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/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | exact ⟨id, this⟩ | α : Sort u_1
this : isInv id id
⊢ invertible id | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | apply And.intro | α : Sort u_1
⊢ isInv id id | case left
α : Sort u_1
⊢ ∀ (x : α), id (id x) = x
case right
α : Sort u_1
⊢ ∀ (y : α), id (id y) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | . intro x
simp | case left
α : Sort u_1
⊢ ∀ (x : α), id (id x) = x
case right
α : Sort u_1
⊢ ∀ (y : α), id (id y) = y | case right
α : Sort u_1
⊢ ∀ (y : α), id (id y) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | . intro y
simp | case right
α : Sort u_1
⊢ ∀ (y : α), id (id y) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | intro x | case left
α : Sort u_1
⊢ ∀ (x : α), id (id x) = x | case left
α : Sort u_1
x : α
⊢ id (id x) = x |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | simp | case left
α : Sort u_1
x : α
⊢ id (id x) = x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | intro y | case right
α : Sort u_1
⊢ ∀ (y : α), id (id y) = y | case right
α : Sort u_1
y : α
⊢ id (id y) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | simp | case right
α : Sort u_1
y : α
⊢ id (id y) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | induction p with
| refl => simp [permuted.to_map, invertible_id]
| step i j fi ij jl p ih =>
let ⟨f', ⟨h1, h2⟩⟩ := ih
let f := permuted.to_map (permuted.step i j fi ij jl p)
show invertible f
let g := fun (k : Fin n) =>
let k' := f' k
if k' = i then j
else if k' = j then i
else k'
have gf : ∀x, g (f x) = x := by
intro x
simp (config := {zeta := false}) [f, permuted.to_map]
cases decEq x i with
| isFalse nxi =>
cases decEq x j with
| isFalse nxj => simp [nxi, nxj, g, h1]
| isTrue xj => simp [nxi, xj, g, h1]
| isTrue xi => simp [xi, g, h1]
have fg : ∀y, f (g y) = y := by
intro y
simp (config := {zeta := false}) [g]
show f (if f' y = i then j else if f' y = j then i else f' y) = y
cases decEq (f' y) i with
| isFalse nfyi =>
cases decEq (f' y) j with
| isFalse nfyj => simp [nfyi, nfyj, permuted.to_map, h2, f]
| isTrue fyj =>
simp [nfyi, fyj, permuted.to_map, f]
rw [←(congrArg p.to_map fyj), h2]
| isTrue fyi =>
simp [fyi, f, permuted.to_map]
have : (if j = i then j else i) = i := by simp
rw [this, ←(congrArg p.to_map fyi), h2]
exact ⟨g, ⟨gf, fg⟩⟩ | n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
p : permuted n first last arr1 arr2
⊢ invertible p.to_map | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [permuted.to_map, invertible_id] | case refl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ : Vec α✝ n
⊢ invertible permuted.refl.to_map | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | let ⟨f', ⟨h1, h2⟩⟩ := ih | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
⊢ invertible (permuted.step i j fi ij jl p).to_map | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
⊢ invertible (permuted.step i j fi ij jl p).to_map |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | let f := permuted.to_map (permuted.step i j fi ij jl p) | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
⊢ invertible (permuted.step i j fi ij jl p).to_map | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
⊢ invertible (permuted.step i j fi ij jl p).to_map |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | show invertible f | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
⊢ invertible (permuted.step i j fi ij jl p).to_map | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
⊢ invertible f |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | let g := fun (k : Fin n) =>
let k' := f' k
if k' = i then j
else if k' = j then i
else k' | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
⊢ invertible f | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
⊢ invertible f |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | have gf : ∀x, g (f x) = x := by
intro x
simp (config := {zeta := false}) [f, permuted.to_map]
cases decEq x i with
| isFalse nxi =>
cases decEq x j with
| isFalse nxj => simp [nxi, nxj, g, h1]
| isTrue xj => simp [nxi, xj, g, h1]
| isTrue xi => simp [xi, g, h1] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
⊢ invertible f | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
⊢ invertible f |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | have fg : ∀y, f (g y) = y := by
intro y
simp (config := {zeta := false}) [g]
show f (if f' y = i then j else if f' y = j then i else f' y) = y
cases decEq (f' y) i with
| isFalse nfyi =>
cases decEq (f' y) j with
| isFalse nfyj => simp [nfyi, nfyj, permuted.to_map, h2, f]
| isTrue fyj =>
simp [nfyi, fyj, permuted.to_map, f]
