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/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
exact inv.2 k mk kl
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
this : swapped[k] = arr'[k]
⊢ ¬arr'[k] < arr'[first]
|
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 last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝² : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝²⟩
this✝¹ : result.fst = mid
this✝ : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
this : swapped[k] = arr'[k]
⊢ ¬arr'[k] < arr'[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply Vec.get_swap_neq
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ swapped[k] = arr'[k]
|
case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ first
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ swapped[k] = arr'[k]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt (Nat.lt_of_le_of_lt mid.property.1 mk)
|
case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ first
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
|
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
|
Please generate a tactic in lean4 to solve the state.
STATE:
case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ first
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
. apply Fin.ne_of_val_ne
exact Nat.ne_of_gt mk
|
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply Fin.ne_of_val_ne
|
case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ first
|
case ki.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑first
|
Please generate a tactic in lean4 to solve the state.
STATE:
case ki
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ first
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
exact Nat.ne_of_gt (Nat.lt_of_le_of_lt mid.property.1 mk)
|
case ki.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑first
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case ki.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑first
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply Fin.ne_of_val_ne
|
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
|
case kj.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑⟨↑mid.val, this✝¹⟩
|
Please generate a tactic in lean4 to solve the state.
STATE:
case kj
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ k ≠ ⟨↑mid.val, this✝¹⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
exact Nat.ne_of_gt mk
|
case kj.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑⟨↑mid.val, this✝¹⟩
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case kj.h
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
k : Fin n
mk : ↑mid.val < ↑k
kl : k ≤ last
⊢ ¬↑k = ↑⟨↑mid.val, this✝¹⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
simp [*]
|
case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]
|
case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply p₁
|
case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply And.intro
|
case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
case right.left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first]
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
. simp [*]
|
case right.left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first]
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first]
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
. simp [*]
apply p₃
|
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
simp [*]
|
case right.left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
simp [*]
|
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
|
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
partition.partition
|
[159, 1]
|
[238, 15]
|
apply p₃
|
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
{ inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
(Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl.get_lt
|
[253, 1]
|
[258, 20]
|
let p := quickSortImpl_permuted arr first last
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
lt : ↑k < first
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
lt : ↑k < first
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
lt : ↑k < first
⊢ (quickSortImpl arr first last)[k] = arr[k]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl.get_lt
|
[253, 1]
|
[258, 20]
|
exact p.get_lt lt
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
lt : ↑k < first
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
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 : Nat
last k : Fin n
lt : ↑k < first
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl.get_gt
|
[260, 1]
|
[265, 20]
|
let p := quickSortImpl_permuted arr first last
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
gt : k > last
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
gt : k > last
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
gt : k > last
⊢ (quickSortImpl arr first last)[k] = arr[k]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl.get_gt
|
[260, 1]
|
[265, 20]
|
exact p.get_gt gt
|
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last k : Fin n
gt : k > last
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
|
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 : Nat
last k : Fin n
gt : k > last
p : permuted n first (↑last) arr (quickSortImpl arr first last) := quickSortImpl_permuted arr first last
⊢ (quickSortImpl arr first last)[k] = arr[k]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
induction arr, first, last using quickSortImpl.induct' with
| base arr first last h =>
simp [*]
intro i j fi ij jl
have : first = last := Nat.le_antisymm (Nat.le_trans (Nat.le_trans fi ij) jl) (Nat.le_of_not_gt h)
subst this
have : i = last := Fin.eq_of_val_eq (Nat.le_antisymm (Nat.le_trans ij jl) fi)
subst this
have : i = j := Fin.eq_of_val_eq (Nat.le_antisymm ij jl)
subst this
apply Order.refl
| step arr first last lt parted eq ih₁ ih₂ =>
have : first < n := Nat.lt_trans lt last.isLt
simp [*]
intro i j fi ij jl
let mid := parted.1
have mn : mid.val < n := Nat.lt_of_le_of_lt mid.property.2 last.isLt
let sorted := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (mid.val + 1) last
let ⟨l, m, r⟩ := partition.partition arr ⟨first, by assumption⟩ last (Nat.le_of_lt lt) parted eq
have getL (k : Fin n) (fk : first ≤ k) (km : k < mid.val) : sorted[k] <o arr[first] := by
rw [quickSortImpl.get_lt (lt := (Nat.lt_trans km (Nat.lt_succ_self ..)))]
