url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
congr
|
case e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (do
let r2 ← g x'
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
do
let b ← g x'
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
case e_a.h.e_a
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (fun r2 => do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
fun b => do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
funext x'
|
case e_a.h.e_a
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (fun r2 => do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
fun b => do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
simp [fish] at IH
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
rewrite [<- IH]
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM f tail >>= mapM g
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
simp[bind_assoc]
|
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM f tail >>= mapM g
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Dialects/LinalgSemantics.lean
|
mapMCommute.mapM_commute
|
[282, 1]
|
[312, 2]
|
apply COMMUTE
|
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
|
no goals
|
https://github.com/opencompl/lean-mlir.git
|
e43d21592801e5e40477b14b7a554e356060c40c
|
MLIR/Examples/IfStatementOptimization.lean
|
SCF_SELECT.equivalent
|
[38, 1]
|
[63, 55]
|
simp [LHS, RHS, INPUT]
|
b : FinInt 1
n m : FinInt 32
⊢ run (denoteOp Δ LHS) (INPUT b n m) = run (denoteOp Δ RHS) (INPUT b n m)
|
b : FinInt 1
n m : FinInt 32
⊢ run
(denoteOp Δ
(Op.mk "scf.if" [(SSAVal.SSAVal "x", MLIRType.int Signedness.Signless 32)]
[(SSAVal.SSAVal "b", MLIRType.int Signedness.Signless 1)]
[Region.mk "entry" []
[Op.mk "scf.yield" [] [(SSAVal.SSAVal "n", MLIRType.int Signedness.Signless 32)] [] (AttrDict.mk [])],
Region.mk "entry" []
[Op.mk "scf.yield" [] [(SSAVal.SSAVal "m", MLIRType.int Signedness.Signless 32)] [] (AttrDict.mk [])]]
(AttrDict.mk [])))
(SSAEnv.One
[(SSAVal.SSAVal "b", { fst := MLIRType.i1, snd := b }), (SSAVal.SSAVal "n", { fst := MLIRType.i32, snd := n }),
(SSAVal.SSAVal "m", { fst := MLIRType.i32, snd := m })]) =
run
(denoteOp Δ
(Op.mk "arith.select" [(SSAVal.SSAVal "x", MLIRType.int Signedness.Signless 32)]
[(SSAVal.SSAVal "b", MLIRType.int Signedness.Signless 1),
(SSAVal.SSAVal "n", MLIRType.int Signedness.Signless 32),
(SSAVal.SSAVal "m", MLIRType.int Signedness.Signless 32)]
[] (AttrDict.mk [])))
(SSAEnv.One
[(SSAVal.SSAVal "b", { fst := MLIRType.i1, snd := b }), (SSAVal.SSAVal "n", { fst := MLIRType.i32, snd := n }),
(SSAVal.SSAVal "m", { fst := MLIRType.i32, snd := m })])
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/02Equality.lean
|
reverse_mkSymm
|
[81, 1]
|
[83, 16]
|
simp [mkSymm]
|
α : Type u_1
xs : List α
⊢ List.reverse (mkSymm xs) = mkSymm xs
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.one_add_add
|
[35, 1]
|
[40, 12]
|
induction n with
| zero => rfl
| succ n ih =>
dsimp
congr 1
|
m n : MyNat
⊢ m + 1 + n = m + n + 1
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.one_add_add
|
[35, 1]
|
[40, 12]
|
rfl
|
case zero
m : MyNat
⊢ m + 1 + zero = m + zero + 1
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.one_add_add
|
[35, 1]
|
[40, 12]
|
dsimp
|
case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + succ n = m + succ n + 1
|
case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + n + 1 = m + n + 1 + 1
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.one_add_add
|
[35, 1]
|
[40, 12]
|
congr 1
|
case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + n + 1 = m + n + 1 + 1
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.zero_add
|
[43, 1]
|
[48, 12]
|
induction m with
| zero => dsimp
| succ m ih =>
dsimp
rw [ih]
|
m : MyNat
⊢ 0 + m = m
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.zero_add
|
[43, 1]
|
[48, 12]
|
dsimp
|
case zero
⊢ 0 + zero = zero
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.zero_add
|
[43, 1]
|
[48, 12]
|
dsimp
|
case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + succ m = succ m
