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/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | contrapose! h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = head? c | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | exact ⟨head c (ToMultiset.not_isEmpty_of_mem hx), MinHeap.head_mem _ _, pc, by intro h'; simp [← h'] at h,
fun h' ↦ (h'.trans_le (head_le c _ hx)).ne rfl⟩ | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | intro h' | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ head c ⋯ ≠ x | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ head c ⋯ ≠ x
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | simp [← h'] at h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.eq_head_of_forall_le | [83, 1] | [87, 57] | apply WithTop.coe_injective | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ x = head c hc | case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ x = head c hc
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.eq_head_of_forall_le | [83, 1] | [87, 57] | simpa using coe_eq_head?_of_forall_le c p hc pc x hx h | case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc)
TACTIC:... |
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | by_cases hc : isEmpty c | C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMulti... | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | simp [head?_eq_top c hc, hc] | case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | apply le_antisymm | case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | have := head?_le c | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | aesop | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | have := head?_mem_toMultiset_map c hc | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | apply Multiset.inf_le | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | aesop | case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/Size.lean | isEmpty_eq_decide_size | [31, 1] | [32, 58] | simp only [← isEmpty_iff_size_eq_zero, Bool.decide_coe] | C : Type u_1
α : Type ?u.565
inst✝ : Size C
c : C
⊢ isEmpty c = decide (size c = 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type ?u.565
inst✝ : Size C
c : C
⊢ isEmpty c = decide (size c = 0)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/Size.lean | isEmpty_eq_size_beq_zero | [34, 1] | [36, 23] | rw [isEmpty_eq_decide_size] | C : Type u_1
α : Type ?u.736
inst✝ : Size C
c : C
⊢ isEmpty c = (size c == 0) | C : Type u_1
α : Type ?u.736
inst✝ : Size C
c : C
⊢ decide (size c = 0) = (size c == 0) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type ?u.736
inst✝ : Size C
c : C
⊢ isEmpty c = (size c == 0)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/Size.lean | isEmpty_eq_size_beq_zero | [34, 1] | [36, 23] | cases size c <;> rfl | C : Type u_1
α : Type ?u.736
inst✝ : Size C
c : C
⊢ decide (size c = 0) = (size c == 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type ?u.736
inst✝ : Size C
c : C
⊢ decide (size c = 0) = (size c == 0)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.head?_isSome | [18, 1] | [19, 38] | simp | C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).head?.isSome = true ↔ head✝ :: tail✝ ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).head?.isSome = true ↔ head✝ :: tail✝ ≠ []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.tail?_isSome | [21, 1] | [22, 38] | simp | C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).tail?.isSome = true ↔ head✝ :: tail✝ ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).tail?.isSome = true ↔ head✝ :: tail✝ ≠ []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_eq_decide_eq_nil | [35, 1] | [36, 29] | cases l <;> simp [isEmpty] | C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l = []) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l = [])
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_eq_decide_length | [38, 1] | [39, 29] | cases l <;> simp [isEmpty] | C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l.length = 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l.length = 0)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_reverse | [41, 1] | [43, 33] | rw [isEmpty_eq_decide_eq_nil, isEmpty_eq_decide_eq_nil] | C : Type u_1
α : Type u_2
l : List α
⊢ l.reverse.isEmpty = l.isEmpty | C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = []) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.reverse.isEmpty = l.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_reverse | [41, 1] | [43, 33] | simp only [reverse_eq_nil_iff] | C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = []) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = [])
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.reverse_dropLast | [45, 1] | [50, 38] | rw [dropLast, reverse_cons x, reverse_cons x, reverse_dropLast (_::_),
tail_append_of_ne_nil] <;> simp | C : Type u_1
α : Type u_2
l✝ l : List α
x head✝ : α
tail✝ : List α
⊢ (x :: head✝ :: tail✝).dropLast.reverse = (x :: head✝ :: tail✝).reverse.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l✝ l : List α
x head✝ : α
tail✝ : List α