rw [←(congrArg p.to_map fyj), h2]
| isTrue fyi =>
simp [fyi, f, permuted.to_map]
have : (if j = i then j else i) = i := by simp
rw [this, ←(congrArg p.to_map fyi), h2] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
⊢ invertible f | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
fg : ∀ (y : Fin n), f (g y) = y
⊢ invertible f |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | exact ⟨g, ⟨gf, fg⟩⟩ | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
fg : ∀ (y : Fin n), f (g y) = y
⊢ invertible f | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | intro x | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
⊢ ∀ (x : Fin n), g (f x) = x | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
⊢ g (f x) = x |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp (config := {zeta := false}) [f, permuted.to_map] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
⊢ g (f x) = x | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | cases decEq x i with
| isFalse nxi =>
cases decEq x j with
| isFalse nxj => simp [nxi, nxj, g, h1]
| isTrue xj => simp [nxi, xj, g, h1]
| isTrue xi => simp [xi, g, h1] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | cases decEq x j with
| isFalse nxj => simp [nxi, nxj, g, h1]
| isTrue xj => simp [nxi, xj, g, h1] | case isFalse
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
nxi : ¬x = i
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [nxi, nxj, g, h1] | case isFalse.isFalse
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
nxi : ¬x = i
nxj : ¬x = j
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [nxi, xj, g, h1] | case isFalse.isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
nxi : ¬x = i
xj : x = j
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [xi, g, h1] | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
x : Fin n
xi : x = i
⊢ g
(let k' := if x = i then j else if x = j then i else x;
p.to_map k') =
x | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | intro y | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
⊢ ∀ (y : Fin n), f (g y) = y | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f (g y) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp (config := {zeta := false}) [g] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f (g y) = y | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f
(let k' := f' y;
if k' = i then j else if k' = j then i else k') =
y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | show f (if f' y = i then j else if f' y = j then i else f' y) = y | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f
(let k' := f' y;
if k' = i then j else if k' = j then i else k') =
y | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | cases decEq (f' y) i with
| isFalse nfyi =>
cases decEq (f' y) j with
| isFalse nfyj => simp [nfyi, nfyj, permuted.to_map, h2, f]
| isTrue fyj =>
simp [nfyi, fyj, permuted.to_map, f]
rw [←(congrArg p.to_map fyj), h2]
| isTrue fyi =>
simp [fyi, f, permuted.to_map]
have : (if j = i then j else i) = i := by simp
rw [this, ←(congrArg p.to_map fyi), h2] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | cases decEq (f' y) j with
| isFalse nfyj => simp [nfyi, nfyj, permuted.to_map, h2, f]
| isTrue fyj =>
simp [nfyi, fyj, permuted.to_map, f]
rw [←(congrArg p.to_map fyj), h2] | case isFalse
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
nfyi : ¬f' y = i
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [nfyi, nfyj, permuted.to_map, h2, f] | case isFalse.isFalse
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
nfyi : ¬f' y = i
nfyj : ¬f' y = j
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [nfyi, fyj, permuted.to_map, f] | case isFalse.isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
nfyi : ¬f' y = i
fyj : f' y = j
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y | case isFalse.isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
nfyi : ¬f' y = i
fyj : f' y = j
⊢ p.to_map j = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | rw [←(congrArg p.to_map fyj), h2] | case isFalse.isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
nfyi : ¬f' y = i
fyj : f' y = j
⊢ p.to_map j = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp [fyi, f, permuted.to_map] | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
⊢ f (if f' y = i then j else if f' y = j then i else f' y) = y | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
⊢ p.to_map (if j = i then j else i) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | have : (if j = i then j else i) = i := by simp | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
⊢ p.to_map (if j = i then j else i) = y | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