|
case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + m + 1 = m + 1
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.zero_add
|
[43, 1]
|
[48, 12]
|
rw [ih]
|
case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + m + 1 = m + 1
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_assoc
|
[50, 1]
|
[53, 27]
|
induction o with
| zero => simp
| succ o ih => simp [ih]
|
m n o : MyNat
⊢ m + (n + o) = m + n + o
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_assoc
|
[50, 1]
|
[53, 27]
|
simp
|
case zero
m n : MyNat
⊢ m + (n + zero) = m + n + zero
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_assoc
|
[50, 1]
|
[53, 27]
|
simp [ih]
|
case succ
m n o : MyNat
ih : m + (n + o) = m + n + o
⊢ m + (n + succ o) = m + n + succ o
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
induction n with
| zero =>
dsimp
rw [zero_add]
| succ n ih =>
dsimp
rw [one_add_add]
rw [ih]
|
m n : MyNat
⊢ m + n = n + m
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
dsimp
|
case zero
m : MyNat
⊢ m + zero = zero + m
|
case zero
m : MyNat
⊢ m = 0 + m
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
rw [zero_add]
|
case zero
m : MyNat
⊢ m = 0 + m
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
dsimp
|
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + succ n = succ n + m
|
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + 1 + m
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
rw [one_add_add]
|
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + 1 + m
|
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + m + 1
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/06Induction.lean
|
MyNat.add_comm
|
[55, 1]
|
[63, 12]
|
rw [ih]
|
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + m + 1
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/10Aesop.lean
|
ne_nil_of_mem
|
[8, 1]
|
[9, 8]
|
aesop
|
α : Type u_1
a : α
l : List α
h : a ∈ l
⊢ l ≠ []
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/10Aesop.lean
|
not_mem_cons_of_ne_of_not_mem
|
[11, 1]
|
[13, 8]
|
aesop
|
α : Type u_1
a y : α
l : List α
⊢ a ≠ y → ¬a ∈ l → ¬a ∈ y :: l
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/10Aesop.lean
|
mem_map
|
[15, 9]
|
[17, 24]
|
induction l <;> aesop
|
α : Type u_1
β : Type u_2
f : α → β
b : β
l : List α
⊢ b ∈ List.map f l ↔ ∃ a, a ∈ l ∧ f a = b
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/10Aesop.lean
|
mem_map_of_mem
|
[20, 1]
|
[22, 8]
|
aesop
|
α : Type u_1
β : Type u_2
f : α → β
a : α
l : List α
h : a ∈ l
⊢ f a ∈ List.map f l
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/10Aesop.lean
|
eq_nil_of_subset_nil
|
[25, 1]
|
[26, 28]
|
aesop (add 1% cases List)
|
α : Type u_1
l : List α
⊢ l ⊆ [] → l = []
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.foo
|
[57, 1]
|
[62, 26]
|
intro legal
|
p : Piece
n✝ : ℕ
start stop : Pos n✝
⊢ Piece.LegalMove start stop p → start ≠ stop
|
p : Piece
n✝ : ℕ
start stop : Pos n✝
legal : Piece.LegalMove start stop p
⊢ start ≠ stop
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.foo
|
[57, 1]
|
[62, 26]
|
cases legal with
| rookHorizontal h _ => exact h
| rookVertical h _ => exact h
| king h _ _ => exact h
|
p : Piece
n✝ : ℕ
start stop : Pos n✝
legal : Piece.LegalMove start stop p
⊢ start ≠ stop
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.foo
|
[57, 1]
|
[62, 26]
|
exact h
|
case rookHorizontal
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.y = stop.y
⊢ start ≠ stop
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.foo
|
[57, 1]
|
[62, 26]
|
exact h
|
case rookVertical
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.x = stop.x
⊢ start ≠ stop
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.foo
|
[57, 1]
|
[62, 26]
|
exact h
|
case king
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝¹ : Nat.dist ↑start.x ↑stop.x ≤ 1
a✝ : Nat.dist ↑start.y ↑stop.y ≤ 1
⊢ start ≠ stop
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.Board.set_same
|
[105, 1]
|
[106, 13]
|
simp [set]
|
n : ℕ
pos : Pos n
p? : Option PPiece
b : Board n
⊢ set b pos p? pos = p?