⊢ (x :: head✝ :: tail✝).dropLast.reverse = (x :: head✝ :: tail✝).reverse.tail
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.back?_toArray | [52, 1] | [55, 7] | rw [Array.back?, Array.get?_eq_data_get?, getLast?_eq_get?] | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).back? = l.getLast? | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).back? = l.getLast?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.back?_toArray | [52, 1] | [55, 7] | simp | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_data | [62, 1] | [63, 46] | rw [List.isEmpty_eq_decide_length, isEmpty] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.isEmpty = a.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_toList | [65, 1] | [66, 31] | rw [isEmpty_data, toList_eq] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.isEmpty = a.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_toListRev | [68, 1] | [69, 56] | rw [toListRev_eq, List.isEmpty_reverse, isEmpty_data] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.isEmpty = a.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.toList_length | [71, 1] | [72, 30] | rw [toList_eq, data_length] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.length = a.size | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.length = a.size
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.get?_data | [74, 1] | [77, 31] | ext i | C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.get? = a.get? | case h.a
C : Type u_1
α : Type u_2
a : Array α
i : ℕ
a✝ : α
⊢ a✝ ∈ a.data.get? i ↔ a✝ ∈ a.get? i | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.get? = a.get?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.get?_data | [74, 1] | [77, 31] | rw [Array.get?_eq_data_get?] | case h.a
C : Type u_1
α : Type u_2
a : Array α
i : ℕ
a✝ : α
⊢ a✝ ∈ a.data.get? i ↔ a✝ ∈ a.get? i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
C : Type u_1
α : Type u_2
a : Array α
i : ℕ
a✝ : α
⊢ a✝ ∈ a.data.get? i ↔ a✝ ∈ a.get? i
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.get?_toList | [79, 1] | [80, 28] | rw [toList_eq, get?_data] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.get? = a.get? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.get? = a.get?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.get?_toListRev | [82, 1] | [84, 75] | rw [toListRev_eq, List.get?_reverse _ (by rwa [data_length]), get?_data] | C : Type u_1
α : Type u_2
a : Array α
i : ℕ
h : i < a.size
⊢ a.toListRev.get? i = a.get? (a.size - 1 - i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
i : ℕ
h : i < a.size
⊢ a.toListRev.get? i = a.get? (a.size - 1 - i)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.get?_toListRev | [82, 1] | [84, 75] | rwa [data_length] | C : Type u_1
α : Type u_2
a : Array α
i : ℕ
h : i < a.size
⊢ i < a.data.length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
i : ℕ
h : i < a.size
⊢ i < a.data.length
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | cases a.size.eq_zero_or_pos | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.head? = a.back? | case inl
C : Type u_1
α : Type u_2
a : Array α
h✝ : a.size = 0
⊢ a.toListRev.head? = a.back?
case inr
C : Type u_1
α : Type u_2
a : Array α
h✝ : a.size > 0
⊢ a.toListRev.head? = a.back? | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.head? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | case inl h =>
rw [toListRev_eq, back?, ← get?_data]
simp [List.length_eq_zero.mp h] | C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.toListRev.head? = a.back? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.toListRev.head? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | case inr h =>
rw [← List.get?_zero, get?_toListRev _ _ h, Nat.sub_zero]
rfl | C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.toListRev.head? = a.back? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.toListRev.head? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | rw [toListRev_eq, back?, ← get?_data] | C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.toListRev.head? = a.back? | C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.data.reverse.head? = a.data.get? (a.size - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.toListRev.head? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | simp [List.length_eq_zero.mp h] | C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.data.reverse.head? = a.data.get? (a.size - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size = 0
⊢ a.data.reverse.head? = a.data.get? (a.size - 1)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | rw [← List.get?_zero, get?_toListRev _ _ h, Nat.sub_zero] | C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.toListRev.head? = a.back? | C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.get? (a.size - 1) = a.back? | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.toListRev.head? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.head?_toListRev | [86, 1] | [93, 8] | rfl | C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.get? (a.size - 1) = a.back? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