this : (if j = i then j else i) = i
⊢ p.to_map (if j = i then j else i) = y |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | rw [this, ←(congrArg p.to_map fyi), h2] | case isTrue
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
this : (if j = i then j else i) = i
⊢ p.to_map (if j = i then j else i) = y | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_invertible | [51, 1] | [87, 24] | simp | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : invertible p.to_map
f' : Fin n → Fin n
h1 : ∀ (x : Fin n), f' (p.to_map x) = x
h2 : ∀ (y : Fin n), p.to_map (f' y) = y
f : Fin n → Fin n := (permuted.step i j fi ij jl p).to_map
g : Fin n → Fin n :=
fun k =>
let k' := f' k;
if k' = i then j else if k' = j then i else k'
gf : ∀ (x : Fin n), g (f x) = x
y : Fin n
fyi : f' y = i
⊢ (if j = i then j else i) = i | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | induction p generalizing k with
| refl => simp [permuted.to_map]
| step i j fi ij jl p ih =>
simp [permuted.to_map]
rw [←ih]
split
case step.inl ki => simp [ki, Vec.get_swap_right]
case step.inr nki =>
split
case inl kj => simp [kj, Vec.get_swap_left]
case inr nkj => rw [Vec.get_swap_neq _ i j k nki nkj] | n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
p : permuted n first last arr1 arr2
k : Fin n
⊢ arr1[k] = arr2[p.to_map k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | simp [permuted.to_map] | case refl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ : Vec α✝ n
k : Fin n
⊢ arr✝[k] = arr✝[permuted.refl.to_map k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | simp [permuted.to_map] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
⊢ arr✝[k] = arr'✝[(permuted.step i j fi ij jl p).to_map k] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
⊢ arr✝[k] = arr'✝[p.to_map (if k = i then j else if k = j then i else k)] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | rw [←ih] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
⊢ arr✝[k] = arr'✝[p.to_map (if k = i then j else if k = j then i else k)] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
⊢ arr✝[k] = (arr✝.swap i j)[if k = i then j else if k = j then i else k] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | split | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
⊢ arr✝[k] = (arr✝.swap i j)[if k = i then j else if k = j then i else k] | case step.inl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
h✝ : k = i
⊢ arr✝[k] = (arr✝.swap i j)[j]
case step.inr
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
h✝ : ¬k = i
⊢ arr✝[k] = (arr✝.swap i j)[if k = j then i else k] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | case step.inl ki => simp [ki, Vec.get_swap_right] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
ki : k = i
⊢ arr✝[k] = (arr✝.swap i j)[j] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | case step.inr nki =>
split
case inl kj => simp [kj, Vec.get_swap_left]
case inr nkj => rw [Vec.get_swap_neq _ i j k nki nkj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
⊢ arr✝[k] = (arr✝.swap i j)[if k = j then i else k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | simp [ki, Vec.get_swap_right] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
ki : k = i
⊢ arr✝[k] = (arr✝.swap i j)[j] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | split | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
⊢ arr✝[k] = (arr✝.swap i j)[if k = j then i else k] | case inl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
h✝ : k = j
⊢ arr✝[k] = (arr✝.swap i j)[i]
case inr
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
h✝ : ¬k = j
⊢ arr✝[k] = (arr✝.swap i j)[k] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | case inl kj => simp [kj, Vec.get_swap_left] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
kj : k = j
⊢ arr✝[k] = (arr✝.swap i j)[i] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | case inr nkj => rw [Vec.get_swap_neq _ i j k nki nkj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
nkj : ¬k = j
⊢ arr✝[k] = (arr✝.swap i j)[k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | simp [kj, Vec.get_swap_left] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
kj : k = j
⊢ arr✝[k] = (arr✝.swap i j)[i] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index | [89, 1] | [101, 60] | rw [Vec.get_swap_neq _ i j k nki nkj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), (arr✝.swap i j)[k] = arr'✝[p.to_map k]
k : Fin n
nki : ¬k = i
nkj : ¬k = j
