (sum27 A B); left27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec β
left27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right case zero suc rec)
right27 : β{Ξ A B} β Tm27 Ξ B β Tm27 Ξ (sum27 A B); right27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec β
right27 _ _ _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case zero suc rec)
case27 : β{Ξ A B C} β Tm27 Ξ (sum27 A B) β Tm27 Ξ (arr27 A C) β Tm27 Ξ (arr27 B C) β Tm27 Ξ C; case27
= Ξ» t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec β
case27 _ _ _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero suc rec)
zero27 : β{Ξ} β Tm27 Ξ nat27; zero27
= Ξ» Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc rec β zero27 _
suc27 : β{Ξ} β Tm27 Ξ nat27 β Tm27 Ξ nat27; suc27
= Ξ» t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec β
suc27 _ (t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec)
rec27 : β{Ξ A} β Tm27 Ξ nat27 β Tm27 Ξ (arr27 nat27 (arr27 A A)) β Tm27 Ξ A β Tm27 Ξ A; rec27
= Ξ» t u v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27 β
rec27 _ _
(t Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(u Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
(v Tm27 var27 lam27 app27 tt27 pair27 fst27 snd27 left27 right27 case27 zero27 suc27 rec27)
v027 : β{Ξ A} β Tm27 (snoc27 Ξ A) A; v027
= var27 vz27
v127 : β{Ξ A B} β Tm27 (snoc27 (snoc27 Ξ A) B) A; v127
= var27 (vs27 vz27)
v227 : β{Ξ A B C} β Tm27 (snoc27 (snoc27 (snoc27 Ξ A) B) C) A; v227
= var27 (vs27 (vs27 vz27))
v327 : β{Ξ A B C D} β Tm27 (snoc27 (snoc27 (snoc27 (snoc27 Ξ A) B) C) D) A; v327
= var27 (vs27 (vs27 (vs27 vz27)))
tbool27 : Ty27; tbool27
= sum27 top27 top27
true27 : β{Ξ} β Tm27 Ξ tbool27; true27
= left27 tt27
tfalse27 : β{Ξ} β Tm27 Ξ tbool27; tfalse27
= right27 tt27
ifthenelse27 : β{Ξ A} β Tm27 Ξ (arr27 tbool27 (arr27 A (arr27 A A))); ifthenelse27
= lam27 (lam27 (lam27 (case27 v227 (lam27 v227) (lam27 v127))))
times427 : β{Ξ A} β Tm27 Ξ (arr27 (arr27 A A) (arr27 A A)); times427
= lam27 (lam27 (app27 v127 (app27 v127 (app27 v127 (app27 v127 v027)))))
add27 : β{Ξ} β Tm27 Ξ (arr27 nat27 (arr27 nat27 nat27)); add27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (suc27 (app27 v127 v027)))))
(lam27 v027))
mul27 : β{Ξ} β Tm27 Ξ (arr27 nat27 (arr27 nat27 nat27)); mul27
= lam27 (rec27 v027
(lam27 (lam27 (lam27 (app27 (app27 add27 (app27 v127 v027)) v027))))
(lam27 zero27))
fact27 : β{Ξ} β Tm27 Ξ (arr27 nat27 nat27); fact27
= lam27 (rec27 v027 (lam27 (lam27 (app27 (app27 mul27 (suc27 v127)) v027)))
(suc27 zero27))
{-# OPTIONS --type-in-type #-}
Ty28 : Set
Ty28 =
(Ty28 : Set)
(nat top bot : Ty28)
(arr prod sum : Ty28 β Ty28 β Ty28)
β Ty28
nat28 : Ty28; nat28 = Ξ» _ nat28 _ _ _ _ _ β nat28
top28 : Ty28; top28 = Ξ» _ _ top28 _ _ _ _ β top28
bot28 : Ty28; bot28 = Ξ» _ _ _ bot28 _ _ _ β bot28
arr28 : Ty28 β Ty28 β Ty28; arr28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod sum β
arr28 (A Ty28 nat28 top28 bot28 arr28 prod sum) (B Ty28 nat28 top28 bot28 arr28 prod sum)
prod28 : Ty28 β Ty28 β Ty28; prod28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod28 sum β
prod28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum) (B Ty28 nat28 top28 bot28 arr28 prod28 sum)
sum28 : Ty28 β Ty28 β Ty28; sum28
= Ξ» A B Ty28 nat28 top28 bot28 arr28 prod28 sum28 β
sum28 (A Ty28 nat28 top28 bot28 arr28 prod28 sum28) (B Ty28 nat28 top28 bot28 arr28 prod28 sum28)
Con28 : Set; Con28
= (Con28 : Set)
(nil : Con28)
(snoc : Con28 β Ty28 β Con28)
β Con28
nil28 : Con28; nil28
= Ξ» Con28 nil28 snoc β nil28
snoc28 : Con28 β Ty28 β Con28; snoc28
= Ξ» Ξ A Con28 nil28 snoc28 β snoc28 (Ξ Con28 nil28 snoc28) A
Var28 : Con28 β Ty28 β Set; Var28
= Ξ» Ξ A β
(Var28 : Con28 β Ty28 β Set)
(vz : β Ξ A β Var28 (snoc28 Ξ A) A)
(vs : β Ξ B A β Var28 Ξ A β Var28 (snoc28 Ξ B) A)
β Var28 Ξ A
vz28 : β{Ξ A} β Var28 (snoc28 Ξ A) A; vz28
= Ξ» Var28 vz28 vs β vz28 _ _
vs28 : β{Ξ B A} β Var28 Ξ A β Var28 (snoc28 Ξ B) A; vs28
= Ξ» x Var28 vz28 vs28 β vs28 _ _ _ (x Var28 vz28 vs28)
Tm28 : Con28 β Ty28 β Set; Tm28
= Ξ» Ξ A β
(Tm28 : Con28 β Ty28 β Set)
(var : β Ξ A β Var28 Ξ A β Tm28 Ξ A)
(lam : β Ξ A B β Tm28 (snoc28 Ξ A) B β Tm28 Ξ (arr28 A B))
(app : β Ξ A B β Tm28 Ξ (arr28 A B) β Tm28 Ξ A β Tm28 Ξ B)
(tt : β Ξ β Tm28 Ξ top28)
(pair : β Ξ A B β Tm28 Ξ A β Tm28 Ξ B β Tm28 Ξ (prod28 A B))
(fst : β Ξ A B β Tm28 Ξ (prod28 A B) β Tm28 Ξ A)
(snd : β Ξ A B β Tm28 Ξ (prod28 A B) β Tm28 Ξ B)
(left : β Ξ A B β Tm28 Ξ A β Tm28 Ξ (sum28 A B))
(right : β Ξ A B β Tm28 Ξ B β Tm28 Ξ (sum28 A B))
(case : β Ξ A B C β Tm28 Ξ (sum28 A B) β Tm28 Ξ (arr28 A C) β Tm28 Ξ (arr28 B C) β Tm28 Ξ C)
(zero : β Ξ β Tm28 Ξ nat28)
(suc : β Ξ β Tm28 Ξ nat28 β Tm28 Ξ nat28)
(rec : β Ξ A β Tm28 Ξ nat28 β Tm28 Ξ (arr28 nat28 (arr28 A A)) β Tm28 Ξ A β Tm28 Ξ A)
β Tm28 Ξ A
var28 : β{Ξ A} β Var28 Ξ A β Tm28 Ξ A; var28
= Ξ» x Tm28 var28 lam app tt pair fst snd left right case zero suc rec β
var28 _ _ x
lam28 : β{Ξ A B} β Tm28 (snoc28 Ξ A) B β Tm28 Ξ (arr28 A B); lam28
= Ξ» t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec β
lam28 _ _ _ (t Tm28 var28 lam28 app tt pair fst snd left right case zero suc rec)
app28 : β{Ξ A B} β Tm28 Ξ (arr28 A B) β Tm28 Ξ A β Tm28 Ξ B; app28
= Ξ» t u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec β
app28 _ _ _ (t Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt pair fst snd left right case zero suc rec)
tt28 : β{Ξ} β Tm28 Ξ top28; tt28
= Ξ» Tm28 var28 lam28 app28 tt28 pair fst snd left right case zero suc rec β tt28 _
pair28 : β{Ξ A B} β Tm28 Ξ A β Tm28 Ξ B β Tm28 Ξ (prod28 A B); pair28
= Ξ» t u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec β
pair28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst snd left right case zero suc rec)
fst28 : β{Ξ A B} β Tm28 Ξ (prod28 A B) β Tm28 Ξ A; fst28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec β
fst28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd left right case zero suc rec)
snd28 : β{Ξ A B} β Tm28 Ξ (prod28 A B) β Tm28 Ξ B; snd28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec β
snd28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left right case zero suc rec)
left28 : β{Ξ A B} β Tm28 Ξ A β Tm28 Ξ (sum28 A B); left28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec β
left28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right case zero suc rec)
right28 : β{Ξ A B} β Tm28 Ξ B β Tm28 Ξ (sum28 A B); right28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec β
right28 _ _ _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case zero suc rec)
case28 : β{Ξ A B C} β Tm28 Ξ (sum28 A B) β Tm28 Ξ (arr28 A C) β Tm28 Ξ (arr28 B C) β Tm28 Ξ C; case28
= Ξ» t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec β
case28 _ _ _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero suc rec)
zero28 : β{Ξ} β Tm28 Ξ nat28; zero28
= Ξ» Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc rec β zero28 _
suc28 : β{Ξ} β Tm28 Ξ nat28 β Tm28 Ξ nat28; suc28
= Ξ» t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec β
suc28 _ (t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec)
rec28 : β{Ξ A} β Tm28 Ξ nat28 β Tm28 Ξ (arr28 nat28 (arr28 A A)) β Tm28 Ξ A β Tm28 Ξ A; rec28
= Ξ» t u v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28 β
rec28 _ _
(t Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(u Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
(v Tm28 var28 lam28 app28 tt28 pair28 fst28 snd28 left28 right28 case28 zero28 suc28 rec28)
v028 : β{Ξ A} β Tm28 (snoc28 Ξ A) A; v028
= var28 vz28
v128 : β{Ξ A B} β Tm28 (snoc28 (snoc28 Ξ A) B) A; v128
= var28 (vs28 vz28)
v228 : β{Ξ A B C} β Tm28 (snoc28 (snoc28 (snoc28 Ξ A) B) C) A; v228
= var28 (vs28 (vs28 vz28))
v328 : β{Ξ A B C D} β Tm28 (snoc28 (snoc28 (snoc28 (snoc28 Ξ A) B) C) D) A; v328
= var28 (vs28 (vs28 (vs28 vz28)))
tbool28 : Ty28; tbool28
= sum28 top28 top28
true28 : β{Ξ} β Tm28 Ξ tbool28; true28
= left28 tt28
tfalse28 : β{Ξ} β Tm28 Ξ tbool28; tfalse28
= right28 tt28
ifthenelse28 : β{Ξ A} β Tm28 Ξ (arr28 tbool28 (arr28 A (arr28 A A))); ifthenelse28
= lam28 (lam28 (lam28 (case28 v228 (lam28 v228) (lam28 v128))))
times428 : β{Ξ A} β Tm28 Ξ (arr28 (arr28 A A) (arr28 A A)); times428
= lam28 (lam28 (app28 v128 (app28 v128 (app28 v128 (app28 v128 v028)))))
add28 : β{Ξ} β Tm28 Ξ (arr28 nat28 (arr28 nat28 nat28)); add28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (suc28 (app28 v128 v028)))))
(lam28 v028))
mul28 : β{Ξ} β Tm28 Ξ (arr28 nat28 (arr28 nat28 nat28)); mul28
= lam28 (rec28 v028
(lam28 (lam28 (lam28 (app28 (app28 add28 (app28 v128 v028)) v028))))
(lam28 zero28))
fact28 : β{Ξ} β Tm28 Ξ (arr28 nat28 nat28); fact28
= lam28 (rec28 v028 (lam28 (lam28 (app28 (app28 mul28 (suc28 v128)) v028)))
(suc28 zero28))
{-# OPTIONS --type-in-type #-}
Ty29 : Set
Ty29 =
(Ty29 : Set)
(nat top bot : Ty29)
(arr prod sum : Ty29 β Ty29 β Ty29)
β Ty29
nat29 : Ty29; nat29 = Ξ» _ nat29 _ _ _ _ _ β nat29
top29 : Ty29; top29 = Ξ» _ _ top29 _ _ _ _ β top29
bot29 : Ty29; bot29 = Ξ» _ _ _ bot29 _ _ _ β bot29
arr29 : Ty29 β Ty29 β Ty29; arr29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod sum β
arr29 (A Ty29 nat29 top29 bot29 arr29 prod sum) (B Ty29 nat29 top29 bot29 arr29 prod sum)
prod29 : Ty29 β Ty29 β Ty29; prod29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod29 sum β
prod29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum) (B Ty29 nat29 top29 bot29 arr29 prod29 sum)
sum29 : Ty29 β Ty29 β Ty29; sum29
= Ξ» A B Ty29 nat29 top29 bot29 arr29 prod29 sum29 β
sum29 (A Ty29 nat29 top29 bot29 arr29 prod29 sum29) (B Ty29 nat29 top29 bot29 arr29 prod29 sum29)