let p := quickSortImpl_permuted parted.2 first mid.val.prev
let ⟨k', ⟨index, fk', km'⟩⟩ := permuted_map_index_in_range_inv p k fk (Nat.le_sub_one_of_lt km)
let inv := index ▸ permuted_map_index p k'
rw [←inv]
apply l k' fk' (Nat.lt_of_le_of_lt km' (Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt km)))
have getM (k : Fin n) (eq : mid.val = k.val): sorted[k] = arr[first] := by
let mid' : Fin n := ⟨mid.val, mn⟩
have : k = mid' := by
apply Fin.eq_of_val_eq
simp [eq]
subst this
rw [quickSortImpl.get_lt (k := mid') (lt := Nat.lt_succ_self ..)]
cases Nat.eq_zero_or_pos mid.val with
| inl zero =>
have zero : mid.val = Fin.mk 0 mid.val.pos := by
apply Fin.eq_of_val_eq
simp [zero]
simp [zero] at *
exact m
| inr pos =>
rw [quickSortImpl.get_gt (k := mid') (gt := Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt pos))]
exact m
have getR (k : Fin n) (mk : mid.val < k) (kl : k ≤ last) : ¬sorted[k] <o arr[first] := by
let p := quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (mid.val + 1) last
let ⟨k', ⟨index, mk', kl'⟩⟩ := permuted_map_index_in_range_inv p k mk kl
let inv := index ▸ permuted_map_index p k'
rw [←inv]
have : mid.val.prev.val ≤ mid.val := by
simp
exact (Nat.sub_le ..)
rw [quickSortImpl.get_gt (gt := Nat.lt_of_le_of_lt this mk')]
apply r k' mk' kl'
have getLM (k : Fin n) (fk : first ≤ k) (km : k ≤ mid.val) : sorted[k] ≤o arr[first] := by
cases Nat.eq_or_lt_of_le km with
| inl km =>
rw [getM k km.symm]
apply Order.refl
| inr km => exact Order.le_of_lt (getL k fk km)
have getMR (k : Fin n) (mk : mid.val ≤ k) (kl : k ≤ last) : ¬sorted[k] <o arr[first] := by
match Nat.eq_or_lt_of_le mk with
| .inl mk =>
rw [getM k mk]
apply Order.irrefl
| .inr mk => exact getR k mk kl
cases Nat.range_split i j mid ij with
| lt im jm =>
rw [quickSortImpl.get_lt (lt := Nat.lt_trans im (Nat.lt_succ_self ..))]
rw [quickSortImpl.get_lt (lt := Nat.lt_trans jm (Nat.lt_succ_self ..))]
exact ih₁ i j fi ij (Nat.le_sub_of_add_le jm)
| ge mi mj => exact ih₂ i j mi ij jl
| split im mj =>
have h₁ : sorted[i] ≤o arr[first] := getLM i fi im
have h₂ : arr[first] ≤o sorted[j] := Order.le_of_not_lt (getMR j mj jl)
apply Order.trans h₁ h₂
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
⊢ sortedRange (quickSortImpl arr first last) first ↑last
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
⊢ sortedRange (quickSortImpl arr first last) first ↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp [*]
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ sortedRange (quickSortImpl arr first last) first ↑last
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ sortedRange arr first ↑last
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ sortedRange (quickSortImpl arr first last) first ↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
intro i j fi ij jl
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ sortedRange arr first ↑last
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ sortedRange arr first ↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : first = last := Nat.le_antisymm (Nat.le_trans (Nat.le_trans fi ij) jl) (Nat.le_of_not_gt h)
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
this : first = ↑last
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
subst this
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
this : first = ↑last
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
this : first = ↑last
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : i = last := Fin.eq_of_val_eq (Nat.le_antisymm (Nat.le_trans ij jl) fi)
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
this : i = last
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
subst this
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
this : i = last
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
last i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑last
h : ¬↑last < ↑last
fi : ↑last ≤ ↑i
this : i = last
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : i = j := Fin.eq_of_val_eq (Nat.le_antisymm ij jl)
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
this : i = j
⊢ (compare arr[i] arr[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
subst this
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
this : i = j
⊢ (compare arr[i] arr[j]).isLE = true
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i : Fin n
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
ij : i ≤ i
jl : ↑i ≤ ↑i