⊢ arr✝[k] = (arr✝.swap i j)[k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | exists p.to_map | α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∃ f, invertible f ∧ ∀ (i : Fin n), arr[i] = arr'[f i] | α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ invertible p.to_map ∧ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | apply And.intro | α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ invertible p.to_map ∧ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] | case left
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ invertible p.to_map
case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | . exact permuted_map_invertible p | case left
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ invertible p.to_map
case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] | case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | . intro i
exact permuted_map_index p i | case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | exact permuted_map_invertible p | case left
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ invertible p.to_map | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | intro i | case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
⊢ ∀ (i : Fin n), arr[i] = arr'[p.to_map i] | case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
i : Fin n
⊢ arr[i] = arr'[p.to_map i] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_invertible | [103, 1] | [110, 33] | exact permuted_map_index p i | case right
α : Type
n first last : Nat
arr arr' : Vec α n
p : permuted n first last arr arr'
i : Fin n
⊢ arr[i] = arr'[p.to_map i] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | induction p generalizing k with
| refl => simp [permuted.to_map, fk, kl]
| step i j fi ij jl p ih =>
simp [permuted.to_map]
let k' := if k = i then j else if k = j then i else k
have : first ≤ k'.val ∧ k'.val ≤ last := by
match decEq k i, decEq k j with
| isTrue ki, _ =>
simp [k', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k', ki, kj]
exact ⟨fk, kl⟩
exact ih k' this.1 this.2 | n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
p : permuted n first last arr1 arr2
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [permuted.to_map, fk, kl] | case refl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ : Vec α✝ n
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ first ≤ ↑(permuted.refl.to_map k) ∧ ↑(permuted.refl.to_map k) ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [permuted.to_map] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ first ≤ ↑((permuted.step i j fi ij jl p).to_map k) ∧ ↑((permuted.step i j fi ij jl p).to_map k) ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | let k' := if k = i then j else if k = j then i else k | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | have : first ≤ k'.val ∧ k'.val ≤ last := by
match decEq k i, decEq k j with
| isTrue ki, _ =>
simp [k', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k', ki, kj]
exact ⟨fk, kl⟩ | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
this : first ≤ ↑k' ∧ ↑k' ≤ last
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | exact ih k' this.1 this.2 | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
this : first ≤ ↑k' ∧ ↑k' ≤ last
⊢ first ≤ ↑(p.to_map (if k = i then j else if k = j then i else k)) ∧
↑(p.to_map (if k = i then j else if k = j then i else k)) ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | match decEq k i, decEq k j with
| isTrue ki, _ =>
simp [k', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k', ki, kj]
exact ⟨fk, kl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
⊢ first ≤ ↑k' ∧ ↑k' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [k', ki] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : k = i
x✝ : Decidable (k = j)
⊢ first ≤ ↑k' ∧ ↑k' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : k = i
x✝ : Decidable (k = j)
⊢ first ≤ ↑j ∧ ↑j ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | exact ⟨Nat.le_trans fi ij, jl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : k = i
x✝ : Decidable (k = j)
⊢ first ≤ ↑j ∧ ↑j ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [k', ki] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : k = j
⊢ first ≤ ↑k' ∧ ↑k' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : k = j
⊢ first ≤ ↑(if k = j then i else k) ∧ ↑(if k = j then i else k) ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : k = j
⊢ first ≤ ↑(if k = j then i else k) ∧ ↑(if k = j then i else k) ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : k = j
⊢ first ≤ ↑i ∧ ↑i ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | exact ⟨fi, Nat.le_trans ij jl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : k = j
⊢ first ≤ ↑i ∧ ↑i ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | simp [k', ki, kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : ¬k = j