Con29 : Set; Con29
= (Con29 : Set)
(nil : Con29)
(snoc : Con29 β Ty29 β Con29)
β Con29
nil29 : Con29; nil29
= Ξ» Con29 nil29 snoc β nil29
snoc29 : Con29 β Ty29 β Con29; snoc29
= Ξ» Ξ A Con29 nil29 snoc29 β snoc29 (Ξ Con29 nil29 snoc29) A
Var29 : Con29 β Ty29 β Set; Var29
= Ξ» Ξ A β
(Var29 : Con29 β Ty29 β Set)
(vz : β Ξ A β Var29 (snoc29 Ξ A) A)
(vs : β Ξ B A β Var29 Ξ A β Var29 (snoc29 Ξ B) A)
β Var29 Ξ A
vz29 : β{Ξ A} β Var29 (snoc29 Ξ A) A; vz29
= Ξ» Var29 vz29 vs β vz29 _ _
vs29 : β{Ξ B A} β Var29 Ξ A β Var29 (snoc29 Ξ B) A; vs29
= Ξ» x Var29 vz29 vs29 β vs29 _ _ _ (x Var29 vz29 vs29)
Tm29 : Con29 β Ty29 β Set; Tm29
= Ξ» Ξ A β
(Tm29 : Con29 β Ty29 β Set)
(var : β Ξ A β Var29 Ξ A β Tm29 Ξ A)
(lam : β Ξ A B β Tm29 (snoc29 Ξ A) B β Tm29 Ξ (arr29 A B))
(app : β Ξ A B β Tm29 Ξ (arr29 A B) β Tm29 Ξ A β Tm29 Ξ B)
(tt : β Ξ β Tm29 Ξ top29)
(pair : β Ξ A B β Tm29 Ξ A β Tm29 Ξ B β Tm29 Ξ (prod29 A B))
(fst : β Ξ A B β Tm29 Ξ (prod29 A B) β Tm29 Ξ A)
(snd : β Ξ A B β Tm29 Ξ (prod29 A B) β Tm29 Ξ B)
(left : β Ξ A B β Tm29 Ξ A β Tm29 Ξ (sum29 A B))
(right : β Ξ A B β Tm29 Ξ B β Tm29 Ξ (sum29 A B))
(case : β Ξ A B C β Tm29 Ξ (sum29 A B) β Tm29 Ξ (arr29 A C) β Tm29 Ξ (arr29 B C) β Tm29 Ξ C)
(zero : β Ξ β Tm29 Ξ nat29)
(suc : β Ξ β Tm29 Ξ nat29 β Tm29 Ξ nat29)
(rec : β Ξ A β Tm29 Ξ nat29 β Tm29 Ξ (arr29 nat29 (arr29 A A)) β Tm29 Ξ A β Tm29 Ξ A)
β Tm29 Ξ A
var29 : β{Ξ A} β Var29 Ξ A β Tm29 Ξ A; var29
= Ξ» x Tm29 var29 lam app tt pair fst snd left right case zero suc rec β
var29 _ _ x
lam29 : β{Ξ A B} β Tm29 (snoc29 Ξ A) B β Tm29 Ξ (arr29 A B); lam29
= Ξ» t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec β
lam29 _ _ _ (t Tm29 var29 lam29 app tt pair fst snd left right case zero suc rec)
app29 : β{Ξ A B} β Tm29 Ξ (arr29 A B) β Tm29 Ξ A β Tm29 Ξ B; app29
= Ξ» t u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec β
app29 _ _ _ (t Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt pair fst snd left right case zero suc rec)
tt29 : β{Ξ} β Tm29 Ξ top29; tt29
= Ξ» Tm29 var29 lam29 app29 tt29 pair fst snd left right case zero suc rec β tt29 _
pair29 : β{Ξ A B} β Tm29 Ξ A β Tm29 Ξ B β Tm29 Ξ (prod29 A B); pair29
= Ξ» t u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec β
pair29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst snd left right case zero suc rec)
fst29 : β{Ξ A B} β Tm29 Ξ (prod29 A B) β Tm29 Ξ A; fst29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec β
fst29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd left right case zero suc rec)
snd29 : β{Ξ A B} β Tm29 Ξ (prod29 A B) β Tm29 Ξ B; snd29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec β
snd29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left right case zero suc rec)
left29 : β{Ξ A B} β Tm29 Ξ A β Tm29 Ξ (sum29 A B); left29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec β
left29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right case zero suc rec)
right29 : β{Ξ A B} β Tm29 Ξ B β Tm29 Ξ (sum29 A B); right29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec β
right29 _ _ _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case zero suc rec)
case29 : β{Ξ A B C} β Tm29 Ξ (sum29 A B) β Tm29 Ξ (arr29 A C) β Tm29 Ξ (arr29 B C) β Tm29 Ξ C; case29
= Ξ» t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec β
case29 _ _ _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero suc rec)
zero29 : β{Ξ} β Tm29 Ξ nat29; zero29
= Ξ» Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc rec β zero29 _
suc29 : β{Ξ} β Tm29 Ξ nat29 β Tm29 Ξ nat29; suc29
= Ξ» t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec β
suc29 _ (t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec)
rec29 : β{Ξ A} β Tm29 Ξ nat29 β Tm29 Ξ (arr29 nat29 (arr29 A A)) β Tm29 Ξ A β Tm29 Ξ A; rec29
= Ξ» t u v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29 β
rec29 _ _
(t Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(u Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
(v Tm29 var29 lam29 app29 tt29 pair29 fst29 snd29 left29 right29 case29 zero29 suc29 rec29)
v029 : β{Ξ A} β Tm29 (snoc29 Ξ A) A; v029
= var29 vz29
v129 : β{Ξ A B} β Tm29 (snoc29 (snoc29 Ξ A) B) A; v129
= var29 (vs29 vz29)
v229 : β{Ξ A B C} β Tm29 (snoc29 (snoc29 (snoc29 Ξ A) B) C) A; v229
= var29 (vs29 (vs29 vz29))
v329 : β{Ξ A B C D} β Tm29 (snoc29 (snoc29 (snoc29 (snoc29 Ξ A) B) C) D) A; v329
= var29 (vs29 (vs29 (vs29 vz29)))
tbool29 : Ty29; tbool29
= sum29 top29 top29
true29 : β{Ξ} β Tm29 Ξ tbool29; true29
= left29 tt29
tfalse29 : β{Ξ} β Tm29 Ξ tbool29; tfalse29
= right29 tt29
ifthenelse29 : β{Ξ A} β Tm29 Ξ (arr29 tbool29 (arr29 A (arr29 A A))); ifthenelse29
= lam29 (lam29 (lam29 (case29 v229 (lam29 v229) (lam29 v129))))
times429 : β{Ξ A} β Tm29 Ξ (arr29 (arr29 A A) (arr29 A A)); times429
= lam29 (lam29 (app29 v129 (app29 v129 (app29 v129 (app29 v129 v029)))))
add29 : β{Ξ} β Tm29 Ξ (arr29 nat29 (arr29 nat29 nat29)); add29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (suc29 (app29 v129 v029)))))
(lam29 v029))
mul29 : β{Ξ} β Tm29 Ξ (arr29 nat29 (arr29 nat29 nat29)); mul29
= lam29 (rec29 v029
(lam29 (lam29 (lam29 (app29 (app29 add29 (app29 v129 v029)) v029))))
(lam29 zero29))
fact29 : β{Ξ} β Tm29 Ξ (arr29 nat29 nat29); fact29
= lam29 (rec29 v029 (lam29 (lam29 (app29 (app29 mul29 (suc29 v129)) v029)))
(suc29 zero29))
{-# OPTIONS --type-in-type #-}
Ty30 : Set
Ty30 =
(Ty30 : Set)
(nat top bot : Ty30)
(arr prod sum : Ty30 β Ty30 β Ty30)
β Ty30
nat30 : Ty30; nat30 = Ξ» _ nat30 _ _ _ _ _ β nat30
top30 : Ty30; top30 = Ξ» _ _ top30 _ _ _ _ β top30
bot30 : Ty30; bot30 = Ξ» _ _ _ bot30 _ _ _ β bot30
arr30 : Ty30 β Ty30 β Ty30; arr30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod sum β
arr30 (A Ty30 nat30 top30 bot30 arr30 prod sum) (B Ty30 nat30 top30 bot30 arr30 prod sum)
prod30 : Ty30 β Ty30 β Ty30; prod30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod30 sum β
prod30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum) (B Ty30 nat30 top30 bot30 arr30 prod30 sum)
sum30 : Ty30 β Ty30 β Ty30; sum30
= Ξ» A B Ty30 nat30 top30 bot30 arr30 prod30 sum30 β
sum30 (A Ty30 nat30 top30 bot30 arr30 prod30 sum30) (B Ty30 nat30 top30 bot30 arr30 prod30 sum30)
Con30 : Set; Con30
= (Con30 : Set)
(nil : Con30)
(snoc : Con30 β Ty30 β Con30)
β Con30
nil30 : Con30; nil30
= Ξ» Con30 nil30 snoc β nil30
snoc30 : Con30 β Ty30 β Con30; snoc30
= Ξ» Ξ A Con30 nil30 snoc30 β snoc30 (Ξ Con30 nil30 snoc30) A
Var30 : Con30 β Ty30 β Set; Var30
= Ξ» Ξ A β
(Var30 : Con30 β Ty30 β Set)
(vz : β Ξ A β Var30 (snoc30 Ξ A) A)
(vs : β Ξ B A β Var30 Ξ A β Var30 (snoc30 Ξ B) A)
β Var30 Ξ A
vz30 : β{Ξ A} β Var30 (snoc30 Ξ A) A; vz30
= Ξ» Var30 vz30 vs β vz30 _ _
vs30 : β{Ξ B A} β Var30 Ξ A β Var30 (snoc30 Ξ B) A; vs30
= Ξ» x Var30 vz30 vs30 β vs30 _ _ _ (x Var30 vz30 vs30)
Tm30 : Con30 β Ty30 β Set; Tm30
= Ξ» Ξ A β
(Tm30 : Con30 β Ty30 β Set)
(var : β Ξ A β Var30 Ξ A β Tm30 Ξ A)
(lam : β Ξ A B β Tm30 (snoc30 Ξ A) B β Tm30 Ξ (arr30 A B))
(app : β Ξ A B β Tm30 Ξ (arr30 A B) β Tm30 Ξ A β Tm30 Ξ B)
(tt : β Ξ β Tm30 Ξ top30)
(pair : β Ξ A B β Tm30 Ξ A β Tm30 Ξ B β Tm30 Ξ (prod30 A B))
(fst : β Ξ A B β Tm30 Ξ (prod30 A B) β Tm30 Ξ A)
(snd : β Ξ A B β Tm30 Ξ (prod30 A B) β Tm30 Ξ B)
(left : β Ξ A B β Tm30 Ξ A β Tm30 Ξ (sum30 A B))
(right : β Ξ A B β Tm30 Ξ B β Tm30 Ξ (sum30 A B))
(case : β Ξ A B C β Tm30 Ξ (sum30 A B) β Tm30 Ξ (arr30 A C) β Tm30 Ξ (arr30 B C) β Tm30 Ξ C)
(zero : β Ξ β Tm30 Ξ nat30)
(suc : β Ξ β Tm30 Ξ nat30 β Tm30 Ξ nat30)
(rec : β Ξ A β Tm30 Ξ nat30 β Tm30 Ξ (arr30 nat30 (arr30 A A)) β Tm30 Ξ A β Tm30 Ξ A)
β Tm30 Ξ A
var30 : β{Ξ A} β Var30 Ξ A β Tm30 Ξ A; var30
= Ξ» x Tm30 var30 lam app tt pair fst snd left right case zero suc rec β
var30 _ _ x
lam30 : β{Ξ A B} β Tm30 (snoc30 Ξ A) B β Tm30 Ξ (arr30 A B); lam30
= Ξ» t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec β
lam30 _ _ _ (t Tm30 var30 lam30 app tt pair fst snd left right case zero suc rec)
app30 : β{Ξ A B} β Tm30 Ξ (arr30 A B) β Tm30 Ξ A β Tm30 Ξ B; app30
= Ξ» t u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec β
app30 _ _ _ (t Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt pair fst snd left right case zero suc rec)
tt30 : β{Ξ} β Tm30 Ξ top30; tt30
= Ξ» Tm30 var30 lam30 app30 tt30 pair fst snd left right case zero suc rec β tt30 _
pair30 : β{Ξ A B} β Tm30 Ξ A β Tm30 Ξ B β Tm30 Ξ (prod30 A B); pair30
= Ξ» t u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec β
pair30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst snd left right case zero suc rec)
fst30 : β{Ξ A B} β Tm30 Ξ (prod30 A B) β Tm30 Ξ A; fst30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec β
fst30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd left right case zero suc rec)
snd30 : β{Ξ A B} β Tm30 Ξ (prod30 A B) β Tm30 Ξ B; snd30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec β
snd30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left right case zero suc rec)
left30 : β{Ξ A B} β Tm30 Ξ A β Tm30 Ξ (sum30 A B); left30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec β
left30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right case zero suc rec)
right30 : β{Ξ A B} β Tm30 Ξ B β Tm30 Ξ (sum30 A B); right30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec β
right30 _ _ _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case zero suc rec)
case30 : β{Ξ A B C} β Tm30 Ξ (sum30 A B) β Tm30 Ξ (arr30 A C) β Tm30 Ξ (arr30 B C) β Tm30 Ξ C; case30
= Ξ» t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec β
case30 _ _ _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero suc rec)
zero30 : β{Ξ} β Tm30 Ξ nat30; zero30