h : a.size > 0
⊢ a.get? (a.size - 1) = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.getLast?_toList | [95, 1] | [97, 7] | rw [back?, get?_eq_data_get?, List.getLast?_eq_get?] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.getLast? = a.back? | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.get? (a.toList.length - 1) = a.data.get? (a.size - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.getLast? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.getLast?_toList | [95, 1] | [97, 7] | simp | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.get? (a.toList.length - 1) = a.data.get? (a.size - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.get? (a.toList.length - 1) = a.data.get? (a.size - 1)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.getLast?_data | [99, 1] | [101, 27] | simp [← getLast?_toList] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.getLast? = a.back? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.getLast? = a.back?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.dropLast_toList | [103, 1] | [104, 7] | simp | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.dropLast = a.pop.toList | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.dropLast = a.pop.toList
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | ToList.lawfulEmptyCollection_iff | [127, 1] | [130, 20] | rw [ToMultiset.lawfulEmptyCollection_iff] | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
⊢ LawfulEmptyCollection C α ↔ toList ∅ = [] | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toList ∅ = [] | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
⊢ LawfulEmptyCollection C α ↔ toList ∅ = []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | ToList.lawfulEmptyCollection_iff | [127, 1] | [130, 20] | simp [toMultiset] | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toList ∅ = [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toList ∅ = []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | toList_empty | [134, 1] | [137, 49] | rwa [ToList.lawfulEmptyCollection_iff] at inst | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
inst : LawfulEmptyCollection C α
⊢ toList ∅ = [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : EmptyCollection C
inst : LawfulEmptyCollection C α
⊢ toList ∅ = []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | size_eq_size_toArray | [143, 1] | [144, 31] | simp [size_eq_length_toList] | C : Type u_1
α : Type u_2
inst✝ : ToList C α
c : C
⊢ size c = (toArray c).size | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝ : ToList C α
c : C
⊢ size c = (toArray c).size
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | isEmpty_toList | [155, 1] | [156, 84] | rw [isEmpty_eq_decide_size, List.isEmpty_eq_decide_length, size_eq_length_toList] | C : Type u_1
α : Type u_2
inst✝ : ToList C α
c : C
⊢ (toList c).isEmpty = isEmpty c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝ : ToList C α
c : C
⊢ (toList c).isEmpty = isEmpty c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | front?_isSome | [179, 1] | [181, 85] | rwa [front?_def, List.head?_isSome, ← List.not_isEmpty_iff_ne_nil, isEmpty_toList] | C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : Front C α
c : C
h : ¬isEmpty c = true
⊢ (front? c).isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : Front C α
c : C
h : ¬isEmpty c = true
⊢ (front? c).isSome = true
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | front_def | [183, 1] | [186, 64] | simpa using (front_mem c h).symm | C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : Front C α
c : C
h : ¬isEmpty c = true
⊢ some (front c h) = some ((front? c).get ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : Front C α
c : C
h : ¬isEmpty c = true
⊢ some (front c h) = some ((front? c).get ⋯)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | back?_isSome | [202, 1] | [204, 87] | rwa [back?_def, List.getLast?_isSome, ← List.not_isEmpty_iff_ne_nil, isEmpty_toList] | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : Back C α
c : C
h : ¬isEmpty c = true
⊢ (back? c).isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : Back C α
c : C
h : ¬isEmpty c = true
⊢ (back? c).isSome = true
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | back_def | [206, 1] | [209, 63] | simpa using (back_mem c h).symm | C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : Back C α
c : C
h : ¬isEmpty c = true
⊢ some (back c h) = some ((back? c).get ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : ToList C α
inst✝ : Back C α
c : C
h : ¬isEmpty c = true
⊢ some (back c h) = some ((back? c).get ⋯)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | ToList.RandomAccess.get_toArray | [258, 1] | [260, 31] | rw [get_eq_get_toArray] | C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : RandomAccess C α
c : C
i : Fin (toArray c).size
⊢ (toArray c).get i = get c (Fin.cast ⋯ i) | C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : RandomAccess C α
c : C
i : Fin (toArray c).size
⊢ (toArray c).get i = (toArray c).get (Fin.cast ⋯ (Fin.cast ⋯ i)) | Please generate a tactic in lean4 to solve the state.
