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101
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stringlengths 6
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https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have ij : i.val ≤ j.val - 1 := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ij | case inv₃
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
this : ↑j - 1 < ↑j
⊢ swapped[j.prev] < swapped[first] | case inv₃
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝¹ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij : ↑i ≤ ↑j - 1
⊢ swapped[j.prev] < swapped[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | cases Nat.eq_or_lt_of_le ij with
| inl ij =>
have : swapped[j.prev] = arr[j] := by
simp [ij.symm, Fin.prev]
apply Vec.get_swap_left
rw [this, sf]
have : i < j := Nat.lt_of_le_of_lt (by assumption : i.val ≤ j.val - 1) (by assumption)
exact inv.3 this
| inr ij =>
have : swapped[j.prev] = arr[j.prev] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt ij
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (by assumption)
rw [this, sf]
apply inv.1 j.prev ij (by assumption) | case inv₃
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝¹ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij : ↑i ≤ ↑j - 1
⊢ swapped[j.prev] < swapped[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | show j.val - 1 + 1 ≤ j.val | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑j - 1 < ↑j | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑j - 1 + 1 ≤ ↑j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [Nat.sub_add_cancel (Nat.zero_lt_of_lt (Nat.lt_of_lt_of_le fi (by assumption)))] | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑j - 1 + 1 ≤ ↑j | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑j ≤ ↑j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Nat.le_refl | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑j ≤ ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
⊢ ↑i ≤ ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)] | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
this : ↑j - 1 < ↑j
⊢ ↑i ≤ ↑j - 1 | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
this : ↑j - 1 < ↑j
⊢ ↑i - Nat.succ 0 + Nat.succ 0 ≤ ↑j - 1 |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact ij | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝ : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij : i.prev < j.prev
this : ↑j - 1 < ↑j
⊢ ↑i - Nat.succ 0 + Nat.succ 0 ≤ ↑j - 1 | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : swapped[j.prev] = arr[j] := by
simp [ij.symm, Fin.prev]
apply Vec.get_swap_left | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
⊢ swapped[j.prev] < swapped[first] | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ swapped[j.prev] < swapped[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ swapped[j.prev] < swapped[first] | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ arr[j] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : i < j := Nat.lt_of_le_of_lt (by assumption : i.val ≤ j.val - 1) (by assumption) | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ arr[j] < arr[first] | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝¹ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this✝ : swapped[j.prev] = arr[j]
this : i < j
⊢ arr[j] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.3 this | case inv₃.inl
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝¹ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this✝ : swapped[j.prev] = arr[j]
this : i < j
⊢ arr[j] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [ij.symm, Fin.prev] | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
⊢ swapped[j.prev] = arr[j] | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
⊢ swapped[i] = arr[j] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Vec.get_swap_left | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
⊢ swapped[i] = arr[j] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ ↑i ≤ ↑j - 1 | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i = ↑j - 1
this : swapped[j.prev] = arr[j]
⊢ ↑j - 1 < ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : swapped[j.prev] = arr[j.prev] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt ij
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (by assumption) | case inv₃.inr
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ swapped[j.prev] < swapped[first] | case inv₃.inr
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
this : swapped[j.prev] = arr[j.prev]
⊢ swapped[j.prev] < swapped[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₃.inr
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
this : swapped[j.prev] = arr[j.prev]
⊢ swapped[j.prev] < swapped[first] | case inv₃.inr
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
this : swapped[j.prev] = arr[j.prev]
⊢ arr[j.prev] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply inv.1 j.prev ij (by assumption) | case inv₃.inr
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
this : swapped[j.prev] = arr[j.prev]
⊢ arr[j.prev] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Vec.get_swap_neq | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ swapped[j.prev] = arr[j.prev] | case ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ i
case kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_gt ij | case ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ i
case kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ j | case kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt (by assumption) | case kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Fin.ne_of_val_ne | case ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ i | case ki.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ ¬↑j.prev = ↑i |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_gt ij | case ki.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ ¬↑j.prev = ↑i | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Fin.ne_of_val_ne | case kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ j.prev ≠ j | case kj.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ ¬↑j.prev = ↑j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_lt (by assumption) | case kj.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ ¬↑j.prev = ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