⊢ (compare arr[i] arr[i]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i j : Fin n
ij : i ≤ j
jl : ↑j ≤ ↑i
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
this : i = j
⊢ (compare arr[i] arr[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Order.refl
|
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i : Fin n
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
ij : i ≤ i
jl : ↑i ≤ ↑i
⊢ (compare arr[i] arr[i]).isLE = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case base
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
i : Fin n
h : ¬↑i < ↑i
fi : ↑i ≤ ↑i
ij : i ≤ i
jl : ↑i ≤ ↑i
⊢ (compare arr[i] arr[i]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : first < n := Nat.lt_trans lt last.isLt
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
⊢ sortedRange (quickSortImpl arr first last) first ↑last
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl arr first last) first ↑last
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
⊢ sortedRange (quickSortImpl arr first last) first ↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp [*]
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl arr first last) first ↑last
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last) first
↑last
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl arr first last) first ↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
intro i j fi ij jl
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last) first
↑last
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
⊢ sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last) first
↑last
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let mid := parted.1
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have mn : mid.val < n := Nat.lt_of_le_of_lt mid.property.2 last.isLt
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let sorted := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (mid.val + 1) last
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let ⟨l, m, r⟩ := partition.partition arr ⟨first, by assumption⟩ last (Nat.le_of_lt lt) parted eq
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have getL (k : Fin n) (fk : first ≤ k) (km : k < mid.val) : sorted[k] <o arr[first] := by
rw [quickSortImpl.get_lt (lt := (Nat.lt_trans km (Nat.lt_succ_self ..)))]
let p := quickSortImpl_permuted parted.2 first mid.val.prev
let ⟨k', ⟨index, fk', km'⟩⟩ := permuted_map_index_in_range_inv p k fk (Nat.le_sub_one_of_lt km)
let inv := index ▸ permuted_map_index p k'
rw [←inv]
apply l k' fk' (Nat.lt_of_le_of_lt km' (Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt km)))
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have getM (k : Fin n) (eq : mid.val = k.val): sorted[k] = arr[first] := by
let mid' : Fin n := ⟨mid.val, mn⟩
have : k = mid' := by
apply Fin.eq_of_val_eq
simp [eq]
subst this
rw [quickSortImpl.get_lt (k := mid') (lt := Nat.lt_succ_self ..)]
cases Nat.eq_zero_or_pos mid.val with
| inl zero =>
have zero : mid.val = Fin.mk 0 mid.val.pos := by
apply Fin.eq_of_val_eq
simp [zero]
simp [zero] at *
exact m
| inr pos =>
rw [quickSortImpl.get_gt (k := mid') (gt := Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt pos))]
exact m
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have getR (k : Fin n) (mk : mid.val < k) (kl : k ≤ last) : ¬sorted[k] <o arr[first] := by
let p := quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (mid.val + 1) last
let ⟨k', ⟨index, mk', kl'⟩⟩ := permuted_map_index_in_range_inv p k mk kl
let inv := index ▸ permuted_map_index p k'
rw [←inv]
have : mid.val.prev.val ≤ mid.val := by
simp
exact (Nat.sub_le ..)
rw [quickSortImpl.get_gt (gt := Nat.lt_of_le_of_lt this mk')]
apply r k' mk' kl'
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have getLM (k : Fin n) (fk : first ≤ k) (km : k ≤ mid.val) : sorted[k] ≤o arr[first] := by
cases Nat.eq_or_lt_of_le km with
| inl km =>
rw [getM k km.symm]
apply Order.refl
| inr km => exact Order.le_of_lt (getL k fk km)
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have getMR (k : Fin n) (mk : mid.val ≤ k) (kl : k ≤ last) : ¬sorted[k] <o arr[first] := by
match Nat.eq_or_lt_of_le mk with
| .inl mk =>
rw [getM k mk]
apply Order.irrefl
| .inr mk => exact getR k mk kl
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
cases Nat.range_split i j mid ij with
| lt im jm =>
rw [quickSortImpl.get_lt (lt := Nat.lt_trans im (Nat.lt_succ_self ..))]