= Ξ» Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc rec β zero30 _
suc30 : β{Ξ} β Tm30 Ξ nat30 β Tm30 Ξ nat30; suc30
= Ξ» t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec β
suc30 _ (t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec)
rec30 : β{Ξ A} β Tm30 Ξ nat30 β Tm30 Ξ (arr30 nat30 (arr30 A A)) β Tm30 Ξ A β Tm30 Ξ A; rec30
= Ξ» t u v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30 β
rec30 _ _
(t Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(u Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
(v Tm30 var30 lam30 app30 tt30 pair30 fst30 snd30 left30 right30 case30 zero30 suc30 rec30)
v030 : β{Ξ A} β Tm30 (snoc30 Ξ A) A; v030
= var30 vz30
v130 : β{Ξ A B} β Tm30 (snoc30 (snoc30 Ξ A) B) A; v130
= var30 (vs30 vz30)
v230 : β{Ξ A B C} β Tm30 (snoc30 (snoc30 (snoc30 Ξ A) B) C) A; v230
= var30 (vs30 (vs30 vz30))
v330 : β{Ξ A B C D} β Tm30 (snoc30 (snoc30 (snoc30 (snoc30 Ξ A) B) C) D) A; v330
= var30 (vs30 (vs30 (vs30 vz30)))
tbool30 : Ty30; tbool30
= sum30 top30 top30
true30 : β{Ξ} β Tm30 Ξ tbool30; true30
= left30 tt30
tfalse30 : β{Ξ} β Tm30 Ξ tbool30; tfalse30
= right30 tt30
ifthenelse30 : β{Ξ A} β Tm30 Ξ (arr30 tbool30 (arr30 A (arr30 A A))); ifthenelse30
= lam30 (lam30 (lam30 (case30 v230 (lam30 v230) (lam30 v130))))
times430 : β{Ξ A} β Tm30 Ξ (arr30 (arr30 A A) (arr30 A A)); times430
= lam30 (lam30 (app30 v130 (app30 v130 (app30 v130 (app30 v130 v030)))))
add30 : β{Ξ} β Tm30 Ξ (arr30 nat30 (arr30 nat30 nat30)); add30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (suc30 (app30 v130 v030)))))
(lam30 v030))
mul30 : β{Ξ} β Tm30 Ξ (arr30 nat30 (arr30 nat30 nat30)); mul30
= lam30 (rec30 v030
(lam30 (lam30 (lam30 (app30 (app30 add30 (app30 v130 v030)) v030))))
(lam30 zero30))
fact30 : β{Ξ} β Tm30 Ξ (arr30 nat30 nat30); fact30
= lam30 (rec30 v030 (lam30 (lam30 (app30 (app30 mul30 (suc30 v130)) v030)))
(suc30 zero30))
{-# OPTIONS --type-in-type #-}
Ty31 : Set
Ty31 =
(Ty31 : Set)
(nat top bot : Ty31)
(arr prod sum : Ty31 β Ty31 β Ty31)
β Ty31
nat31 : Ty31; nat31 = Ξ» _ nat31 _ _ _ _ _ β nat31
top31 : Ty31; top31 = Ξ» _ _ top31 _ _ _ _ β top31
bot31 : Ty31; bot31 = Ξ» _ _ _ bot31 _ _ _ β bot31
arr31 : Ty31 β Ty31 β Ty31; arr31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod sum β
arr31 (A Ty31 nat31 top31 bot31 arr31 prod sum) (B Ty31 nat31 top31 bot31 arr31 prod sum)
prod31 : Ty31 β Ty31 β Ty31; prod31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod31 sum β
prod31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum) (B Ty31 nat31 top31 bot31 arr31 prod31 sum)
sum31 : Ty31 β Ty31 β Ty31; sum31
= Ξ» A B Ty31 nat31 top31 bot31 arr31 prod31 sum31 β
sum31 (A Ty31 nat31 top31 bot31 arr31 prod31 sum31) (B Ty31 nat31 top31 bot31 arr31 prod31 sum31)
Con31 : Set; Con31
= (Con31 : Set)
(nil : Con31)
(snoc : Con31 β Ty31 β Con31)
β Con31
nil31 : Con31; nil31
= Ξ» Con31 nil31 snoc β nil31
snoc31 : Con31 β Ty31 β Con31; snoc31
= Ξ» Ξ A Con31 nil31 snoc31 β snoc31 (Ξ Con31 nil31 snoc31) A
Var31 : Con31 β Ty31 β Set; Var31
= Ξ» Ξ A β
(Var31 : Con31 β Ty31 β Set)
(vz : β Ξ A β Var31 (snoc31 Ξ A) A)
(vs : β Ξ B A β Var31 Ξ A β Var31 (snoc31 Ξ B) A)
β Var31 Ξ A
vz31 : β{Ξ A} β Var31 (snoc31 Ξ A) A; vz31
= Ξ» Var31 vz31 vs β vz31 _ _
vs31 : β{Ξ B A} β Var31 Ξ A β Var31 (snoc31 Ξ B) A; vs31
= Ξ» x Var31 vz31 vs31 β vs31 _ _ _ (x Var31 vz31 vs31)
Tm31 : Con31 β Ty31 β Set; Tm31
= Ξ» Ξ A β
(Tm31 : Con31 β Ty31 β Set)
(var : β Ξ A β Var31 Ξ A β Tm31 Ξ A)
(lam : β Ξ A B β Tm31 (snoc31 Ξ A) B β Tm31 Ξ (arr31 A B))
(app : β Ξ A B β Tm31 Ξ (arr31 A B) β Tm31 Ξ A β Tm31 Ξ B)
(tt : β Ξ β Tm31 Ξ top31)
(pair : β Ξ A B β Tm31 Ξ A β Tm31 Ξ B β Tm31 Ξ (prod31 A B))
(fst : β Ξ A B β Tm31 Ξ (prod31 A B) β Tm31 Ξ A)
(snd : β Ξ A B β Tm31 Ξ (prod31 A B) β Tm31 Ξ B)
(left : β Ξ A B β Tm31 Ξ A β Tm31 Ξ (sum31 A B))
(right : β Ξ A B β Tm31 Ξ B β Tm31 Ξ (sum31 A B))
(case : β Ξ A B C β Tm31 Ξ (sum31 A B) β Tm31 Ξ (arr31 A C) β Tm31 Ξ (arr31 B C) β Tm31 Ξ C)
(zero : β Ξ β Tm31 Ξ nat31)
(suc : β Ξ β Tm31 Ξ nat31 β Tm31 Ξ nat31)
(rec : β Ξ A β Tm31 Ξ nat31 β Tm31 Ξ (arr31 nat31 (arr31 A A)) β Tm31 Ξ A β Tm31 Ξ A)
β Tm31 Ξ A
var31 : β{Ξ A} β Var31 Ξ A β Tm31 Ξ A; var31
= Ξ» x Tm31 var31 lam app tt pair fst snd left right case zero suc rec β
var31 _ _ x
lam31 : β{Ξ A B} β Tm31 (snoc31 Ξ A) B β Tm31 Ξ (arr31 A B); lam31
= Ξ» t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec β
lam31 _ _ _ (t Tm31 var31 lam31 app tt pair fst snd left right case zero suc rec)
app31 : β{Ξ A B} β Tm31 Ξ (arr31 A B) β Tm31 Ξ A β Tm31 Ξ B; app31
= Ξ» t u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec β
app31 _ _ _ (t Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt pair fst snd left right case zero suc rec)
tt31 : β{Ξ} β Tm31 Ξ top31; tt31
= Ξ» Tm31 var31 lam31 app31 tt31 pair fst snd left right case zero suc rec β tt31 _
pair31 : β{Ξ A B} β Tm31 Ξ A β Tm31 Ξ B β Tm31 Ξ (prod31 A B); pair31
= Ξ» t u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec β
pair31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst snd left right case zero suc rec)
fst31 : β{Ξ A B} β Tm31 Ξ (prod31 A B) β Tm31 Ξ A; fst31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec β
fst31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd left right case zero suc rec)
snd31 : β{Ξ A B} β Tm31 Ξ (prod31 A B) β Tm31 Ξ B; snd31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec β
snd31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left right case zero suc rec)
left31 : β{Ξ A B} β Tm31 Ξ A β Tm31 Ξ (sum31 A B); left31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec β
left31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right case zero suc rec)
right31 : β{Ξ A B} β Tm31 Ξ B β Tm31 Ξ (sum31 A B); right31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec β
right31 _ _ _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case zero suc rec)
case31 : β{Ξ A B C} β Tm31 Ξ (sum31 A B) β Tm31 Ξ (arr31 A C) β Tm31 Ξ (arr31 B C) β Tm31 Ξ C; case31
= Ξ» t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec β
case31 _ _ _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero suc rec)
zero31 : β{Ξ} β Tm31 Ξ nat31; zero31
= Ξ» Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc rec β zero31 _
suc31 : β{Ξ} β Tm31 Ξ nat31 β Tm31 Ξ nat31; suc31
= Ξ» t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec β
suc31 _ (t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec)
rec31 : β{Ξ A} β Tm31 Ξ nat31 β Tm31 Ξ (arr31 nat31 (arr31 A A)) β Tm31 Ξ A β Tm31 Ξ A; rec31
= Ξ» t u v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31 β
rec31 _ _
(t Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(u Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
(v Tm31 var31 lam31 app31 tt31 pair31 fst31 snd31 left31 right31 case31 zero31 suc31 rec31)
v031 : β{Ξ A} β Tm31 (snoc31 Ξ A) A; v031
= var31 vz31
v131 : β{Ξ A B} β Tm31 (snoc31 (snoc31 Ξ A) B) A; v131
= var31 (vs31 vz31)
v231 : β{Ξ A B C} β Tm31 (snoc31 (snoc31 (snoc31 Ξ A) B) C) A; v231
= var31 (vs31 (vs31 vz31))
v331 : β{Ξ A B C D} β Tm31 (snoc31 (snoc31 (snoc31 (snoc31 Ξ A) B) C) D) A; v331
= var31 (vs31 (vs31 (vs31 vz31)))
tbool31 : Ty31; tbool31
= sum31 top31 top31
true31 : β{Ξ} β Tm31 Ξ tbool31; true31
= left31 tt31
tfalse31 : β{Ξ} β Tm31 Ξ tbool31; tfalse31
= right31 tt31
ifthenelse31 : β{Ξ A} β Tm31 Ξ (arr31 tbool31 (arr31 A (arr31 A A))); ifthenelse31
= lam31 (lam31 (lam31 (case31 v231 (lam31 v231) (lam31 v131))))
times431 : β{Ξ A} β Tm31 Ξ (arr31 (arr31 A A) (arr31 A A)); times431
= lam31 (lam31 (app31 v131 (app31 v131 (app31 v131 (app31 v131 v031)))))
add31 : β{Ξ} β Tm31 Ξ (arr31 nat31 (arr31 nat31 nat31)); add31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (suc31 (app31 v131 v031)))))
(lam31 v031))
mul31 : β{Ξ} β Tm31 Ξ (arr31 nat31 (arr31 nat31 nat31)); mul31
= lam31 (rec31 v031
(lam31 (lam31 (lam31 (app31 (app31 add31 (app31 v131 v031)) v031))))
(lam31 zero31))
fact31 : β{Ξ} β Tm31 Ξ (arr31 nat31 nat31); fact31
= lam31 (rec31 v031 (lam31 (lam31 (app31 (app31 mul31 (suc31 v131)) v031)))
(suc31 zero31))
{-# OPTIONS --type-in-type #-}
Ty32 : Set
Ty32 =
(Ty32 : Set)
(nat top bot : Ty32)
(arr prod sum : Ty32 β Ty32 β Ty32)
β Ty32
nat32 : Ty32; nat32 = Ξ» _ nat32 _ _ _ _ _ β nat32
top32 : Ty32; top32 = Ξ» _ _ top32 _ _ _ _ β top32
bot32 : Ty32; bot32 = Ξ» _ _ _ bot32 _ _ _ β bot32
arr32 : Ty32 β Ty32 β Ty32; arr32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod sum β
arr32 (A Ty32 nat32 top32 bot32 arr32 prod sum) (B Ty32 nat32 top32 bot32 arr32 prod sum)
prod32 : Ty32 β Ty32 β Ty32; prod32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod32 sum β
prod32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum) (B Ty32 nat32 top32 bot32 arr32 prod32 sum)
sum32 : Ty32 β Ty32 β Ty32; sum32
= Ξ» A B Ty32 nat32 top32 bot32 arr32 prod32 sum32 β
sum32 (A Ty32 nat32 top32 bot32 arr32 prod32 sum32) (B Ty32 nat32 top32 bot32 arr32 prod32 sum32)
Con32 : Set; Con32
= (Con32 : Set)
(nil : Con32)
(snoc : Con32 β Ty32 β Con32)
β Con32
nil32 : Con32; nil32
= Ξ» Con32 nil32 snoc β nil32
snoc32 : Con32 β Ty32 β Con32; snoc32
= Ξ» Ξ A Con32 nil32 snoc32 β snoc32 (Ξ Con32 nil32 snoc32) A
Var32 : Con32 β Ty32 β Set; Var32
= Ξ» Ξ A β
(Var32 : Con32 β Ty32 β Set)
(vz : β Ξ A β Var32 (snoc32 Ξ A) A)
(vs : β Ξ B A β Var32 Ξ A β Var32 (snoc32 Ξ B) A)
β Var32 Ξ A
vz32 : β{Ξ A} β Var32 (snoc32 Ξ A) A; vz32
= Ξ» Var32 vz32 vs β vz32 _ _
vs32 : β{Ξ B A} β Var32 Ξ A β Var32 (snoc32 Ξ B) A; vs32
= Ξ» x Var32 vz32 vs32 β vs32 _ _ _ (x Var32 vz32 vs32)
Tm32 : Con32 β Ty32 β Set; Tm32
= Ξ» Ξ A β