|
no goals
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.Board.set_different
|
[108, 1]
|
[112, 11]
|
intro h
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
⊢ pos' ≠ pos → set b pos p? pos' = b pos'
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ set b pos p? pos' = b pos'
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.Board.set_different
|
[108, 1]
|
[112, 11]
|
simp [set]
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ set b pos p? pos' = b pos'
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ pos = pos' → p? = b pos'
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.Board.set_different
|
[108, 1]
|
[112, 11]
|
intros
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ pos = pos' → p? = b pos'
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
a✝ : pos = pos'
⊢ p? = b pos'
|
https://github.com/JLimperg/regensburg-itp-school-2023.git
|
7307618fa6d85c090896d29021541cd67920d750
|
Talk/11MiniChess.lean
|
MiniChess.Board.set_different
|
[108, 1]
|
[112, 11]
|
simp_all
|
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
a✝ : pos = pos'
⊢ p? = b pos'
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.no_nbhrs
|
[50, 1]
|
[56, 40]
|
contrapose! hA
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA : ¬v ∈ M.A
⊢ ¬Adj (mp M) v w
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA :
∃ i,
i ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
i ≠ j ∧ (v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i)
⊢ v ∈ M.A
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.no_nbhrs
|
[50, 1]
|
[56, 40]
|
obtain ⟨i, hi, j, hj, _, hv⟩ := hA
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA :
∃ i,
i ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
i ≠ j ∧ (v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i)
⊢ v ∈ M.A
|
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.no_nbhrs
|
[50, 1]
|
[56, 40]
|
cases hv with
| inl hv => exact (sub_part hi) hv.1
| inr hv => exact (sub_part hj) hv.1
|
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.no_nbhrs
|
[50, 1]
|
[56, 40]
|
exact (sub_part hi) hv.1
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
⊢ v ∈ M.A
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.no_nbhrs
|
[50, 1]
|
[56, 40]
|
exact (sub_part hj) hv.1
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhrs_imp
|
[59, 1]
|
[62, 22]
|
intro h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
⊢ Adj (mp M) v w → v ∈ M.A
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
⊢ v ∈ M.A
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhrs_imp
|
[59, 1]
|
[62, 22]
|
by_contra h
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
⊢ v ∈ M.A
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
h : ¬v ∈ M.A
⊢ False
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhrs_imp
|
[59, 1]
|
[62, 22]
|
exact no_nbhrs h h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
h : ¬v ∈ M.A
⊢ False
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
intro h
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w → i ≠ j
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : Adj (mp M) v w
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
cases' h with a ha
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : Adj (mp M) v w
⊢ i ≠ j
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
ha :
a ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
a ≠ j ∧ (v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M a)
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
obtain ⟨har, b, hbr, abne, ab⟩ := ha
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
ha :
a ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
a ≠ j ∧ (v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M a)
⊢ i ≠ j
|
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b ∨ v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
cases ab with
| inl ab =>
have ai := uniq_part hi har hvi ab.1; have bj := uniq_part hj hbr hwj ab.2
rwa [← ai, ← bj] at abne
| inr ab =>
have aj := uniq_part hj har hwj ab.2; have bi := uniq_part hi hbr hvi ab.1
rw [← aj, ← bi] at abne
exact abne.symm
|
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b ∨ v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
have ai := uniq_part hi har hvi ab.1
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
⊢ i ≠ j
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
have bj := uniq_part hj hbr hwj ab.2
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
⊢ i ≠ j
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
bj : j = b
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
rwa [← ai, ← bj] at abne
|
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
bj : j = b
⊢ i ≠ j
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
have aj := uniq_part hj har hwj ab.2
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
have bi := uniq_part hi hbr hvi ab.1
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
⊢ i ≠ j
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
rw [← aj, ← bi] at abne
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : j ≠ i
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_adj_imp
|
[65, 1]
|
[77, 20]
|
exact abne.symm
|
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : j ≠ i
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_imp_adj
|
[80, 1]
|
[82, 64]
|
intro h
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ i ≠ j → Adj (mp M) v w
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_imp_adj
|
[80, 1]
|
[82, 64]
|
refine' ⟨i, hi, j, hj, h, _⟩
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ Adj (mp M) v w
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_imp_adj
|
[80, 1]
|
[82, 64]
|
left
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
|
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_imp_adj
|
[80, 1]
|
[82, 64]
|
exact ⟨hvi, hwj⟩
|
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part
|
[85, 1]
|
[89, 38]
|
intro h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
⊢ Adj (mp M) v w → ¬w ∈ MultiPart.P M i
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ ¬w ∈ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part
|
[85, 1]
|
[89, 38]
|
by_contra h
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ ¬w ∈ MultiPart.P M i
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ False
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part
|
[85, 1]
|
[89, 38]
|
apply mp_adj_imp hi hi hv h h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ False
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ i = i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part
|
[85, 1]
|
[89, 38]
|
rfl
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ i = i
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part'
|
[92, 1]
|
[96, 36]
|
intro h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
⊢ ¬Adj (mp M) v w
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ False
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part'