rw [quickSortImpl.get_lt (lt := Nat.lt_trans jm (Nat.lt_succ_self ..))]
exact ih₁ i j fi ij (Nat.le_sub_of_add_le jm)
| ge mi mj => exact ih₂ i j mi ij jl
| split im mj =>
have h₁ : sorted[i] ≤o arr[first] := getLM i fi im
have h₂ : arr[first] ≤o sorted[j] := Order.le_of_not_lt (getMR j mj jl)
apply Order.trans h₁ h₂
|
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
assumption
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ first < n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ first < n
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_lt (lt := (Nat.lt_trans km (Nat.lt_succ_self ..)))]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
⊢ sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
⊢ sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let p := quickSortImpl_permuted parted.2 first mid.val.prev
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let ⟨k', ⟨index, fk', km'⟩⟩ := permuted_map_index_in_range_inv p k fk (Nat.le_sub_one_of_lt km)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let inv := index ▸ permuted_map_index p k'
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [←inv]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ parted.snd[k'] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ (quickSortImpl parted.snd first mid.val.prev)[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply l k' fk' (Nat.lt_of_le_of_lt km' (Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt km)))
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ parted.snd[k'] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ parted.snd[k'] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
decide
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ 0 < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
k : Fin n
fk : first ≤ ↑k
km : k < mid.val
p : permuted n first (↑mid.val.prev) parted.snd (quickSortImpl parted.snd first mid.val.prev) :=
quickSortImpl_permuted parted.snd first mid.val.prev
k' : Fin n
index : p.to_map k' = k
fk' : first ≤ ↑k'
km' : ↑k' ≤ ↑mid.val.prev
inv : parted.snd[k'] = (quickSortImpl parted.snd first mid.val.prev)[k] := index ▸ permuted_map_index p k'
⊢ 0 < 1
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let mid' : Fin n := ⟨mid.val, mn⟩
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
⊢ sorted[k] = arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ sorted[k] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
⊢ sorted[k] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : k = mid' := by
apply Fin.eq_of_val_eq
simp [eq]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ sorted[k] = arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
this : k = mid'
⊢ sorted[k] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ sorted[k] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
subst this
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
this : k = mid'
⊢ sorted[k] = arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ sorted[mid'] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
this : k = mid'
⊢ sorted[k] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_lt (k := mid') (lt := Nat.lt_succ_self ..)]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ sorted[mid'] = arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ sorted[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
cases Nat.eq_zero_or_pos mid.val with
| inl zero =>
have zero : mid.val = Fin.mk 0 mid.val.pos := by
apply Fin.eq_of_val_eq
simp [zero]
simp [zero] at *
exact m
| inr pos =>
rw [quickSortImpl.get_gt (k := mid') (gt := Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt pos))]
exact m
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Fin.eq_of_val_eq
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ k = mid'
|
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ ↑k = ↑mid'
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ k = mid'
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp [eq]
|
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ ↑k = ↑mid'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
k : Fin n
eq : ↑mid.val = ↑k
mid' : Fin n := ⟨↑mid.val, mn⟩
⊢ ↑k = ↑mid'
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have zero : mid.val = Fin.mk 0 mid.val.pos := by
apply Fin.eq_of_val_eq
simp [zero]
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero✝ : ↑mid.val = 0
zero : mid.val = ⟨0, ⋯⟩
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp [zero] at *
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero✝ : ↑mid.val = 0
zero : mid.val = ⟨0, ⋯⟩
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first (↑parted.fst.val - 1)
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
mid' : Fin n := ⟨↑mid.val, mn⟩