(Tm32 : Con32 β Ty32 β Set)
(var : β Ξ A β Var32 Ξ A β Tm32 Ξ A)
(lam : β Ξ A B β Tm32 (snoc32 Ξ A) B β Tm32 Ξ (arr32 A B))
(app : β Ξ A B β Tm32 Ξ (arr32 A B) β Tm32 Ξ A β Tm32 Ξ B)
(tt : β Ξ β Tm32 Ξ top32)
(pair : β Ξ A B β Tm32 Ξ A β Tm32 Ξ B β Tm32 Ξ (prod32 A B))
(fst : β Ξ A B β Tm32 Ξ (prod32 A B) β Tm32 Ξ A)
(snd : β Ξ A B β Tm32 Ξ (prod32 A B) β Tm32 Ξ B)
(left : β Ξ A B β Tm32 Ξ A β Tm32 Ξ (sum32 A B))
(right : β Ξ A B β Tm32 Ξ B β Tm32 Ξ (sum32 A B))
(case : β Ξ A B C β Tm32 Ξ (sum32 A B) β Tm32 Ξ (arr32 A C) β Tm32 Ξ (arr32 B C) β Tm32 Ξ C)
(zero : β Ξ β Tm32 Ξ nat32)
(suc : β Ξ β Tm32 Ξ nat32 β Tm32 Ξ nat32)
(rec : β Ξ A β Tm32 Ξ nat32 β Tm32 Ξ (arr32 nat32 (arr32 A A)) β Tm32 Ξ A β Tm32 Ξ A)
β Tm32 Ξ A
var32 : β{Ξ A} β Var32 Ξ A β Tm32 Ξ A; var32
= Ξ» x Tm32 var32 lam app tt pair fst snd left right case zero suc rec β
var32 _ _ x
lam32 : β{Ξ A B} β Tm32 (snoc32 Ξ A) B β Tm32 Ξ (arr32 A B); lam32
= Ξ» t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec β
lam32 _ _ _ (t Tm32 var32 lam32 app tt pair fst snd left right case zero suc rec)
app32 : β{Ξ A B} β Tm32 Ξ (arr32 A B) β Tm32 Ξ A β Tm32 Ξ B; app32
= Ξ» t u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec β
app32 _ _ _ (t Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt pair fst snd left right case zero suc rec)
tt32 : β{Ξ} β Tm32 Ξ top32; tt32
= Ξ» Tm32 var32 lam32 app32 tt32 pair fst snd left right case zero suc rec β tt32 _
pair32 : β{Ξ A B} β Tm32 Ξ A β Tm32 Ξ B β Tm32 Ξ (prod32 A B); pair32
= Ξ» t u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec β
pair32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst snd left right case zero suc rec)
fst32 : β{Ξ A B} β Tm32 Ξ (prod32 A B) β Tm32 Ξ A; fst32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec β
fst32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd left right case zero suc rec)
snd32 : β{Ξ A B} β Tm32 Ξ (prod32 A B) β Tm32 Ξ B; snd32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec β
snd32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left right case zero suc rec)
left32 : β{Ξ A B} β Tm32 Ξ A β Tm32 Ξ (sum32 A B); left32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec β
left32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right case zero suc rec)
right32 : β{Ξ A B} β Tm32 Ξ B β Tm32 Ξ (sum32 A B); right32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec β
right32 _ _ _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case zero suc rec)
case32 : β{Ξ A B C} β Tm32 Ξ (sum32 A B) β Tm32 Ξ (arr32 A C) β Tm32 Ξ (arr32 B C) β Tm32 Ξ C; case32
= Ξ» t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec β
case32 _ _ _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero suc rec)
zero32 : β{Ξ} β Tm32 Ξ nat32; zero32
= Ξ» Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc rec β zero32 _
suc32 : β{Ξ} β Tm32 Ξ nat32 β Tm32 Ξ nat32; suc32
= Ξ» t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec β
suc32 _ (t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec)
rec32 : β{Ξ A} β Tm32 Ξ nat32 β Tm32 Ξ (arr32 nat32 (arr32 A A)) β Tm32 Ξ A β Tm32 Ξ A; rec32
= Ξ» t u v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32 β
rec32 _ _
(t Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(u Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
(v Tm32 var32 lam32 app32 tt32 pair32 fst32 snd32 left32 right32 case32 zero32 suc32 rec32)
v032 : β{Ξ A} β Tm32 (snoc32 Ξ A) A; v032
= var32 vz32
v132 : β{Ξ A B} β Tm32 (snoc32 (snoc32 Ξ A) B) A; v132
= var32 (vs32 vz32)
v232 : β{Ξ A B C} β Tm32 (snoc32 (snoc32 (snoc32 Ξ A) B) C) A; v232
= var32 (vs32 (vs32 vz32))
v332 : β{Ξ A B C D} β Tm32 (snoc32 (snoc32 (snoc32 (snoc32 Ξ A) B) C) D) A; v332
= var32 (vs32 (vs32 (vs32 vz32)))
tbool32 : Ty32; tbool32
= sum32 top32 top32
true32 : β{Ξ} β Tm32 Ξ tbool32; true32
= left32 tt32
tfalse32 : β{Ξ} β Tm32 Ξ tbool32; tfalse32
= right32 tt32
ifthenelse32 : β{Ξ A} β Tm32 Ξ (arr32 tbool32 (arr32 A (arr32 A A))); ifthenelse32
= lam32 (lam32 (lam32 (case32 v232 (lam32 v232) (lam32 v132))))
times432 : β{Ξ A} β Tm32 Ξ (arr32 (arr32 A A) (arr32 A A)); times432
= lam32 (lam32 (app32 v132 (app32 v132 (app32 v132 (app32 v132 v032)))))
add32 : β{Ξ} β Tm32 Ξ (arr32 nat32 (arr32 nat32 nat32)); add32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (suc32 (app32 v132 v032)))))
(lam32 v032))
mul32 : β{Ξ} β Tm32 Ξ (arr32 nat32 (arr32 nat32 nat32)); mul32
= lam32 (rec32 v032
(lam32 (lam32 (lam32 (app32 (app32 add32 (app32 v132 v032)) v032))))
(lam32 zero32))
fact32 : β{Ξ} β Tm32 Ξ (arr32 nat32 nat32); fact32
= lam32 (rec32 v032 (lam32 (lam32 (app32 (app32 mul32 (suc32 v132)) v032)))
(suc32 zero32))
{-# OPTIONS --type-in-type #-}
Ty33 : Set
Ty33 =
(Ty33 : Set)
(nat top bot : Ty33)
(arr prod sum : Ty33 β Ty33 β Ty33)
β Ty33
nat33 : Ty33; nat33 = Ξ» _ nat33 _ _ _ _ _ β nat33
top33 : Ty33; top33 = Ξ» _ _ top33 _ _ _ _ β top33
bot33 : Ty33; bot33 = Ξ» _ _ _ bot33 _ _ _ β bot33
arr33 : Ty33 β Ty33 β Ty33; arr33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod sum β
arr33 (A Ty33 nat33 top33 bot33 arr33 prod sum) (B Ty33 nat33 top33 bot33 arr33 prod sum)
prod33 : Ty33 β Ty33 β Ty33; prod33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod33 sum β
prod33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum) (B Ty33 nat33 top33 bot33 arr33 prod33 sum)
sum33 : Ty33 β Ty33 β Ty33; sum33
= Ξ» A B Ty33 nat33 top33 bot33 arr33 prod33 sum33 β
sum33 (A Ty33 nat33 top33 bot33 arr33 prod33 sum33) (B Ty33 nat33 top33 bot33 arr33 prod33 sum33)
Con33 : Set; Con33
= (Con33 : Set)
(nil : Con33)
(snoc : Con33 β Ty33 β Con33)
β Con33
nil33 : Con33; nil33
= Ξ» Con33 nil33 snoc β nil33
snoc33 : Con33 β Ty33 β Con33; snoc33
= Ξ» Ξ A Con33 nil33 snoc33 β snoc33 (Ξ Con33 nil33 snoc33) A
Var33 : Con33 β Ty33 β Set; Var33
= Ξ» Ξ A β
(Var33 : Con33 β Ty33 β Set)
(vz : β Ξ A β Var33 (snoc33 Ξ A) A)
(vs : β Ξ B A β Var33 Ξ A β Var33 (snoc33 Ξ B) A)
β Var33 Ξ A
vz33 : β{Ξ A} β Var33 (snoc33 Ξ A) A; vz33
= Ξ» Var33 vz33 vs β vz33 _ _
vs33 : β{Ξ B A} β Var33 Ξ A β Var33 (snoc33 Ξ B) A; vs33
= Ξ» x Var33 vz33 vs33 β vs33 _ _ _ (x Var33 vz33 vs33)
Tm33 : Con33 β Ty33 β Set; Tm33
= Ξ» Ξ A β
(Tm33 : Con33 β Ty33 β Set)
(var : β Ξ A β Var33 Ξ A β Tm33 Ξ A)
(lam : β Ξ A B β Tm33 (snoc33 Ξ A) B β Tm33 Ξ (arr33 A B))
(app : β Ξ A B β Tm33 Ξ (arr33 A B) β Tm33 Ξ A β Tm33 Ξ B)
(tt : β Ξ β Tm33 Ξ top33)
(pair : β Ξ A B β Tm33 Ξ A β Tm33 Ξ B β Tm33 Ξ (prod33 A B))
(fst : β Ξ A B β Tm33 Ξ (prod33 A B) β Tm33 Ξ A)
(snd : β Ξ A B β Tm33 Ξ (prod33 A B) β Tm33 Ξ B)
(left : β Ξ A B β Tm33 Ξ A β Tm33 Ξ (sum33 A B))
(right : β Ξ A B β Tm33 Ξ B β Tm33 Ξ (sum33 A B))
(case : β Ξ A B C β Tm33 Ξ (sum33 A B) β Tm33 Ξ (arr33 A C) β Tm33 Ξ (arr33 B C) β Tm33 Ξ C)
(zero : β Ξ β Tm33 Ξ nat33)
(suc : β Ξ β Tm33 Ξ nat33 β Tm33 Ξ nat33)
(rec : β Ξ A β Tm33 Ξ nat33 β Tm33 Ξ (arr33 nat33 (arr33 A A)) β Tm33 Ξ A β Tm33 Ξ A)
β Tm33 Ξ A
var33 : β{Ξ A} β Var33 Ξ A β Tm33 Ξ A; var33
= Ξ» x Tm33 var33 lam app tt pair fst snd left right case zero suc rec β
var33 _ _ x
lam33 : β{Ξ A B} β Tm33 (snoc33 Ξ A) B β Tm33 Ξ (arr33 A B); lam33
= Ξ» t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec β
lam33 _ _ _ (t Tm33 var33 lam33 app tt pair fst snd left right case zero suc rec)
app33 : β{Ξ A B} β Tm33 Ξ (arr33 A B) β Tm33 Ξ A β Tm33 Ξ B; app33
= Ξ» t u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec β
app33 _ _ _ (t Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt pair fst snd left right case zero suc rec)
tt33 : β{Ξ} β Tm33 Ξ top33; tt33
= Ξ» Tm33 var33 lam33 app33 tt33 pair fst snd left right case zero suc rec β tt33 _
pair33 : β{Ξ A B} β Tm33 Ξ A β Tm33 Ξ B β Tm33 Ξ (prod33 A B); pair33
= Ξ» t u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec β
pair33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst snd left right case zero suc rec)
fst33 : β{Ξ A B} β Tm33 Ξ (prod33 A B) β Tm33 Ξ A; fst33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec β
fst33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd left right case zero suc rec)
snd33 : β{Ξ A B} β Tm33 Ξ (prod33 A B) β Tm33 Ξ B; snd33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec β
snd33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left right case zero suc rec)
left33 : β{Ξ A B} β Tm33 Ξ A β Tm33 Ξ (sum33 A B); left33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec β
left33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right case zero suc rec)
right33 : β{Ξ A B} β Tm33 Ξ B β Tm33 Ξ (sum33 A B); right33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec β
right33 _ _ _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case zero suc rec)
case33 : β{Ξ A B C} β Tm33 Ξ (sum33 A B) β Tm33 Ξ (arr33 A C) β Tm33 Ξ (arr33 B C) β Tm33 Ξ C; case33
= Ξ» t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec β
case33 _ _ _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero suc rec)
zero33 : β{Ξ} β Tm33 Ξ nat33; zero33
= Ξ» Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc rec β zero33 _
suc33 : β{Ξ} β Tm33 Ξ nat33 β Tm33 Ξ nat33; suc33
= Ξ» t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec β
suc33 _ (t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec)
rec33 : β{Ξ A} β Tm33 Ξ nat33 β Tm33 Ξ (arr33 nat33 (arr33 A A)) β Tm33 Ξ A β Tm33 Ξ A; rec33
= Ξ» t u v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33 β
rec33 _ _
(t Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(u Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
(v Tm33 var33 lam33 app33 tt33 pair33 fst33 snd33 left33 right33 case33 zero33 suc33 rec33)