⊢ ↑j.prev < ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij✝² : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
ij✝¹ : i.prev < j.prev
this✝ : ↑j - 1 < ↑j
ij✝ : ↑i ≤ ↑j - 1
ij : ↑i < ↑j - 1
this : swapped[j.prev] = arr[j.prev]
⊢ j.prev < j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rfl | α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
LoopInvariant (arr.swap i j) first i.prev j.prev last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv✝ : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
inv : LoopInvariant swapped first i.prev j.prev last
⊢ partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | induction arr, first, i, j, fi, ij using partitionImpl.induct' with
| base => simp [*]
| step_lt => simp [*]
| step_ge arr first i j _ ij fi =>
simp [*]
have fj : first < j := Nat.lt_of_lt_of_le fi ij
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt fi
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt fj | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
⊢ (partitionImpl arr first i j fi ij).snd[first] = arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | simp [*] | case base
α : Type
inst✝ : Ord α
n : Nat
first i j : Fin n
arr✝ : Vec α n
first✝ i✝ j✝ : Fin n
fi✝ : first✝ ≤ i✝
ij✝ : i✝ ≤ j✝
x✝ : ¬first✝ < i✝
⊢ (partitionImpl arr✝ first✝ i✝ j✝ fi✝ ij✝).snd[first✝] = arr✝[first✝] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | simp [*] | case step_lt
α : Type
inst✝ : Ord α
n : Nat
first i j : Fin n
arr✝ : Vec α n
first✝ i✝ j✝ : Fin n
fi✝ : first✝ ≤ i✝
ij✝ : i✝ ≤ j✝
x✝³ : first✝ < i✝
x✝² : ↑first✝ ≤ ↑i✝ - 1
x✝¹ : arr✝[i✝] < arr✝[first✝]
x✝ : ↑i✝ - 1 ≤ ↑j✝
ih✝ : (partitionImpl arr✝ first✝ i✝.prev j✝ x✝² x✝).snd[first✝] = arr✝[first✝]
⊢ (partitionImpl arr✝ first✝ i✝ j✝ fi✝ ij✝).snd[first✝] = arr✝[first✝] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | simp [*] | case step_ge
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
⊢ (partitionImpl arr first i j fi✝ ij).snd[first] = arr[first] | case step_ge
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
⊢ (arr.swap i j)[first] = arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | have fj : first < j := Nat.lt_of_lt_of_le fi ij | case step_ge
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
⊢ (arr.swap i j)[first] = arr[first] | case step_ge
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ (arr.swap i j)[first] = arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | apply Vec.get_swap_neq | case step_ge
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ (arr.swap i j)[first] = arr[first] | case step_ge.ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ i
case step_ge.kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt fi | case step_ge.ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ i
case step_ge.kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ j | case step_ge.kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt fj | case step_ge.kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | apply Fin.ne_of_val_ne | case step_ge.ki
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ i | case step_ge.ki.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ ¬↑first = ↑i |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | exact Nat.ne_of_lt fi | case step_ge.ki.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ ¬↑first = ↑i | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | apply Fin.ne_of_val_ne | case step_ge.kj
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ first ≠ j | case step_ge.kj.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ ¬↑first = ↑j |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.first_eq | [143, 1] | [157, 28] | exact Nat.ne_of_lt fj | case step_ge.kj.h
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih✝ : (partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝).snd[first] = (arr.swap i j)[first]
fj : first < j
⊢ ¬↑first = ↑j | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let afterLoop := partitionImpl arr first last last fl (Nat.le_refl _) | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let mid := afterLoop.1 | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let arr' := afterLoop.2 | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : mid < n := Nat.lt_of_le_of_lt mid.property.2 last.isLt | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let swapped := arr'.swap first ⟨mid, by assumption⟩ | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : result.1 = mid := by
rw [←eq]
unfold partition
simp [afterLoop] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : result.2 = swapped := by
rw [←eq]
unfold partition
simp [afterLoop, dbgTraceIfShared] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have first_eq : arr'[first] = arr[first] := by
apply partitionImpl.first_eq | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let inv₀ : partitionImpl.LoopInvariant arr first last last last := by
apply partitionImpl.LoopInvariant.intro
. intro k lk kl
exact (Nat.lt_irrefl k (Nat.lt_trans kl lk)).elim
. intro k lk kl
exact (Nat.lt_irrefl k (Nat.lt_of_le_of_lt kl lk)).elim
. intro ll
exact (Nat.lt_irrefl last ll).elim | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | let inv := partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl _) inv₀ afterLoop (by rfl) | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have p₁ (k : Fin n) (fk : first ≤ k) (km : k.val < mid) : swapped[k] <o arr[first] := by
cases Nat.eq_or_lt_of_le fk with
| inl fk =>
have : swapped[k] = swapped[first] := by
simp [Fin.eq_of_val_eq fk]
rw [this]
have : swapped[first] = arr'[mid.val] := by
apply Vec.get_swap_left
rw [this, ←first_eq]
exact inv.3 (Fin.eq_of_val_eq fk ▸ km)
| inr fk =>
have : swapped[k] = arr'[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt fk
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt km
rw [this, ←first_eq]
exact inv.1 k fk km | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have p₂ : swapped[mid.val] = arr[first] := by
have : swapped[mid.val] = arr'[first] := by
apply Vec.get_swap_right
rw [this]