zero✝¹ : mid.val = ⟨0, ⋯⟩
getL : ∀ (k : Fin n), first ≤ ↑k → k < ⟨0, ⋯⟩ → sorted[k] < arr[first]
eq zero✝ zero : True
⊢ parted.snd[mid'] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero✝ : ↑mid.val = 0
zero : mid.val = ⟨0, ⋯⟩
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact m
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first (↑parted.fst.val - 1)
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
mid' : Fin n := ⟨↑mid.val, mn⟩
zero✝¹ : mid.val = ⟨0, ⋯⟩
getL : ∀ (k : Fin n), first ≤ ↑k → k < ⟨0, ⋯⟩ → sorted[k] < arr[first]
eq zero✝ zero : True
⊢ parted.snd[mid'] = arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first (↑parted.fst.val - 1)
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
mid' : Fin n := ⟨↑mid.val, mn⟩
zero✝¹ : mid.val = ⟨0, ⋯⟩
getL : ∀ (k : Fin n), first ≤ ↑k → k < ⟨0, ⋯⟩ → sorted[k] < arr[first]
eq zero✝ zero : True
⊢ parted.snd[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Fin.eq_of_val_eq
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ mid.val = ⟨0, ⋯⟩
|
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ ↑mid.val = ↑⟨0, ⋯⟩
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ mid.val = ⟨0, ⋯⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp [zero]
|
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ ↑mid.val = ↑⟨0, ⋯⟩
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
zero : ↑mid.val = 0
⊢ ↑mid.val = ↑⟨0, ⋯⟩
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_gt (k := mid') (gt := Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt pos))]
|
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
|
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
⊢ parted.snd[mid'] = arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
⊢ (quickSortImpl parted.snd first mid.val.prev)[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact m
|
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
⊢ parted.snd[mid'] = arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
⊢ parted.snd[mid'] = arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
decide
|
α : Type
inst✝ : Order α
n : Nat
arr✝ : Vec α n
first✝ : Nat
last : Fin n
lt : first✝ < ↑last
parted : { mid // first✝ ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr✝ ⟨first✝, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first✝ parted.fst.val.prev) first✝ ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first✝ parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first✝ < n
i j : Fin n
fi : first✝ ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first✝ ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first✝ mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first✝, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr✝[⟨first✝, this⟩]
m : parted.snd[parted.fst.val] = arr✝[⟨first✝, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr✝[⟨first✝, this⟩]
getL : ∀ (k : Fin n), first✝ ≤ ↑k → k < mid.val → sorted[k] < arr✝[first✝]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
arr : Vec α n
first : Nat
⊢ 0 < Nat.succ 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr✝ : Vec α n
first✝ : Nat
last : Fin n
lt : first✝ < ↑last
parted : { mid // first✝ ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq✝ : partition arr✝ ⟨first✝, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first✝ parted.fst.val.prev) first✝ ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first✝ parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first✝ < n
i j : Fin n
fi : first✝ ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first✝ ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first✝ mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first✝, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr✝[⟨first✝, this⟩]
m : parted.snd[parted.fst.val] = arr✝[⟨first✝, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr✝[⟨first✝, this⟩]
getL : ∀ (k : Fin n), first✝ ≤ ↑k → k < mid.val → sorted[k] < arr✝[first✝]
mid' : Fin n := ⟨↑mid.val, mn⟩
eq : ↑mid.val = ↑mid'
pos : ↑mid.val > 0
arr : Vec α n
first : Nat
⊢ 0 < Nat.succ 0
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let p := quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (mid.val + 1) last
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
⊢ ¬sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ ¬sorted[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let ⟨k', ⟨index, mk', kl'⟩⟩ := permuted_map_index_in_range_inv p k mk kl
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ ¬sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
⊢ ¬sorted[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
let inv := index ▸ permuted_map_index p k'
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