v033 : β{Ξ A} β Tm33 (snoc33 Ξ A) A; v033
= var33 vz33
v133 : β{Ξ A B} β Tm33 (snoc33 (snoc33 Ξ A) B) A; v133
= var33 (vs33 vz33)
v233 : β{Ξ A B C} β Tm33 (snoc33 (snoc33 (snoc33 Ξ A) B) C) A; v233
= var33 (vs33 (vs33 vz33))
v333 : β{Ξ A B C D} β Tm33 (snoc33 (snoc33 (snoc33 (snoc33 Ξ A) B) C) D) A; v333
= var33 (vs33 (vs33 (vs33 vz33)))
tbool33 : Ty33; tbool33
= sum33 top33 top33
true33 : β{Ξ} β Tm33 Ξ tbool33; true33
= left33 tt33
tfalse33 : β{Ξ} β Tm33 Ξ tbool33; tfalse33
= right33 tt33
ifthenelse33 : β{Ξ A} β Tm33 Ξ (arr33 tbool33 (arr33 A (arr33 A A))); ifthenelse33
= lam33 (lam33 (lam33 (case33 v233 (lam33 v233) (lam33 v133))))
times433 : β{Ξ A} β Tm33 Ξ (arr33 (arr33 A A) (arr33 A A)); times433
= lam33 (lam33 (app33 v133 (app33 v133 (app33 v133 (app33 v133 v033)))))
add33 : β{Ξ} β Tm33 Ξ (arr33 nat33 (arr33 nat33 nat33)); add33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (suc33 (app33 v133 v033)))))
(lam33 v033))
mul33 : β{Ξ} β Tm33 Ξ (arr33 nat33 (arr33 nat33 nat33)); mul33
= lam33 (rec33 v033
(lam33 (lam33 (lam33 (app33 (app33 add33 (app33 v133 v033)) v033))))
(lam33 zero33))
fact33 : β{Ξ} β Tm33 Ξ (arr33 nat33 nat33); fact33
= lam33 (rec33 v033 (lam33 (lam33 (app33 (app33 mul33 (suc33 v133)) v033)))
(suc33 zero33))
{-# OPTIONS --type-in-type #-}
Ty34 : Set
Ty34 =
(Ty34 : Set)
(nat top bot : Ty34)
(arr prod sum : Ty34 β Ty34 β Ty34)
β Ty34
nat34 : Ty34; nat34 = Ξ» _ nat34 _ _ _ _ _ β nat34
top34 : Ty34; top34 = Ξ» _ _ top34 _ _ _ _ β top34
bot34 : Ty34; bot34 = Ξ» _ _ _ bot34 _ _ _ β bot34
arr34 : Ty34 β Ty34 β Ty34; arr34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod sum β
arr34 (A Ty34 nat34 top34 bot34 arr34 prod sum) (B Ty34 nat34 top34 bot34 arr34 prod sum)
prod34 : Ty34 β Ty34 β Ty34; prod34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod34 sum β
prod34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum) (B Ty34 nat34 top34 bot34 arr34 prod34 sum)
sum34 : Ty34 β Ty34 β Ty34; sum34
= Ξ» A B Ty34 nat34 top34 bot34 arr34 prod34 sum34 β
sum34 (A Ty34 nat34 top34 bot34 arr34 prod34 sum34) (B Ty34 nat34 top34 bot34 arr34 prod34 sum34)
Con34 : Set; Con34
= (Con34 : Set)
(nil : Con34)
(snoc : Con34 β Ty34 β Con34)
β Con34
nil34 : Con34; nil34
= Ξ» Con34 nil34 snoc β nil34
snoc34 : Con34 β Ty34 β Con34; snoc34
= Ξ» Ξ A Con34 nil34 snoc34 β snoc34 (Ξ Con34 nil34 snoc34) A
Var34 : Con34 β Ty34 β Set; Var34
= Ξ» Ξ A β
(Var34 : Con34 β Ty34 β Set)
(vz : β Ξ A β Var34 (snoc34 Ξ A) A)
(vs : β Ξ B A β Var34 Ξ A β Var34 (snoc34 Ξ B) A)
β Var34 Ξ A
vz34 : β{Ξ A} β Var34 (snoc34 Ξ A) A; vz34
= Ξ» Var34 vz34 vs β vz34 _ _
vs34 : β{Ξ B A} β Var34 Ξ A β Var34 (snoc34 Ξ B) A; vs34
= Ξ» x Var34 vz34 vs34 β vs34 _ _ _ (x Var34 vz34 vs34)
Tm34 : Con34 β Ty34 β Set; Tm34
= Ξ» Ξ A β
(Tm34 : Con34 β Ty34 β Set)
(var : β Ξ A β Var34 Ξ A β Tm34 Ξ A)
(lam : β Ξ A B β Tm34 (snoc34 Ξ A) B β Tm34 Ξ (arr34 A B))
(app : β Ξ A B β Tm34 Ξ (arr34 A B) β Tm34 Ξ A β Tm34 Ξ B)
(tt : β Ξ β Tm34 Ξ top34)
(pair : β Ξ A B β Tm34 Ξ A β Tm34 Ξ B β Tm34 Ξ (prod34 A B))
(fst : β Ξ A B β Tm34 Ξ (prod34 A B) β Tm34 Ξ A)
(snd : β Ξ A B β Tm34 Ξ (prod34 A B) β Tm34 Ξ B)
(left : β Ξ A B β Tm34 Ξ A β Tm34 Ξ (sum34 A B))
(right : β Ξ A B β Tm34 Ξ B β Tm34 Ξ (sum34 A B))
(case : β Ξ A B C β Tm34 Ξ (sum34 A B) β Tm34 Ξ (arr34 A C) β Tm34 Ξ (arr34 B C) β Tm34 Ξ C)
(zero : β Ξ β Tm34 Ξ nat34)
(suc : β Ξ β Tm34 Ξ nat34 β Tm34 Ξ nat34)
(rec : β Ξ A β Tm34 Ξ nat34 β Tm34 Ξ (arr34 nat34 (arr34 A A)) β Tm34 Ξ A β Tm34 Ξ A)
β Tm34 Ξ A
var34 : β{Ξ A} β Var34 Ξ A β Tm34 Ξ A; var34
= Ξ» x Tm34 var34 lam app tt pair fst snd left right case zero suc rec β
var34 _ _ x
lam34 : β{Ξ A B} β Tm34 (snoc34 Ξ A) B β Tm34 Ξ (arr34 A B); lam34
= Ξ» t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec β
lam34 _ _ _ (t Tm34 var34 lam34 app tt pair fst snd left right case zero suc rec)
app34 : β{Ξ A B} β Tm34 Ξ (arr34 A B) β Tm34 Ξ A β Tm34 Ξ B; app34
= Ξ» t u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec β
app34 _ _ _ (t Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt pair fst snd left right case zero suc rec)
tt34 : β{Ξ} β Tm34 Ξ top34; tt34
= Ξ» Tm34 var34 lam34 app34 tt34 pair fst snd left right case zero suc rec β tt34 _
pair34 : β{Ξ A B} β Tm34 Ξ A β Tm34 Ξ B β Tm34 Ξ (prod34 A B); pair34
= Ξ» t u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec β
pair34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst snd left right case zero suc rec)
fst34 : β{Ξ A B} β Tm34 Ξ (prod34 A B) β Tm34 Ξ A; fst34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec β
fst34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd left right case zero suc rec)
snd34 : β{Ξ A B} β Tm34 Ξ (prod34 A B) β Tm34 Ξ B; snd34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec β
snd34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left right case zero suc rec)
left34 : β{Ξ A B} β Tm34 Ξ A β Tm34 Ξ (sum34 A B); left34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec β
left34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right case zero suc rec)
right34 : β{Ξ A B} β Tm34 Ξ B β Tm34 Ξ (sum34 A B); right34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec β
right34 _ _ _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case zero suc rec)
case34 : β{Ξ A B C} β Tm34 Ξ (sum34 A B) β Tm34 Ξ (arr34 A C) β Tm34 Ξ (arr34 B C) β Tm34 Ξ C; case34
= Ξ» t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec β
case34 _ _ _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero suc rec)
zero34 : β{Ξ} β Tm34 Ξ nat34; zero34
= Ξ» Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc rec β zero34 _
suc34 : β{Ξ} β Tm34 Ξ nat34 β Tm34 Ξ nat34; suc34
= Ξ» t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec β
suc34 _ (t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec)
rec34 : β{Ξ A} β Tm34 Ξ nat34 β Tm34 Ξ (arr34 nat34 (arr34 A A)) β Tm34 Ξ A β Tm34 Ξ A; rec34
= Ξ» t u v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34 β
rec34 _ _
(t Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(u Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
(v Tm34 var34 lam34 app34 tt34 pair34 fst34 snd34 left34 right34 case34 zero34 suc34 rec34)
v034 : β{Ξ A} β Tm34 (snoc34 Ξ A) A; v034
= var34 vz34
v134 : β{Ξ A B} β Tm34 (snoc34 (snoc34 Ξ A) B) A; v134
= var34 (vs34 vz34)
v234 : β{Ξ A B C} β Tm34 (snoc34 (snoc34 (snoc34 Ξ A) B) C) A; v234
= var34 (vs34 (vs34 vz34))
v334 : β{Ξ A B C D} β Tm34 (snoc34 (snoc34 (snoc34 (snoc34 Ξ A) B) C) D) A; v334
= var34 (vs34 (vs34 (vs34 vz34)))
tbool34 : Ty34; tbool34
= sum34 top34 top34
true34 : β{Ξ} β Tm34 Ξ tbool34; true34
= left34 tt34
tfalse34 : β{Ξ} β Tm34 Ξ tbool34; tfalse34
= right34 tt34
ifthenelse34 : β{Ξ A} β Tm34 Ξ (arr34 tbool34 (arr34 A (arr34 A A))); ifthenelse34
= lam34 (lam34 (lam34 (case34 v234 (lam34 v234) (lam34 v134))))
times434 : β{Ξ A} β Tm34 Ξ (arr34 (arr34 A A) (arr34 A A)); times434
= lam34 (lam34 (app34 v134 (app34 v134 (app34 v134 (app34 v134 v034)))))
add34 : β{Ξ} β Tm34 Ξ (arr34 nat34 (arr34 nat34 nat34)); add34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (suc34 (app34 v134 v034)))))
(lam34 v034))
mul34 : β{Ξ} β Tm34 Ξ (arr34 nat34 (arr34 nat34 nat34)); mul34
= lam34 (rec34 v034
(lam34 (lam34 (lam34 (app34 (app34 add34 (app34 v134 v034)) v034))))
(lam34 zero34))
fact34 : β{Ξ} β Tm34 Ξ (arr34 nat34 nat34); fact34
= lam34 (rec34 v034 (lam34 (lam34 (app34 (app34 mul34 (suc34 v134)) v034)))
(suc34 zero34))
{-# OPTIONS --type-in-type #-}
Ty35 : Set
Ty35 =
(Ty35 : Set)
(nat top bot : Ty35)
(arr prod sum : Ty35 β Ty35 β Ty35)
β Ty35
nat35 : Ty35; nat35 = Ξ» _ nat35 _ _ _ _ _ β nat35
top35 : Ty35; top35 = Ξ» _ _ top35 _ _ _ _ β top35
bot35 : Ty35; bot35 = Ξ» _ _ _ bot35 _ _ _ β bot35
arr35 : Ty35 β Ty35 β Ty35; arr35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod sum β
arr35 (A Ty35 nat35 top35 bot35 arr35 prod sum) (B Ty35 nat35 top35 bot35 arr35 prod sum)
prod35 : Ty35 β Ty35 β Ty35; prod35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod35 sum β
prod35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum) (B Ty35 nat35 top35 bot35 arr35 prod35 sum)
sum35 : Ty35 β Ty35 β Ty35; sum35
= Ξ» A B Ty35 nat35 top35 bot35 arr35 prod35 sum35 β
sum35 (A Ty35 nat35 top35 bot35 arr35 prod35 sum35) (B Ty35 nat35 top35 bot35 arr35 prod35 sum35)
Con35 : Set; Con35
= (Con35 : Set)
(nil : Con35)
(snoc : Con35 β Ty35 β Con35)
β Con35
nil35 : Con35; nil35
= Ξ» Con35 nil35 snoc β nil35
snoc35 : Con35 β Ty35 β Con35; snoc35
= Ξ» Ξ A Con35 nil35 snoc35 β snoc35 (Ξ Con35 nil35 snoc35) A
Var35 : Con35 β Ty35 β Set; Var35
= Ξ» Ξ A β
(Var35 : Con35 β Ty35 β Set)
(vz : β Ξ A β Var35 (snoc35 Ξ A) A)
(vs : β Ξ B A β Var35 Ξ A β Var35 (snoc35 Ξ B) A)
β Var35 Ξ A
vz35 : β{Ξ A} β Var35 (snoc35 Ξ A) A; vz35
= Ξ» Var35 vz35 vs β vz35 _ _
vs35 : β{Ξ B A} β Var35 Ξ A β Var35 (snoc35 Ξ B) A; vs35
= Ξ» x Var35 vz35 vs35 β vs35 _ _ _ (x Var35 vz35 vs35)
Tm35 : Con35 β Ty35 β Set; Tm35
= Ξ» Ξ A β
(Tm35 : Con35 β Ty35 β Set)
(var : β Ξ A β Var35 Ξ A β Tm35 Ξ A)
(lam : β Ξ A B β Tm35 (snoc35 Ξ A) B β Tm35 Ξ (arr35 A B))
(app : β Ξ A B β Tm35 Ξ (arr35 A B) β Tm35 Ξ A β Tm35 Ξ B)
(tt : β Ξ β Tm35 Ξ top35)
(pair : β Ξ A B β Tm35 Ξ A β Tm35 Ξ B β Tm35 Ξ (prod35 A B))
(fst : β Ξ A B β Tm35 Ξ (prod35 A B) β Tm35 Ξ A)
(snd : β Ξ A B β Tm35 Ξ (prod35 A B) β Tm35 Ξ B)
(left : β Ξ A B β Tm35 Ξ A β Tm35 Ξ (sum35 A B))
(right : β Ξ A B β Tm35 Ξ B β Tm35 Ξ (sum35 A B))
(case : β Ξ A B C β Tm35 Ξ (sum35 A B) β Tm35 Ξ (arr35 A C) β Tm35 Ξ (arr35 B C) β Tm35 Ξ C)
(zero : β Ξ β Tm35 Ξ nat35)
(suc : β Ξ β Tm35 Ξ nat35 β Tm35 Ξ nat35)