assumption | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have p₃ (k : Fin n) (mk : mid < k.val) (kl : k ≤ last) : ¬swapped[k] <o arr[first] := by
have : swapped[k] = arr'[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt (Nat.lt_of_le_of_lt mid.property.1 mk)
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt mk
rw [this, ←first_eq]
exact inv.2 k mk kl | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply And.intro | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
result.snd[result.fst.val] = arr[first] ∧
∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]
case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . simp [*]
apply p₁ | case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]
case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . apply And.intro
. simp [*]
. simp [*]
apply p₃ | case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | assumption | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
⊢ ↑mid.val < n | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [←eq] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ result.fst = mid | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (_root_.partition arr first last fl).fst = mid |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | unfold partition | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (_root_.partition arr first last fl).fst = mid | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (let result := partitionImpl arr first last last fl ⋯;
let mid := result.fst;
let arr_1 := result.snd;
(mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).fst =
mid |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | simp [afterLoop] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (let result := partitionImpl arr first last last fl ⋯;
let mid := result.fst;
let arr_1 := result.snd;
(mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).fst =
mid | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [←eq] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ result.snd = swapped | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (_root_.partition arr first last fl).snd = swapped |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | unfold partition | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (_root_.partition arr first last fl).snd = swapped | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (let result := partitionImpl arr first last last fl ⋯;
let mid := result.fst;
let arr_1 := result.snd;
(mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).snd =
swapped |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | simp [afterLoop, dbgTraceIfShared] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (let result := partitionImpl arr first last last fl ⋯;
let mid := result.fst;
let arr_1 := result.snd;
(mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).snd =
swapped | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply partitionImpl.first_eq | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
⊢ arr'[first] = arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply partitionImpl.LoopInvariant.intro | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ partitionImpl.LoopInvariant arr first last last last | case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first]
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . intro k lk kl
exact (Nat.lt_irrefl k (Nat.lt_trans kl lk)).elim | case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first]
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] | case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . intro k lk kl
exact (Nat.lt_irrefl k (Nat.lt_of_le_of_lt kl lk)).elim | case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] | case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . intro ll
exact (Nat.lt_irrefl last ll).elim | case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | intro k lk kl | case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first] | case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k < last
⊢ arr[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact (Nat.lt_irrefl k (Nat.lt_trans kl lk)).elim | case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k < last
⊢ arr[k] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | intro k lk kl | case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first] | case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k ≤ last
⊢ ¬arr[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact (Nat.lt_irrefl k (Nat.lt_of_le_of_lt kl lk)).elim | case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k ≤ last
⊢ ¬arr[k] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | intro ll | case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first] | case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
ll : last < last
⊢ arr[last] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact (Nat.lt_irrefl last ll).elim | case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
ll : last < last
⊢ arr[last] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rfl | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
⊢ partitionImpl arr first last last fl ⋯ = afterLoop | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | cases Nat.eq_or_lt_of_le fk with
| inl fk =>
have : swapped[k] = swapped[first] := by
simp [Fin.eq_of_val_eq fk]
rw [this]
have : swapped[first] = arr'[mid.val] := by
apply Vec.get_swap_left
rw [this, ←first_eq]
exact inv.3 (Fin.eq_of_val_eq fk ▸ km)
| inr fk =>
have : swapped[k] = arr'[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt fk
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt km
rw [this, ←first_eq]
exact inv.1 k fk km | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk : first ≤ k
km : ↑k < ↑mid.val
⊢ swapped[k] < arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : swapped[k] = swapped[first] := by
simp [Fin.eq_of_val_eq fk] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
⊢ swapped[k] < arr[first] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this : swapped[k] = swapped[first]
⊢ swapped[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [this] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this : swapped[k] = swapped[first]