⊢ ¬sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬sorted[k] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [←inv]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have : mid.val.prev.val ≤ mid.val := by
simp
exact (Nat.sub_le ..)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_gt (gt := Nat.lt_of_le_of_lt this mk')]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬parted.snd[k'] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬(quickSortImpl parted.snd first mid.val.prev)[k'] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply r k' mk' kl'
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬parted.snd[k'] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this✝ : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this✝⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this✝⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this✝⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this✝⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
this : ↑mid.val.prev ≤ ↑mid.val
⊢ ¬parted.snd[k'] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
simp
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ↑mid.val.prev ≤ ↑mid.val
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ↑mid.val - 1 ≤ ↑mid.val
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ↑mid.val.prev ≤ ↑mid.val
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact (Nat.sub_le ..)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ↑mid.val - 1 ≤ ↑mid.val
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
k : Fin n
mk : mid.val < k
kl : k ≤ last
p : permuted n (↑mid.val + 1) (↑last) (quickSortImpl parted.snd first mid.val.prev)
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last) :=
quickSortImpl_permuted (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
k' : Fin n
index : p.to_map k' = k
mk' : ↑mid.val + 1 ≤ ↑k'
kl' : ↑k' ≤ ↑last
inv : (quickSortImpl parted.snd first mid.val.prev)[k'] =
(quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last)[k] :=
index ▸ permuted_map_index p k'
⊢ ↑mid.val - 1 ≤ ↑mid.val
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
cases Nat.eq_or_lt_of_le km with
| inl km =>
rw [getM k km.symm]
apply Order.refl
| inr km => exact Order.le_of_lt (getL k fk km)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km : k ≤ mid.val
⊢ sorted[k] ≤ arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km : k ≤ mid.val
⊢ sorted[k] ≤ arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [getM k km.symm]
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k = ↑mid.val
⊢ sorted[k] ≤ arr[first]
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k = ↑mid.val
⊢ arr[first] ≤ arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k = ↑mid.val
⊢ sorted[k] ≤ arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Order.refl
|
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k = ↑mid.val
⊢ arr[first] ≤ arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k = ↑mid.val
⊢ arr[first] ≤ arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact Order.le_of_lt (getL k fk km)
|
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k < ↑mid.val
⊢ sorted[k] ≤ arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
k : Fin n
fk : first ≤ ↑k
km✝ : k ≤ mid.val
km : ↑k < ↑mid.val
⊢ sorted[k] ≤ arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
match Nat.eq_or_lt_of_le mk with
| .inl mk =>
rw [getM k mk]
apply Order.irrefl
| .inr mk => exact getR k mk kl
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk : mid.val ≤ k
kl : k ≤ last
⊢ ¬sorted[k] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk : mid.val ≤ k
kl : k ≤ last
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [getM k mk]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val = ↑k
⊢ ¬sorted[k] < arr[first]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val = ↑k
⊢ ¬arr[first] < arr[first]
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val = ↑k
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Order.irrefl
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val = ↑k
⊢ ¬arr[first] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val = ↑k
⊢ ¬arr[first] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact getR k mk kl
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val < ↑k
⊢ ¬sorted[k] < arr[first]
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
k : Fin n
mk✝ : mid.val ≤ k
kl : k ≤ last
mk : ↑mid.val < ↑k
⊢ ¬sorted[k] < arr[first]
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_lt (lt := Nat.lt_trans im (Nat.lt_succ_self ..))]
|
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
rw [quickSortImpl.get_lt (lt := Nat.lt_trans jm (Nat.lt_succ_self ..))]