STATE:
C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : RandomAccess C α
c : C
i : Fin (toArray c).size
⊢ (toArray c).get i = get c (Fin.cast ⋯ i)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | ToList.RandomAccess.get_toArray | [258, 1] | [260, 31] | rfl | C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : RandomAccess C α
c : C
i : Fin (toArray c).size
⊢ (toArray c).get i = (toArray c).get (Fin.cast ⋯ (Fin.cast ⋯ i)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C✝ : Type u_1
α✝ : Type u_2
C : Type u_3
α : Type u_4
inst✝¹ : ToList C α
inst✝ : RandomAccess C α
c : C
i : Fin (toArray c).size
⊢ (toArray c).get i = (toArray c).get (Fin.cast ⋯ (Fin.cast ⋯ i))
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | Multiset.inf_eq_top | [13, 1] | [15, 7] | rw [le_antisymm_iff] | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf = ⊤ ↔ ∀ a ∈ m, a = ⊤ | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf = ⊤ ↔ ∀ a ∈ m, a = ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | Multiset.inf_eq_top | [13, 1] | [15, 7] | simp | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | intro hc | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head? c ≠ ⊤ | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head? c ≠ ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | have := hc ▸ (head?_mem_toMultiset_map c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | simp at this | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_def | [58, 1] | [61, 71] | simpa using MinHeap.coe_head_eq_head? c h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ ↑(head c h) = ↑((head? c).untop ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ ↑(head c h) = ↑((head? c).untop ⋯)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem_toMultiset | [63, 1] | [66, 20] | obtain ⟨x, hx₁, hx₂⟩ := Multiset.mem_map.mp (head?_mem_toMultiset_map c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ toMultiset c | case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ toMultiset c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem_toMultiset | [63, 1] | [66, 20] | simp [hx₁, ← hx₂] | case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem | [68, 1] | [70, 52] | exact mem_toMultiset.mp (head_mem_toMultiset c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_le | [72, 1] | [74, 53] | simpa [WithTop.untop_le_iff] using head?_le c x hx | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
x : α
hx : x ∈ c
⊢ head c ⋯ ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
x : α
hx : x ∈ c
⊢ head c ⋯ ≤ x
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | contrapose! h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = head? c | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | exact ⟨head c (ToMultiset.not_isEmpty_of_mem hx), MinHeap.head_mem _ _, pc, by intro h'; simp [← h'] at h,
fun h' ↦ (h'.trans_le (head_le c _ hx)).ne rfl⟩ | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ ∃ y ∈ c, p y ∧ y ≠ x ∧ ¬x < y
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | intro h' | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ head c ⋯ ≠ x | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
⊢ head c ⋯ ≠ x
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.coe_eq_head?_of_forall_le | [76, 1] | [81, 52] | simp [← h'] at h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ↑x ≠ head? c
h' : head c ⋯ = x
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.eq_head_of_forall_le | [83, 1] | [87, 57] | apply WithTop.coe_injective | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ x = head c hc | case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ x = head c hc
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.eq_head_of_forall_le | [83, 1] | [87, 57] | simpa using coe_eq_head?_of_forall_le c p hc pc x hx h | case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
p : α → Prop
hc : ¬isEmpty c = true
pc : p (head c hc)
x : α
hx : x ∈ c
h : ∀ y ∈ c, p y → y ≠ x → x < y
⊢ ↑x = ↑(head c hc)
TACTIC:... |
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | by_cases hc : isEmpty c | C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMulti... | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | simp [head?_eq_top c hc, hc] | case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | apply le_antisymm | case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c = (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | have := head?_le c | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | aesop | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : ∀ x ∈ c, head? c ≤ ↑x
⊢ head? c ≤ (Multiset.map WithTop.some (toMultiset c)).inf
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | have := head?_mem_toMultiset_map c hc | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | apply Multiset.inf_le | case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c | case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ (Multiset.map WithTop.some (toMultiset c)).inf ≤ head? c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_def | [95, 1] | [104, 12] | aesop | case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
C : Type u_1
α : Type u_2
inst✝¹ : LinearOrder α
inst✝ : MinHeap C α
c : C
hc : ¬isEmpty c = true
this : head? c ∈ Multiset.map WithTop.some (toMultiset c)