(rec : β Ξ A β Tm35 Ξ nat35 β Tm35 Ξ (arr35 nat35 (arr35 A A)) β Tm35 Ξ A β Tm35 Ξ A)
β Tm35 Ξ A
var35 : β{Ξ A} β Var35 Ξ A β Tm35 Ξ A; var35
= Ξ» x Tm35 var35 lam app tt pair fst snd left right case zero suc rec β
var35 _ _ x
lam35 : β{Ξ A B} β Tm35 (snoc35 Ξ A) B β Tm35 Ξ (arr35 A B); lam35
= Ξ» t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec β
lam35 _ _ _ (t Tm35 var35 lam35 app tt pair fst snd left right case zero suc rec)
app35 : β{Ξ A B} β Tm35 Ξ (arr35 A B) β Tm35 Ξ A β Tm35 Ξ B; app35
= Ξ» t u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec β
app35 _ _ _ (t Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt pair fst snd left right case zero suc rec)
tt35 : β{Ξ} β Tm35 Ξ top35; tt35
= Ξ» Tm35 var35 lam35 app35 tt35 pair fst snd left right case zero suc rec β tt35 _
pair35 : β{Ξ A B} β Tm35 Ξ A β Tm35 Ξ B β Tm35 Ξ (prod35 A B); pair35
= Ξ» t u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec β
pair35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst snd left right case zero suc rec)
fst35 : β{Ξ A B} β Tm35 Ξ (prod35 A B) β Tm35 Ξ A; fst35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec β
fst35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd left right case zero suc rec)
snd35 : β{Ξ A B} β Tm35 Ξ (prod35 A B) β Tm35 Ξ B; snd35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec β
snd35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left right case zero suc rec)
left35 : β{Ξ A B} β Tm35 Ξ A β Tm35 Ξ (sum35 A B); left35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec β
left35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right case zero suc rec)
right35 : β{Ξ A B} β Tm35 Ξ B β Tm35 Ξ (sum35 A B); right35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec β
right35 _ _ _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case zero suc rec)
case35 : β{Ξ A B C} β Tm35 Ξ (sum35 A B) β Tm35 Ξ (arr35 A C) β Tm35 Ξ (arr35 B C) β Tm35 Ξ C; case35
= Ξ» t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec β
case35 _ _ _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero suc rec)
zero35 : β{Ξ} β Tm35 Ξ nat35; zero35
= Ξ» Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc rec β zero35 _
suc35 : β{Ξ} β Tm35 Ξ nat35 β Tm35 Ξ nat35; suc35
= Ξ» t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec β
suc35 _ (t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec)
rec35 : β{Ξ A} β Tm35 Ξ nat35 β Tm35 Ξ (arr35 nat35 (arr35 A A)) β Tm35 Ξ A β Tm35 Ξ A; rec35
= Ξ» t u v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35 β
rec35 _ _
(t Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(u Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
(v Tm35 var35 lam35 app35 tt35 pair35 fst35 snd35 left35 right35 case35 zero35 suc35 rec35)
v035 : β{Ξ A} β Tm35 (snoc35 Ξ A) A; v035
= var35 vz35
v135 : β{Ξ A B} β Tm35 (snoc35 (snoc35 Ξ A) B) A; v135
= var35 (vs35 vz35)
v235 : β{Ξ A B C} β Tm35 (snoc35 (snoc35 (snoc35 Ξ A) B) C) A; v235
= var35 (vs35 (vs35 vz35))
v335 : β{Ξ A B C D} β Tm35 (snoc35 (snoc35 (snoc35 (snoc35 Ξ A) B) C) D) A; v335
= var35 (vs35 (vs35 (vs35 vz35)))
tbool35 : Ty35; tbool35
= sum35 top35 top35
true35 : β{Ξ} β Tm35 Ξ tbool35; true35
= left35 tt35
tfalse35 : β{Ξ} β Tm35 Ξ tbool35; tfalse35
= right35 tt35
ifthenelse35 : β{Ξ A} β Tm35 Ξ (arr35 tbool35 (arr35 A (arr35 A A))); ifthenelse35
= lam35 (lam35 (lam35 (case35 v235 (lam35 v235) (lam35 v135))))
times435 : β{Ξ A} β Tm35 Ξ (arr35 (arr35 A A) (arr35 A A)); times435
= lam35 (lam35 (app35 v135 (app35 v135 (app35 v135 (app35 v135 v035)))))
add35 : β{Ξ} β Tm35 Ξ (arr35 nat35 (arr35 nat35 nat35)); add35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (suc35 (app35 v135 v035)))))
(lam35 v035))
mul35 : β{Ξ} β Tm35 Ξ (arr35 nat35 (arr35 nat35 nat35)); mul35
= lam35 (rec35 v035
(lam35 (lam35 (lam35 (app35 (app35 add35 (app35 v135 v035)) v035))))
(lam35 zero35))
fact35 : β{Ξ} β Tm35 Ξ (arr35 nat35 nat35); fact35
= lam35 (rec35 v035 (lam35 (lam35 (app35 (app35 mul35 (suc35 v135)) v035)))
(suc35 zero35))
{-# OPTIONS --type-in-type #-}
Ty36 : Set
Ty36 =
(Ty36 : Set)
(nat top bot : Ty36)
(arr prod sum : Ty36 β Ty36 β Ty36)
β Ty36
nat36 : Ty36; nat36 = Ξ» _ nat36 _ _ _ _ _ β nat36
top36 : Ty36; top36 = Ξ» _ _ top36 _ _ _ _ β top36
bot36 : Ty36; bot36 = Ξ» _ _ _ bot36 _ _ _ β bot36
arr36 : Ty36 β Ty36 β Ty36; arr36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod sum β
arr36 (A Ty36 nat36 top36 bot36 arr36 prod sum) (B Ty36 nat36 top36 bot36 arr36 prod sum)
prod36 : Ty36 β Ty36 β Ty36; prod36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod36 sum β
prod36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum) (B Ty36 nat36 top36 bot36 arr36 prod36 sum)
sum36 : Ty36 β Ty36 β Ty36; sum36
= Ξ» A B Ty36 nat36 top36 bot36 arr36 prod36 sum36 β
sum36 (A Ty36 nat36 top36 bot36 arr36 prod36 sum36) (B Ty36 nat36 top36 bot36 arr36 prod36 sum36)
Con36 : Set; Con36
= (Con36 : Set)
(nil : Con36)
(snoc : Con36 β Ty36 β Con36)
β Con36
nil36 : Con36; nil36
= Ξ» Con36 nil36 snoc β nil36
snoc36 : Con36 β Ty36 β Con36; snoc36
= Ξ» Ξ A Con36 nil36 snoc36 β snoc36 (Ξ Con36 nil36 snoc36) A
Var36 : Con36 β Ty36 β Set; Var36
= Ξ» Ξ A β
(Var36 : Con36 β Ty36 β Set)
(vz : β Ξ A β Var36 (snoc36 Ξ A) A)
(vs : β Ξ B A β Var36 Ξ A β Var36 (snoc36 Ξ B) A)
β Var36 Ξ A
vz36 : β{Ξ A} β Var36 (snoc36 Ξ A) A; vz36
= Ξ» Var36 vz36 vs β vz36 _ _
vs36 : β{Ξ B A} β Var36 Ξ A β Var36 (snoc36 Ξ B) A; vs36
= Ξ» x Var36 vz36 vs36 β vs36 _ _ _ (x Var36 vz36 vs36)
Tm36 : Con36 β Ty36 β Set; Tm36
= Ξ» Ξ A β
(Tm36 : Con36 β Ty36 β Set)
(var : β Ξ A β Var36 Ξ A β Tm36 Ξ A)
(lam : β Ξ A B β Tm36 (snoc36 Ξ A) B β Tm36 Ξ (arr36 A B))
(app : β Ξ A B β Tm36 Ξ (arr36 A B) β Tm36 Ξ A β Tm36 Ξ B)
(tt : β Ξ β Tm36 Ξ top36)
(pair : β Ξ A B β Tm36 Ξ A β Tm36 Ξ B β Tm36 Ξ (prod36 A B))
(fst : β Ξ A B β Tm36 Ξ (prod36 A B) β Tm36 Ξ A)
(snd : β Ξ A B β Tm36 Ξ (prod36 A B) β Tm36 Ξ B)
(left : β Ξ A B β Tm36 Ξ A β Tm36 Ξ (sum36 A B))
(right : β Ξ A B β Tm36 Ξ B β Tm36 Ξ (sum36 A B))
(case : β Ξ A B C β Tm36 Ξ (sum36 A B) β Tm36 Ξ (arr36 A C) β Tm36 Ξ (arr36 B C) β Tm36 Ξ C)
(zero : β Ξ β Tm36 Ξ nat36)
(suc : β Ξ β Tm36 Ξ nat36 β Tm36 Ξ nat36)
(rec : β Ξ A β Tm36 Ξ nat36 β Tm36 Ξ (arr36 nat36 (arr36 A A)) β Tm36 Ξ A β Tm36 Ξ A)
β Tm36 Ξ A
var36 : β{Ξ A} β Var36 Ξ A β Tm36 Ξ A; var36
= Ξ» x Tm36 var36 lam app tt pair fst snd left right case zero suc rec β
var36 _ _ x
lam36 : β{Ξ A B} β Tm36 (snoc36 Ξ A) B β Tm36 Ξ (arr36 A B); lam36
= Ξ» t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec β
lam36 _ _ _ (t Tm36 var36 lam36 app tt pair fst snd left right case zero suc rec)
app36 : β{Ξ A B} β Tm36 Ξ (arr36 A B) β Tm36 Ξ A β Tm36 Ξ B; app36
= Ξ» t u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec β
app36 _ _ _ (t Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt pair fst snd left right case zero suc rec)
tt36 : β{Ξ} β Tm36 Ξ top36; tt36
= Ξ» Tm36 var36 lam36 app36 tt36 pair fst snd left right case zero suc rec β tt36 _
pair36 : β{Ξ A B} β Tm36 Ξ A β Tm36 Ξ B β Tm36 Ξ (prod36 A B); pair36
= Ξ» t u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec β
pair36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst snd left right case zero suc rec)
fst36 : β{Ξ A B} β Tm36 Ξ (prod36 A B) β Tm36 Ξ A; fst36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec β
fst36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd left right case zero suc rec)
snd36 : β{Ξ A B} β Tm36 Ξ (prod36 A B) β Tm36 Ξ B; snd36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec β
snd36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left right case zero suc rec)
left36 : β{Ξ A B} β Tm36 Ξ A β Tm36 Ξ (sum36 A B); left36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec β
left36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right case zero suc rec)
right36 : β{Ξ A B} β Tm36 Ξ B β Tm36 Ξ (sum36 A B); right36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec β
right36 _ _ _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case zero suc rec)
case36 : β{Ξ A B C} β Tm36 Ξ (sum36 A B) β Tm36 Ξ (arr36 A C) β Tm36 Ξ (arr36 B C) β Tm36 Ξ C; case36
= Ξ» t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec β
case36 _ _ _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero suc rec)
zero36 : β{Ξ} β Tm36 Ξ nat36; zero36
= Ξ» Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc rec β zero36 _
suc36 : β{Ξ} β Tm36 Ξ nat36 β Tm36 Ξ nat36; suc36
= Ξ» t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec β
suc36 _ (t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec)
rec36 : β{Ξ A} β Tm36 Ξ nat36 β Tm36 Ξ (arr36 nat36 (arr36 A A)) β Tm36 Ξ A β Tm36 Ξ A; rec36
= Ξ» t u v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36 β
rec36 _ _
(t Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(u Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
(v Tm36 var36 lam36 app36 tt36 pair36 fst36 snd36 left36 right36 case36 zero36 suc36 rec36)
v036 : β{Ξ A} β Tm36 (snoc36 Ξ A) A; v036
= var36 vz36
v136 : β{Ξ A B} β Tm36 (snoc36 (snoc36 Ξ A) B) A; v136
= var36 (vs36 vz36)
v236 : β{Ξ A B C} β Tm36 (snoc36 (snoc36 (snoc36 Ξ A) B) C) A; v236
= var36 (vs36 (vs36 vz36))
v336 : β{Ξ A B C D} β Tm36 (snoc36 (snoc36 (snoc36 (snoc36 Ξ A) B) C) D) A; v336
= var36 (vs36 (vs36 (vs36 vz36)))
tbool36 : Ty36; tbool36
= sum36 top36 top36
true36 : β{Ξ} β Tm36 Ξ tbool36; true36
= left36 tt36
tfalse36 : β{Ξ} β Tm36 Ξ tbool36; tfalse36
= right36 tt36
ifthenelse36 : β{Ξ A} β Tm36 Ξ (arr36 tbool36 (arr36 A (arr36 A A))); ifthenelse36
= lam36 (lam36 (lam36 (case36 v236 (lam36 v236) (lam36 v136))))
