url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.not_lt_of_le | [53, 1] | [61, 18] | simp [leOfOrd, cmp] at h | α : Type
inst✝ : Order α
x y : α
h : x ≤ y
cmp : compare x y = Ordering.gt
⊢ ¬y < x | α : Type
inst✝ : Order α
x y : α
cmp : compare x y = Ordering.gt
h : Ordering.gt.isLE = true
⊢ ¬y < x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : x ≤ y
cmp : compare x y = Ordering.gt
⊢ ¬y < x
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.not_lt_of_le | [53, 1] | [61, 18] | contradiction | α : Type
inst✝ : Order α
x y : α
cmp : compare x y = Ordering.gt
h : Ordering.gt.isLE = true
⊢ ¬y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
cmp : compare x y = Ordering.gt
h : Ordering.gt.isLE = true
⊢ ¬y < x
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | show (compare x y).isLE | α : Type
inst✝ : Order α
x y : α
h : x < y
⊢ x ≤ y | α : Type
inst✝ : Order α
x y : α
h : x < y
⊢ (compare x y).isLE = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : x < y
⊢ x ≤ y
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | simp [ltOfOrd] at h | α : Type
inst✝ : Order α
x y : α
h : x < y
⊢ (compare x y).isLE = true | α : Type
inst✝ : Order α
x y : α
h : compare x y = Ordering.lt
⊢ (compare x y).isLE = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : x < y
⊢ (compare x y).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | match cmp : compare x y with
| .lt => decide
| .eq | .gt =>
rw [cmp] at h
contradiction | α : Type
inst✝ : Order α
x y : α
h : compare x y = Ordering.lt
⊢ (compare x y).isLE = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : compare x y = Ordering.lt
⊢ (compare x y).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | decide | α : Type
inst✝ : Order α
x y : α
h cmp : compare x y = Ordering.lt
⊢ Ordering.lt.isLE = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h cmp : compare x y = Ordering.lt
⊢ Ordering.lt.isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | rw [cmp] at h | α : Type
inst✝ : Order α
x y : α
h : compare x y = Ordering.lt
cmp : compare x y = Ordering.gt
⊢ Ordering.gt.isLE = true | α : Type
inst✝ : Order α
x y : α
h : Ordering.gt = Ordering.lt
cmp : compare x y = Ordering.gt
⊢ Ordering.gt.isLE = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : compare x y = Ordering.lt
cmp : compare x y = Ordering.gt
⊢ Ordering.gt.isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Order.lean | Order.le_of_lt | [63, 1] | [70, 18] | contradiction | α : Type
inst✝ : Order α
x y : α
h : Ordering.gt = Ordering.lt
cmp : compare x y = Ordering.gt
⊢ Ordering.gt.isLE = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
x y : α
h : Ordering.gt = Ordering.lt
cmp : compare x y = Ordering.gt
⊢ Ordering.gt.isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | induction arr, first, i, j, fi, ij using partitionImpl.induct' with
| base arr first i j fi _ h =>
revert result
simp [*]
have : i = first := Fin.eq_of_val_eq (Nat.le_antisymm (Nat.le_of_not_lt h) fi)
exact this ▸ inv
| step_lt arr first i j _ _ fi _ lt _ ih =>
have inv : LoopInvariant arr first i.prev j last... | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j last : Fin n
fi : first ≤ i
ij : i ≤ j
inv : LoopInvariant arr first i j last
result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n
eq : partitionImpl arr first i j fi ij = result
⊢ LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j last : Fin n
fi : first ≤ i
ij : i ≤ j
inv : LoopInvariant arr first i j last
result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n
eq : partitionImpl arr first i j fi ij = result
⊢ LoopInvariant result.snd... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | revert result | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n
eq : partitionImpl arr first i j fi ij✝ = result
⊢ LoopInvariant result.snd first first ... | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ ∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionImpl arr first i j fi ij✝ = result → LoopInvariant result.snd first ... | Please generate a tactic in lean4 to solve the state.
STATE:
case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n
eq : partitionImpl arr firs... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [*] | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ ∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionImpl arr first i j fi ij✝ = result → LoopInvariant result.snd first ... | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ LoopInvariant arr first first j last | Please generate a tactic in lean4 to solve the state.