|
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl parted.snd first parted.fst.val.prev)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact ih₁ i j fi ij (Nat.le_sub_of_add_le jm)
|
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl parted.snd first parted.fst.val.prev)[j]).isLE =
true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.lt
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i < ↑mid.val
jm : ↑j < ↑mid.val
⊢ (compare (quickSortImpl parted.snd first parted.fst.val.prev)[i]
(quickSortImpl parted.snd first parted.fst.val.prev)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
exact ih₂ i j mi ij jl
|
case step.ge
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
mi : ↑mid.val < ↑i
mj : ↑mid.val < ↑j
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.ge
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
mi : ↑mid.val < ↑i
mj : ↑mid.val < ↑j
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have h₁ : sorted[i] ≤o arr[first] := getLM i fi im
|
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
have h₂ : arr[first] ≤o sorted[j] := Order.le_of_not_lt (getMR j mj jl)
|
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
h₂ : arr[first] ≤ sorted[j]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSortImpl_sortedRange
|
[267, 1]
|
[352, 30]
|
apply Order.trans h₁ h₂
|
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
h₂ : arr[first] ≤ sorted[j]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case step.split
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
(↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
getR : ∀ (k : Fin n), mid.val < k → k ≤ last → ¬sorted[k] < arr[first]
getLM : ∀ (k : Fin n), first ≤ ↑k → k ≤ mid.val → sorted[k] ≤ arr[first]
getMR : ∀ (k : Fin n), mid.val ≤ k → k ≤ last → ¬sorted[k] < arr[first]
im : ↑i ≤ ↑mid.val
mj : ↑mid.val ≤ ↑j
h₁ : sorted[i] ≤ arr[first]
h₂ : arr[first] ≤ sorted[j]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
(quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
if h : n > 0 then
simp [quickSort', h]
apply quickSortImpl_sortedRange
else
intro i
have : n = 0 := Nat.eq_zero_of_not_pos h
subst n
exact (Nat.not_lt_zero i.val i.isLt).elim
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
⊢ sortedRange (quickSort' arr) 0 (n - 1)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
⊢ sortedRange (quickSort' arr) 0 (n - 1)
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
simp [quickSort', h]
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : n > 0
⊢ sortedRange (quickSort' arr) 0 (n - 1)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : n > 0
⊢ sortedRange (quickSortImpl arr 0 ⟨n - 1, ⋯⟩) 0 (n - 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : n > 0
⊢ sortedRange (quickSort' arr) 0 (n - 1)
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
apply quickSortImpl_sortedRange
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : n > 0
⊢ sortedRange (quickSortImpl arr 0 ⟨n - 1, ⋯⟩) 0 (n - 1)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : n > 0
⊢ sortedRange (quickSortImpl arr 0 ⟨n - 1, ⋯⟩) 0 (n - 1)
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
intro i
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
⊢ sortedRange (quickSort' arr) 0 (n - 1)
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
⊢ sortedRange (quickSort' arr) 0 (n - 1)
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
have : n = 0 := Nat.eq_zero_of_not_pos h
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
this : n = 0
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
subst n
|
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
this : n = 0
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
α : Type
inst✝ : Order α
arr : Vec α 0
h : ¬0 > 0
i : Fin 0
⊢ ∀ (j : Fin 0), 0 ≤ ↑i → i ≤ j → ↑j ≤ 0 - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
h : ¬n > 0
i : Fin n
this : n = 0
⊢ ∀ (j : Fin n), 0 ≤ ↑i → i ≤ j → ↑j ≤ n - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort'_sortedRange
|
[354, 1]
|
[363, 46]
|
exact (Nat.not_lt_zero i.val i.isLt).elim
|
α : Type
inst✝ : Order α
arr : Vec α 0
h : ¬0 > 0
i : Fin 0
⊢ ∀ (j : Fin 0), 0 ≤ ↑i → i ≤ j → ↑j ≤ 0 - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
arr : Vec α 0
h : ¬0 > 0
i : Fin 0
⊢ ∀ (j : Fin 0), 0 ≤ ↑i → i ≤ j → ↑j ≤ 0 - 1 → (compare (quickSort' arr)[i] (quickSort' arr)[j]).isLE = true
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort_size
|
[365, 1]
|
[369, 25]
|
let sorted := quickSort' ⟨arr, rfl⟩
|
α : Type
inst✝ : Order α
arr : Array α
⊢ (quickSort arr).size = arr.size
|
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ (quickSort arr).size = arr.size
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
arr : Array α
⊢ (quickSort arr).size = arr.size
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort_size
|
[365, 1]
|
[369, 25]
|
show sorted.val.size = arr.size
|
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ (quickSort arr).size = arr.size
|
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ sorted.val.size = arr.size
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ (quickSort arr).size = arr.size
TACTIC:
|
https://github.com/pandaman64/QuickSortInLean.git
|
ab0aaee0aed280959328844f9a6cd13bf00c5935
|
QuickSortInLean/Sorted.lean
|
quickSort_size
|
[365, 1]
|
[369, 25]
|
simp [sorted.property]
|
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ sorted.val.size = arr.size
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Order α
arr : Array α
sorted : Vec α arr.size := quickSort' ⟨arr, ⋯⟩
⊢ sorted.val.size = arr.size
TACTIC:
|
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