times436 : β{Ξ A} β Tm36 Ξ (arr36 (arr36 A A) (arr36 A A)); times436
= lam36 (lam36 (app36 v136 (app36 v136 (app36 v136 (app36 v136 v036)))))
add36 : β{Ξ} β Tm36 Ξ (arr36 nat36 (arr36 nat36 nat36)); add36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (suc36 (app36 v136 v036)))))
(lam36 v036))
mul36 : β{Ξ} β Tm36 Ξ (arr36 nat36 (arr36 nat36 nat36)); mul36
= lam36 (rec36 v036
(lam36 (lam36 (lam36 (app36 (app36 add36 (app36 v136 v036)) v036))))
(lam36 zero36))
fact36 : β{Ξ} β Tm36 Ξ (arr36 nat36 nat36); fact36
= lam36 (rec36 v036 (lam36 (lam36 (app36 (app36 mul36 (suc36 v136)) v036)))
(suc36 zero36))
{-# OPTIONS --type-in-type #-}
Ty37 : Set
Ty37 =
(Ty37 : Set)
(nat top bot : Ty37)
(arr prod sum : Ty37 β Ty37 β Ty37)
β Ty37
nat37 : Ty37; nat37 = Ξ» _ nat37 _ _ _ _ _ β nat37
top37 : Ty37; top37 = Ξ» _ _ top37 _ _ _ _ β top37
bot37 : Ty37; bot37 = Ξ» _ _ _ bot37 _ _ _ β bot37
arr37 : Ty37 β Ty37 β Ty37; arr37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod sum β
arr37 (A Ty37 nat37 top37 bot37 arr37 prod sum) (B Ty37 nat37 top37 bot37 arr37 prod sum)
prod37 : Ty37 β Ty37 β Ty37; prod37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod37 sum β
prod37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum) (B Ty37 nat37 top37 bot37 arr37 prod37 sum)
sum37 : Ty37 β Ty37 β Ty37; sum37
= Ξ» A B Ty37 nat37 top37 bot37 arr37 prod37 sum37 β
sum37 (A Ty37 nat37 top37 bot37 arr37 prod37 sum37) (B Ty37 nat37 top37 bot37 arr37 prod37 sum37)
Con37 : Set; Con37
= (Con37 : Set)
(nil : Con37)
(snoc : Con37 β Ty37 β Con37)
β Con37
nil37 : Con37; nil37
= Ξ» Con37 nil37 snoc β nil37
snoc37 : Con37 β Ty37 β Con37; snoc37
= Ξ» Ξ A Con37 nil37 snoc37 β snoc37 (Ξ Con37 nil37 snoc37) A
Var37 : Con37 β Ty37 β Set; Var37
= Ξ» Ξ A β
(Var37 : Con37 β Ty37 β Set)
(vz : β Ξ A β Var37 (snoc37 Ξ A) A)
(vs : β Ξ B A β Var37 Ξ A β Var37 (snoc37 Ξ B) A)
β Var37 Ξ A
vz37 : β{Ξ A} β Var37 (snoc37 Ξ A) A; vz37
= Ξ» Var37 vz37 vs β vz37 _ _
vs37 : β{Ξ B A} β Var37 Ξ A β Var37 (snoc37 Ξ B) A; vs37
= Ξ» x Var37 vz37 vs37 β vs37 _ _ _ (x Var37 vz37 vs37)
Tm37 : Con37 β Ty37 β Set; Tm37
= Ξ» Ξ A β
(Tm37 : Con37 β Ty37 β Set)
(var : β Ξ A β Var37 Ξ A β Tm37 Ξ A)
(lam : β Ξ A B β Tm37 (snoc37 Ξ A) B β Tm37 Ξ (arr37 A B))
(app : β Ξ A B β Tm37 Ξ (arr37 A B) β Tm37 Ξ A β Tm37 Ξ B)
(tt : β Ξ β Tm37 Ξ top37)
(pair : β Ξ A B β Tm37 Ξ A β Tm37 Ξ B β Tm37 Ξ (prod37 A B))
(fst : β Ξ A B β Tm37 Ξ (prod37 A B) β Tm37 Ξ A)
(snd : β Ξ A B β Tm37 Ξ (prod37 A B) β Tm37 Ξ B)
(left : β Ξ A B β Tm37 Ξ A β Tm37 Ξ (sum37 A B))
(right : β Ξ A B β Tm37 Ξ B β Tm37 Ξ (sum37 A B))
(case : β Ξ A B C β Tm37 Ξ (sum37 A B) β Tm37 Ξ (arr37 A C) β Tm37 Ξ (arr37 B C) β Tm37 Ξ C)
(zero : β Ξ β Tm37 Ξ nat37)
(suc : β Ξ β Tm37 Ξ nat37 β Tm37 Ξ nat37)
(rec : β Ξ A β Tm37 Ξ nat37 β Tm37 Ξ (arr37 nat37 (arr37 A A)) β Tm37 Ξ A β Tm37 Ξ A)
β Tm37 Ξ A
var37 : β{Ξ A} β Var37 Ξ A β Tm37 Ξ A; var37
= Ξ» x Tm37 var37 lam app tt pair fst snd left right case zero suc rec β
var37 _ _ x
lam37 : β{Ξ A B} β Tm37 (snoc37 Ξ A) B β Tm37 Ξ (arr37 A B); lam37
= Ξ» t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec β
lam37 _ _ _ (t Tm37 var37 lam37 app tt pair fst snd left right case zero suc rec)
app37 : β{Ξ A B} β Tm37 Ξ (arr37 A B) β Tm37 Ξ A β Tm37 Ξ B; app37
= Ξ» t u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec β
app37 _ _ _ (t Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt pair fst snd left right case zero suc rec)
tt37 : β{Ξ} β Tm37 Ξ top37; tt37
= Ξ» Tm37 var37 lam37 app37 tt37 pair fst snd left right case zero suc rec β tt37 _
pair37 : β{Ξ A B} β Tm37 Ξ A β Tm37 Ξ B β Tm37 Ξ (prod37 A B); pair37
= Ξ» t u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec β
pair37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst snd left right case zero suc rec)
fst37 : β{Ξ A B} β Tm37 Ξ (prod37 A B) β Tm37 Ξ A; fst37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec β
fst37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd left right case zero suc rec)
snd37 : β{Ξ A B} β Tm37 Ξ (prod37 A B) β Tm37 Ξ B; snd37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec β
snd37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left right case zero suc rec)
left37 : β{Ξ A B} β Tm37 Ξ A β Tm37 Ξ (sum37 A B); left37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec β
left37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right case zero suc rec)
right37 : β{Ξ A B} β Tm37 Ξ B β Tm37 Ξ (sum37 A B); right37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec β
right37 _ _ _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case zero suc rec)
case37 : β{Ξ A B C} β Tm37 Ξ (sum37 A B) β Tm37 Ξ (arr37 A C) β Tm37 Ξ (arr37 B C) β Tm37 Ξ C; case37
= Ξ» t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec β
case37 _ _ _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero suc rec)
zero37 : β{Ξ} β Tm37 Ξ nat37; zero37
= Ξ» Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc rec β zero37 _
suc37 : β{Ξ} β Tm37 Ξ nat37 β Tm37 Ξ nat37; suc37
= Ξ» t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec β
suc37 _ (t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec)
rec37 : β{Ξ A} β Tm37 Ξ nat37 β Tm37 Ξ (arr37 nat37 (arr37 A A)) β Tm37 Ξ A β Tm37 Ξ A; rec37
= Ξ» t u v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37 β
rec37 _ _
(t Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(u Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
(v Tm37 var37 lam37 app37 tt37 pair37 fst37 snd37 left37 right37 case37 zero37 suc37 rec37)
v037 : β{Ξ A} β Tm37 (snoc37 Ξ A) A; v037
= var37 vz37
v137 : β{Ξ A B} β Tm37 (snoc37 (snoc37 Ξ A) B) A; v137
= var37 (vs37 vz37)
v237 : β{Ξ A B C} β Tm37 (snoc37 (snoc37 (snoc37 Ξ A) B) C) A; v237
= var37 (vs37 (vs37 vz37))
v337 : β{Ξ A B C D} β Tm37 (snoc37 (snoc37 (snoc37 (snoc37 Ξ A) B) C) D) A; v337
= var37 (vs37 (vs37 (vs37 vz37)))
tbool37 : Ty37; tbool37
= sum37 top37 top37
true37 : β{Ξ} β Tm37 Ξ tbool37; true37
= left37 tt37
tfalse37 : β{Ξ} β Tm37 Ξ tbool37; tfalse37
= right37 tt37
ifthenelse37 : β{Ξ A} β Tm37 Ξ (arr37 tbool37 (arr37 A (arr37 A A))); ifthenelse37
= lam37 (lam37 (lam37 (case37 v237 (lam37 v237) (lam37 v137))))
times437 : β{Ξ A} β Tm37 Ξ (arr37 (arr37 A A) (arr37 A A)); times437
= lam37 (lam37 (app37 v137 (app37 v137 (app37 v137 (app37 v137 v037)))))
add37 : β{Ξ} β Tm37 Ξ (arr37 nat37 (arr37 nat37 nat37)); add37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (suc37 (app37 v137 v037)))))
(lam37 v037))
mul37 : β{Ξ} β Tm37 Ξ (arr37 nat37 (arr37 nat37 nat37)); mul37
= lam37 (rec37 v037
(lam37 (lam37 (lam37 (app37 (app37 add37 (app37 v137 v037)) v037))))
(lam37 zero37))
fact37 : β{Ξ} β Tm37 Ξ (arr37 nat37 nat37); fact37
= lam37 (rec37 v037 (lam37 (lam37 (app37 (app37 mul37 (suc37 v137)) v037)))
(suc37 zero37))
{-# OPTIONS --type-in-type #-}
Ty38 : Set
Ty38 =
(Ty38 : Set)
(nat top bot : Ty38)
(arr prod sum : Ty38 β Ty38 β Ty38)
β Ty38
nat38 : Ty38; nat38 = Ξ» _ nat38 _ _ _ _ _ β nat38
top38 : Ty38; top38 = Ξ» _ _ top38 _ _ _ _ β top38
bot38 : Ty38; bot38 = Ξ» _ _ _ bot38 _ _ _ β bot38
arr38 : Ty38 β Ty38 β Ty38; arr38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod sum β
arr38 (A Ty38 nat38 top38 bot38 arr38 prod sum) (B Ty38 nat38 top38 bot38 arr38 prod sum)
prod38 : Ty38 β Ty38 β Ty38; prod38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod38 sum β
prod38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum) (B Ty38 nat38 top38 bot38 arr38 prod38 sum)
sum38 : Ty38 β Ty38 β Ty38; sum38
= Ξ» A B Ty38 nat38 top38 bot38 arr38 prod38 sum38 β
sum38 (A Ty38 nat38 top38 bot38 arr38 prod38 sum38) (B Ty38 nat38 top38 bot38 arr38 prod38 sum38)
Con38 : Set; Con38
= (Con38 : Set)
(nil : Con38)
(snoc : Con38 β Ty38 β Con38)
β Con38
nil38 : Con38; nil38
= Ξ» Con38 nil38 snoc β nil38
snoc38 : Con38 β Ty38 β Con38; snoc38
= Ξ» Ξ A Con38 nil38 snoc38 β snoc38 (Ξ Con38 nil38 snoc38) A
Var38 : Con38 β Ty38 β Set; Var38
= Ξ» Ξ A β
(Var38 : Con38 β Ty38 β Set)
(vz : β Ξ A β Var38 (snoc38 Ξ A) A)
(vs : β Ξ B A β Var38 Ξ A β Var38 (snoc38 Ξ B) A)
β Var38 Ξ A
vz38 : β{Ξ A} β Var38 (snoc38 Ξ A) A; vz38
= Ξ» Var38 vz38 vs β vz38 _ _
vs38 : β{Ξ B A} β Var38 Ξ A β Var38 (snoc38 Ξ B) A; vs38
= Ξ» x Var38 vz38 vs38 β vs38 _ _ _ (x Var38 vz38 vs38)
Tm38 : Con38 β Ty38 β Set; Tm38
= Ξ» Ξ A β
(Tm38 : Con38 β Ty38 β Set)
(var : β Ξ A β Var38 Ξ A β Tm38 Ξ A)
(lam : β Ξ A B β Tm38 (snoc38 Ξ A) B β Tm38 Ξ (arr38 A B))
(app : β Ξ A B β Tm38 Ξ (arr38 A B) β Tm38 Ξ A β Tm38 Ξ B)
(tt : β Ξ β Tm38 Ξ top38)
(pair : β Ξ A B β Tm38 Ξ A β Tm38 Ξ B β Tm38 Ξ (prod38 A B))
(fst : β Ξ A B β Tm38 Ξ (prod38 A B) β Tm38 Ξ A)
(snd : β Ξ A B β Tm38 Ξ (prod38 A B) β Tm38 Ξ B)
(left : β Ξ A B β Tm38 Ξ A β Tm38 Ξ (sum38 A B))
(right : β Ξ A B β Tm38 Ξ B β Tm38 Ξ (sum38 A B))
(case : β Ξ A B C β Tm38 Ξ (sum38 A B) β Tm38 Ξ (arr38 A C) β Tm38 Ξ (arr38 B C) β Tm38 Ξ C)
(zero : β Ξ β Tm38 Ξ nat38)
(suc : β Ξ β Tm38 Ξ nat38 β Tm38 Ξ nat38)
(rec : β Ξ A β Tm38 Ξ nat38 β Tm38 Ξ (arr38 nat38 (arr38 A A)) β Tm38 Ξ A β Tm38 Ξ A)
β Tm38 Ξ A
var38 : β{Ξ A} β Var38 Ξ A β Tm38 Ξ A; var38
= Ξ» x Tm38 var38 lam app tt pair fst snd left right case zero suc rec β
var38 _ _ x
lam38 : β{Ξ A B} β Tm38 (snoc38 Ξ A) B β Tm38 Ξ (arr38 A B); lam38
= Ξ» t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec β
lam38 _ _ _ (t Tm38 var38 lam38 app tt pair fst snd left right case zero suc rec)
app38 : β{Ξ A B} β Tm38 Ξ (arr38 A B) β Tm38 Ξ A β Tm38 Ξ B; app38
= Ξ» t u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec β
app38 _ _ _ (t Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt pair fst snd left right case zero suc rec)