⊢ first ≤ ↑k' ∧ ↑k' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : ¬k = j
⊢ first ≤ ↑k ∧ ↑k ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range | [112, 1] | [132, 30] | exact ⟨fk, kl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → first ≤ ↑(p.to_map k) ∧ ↑(p.to_map k) ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n := if k = i then j else if k = j then i else k
ki : ¬k = i
kj : ¬k = j
⊢ first ≤ ↑k ∧ ↑k ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | induction p generalizing k with
| refl => simp [permuted.to_map, fk, kl]
| step i j fi ij jl p ih =>
simp [permuted.to_map]
let ⟨k', ⟨index, fk', kl'⟩⟩ := ih k fk kl
let k'' := if k' = i then j else if k' = j then i else k'
exists k''
have : (if k'' = i then j else if k'' = j then i else k'') = k' := by
match decEq k' i, decEq k' j with
| isTrue ki, _ => simp [k'', ki]
| _, isTrue kj => simp [k'', kj]
| isFalse ki, isFalse kj => simp [k'',ki, kj]
rw [this]
have : first ≤ k''.val ∧ k''.val ≤ last := by
match decEq k' i, decEq k' j with
| isTrue ki, _ =>
simp [k'', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k'', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k'', ki, kj]
exact ⟨fk', kl'⟩
exact ⟨index, this.1, this.2⟩ | n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
p : permuted n first last arr1 arr2
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [permuted.to_map, fk, kl] | case refl
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ : Vec α✝ n
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ ∃ k', permuted.refl.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [permuted.to_map] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ ∃ k', (permuted.step i j fi ij jl p).to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | let ⟨k', ⟨index, fk', kl'⟩⟩ := ih k fk kl | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | let k'' := if k' = i then j else if k' = j then i else k' | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | exists k'' | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
⊢ ∃ k', p.to_map (if k' = i then j else if k' = j then i else k') = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
⊢ p.to_map (if k'' = i then j else if k'' = j then i else k'') = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | have : (if k'' = i then j else if k'' = j then i else k'') = k' := by
match decEq k' i, decEq k' j with
| isTrue ki, _ => simp [k'', ki]
| _, isTrue kj => simp [k'', kj]
| isFalse ki, isFalse kj => simp [k'',ki, kj] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
⊢ p.to_map (if k'' = i then j else if k'' = j then i else k'') = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
⊢ p.to_map (if k'' = i then j else if k'' = j then i else k'') = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | rw [this] | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
⊢ p.to_map (if k'' = i then j else if k'' = j then i else k'') = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
⊢ p.to_map k' = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | have : first ≤ k''.val ∧ k''.val ≤ last := by
match decEq k' i, decEq k' j with
| isTrue ki, _ =>
simp [k'', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k'', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k'', ki, kj]
exact ⟨fk', kl'⟩ | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
⊢ p.to_map k' = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this✝ : (if k'' = i then j else if k'' = j then i else k'') = k'
this : first ≤ ↑k'' ∧ ↑k'' ≤ last
⊢ p.to_map k' = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | exact ⟨index, this.1, this.2⟩ | case step
n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this✝ : (if k'' = i then j else if k'' = j then i else k'') = k'
this : first ≤ ↑k'' ∧ ↑k'' ≤ last
⊢ p.to_map k' = k ∧ first ≤ ↑k'' ∧ ↑k'' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | match decEq k' i, decEq k' j with
| isTrue ki, _ => simp [k'', ki]
| _, isTrue kj => simp [k'', kj]
| isFalse ki, isFalse kj => simp [k'',ki, kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
⊢ (if k'' = i then j else if k'' = j then i else k'') = k' | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'', ki] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
ki : k' = i
x✝ : Decidable (k' = j)
⊢ (if k'' = i then j else if k'' = j then i else k'') = k' | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'', kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
x✝ : Decidable (k' = i)
kj : k' = j
⊢ (if k'' = i then j else if k'' = j then i else k'') = k' | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'',ki, kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