⊢ swapped[k] < arr[first] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this : swapped[k] = swapped[first]
⊢ swapped[first] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : swapped[first] = arr'[mid.val] := by
apply Vec.get_swap_left | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this : swapped[k] = swapped[first]
⊢ swapped[first] < arr[first] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝³ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝³⟩
this✝² : result.fst = mid
this✝¹ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this✝ : swapped[k] = swapped[first]
this : swapped[first] = arr'[mid.val]
⊢ swapped[first] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [this, ←first_eq] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝³ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝³⟩
this✝² : result.fst = mid
this✝¹ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this✝ : swapped[k] = swapped[first]
this : swapped[first] = arr'[mid.val]
⊢ swapped[first] < arr[first] | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝³ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝³⟩
this✝² : result.fst = mid
this✝¹ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this✝ : swapped[k] = swapped[first]
this : swapped[first] = arr'[mid.val]
⊢ arr'[mid.val] < arr'[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact inv.3 (Fin.eq_of_val_eq fk ▸ km) | case inl
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝³ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝³⟩
this✝² : result.fst = mid
this✝¹ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this✝ : swapped[k] = swapped[first]
this : swapped[first] = arr'[mid.val]
⊢ arr'[mid.val] < arr'[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | simp [Fin.eq_of_val_eq fk] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
⊢ swapped[k] = swapped[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply Vec.get_swap_left | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first = ↑k
this : swapped[k] = swapped[first]
⊢ swapped[first] = arr'[mid.val] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : swapped[k] = arr'[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt fk
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt km | case inr
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ swapped[k] < arr[first] | case inr
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
this : swapped[k] = arr'[k]
⊢ swapped[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [this, ←first_eq] | case inr
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
this : swapped[k] = arr'[k]
⊢ swapped[k] < arr[first] | case inr
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
this : swapped[k] = arr'[k]
⊢ arr'[k] < arr'[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact inv.1 k fk km | case inr
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
this : swapped[k] = arr'[k]
⊢ arr'[k] < arr'[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply Vec.get_swap_neq | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ swapped[k] = arr'[k] | case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ first
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ ⟨↑mid.val, this✝¹⟩ |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_gt fk | case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ first
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ ⟨↑mid.val, this✝¹⟩ | case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ ⟨↑mid.val, this✝¹⟩ |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | . apply Fin.ne_of_val_ne
exact Nat.ne_of_lt km | case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ ⟨↑mid.val, this✝¹⟩ | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply Fin.ne_of_val_ne | case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ first | case ki.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ ¬↑k = ↑first |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact Nat.ne_of_gt fk | case ki.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ ¬↑k = ↑first | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply Fin.ne_of_val_ne | case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ k ≠ ⟨↑mid.val, this✝¹⟩ | case kj.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ ¬↑k = ↑⟨↑mid.val, this✝¹⟩ |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | exact Nat.ne_of_lt km | case kj.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
k : Fin n
fk✝ : first ≤ k
km : ↑k < ↑mid.val
fk : ↑first < ↑k
⊢ ¬↑k = ↑⟨↑mid.val, this✝¹⟩ | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : swapped[mid.val] = arr'[first] := by
apply Vec.get_swap_right | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
⊢ swapped[mid.val] = arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
this : swapped[mid.val] = arr'[first]
⊢ swapped[mid.val] = arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [this] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
this : swapped[mid.val] = arr'[first]
⊢ swapped[mid.val] = arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
this : swapped[mid.val] = arr'[first]
⊢ arr'[first] = arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | assumption | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
this : swapped[mid.val] = arr'[first]
⊢ arr'[first] = arr[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | apply Vec.get_swap_right | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
⊢ swapped[mid.val] = arr'[first] | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | have : swapped[k] = arr'[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt (Nat.lt_of_le_of_lt mid.property.1 mk)
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt mk | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬swapped[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
this : swapped[k] = arr'[k]
⊢ ¬swapped[k] < arr[first] |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partition.partition | [159, 1] | [238, 15] | rw [this, ←first_eq] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
this : swapped[k] = arr'[k]
⊢ ¬swapped[k] < arr[first] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
this : swapped[k] = arr'[k]
⊢ ¬arr'[k] < arr'[first] |
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