|
[92, 1]
|
[96, 36]
|
contrapose hw
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ False
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
hw : ¬False
⊢ ¬w ∈ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.not_nbhr_same_part'
|
[92, 1]
|
[96, 36]
|
exact not_nbhr_same_part hi hv h1
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
hw : ¬False
⊢ ¬w ∈ MultiPart.P M i
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
rw [mem_sdiff] at hw
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A \ MultiPart.P M i
⊢ Adj (mp M) v w
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A ∧ ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
cases' hw with hA hni
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A ∧ ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ M.A
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
rw [M.uni] at hA
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ M.A
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
rw [mem_biUnion] at hA
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : ∃ a, a ∈ range (M.t + 1) ∧ w ∈ MultiPart.P M a
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
obtain ⟨j, hj1, hj2⟩ := hA
|
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : ∃ a, a ∈ range (M.t + 1) ∧ w ∈ MultiPart.P M a
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
refine' mp_imp_adj hi hj1 hv hj2 _
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ i ≠ j
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
intro h
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ i ≠ j
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
rw [h] at hni
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
j : ℕ
hni : ¬w ∈ MultiPart.P M j
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.nbhr_diff_parts
|
[99, 1]
|
[106, 77]
|
exact hni hj2
|
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
j : ℕ
hni : ¬w ∈ MultiPart.P M j
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
ext
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ neighborFinset (mp M) v = M.A \ MultiPart.P M i
|
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v ↔ a✝ ∈ M.A \ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
constructor
|
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v ↔ a✝ ∈ M.A \ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v → a✝ ∈ M.A \ MultiPart.P M i
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → a✝ ∈ neighborFinset (mp M) v
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
rw [mem_neighborFinset]
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v → a✝ ∈ M.A \ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ Adj (mp M) v a✝ → a✝ ∈ M.A \ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
intro h
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ Adj (mp M) v a✝ → a✝ ∈ M.A \ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) v a✝
⊢ a✝ ∈ M.A \ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
rw [adj_comm] at h
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) v a✝
⊢ a✝ ∈ M.A \ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A \ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
rw [mem_sdiff]
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A \ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A ∧ ¬a✝ ∈ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
refine' ⟨nbhrs_imp h, _⟩
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A ∧ ¬a✝ ∈ MultiPart.P M i
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ ¬a✝ ∈ MultiPart.P M i
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
exact not_nbhr_same_part hv.1 hv.2 h.symm
|
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ ¬a✝ ∈ MultiPart.P M i
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
rw [mem_neighborFinset]
|
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → a✝ ∈ neighborFinset (mp M) v
|
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → Adj (mp M) v a✝
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_nbhd
|
[109, 1]
|
[114, 61]
|
exact nbhr_diff_parts hv.1 hv.2
|
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → Adj (mp M) v a✝
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg
|
[121, 1]
|
[122, 76]
|
rw [degree]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card (M.A \ MultiPart.P M i)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (neighborFinset (mp M) v) = card (M.A \ MultiPart.P M i)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg
|
[121, 1]
|
[122, 76]
|
rw [mp_nbhd hv]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (neighborFinset (mp M) v) = card (M.A \ MultiPart.P M i)
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_diff
|
[125, 1]
|
[126, 101]
|
rw [mp_deg hv]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card M.A - card (MultiPart.P M i)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (M.A \ MultiPart.P M i) = card M.A - card (MultiPart.P M i)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_diff
|
[125, 1]
|
[126, 101]
|
exact card_sdiff (sub_part hv.1)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (M.A \ MultiPart.P M i) = card M.A - card (MultiPart.P M i)
|
no goals
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
nth_rw 1 [M.uni]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in M.A, degree (mp M) v = ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i, degree (mp M) v =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
rw [sum_biUnion (pair_disjoint M)]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i, degree (mp M) v =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), ∑ i in MultiPart.P M x, degree (mp M) i =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
apply Finset.sum_congr rfl _
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), ∑ i in MultiPart.P M x, degree (mp M) i =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
intro x hx
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
rw [Finset.card_eq_sum_ones, sum_mul, one_mul]
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = ∑ x_1 in MultiPart.P M x, card (M.A \ MultiPart.P M x)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
apply Finset.sum_congr rfl _
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = ∑ x_1 in MultiPart.P M x, card (M.A \ MultiPart.P M x)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∀ (x_1 : α), x_1 ∈ MultiPart.P M x → degree (mp M) x_1 = card (M.A \ MultiPart.P M x)
|
https://github.com/jt496/Turan_4.git
|
329b6acff8f9b8f41609e3e5758ed80c61047eb5
|
Turan4/Multipartite.lean
|
SimpleGraph.mp_deg_sum
|
[129, 1]
|
[135, 36]
|
intro v hv
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∀ (x_1 : α), x_1 ∈ MultiPart.P M x → degree (mp M) x_1 = card (M.A \ MultiPart.P M x)
|
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
v : α
hv : v ∈ MultiPart.P M x
⊢ degree (mp M) v = card (M.A \ MultiPart.P M x)
|
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