ki : ¬k' = i
kj : ¬k' = j
⊢ (if k'' = i then j else if k'' = j then i else k'') = k' | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | match decEq k' i, decEq k' j with
| isTrue ki, _ =>
simp [k'', ki]
exact ⟨Nat.le_trans fi ij, jl⟩
| isFalse ki, isTrue kj =>
simp [k'', ki]
simp [kj]
exact ⟨fi, Nat.le_trans ij jl⟩
| isFalse ki, isFalse kj =>
simp [k'', ki, kj]
exact ⟨fk', kl'⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
⊢ first ≤ ↑k'' ∧ ↑k'' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'', ki] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : k' = i
x✝ : Decidable (k' = j)
⊢ first ≤ ↑k'' ∧ ↑k'' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : k' = i
x✝ : Decidable (k' = j)
⊢ first ≤ ↑j ∧ ↑j ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | exact ⟨Nat.le_trans fi ij, jl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : k' = i
x✝ : Decidable (k' = j)
⊢ first ≤ ↑j ∧ ↑j ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'', ki] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : k' = j
⊢ first ≤ ↑k'' ∧ ↑k'' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : k' = j
⊢ first ≤ ↑(if k' = j then i else k') ∧ ↑(if k' = j then i else k') ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : k' = j
⊢ first ≤ ↑(if k' = j then i else k') ∧ ↑(if k' = j then i else k') ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : k' = j
⊢ first ≤ ↑i ∧ ↑i ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | exact ⟨fi, Nat.le_trans ij jl⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : k' = j
⊢ first ≤ ↑i ∧ ↑i ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | simp [k'', ki, kj] | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : ¬k' = j
⊢ first ≤ ↑k'' ∧ ↑k'' ≤ last | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : ¬k' = j
⊢ first ≤ ↑k' ∧ ↑k' ≤ last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted_map_index_in_range_inv | [134, 1] | [165, 34] | exact ⟨fk', kl'⟩ | n first last : Nat
α✝ : Type
arr1 arr2 arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : ∀ (k : Fin n), first ≤ ↑k → ↑k ≤ last → ∃ k', p.to_map k' = k ∧ first ≤ ↑k' ∧ ↑k' ≤ last
k : Fin n
fk : first ≤ ↑k
kl : ↑k ≤ last
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
kl' : ↑k' ≤ last
k'' : Fin n := if k' = i then j else if k' = j then i else k'
this : (if k'' = i then j else if k'' = j then i else k'') = k'
ki : ¬k' = i
kj : ¬k' = j
⊢ first ≤ ↑k' ∧ ↑k' ≤ last | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | induction p with
| refl => rfl
| step i j fi ij jl p ih =>
rw [ih]
rw [Vec.get_swap_neq _ i j k]
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (Nat.lt_of_lt_of_le lt fi)
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (Nat.lt_of_lt_of_le lt (Nat.le_trans fi ij)) | n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
p : permuted n first last arr1 arr2
k : Fin n
lt : ↑k < first
⊢ arr2[k] = arr1[k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | rfl | case refl
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ : Vec α✝ n
⊢ arr✝[k] = arr✝[k] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | rw [ih] | case step
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ arr'✝[k] = arr✝[k] | case step
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ (arr✝.swap i j)[k] = arr✝[k] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | rw [Vec.get_swap_neq _ i j k] | case step
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ (arr✝.swap i j)[k] = arr✝[k] | case step.ki
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ i
case step.kj
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (Nat.lt_of_lt_of_le lt fi) | case step.ki
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ i
case step.kj
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ j | case step.kj
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (Nat.lt_of_lt_of_le lt (Nat.le_trans fi ij)) | case step.kj
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | apply Fin.ne_of_val_ne | case step.ki
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ k ≠ i | case step.ki.h
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ ¬↑k = ↑i |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | permuted.get_lt | [167, 1] | [177, 70] | exact Nat.ne_of_lt (Nat.lt_of_lt_of_le lt fi) | case step.ki.h
n first last : Nat
α✝ : Type
arr1 arr2 : Vec α✝ n
k : Fin n
lt : ↑k < first
arr✝ arr'✝ : Vec α✝ n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ last
p : permuted n first last (arr✝.swap i j) arr'✝
ih : arr'✝[k] = (arr✝.swap i j)[k]
⊢ ¬↑k = ↑i | no goals |
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