STATE:
case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ ∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionImpl ar... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : i = first := Fin.eq_of_val_eq (Nat.le_antisymm (Nat.le_of_not_lt h) fi) | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ LoopInvariant arr first first j last | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
this : i = first
⊢ LoopInvariant arr first first j last | Please generate a tactic in lean4 to solve the state.
STATE:
case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
⊢ LoopInvariant arr first first j last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact this ▸ inv | case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
this : i = first
⊢ LoopInvariant arr first first j last | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case base
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
h : ¬first < i
inv : LoopInvariant arr first i j last
this : i = first
⊢ LoopInvariant arr first first j last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have inv : LoopInvariant arr first i.prev j last := by
apply LoopInvariant.intro
. intro k ik kj
have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik
cases Nat.eq_or_lt_of_le ik with
| inl ik =>
simp [Fin.eq_of_val_eq ik] at lt
exact lt
| inr... | case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [*] at eq | case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply ih inv result eq | case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step_lt
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply LoopInvariant.intro | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . intro k ik kj
have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik
cases Nat.eq_or_lt_of_le ik with
| inl ik =>
simp [Fin.eq_of_val_eq ik] at lt
exact lt
| inr ik => exact inv.1 k ik kj | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case inv₂
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . exact inv.2 | case inv₂
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . intro ij
have ij : i.val ≤ j.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ij
cases Nat.eq_or_lt_of_le ij with
| inl ij =>
simp [Fin.eq_of_val_eq ij.symm]
assumption
| inr ij => exact inv.3 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | intro k ik kj | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | cases Nat.eq_or_lt_of_le ik with
| inl ik =>
simp [Fin.eq_of_val_eq ik] at lt
exact lt
| inr ik => exact inv.1 k ik kj | case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact ik | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [Fin.eq_of_val_eq ik] at lt | case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionImpl arr fir... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact lt | case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionImpl arr fir... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // fi... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.1 k ik kj | case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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✝ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.2 | case inv₂
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | intro 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have ij : i.val ≤ j.val := 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ ... |
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 =>
simp [Fin.eq_of_val_eq ij.symm]
assumption
| inr ij => exact inv.3 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✝¹ : i ≤ j
fi : first < i
x✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀... |
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : ... |
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
partitionI... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : ... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [Fin.eq_of_val_eq ij.symm] | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.3 ij | 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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
∀ (result : { mid // first ≤ mid ∧ mid ≤ j } × Vec α n),
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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✝¹ : ↑first ≤ ↑i - 1
lt : arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j
ih :
LoopInvariant arr first i.prev j last →
... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | let swapped := arr.swap i j | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have sf : swapped[first] = arr[first] := by
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 (Nat.lt_of_lt_of_le fi ij) | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | let result := partitionImpl swapped first i.prev j.prev (by assumption) (by assumption) | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | subst eq | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [*] | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have inv : LoopInvariant swapped first i.prev j.prev last := by
apply LoopInvariant.intro
. intro k ik kj
have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik
cases Nat.eq_or_lt_of_le ik with
| inl ik =>
have : swapped[k] = arr[j] := by
simp [F... | case step_ge
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 ... | case step_ge
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact ih inv result (by rfl) | case step_ge
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case step_ge
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.p... |
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 : 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 } × ... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
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 fi | 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 : 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.p... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 (Nat.lt_of_lt_of_le fi ij) | 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 : 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.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_lt fi | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_lt (Nat.lt_of_lt_of_le fi ij) | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply LoopInvariant.intro | 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 } × ... | case inv₁
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . intro k ik kj
have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik
cases Nat.eq_or_lt_of_le ik with
| inl ik =>
have : swapped[k] = arr[j] := by
simp [Fin.eq_of_val_eq ik.symm]
apply Vec.get_swap_left
rw [this, sf]
exact inv.3 (Nat.lt_of_lt_of_le... | case inv₁
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... | case inv₂
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . intro k jk kl
have jk : j.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt (Nat.zero_lt_of_lt (Nat.lt_of_lt_of_le fi ij)))]
exact jk
cases Nat.eq_or_lt_of_le jk with
| inl jk =>
have : swapped[k] = arr[i] := by
simp [Fin.eq_of_val_eq jk.symm]
apply Vec.get_swap_right
rw [... | case inv₂
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | . intro ij
have : j.val - 1 < j.val := by
show j.val - 1 + 1 ≤ j.val
rw [Nat.sub_add_cancel (Nat.zero_lt_of_lt (Nat.lt_of_lt_of_le fi (by assumption)))]
apply Nat.le_refl
have ij : i.val ≤ j.val - 1 := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ij
cases Nat.eq_or_lt_of_le ij with... | 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | intro k ik kj | case inv₁
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... | case inv₁
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have ik : i.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
exact ik | case inv₁
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... | case inv₁
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | cases Nat.eq_or_lt_of_le ik with
| inl ik =>
have : swapped[k] = arr[j] := by
simp [Fin.eq_of_val_eq ik.symm]
apply Vec.get_swap_left
rw [this, sf]
exact inv.3 (Nat.lt_of_lt_of_le (Fin.eq_of_val_eq ik.symm ▸ kj) (Nat.sub_le ..))