tt38 : β{Ξ} β Tm38 Ξ top38; tt38
= Ξ» Tm38 var38 lam38 app38 tt38 pair fst snd left right case zero suc rec β tt38 _
pair38 : β{Ξ A B} β Tm38 Ξ A β Tm38 Ξ B β Tm38 Ξ (prod38 A B); pair38
= Ξ» t u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec β
pair38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst snd left right case zero suc rec)
fst38 : β{Ξ A B} β Tm38 Ξ (prod38 A B) β Tm38 Ξ A; fst38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec β
fst38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd left right case zero suc rec)
snd38 : β{Ξ A B} β Tm38 Ξ (prod38 A B) β Tm38 Ξ B; snd38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec β
snd38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left right case zero suc rec)
left38 : β{Ξ A B} β Tm38 Ξ A β Tm38 Ξ (sum38 A B); left38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec β
left38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right case zero suc rec)
right38 : β{Ξ A B} β Tm38 Ξ B β Tm38 Ξ (sum38 A B); right38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec β
right38 _ _ _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case zero suc rec)
case38 : β{Ξ A B C} β Tm38 Ξ (sum38 A B) β Tm38 Ξ (arr38 A C) β Tm38 Ξ (arr38 B C) β Tm38 Ξ C; case38
= Ξ» t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec β
case38 _ _ _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero suc rec)
zero38 : β{Ξ} β Tm38 Ξ nat38; zero38
= Ξ» Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc rec β zero38 _
suc38 : β{Ξ} β Tm38 Ξ nat38 β Tm38 Ξ nat38; suc38
= Ξ» t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec β
suc38 _ (t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec)
rec38 : β{Ξ A} β Tm38 Ξ nat38 β Tm38 Ξ (arr38 nat38 (arr38 A A)) β Tm38 Ξ A β Tm38 Ξ A; rec38
= Ξ» t u v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38 β
rec38 _ _
(t Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(u Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
(v Tm38 var38 lam38 app38 tt38 pair38 fst38 snd38 left38 right38 case38 zero38 suc38 rec38)
v038 : β{Ξ A} β Tm38 (snoc38 Ξ A) A; v038
= var38 vz38
v138 : β{Ξ A B} β Tm38 (snoc38 (snoc38 Ξ A) B) A; v138
= var38 (vs38 vz38)
v238 : β{Ξ A B C} β Tm38 (snoc38 (snoc38 (snoc38 Ξ A) B) C) A; v238
= var38 (vs38 (vs38 vz38))
v338 : β{Ξ A B C D} β Tm38 (snoc38 (snoc38 (snoc38 (snoc38 Ξ A) B) C) D) A; v338
= var38 (vs38 (vs38 (vs38 vz38)))
tbool38 : Ty38; tbool38
= sum38 top38 top38
true38 : β{Ξ} β Tm38 Ξ tbool38; true38
= left38 tt38
tfalse38 : β{Ξ} β Tm38 Ξ tbool38; tfalse38
= right38 tt38
ifthenelse38 : β{Ξ A} β Tm38 Ξ (arr38 tbool38 (arr38 A (arr38 A A))); ifthenelse38
= lam38 (lam38 (lam38 (case38 v238 (lam38 v238) (lam38 v138))))
times438 : β{Ξ A} β Tm38 Ξ (arr38 (arr38 A A) (arr38 A A)); times438
= lam38 (lam38 (app38 v138 (app38 v138 (app38 v138 (app38 v138 v038)))))
add38 : β{Ξ} β Tm38 Ξ (arr38 nat38 (arr38 nat38 nat38)); add38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (suc38 (app38 v138 v038)))))
(lam38 v038))
mul38 : β{Ξ} β Tm38 Ξ (arr38 nat38 (arr38 nat38 nat38)); mul38
= lam38 (rec38 v038
(lam38 (lam38 (lam38 (app38 (app38 add38 (app38 v138 v038)) v038))))
(lam38 zero38))
fact38 : β{Ξ} β Tm38 Ξ (arr38 nat38 nat38); fact38
= lam38 (rec38 v038 (lam38 (lam38 (app38 (app38 mul38 (suc38 v138)) v038)))
(suc38 zero38))
{-# OPTIONS --type-in-type #-}
Ty39 : Set
Ty39 =
(Ty39 : Set)
(nat top bot : Ty39)
(arr prod sum : Ty39 β Ty39 β Ty39)
β Ty39
nat39 : Ty39; nat39 = Ξ» _ nat39 _ _ _ _ _ β nat39
top39 : Ty39; top39 = Ξ» _ _ top39 _ _ _ _ β top39
bot39 : Ty39; bot39 = Ξ» _ _ _ bot39 _ _ _ β bot39
arr39 : Ty39 β Ty39 β Ty39; arr39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod sum β
arr39 (A Ty39 nat39 top39 bot39 arr39 prod sum) (B Ty39 nat39 top39 bot39 arr39 prod sum)
prod39 : Ty39 β Ty39 β Ty39; prod39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod39 sum β
prod39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum) (B Ty39 nat39 top39 bot39 arr39 prod39 sum)
sum39 : Ty39 β Ty39 β Ty39; sum39
= Ξ» A B Ty39 nat39 top39 bot39 arr39 prod39 sum39 β
sum39 (A Ty39 nat39 top39 bot39 arr39 prod39 sum39) (B Ty39 nat39 top39 bot39 arr39 prod39 sum39)
Con39 : Set; Con39
= (Con39 : Set)
(nil : Con39)
(snoc : Con39 β Ty39 β Con39)
β Con39
nil39 : Con39; nil39
= Ξ» Con39 nil39 snoc β nil39
snoc39 : Con39 β Ty39 β Con39; snoc39
= Ξ» Ξ A Con39 nil39 snoc39 β snoc39 (Ξ Con39 nil39 snoc39) A
Var39 : Con39 β Ty39 β Set; Var39
= Ξ» Ξ A β
(Var39 : Con39 β Ty39 β Set)
(vz : β Ξ A β Var39 (snoc39 Ξ A) A)
(vs : β Ξ B A β Var39 Ξ A β Var39 (snoc39 Ξ B) A)
β Var39 Ξ A
vz39 : β{Ξ A} β Var39 (snoc39 Ξ A) A; vz39
= Ξ» Var39 vz39 vs β vz39 _ _
vs39 : β{Ξ B A} β Var39 Ξ A β Var39 (snoc39 Ξ B) A; vs39
= Ξ» x Var39 vz39 vs39 β vs39 _ _ _ (x Var39 vz39 vs39)
Tm39 : Con39 β Ty39 β Set; Tm39
= Ξ» Ξ A β
(Tm39 : Con39 β Ty39 β Set)
(var : β Ξ A β Var39 Ξ A β Tm39 Ξ A)
(lam : β Ξ A B β Tm39 (snoc39 Ξ A) B β Tm39 Ξ (arr39 A B))
(app : β Ξ A B β Tm39 Ξ (arr39 A B) β Tm39 Ξ A β Tm39 Ξ B)
(tt : β Ξ β Tm39 Ξ top39)
(pair : β Ξ A B β Tm39 Ξ A β Tm39 Ξ B β Tm39 Ξ (prod39 A B))
(fst : β Ξ A B β Tm39 Ξ (prod39 A B) β Tm39 Ξ A)
(snd : β Ξ A B β Tm39 Ξ (prod39 A B) β Tm39 Ξ B)
(left : β Ξ A B β Tm39 Ξ A β Tm39 Ξ (sum39 A B))
(right : β Ξ A B β Tm39 Ξ B β Tm39 Ξ (sum39 A B))
(case : β Ξ A B C β Tm39 Ξ (sum39 A B) β Tm39 Ξ (arr39 A C) β Tm39 Ξ (arr39 B C) β Tm39 Ξ C)
(zero : β Ξ β Tm39 Ξ nat39)
(suc : β Ξ β Tm39 Ξ nat39 β Tm39 Ξ nat39)
(rec : β Ξ A β Tm39 Ξ nat39 β Tm39 Ξ (arr39 nat39 (arr39 A A)) β Tm39 Ξ A β Tm39 Ξ A)
β Tm39 Ξ A
var39 : β{Ξ A} β Var39 Ξ A β Tm39 Ξ A; var39
= Ξ» x Tm39 var39 lam app tt pair fst snd left right case zero suc rec β
var39 _ _ x
lam39 : β{Ξ A B} β Tm39 (snoc39 Ξ A) B β Tm39 Ξ (arr39 A B); lam39
= Ξ» t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec β
lam39 _ _ _ (t Tm39 var39 lam39 app tt pair fst snd left right case zero suc rec)
app39 : β{Ξ A B} β Tm39 Ξ (arr39 A B) β Tm39 Ξ A β Tm39 Ξ B; app39
= Ξ» t u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec β
app39 _ _ _ (t Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt pair fst snd left right case zero suc rec)
tt39 : β{Ξ} β Tm39 Ξ top39; tt39
= Ξ» Tm39 var39 lam39 app39 tt39 pair fst snd left right case zero suc rec β tt39 _
pair39 : β{Ξ A B} β Tm39 Ξ A β Tm39 Ξ B β Tm39 Ξ (prod39 A B); pair39
= Ξ» t u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec β
pair39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst snd left right case zero suc rec)
fst39 : β{Ξ A B} β Tm39 Ξ (prod39 A B) β Tm39 Ξ A; fst39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec β
fst39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd left right case zero suc rec)
snd39 : β{Ξ A B} β Tm39 Ξ (prod39 A B) β Tm39 Ξ B; snd39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec β
snd39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left right case zero suc rec)
left39 : β{Ξ A B} β Tm39 Ξ A β Tm39 Ξ (sum39 A B); left39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec β
left39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right case zero suc rec)
right39 : β{Ξ A B} β Tm39 Ξ B β Tm39 Ξ (sum39 A B); right39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec β
right39 _ _ _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case zero suc rec)
case39 : β{Ξ A B C} β Tm39 Ξ (sum39 A B) β Tm39 Ξ (arr39 A C) β Tm39 Ξ (arr39 B C) β Tm39 Ξ C; case39
= Ξ» t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec β
case39 _ _ _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero suc rec)
zero39 : β{Ξ} β Tm39 Ξ nat39; zero39
= Ξ» Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc rec β zero39 _
suc39 : β{Ξ} β Tm39 Ξ nat39 β Tm39 Ξ nat39; suc39
= Ξ» t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec β
suc39 _ (t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec)
rec39 : β{Ξ A} β Tm39 Ξ nat39 β Tm39 Ξ (arr39 nat39 (arr39 A A)) β Tm39 Ξ A β Tm39 Ξ A; rec39
= Ξ» t u v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39 β
rec39 _ _
(t Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(u Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
(v Tm39 var39 lam39 app39 tt39 pair39 fst39 snd39 left39 right39 case39 zero39 suc39 rec39)
v039 : β{Ξ A} β Tm39 (snoc39 Ξ A) A; v039
= var39 vz39
v139 : β{Ξ A B} β Tm39 (snoc39 (snoc39 Ξ A) B) A; v139
= var39 (vs39 vz39)
v239 : β{Ξ A B C} β Tm39 (snoc39 (snoc39 (snoc39 Ξ A) B) C) A; v239
= var39 (vs39 (vs39 vz39))
v339 : β{Ξ A B C D} β Tm39 (snoc39 (snoc39 (snoc39 (snoc39 Ξ A) B) C) D) A; v339
= var39 (vs39 (vs39 (vs39 vz39)))
tbool39 : Ty39; tbool39
= sum39 top39 top39
true39 : β{Ξ} β Tm39 Ξ tbool39; true39
= left39 tt39
tfalse39 : β{Ξ} β Tm39 Ξ tbool39; tfalse39
= right39 tt39
ifthenelse39 : β{Ξ A} β Tm39 Ξ (arr39 tbool39 (arr39 A (arr39 A A))); ifthenelse39
= lam39 (lam39 (lam39 (case39 v239 (lam39 v239) (lam39 v139))))
times439 : β{Ξ A} β Tm39 Ξ (arr39 (arr39 A A) (arr39 A A)); times439
= lam39 (lam39 (app39 v139 (app39 v139 (app39 v139 (app39 v139 v039)))))
add39 : β{Ξ} β Tm39 Ξ (arr39 nat39 (arr39 nat39 nat39)); add39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (suc39 (app39 v139 v039)))))
(lam39 v039))
mul39 : β{Ξ} β Tm39 Ξ (arr39 nat39 (arr39 nat39 nat39)); mul39
= lam39 (rec39 v039
(lam39 (lam39 (lam39 (app39 (app39 add39 (app39 v139 v039)) v039))))
(lam39 zero39))
fact39 : β{Ξ} β Tm39 Ξ (arr39 nat39 nat39); fact39
= lam39 (rec39 v039 (lam39 (lam39 (app39 (app39 mul39 (suc39 v139)) v039)))
(suc39 zero39))
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