| inr ik =>
have kj : k < j := (Nat.lt_of_lt_of_le kj (Nat.sub_le ..))
ha... | case inv₁
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁
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... |
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 } × ... | 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 } × ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact ik | 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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : swapped[k] = arr[j] := by
simp [Fin.eq_of_val_eq ik.symm]
apply Vec.get_swap_left | case inv₁.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 : 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... | case inv₁.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₁.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 : 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... | case inv₁.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.3 (Nat.lt_of_lt_of_le (Fin.eq_of_val_eq ik.symm ▸ kj) (Nat.sub_le ..)) | case inv₁.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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [Fin.eq_of_val_eq ik.symm] | 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 } × ... | 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 } × ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
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 : 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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have kj : k < j := (Nat.lt_of_lt_of_le kj (Nat.sub_le ..)) | case inv₁.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 : 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... | case inv₁.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : swapped[k] = arr[k] := by
apply Vec.get_swap_neq
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt ik
. apply Fin.ne_of_val_ne
exact Nat.ne_of_lt kj | case inv₁.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 : 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... | case inv₁.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₁.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 : 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... | case inv₁.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.1 k ik kj | case inv₁.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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₁.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 : 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.... |
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 : 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 } × ... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
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 ik | 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 : 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.p... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 kj | 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 : 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.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_gt ik | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_lt kj | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | intro k jk kl | case inv₂
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... | case inv₂
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have jk : j.val ≤ k.val := by
rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt (Nat.zero_lt_of_lt (Nat.lt_of_lt_of_le fi ij)))]
exact jk | case inv₂
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... | case inv₂
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | cases Nat.eq_or_lt_of_le jk with
| inl jk =>
have : swapped[k] = arr[i] := by
simp [Fin.eq_of_val_eq jk.symm]
apply Vec.get_swap_right
rw [this, sf]
assumption
| inr jk =>
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_l... | case inv₂
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂
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... |
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.zero_lt_of_lt (Nat.lt_of_lt_of_le fi 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 } × ... | 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 } × ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact jk | 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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : swapped[k] = arr[i] := by
simp [Fin.eq_of_val_eq jk.symm]
apply Vec.get_swap_right | case inv₂.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 : 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... | case inv₂.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₂.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 : 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... | case inv₂.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | assumption | case inv₂.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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | simp [Fin.eq_of_val_eq jk.symm] | 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 } × ... | 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 } × ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | apply Vec.get_swap_right | 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 } × ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | 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 ij jk)
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt jk | case inv₂.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 : 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... | case inv₂.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | rw [this, sf] | case inv₂.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 : 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... | case inv₂.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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact inv.2 k jk kl | case inv₂.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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inv₂.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 : 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.... |
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 : 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 } × ... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 la... |
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 (Nat.lt_of_le_of_lt ij jk) | 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 : 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.p... | 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 : 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.p... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 jk | 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 : 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.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_gt (Nat.lt_of_le_of_lt ij jk) | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
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 : 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.p... | 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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | exact Nat.ne_of_gt jk | 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 : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | intro 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... | 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 ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | partitionImpl.loop_invariant | [21, 1] | [141, 33] | have : j.val - 1 < j.val := by
show j.val - 1 + 1 ≤ j.val
rw [Nat.sub_add_cancel (Nat.zero_lt_of_lt (Nat.lt_of_lt_of_le fi (by assumption)))]
apply Nat.le_refl | 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 ≤ ... | 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 ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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.pre... |
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