⊢ head? c ∈ Multiset.map WithTop.some (toMultiset c)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToFinset.lean | ToFinset.lawfulEmptyCollection_iff | [34, 1] | [38, 7] | rw [ToMultiset.lawfulEmptyCollection_iff] | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ LawfulEmptyCollection C α ↔ toFinset ∅ = ∅ | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toFinset ∅ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ LawfulEmptyCollection C α ↔ toFinset ∅ = ∅
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToFinset.lean | ToFinset.lawfulEmptyCollection_iff | [34, 1] | [38, 7] | change (toFinset (∅ : C)).val = ∅ ↔ _ | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toFinset ∅ = ∅ | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ (toFinset ∅).val = ∅ ↔ toFinset ∅ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ ToMultiset.toMultiset ∅ = 0 ↔ toFinset ∅ = ∅
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToFinset.lean | ToFinset.lawfulEmptyCollection_iff | [34, 1] | [38, 7] | simp | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ (toFinset ∅).val = ∅ ↔ toFinset ∅ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
⊢ (toFinset ∅).val = ∅ ↔ toFinset ∅ = ∅
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToFinset.lean | toFinset_empty | [42, 1] | [45, 51] | rwa [ToFinset.lawfulEmptyCollection_iff] at inst | α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
inst : LawfulEmptyCollection C α
⊢ toFinset ∅ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
C : Type u_2
α : Type u_3
inst✝⁴ : ToFinset C α
c : C
inst✝³ : EmptyCollection C
inst✝² : LawfulEmptyCollection C α
inst✝¹ : ToFinset C α
inst✝ : EmptyCollection C
inst : LawfulEmptyCollection C α
⊢ toFinset ∅ = ∅
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Forest.lean | Forest.roots_subset_support | [26, 1] | [30, 76] | induction f with
| nil => rfl
| node a c s _ ihs =>
exact Set.insert_subset_insert (Set.subset_union_of_subset_right ihs _) | α : Type u_1
f : Forest α
⊢ f.roots ⊆ f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : Forest α
⊢ f.roots ⊆ f.support
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Forest.lean | Forest.roots_subset_support | [26, 1] | [30, 76] | rfl | case nil
α : Type u_1
⊢ nil.roots ⊆ nil.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
⊢ nil.roots ⊆ nil.support
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Forest.lean | Forest.roots_subset_support | [26, 1] | [30, 76] | exact Set.insert_subset_insert (Set.subset_union_of_subset_right ihs _) | case node
α : Type u_1
a : α
c s : Forest α
child_ih✝ : c.roots ⊆ c.support
ihs : s.roots ⊆ s.support
⊢ (node a c s).roots ⊆ (node a c s).support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case node
α : Type u_1
a : α
c s : Forest α
child_ih✝ : c.roots ⊆ c.support
ihs : s.roots ⊆ s.support
⊢ (node a c s).roots ⊆ (node a c s).support
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.head?_isSome | [18, 1] | [19, 38] | simp | C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).head?.isSome = true ↔ head✝ :: tail✝ ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).head?.isSome = true ↔ head✝ :: tail✝ ≠ []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.tail?_isSome | [21, 1] | [22, 38] | simp | C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).tail?.isSome = true ↔ head✝ :: tail✝ ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
head✝ : α
tail✝ : List α
⊢ (head✝ :: tail✝).tail?.isSome = true ↔ head✝ :: tail✝ ≠ []
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_eq_decide_eq_nil | [35, 1] | [36, 29] | cases l <;> simp [isEmpty] | C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l = []) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l = [])
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_eq_decide_length | [38, 1] | [39, 29] | cases l <;> simp [isEmpty] | C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l.length = 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.isEmpty = decide (l.length = 0)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_reverse | [41, 1] | [43, 33] | rw [isEmpty_eq_decide_eq_nil, isEmpty_eq_decide_eq_nil] | C : Type u_1
α : Type u_2
l : List α
⊢ l.reverse.isEmpty = l.isEmpty | C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = []) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ l.reverse.isEmpty = l.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.isEmpty_reverse | [41, 1] | [43, 33] | simp only [reverse_eq_nil_iff] | C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = []) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ decide (l.reverse = []) = decide (l = [])
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.reverse_dropLast | [45, 1] | [50, 38] | rw [dropLast, reverse_cons x, reverse_cons x, reverse_dropLast (_::_),
tail_append_of_ne_nil] <;> simp | C : Type u_1
α : Type u_2
l✝ l : List α
x head✝ : α
tail✝ : List α
⊢ (x :: head✝ :: tail✝).dropLast.reverse = (x :: head✝ :: tail✝).reverse.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l✝ l : List α
x head✝ : α
tail✝ : List α
⊢ (x :: head✝ :: tail✝).dropLast.reverse = (x :: head✝ :: tail✝).reverse.tail
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.back?_toArray | [52, 1] | [55, 7] | rw [Array.back?, Array.get?_eq_data_get?, getLast?_eq_get?] | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).back? = l.getLast? | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).back? = l.getLast?
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | List.back?_toArray | [52, 1] | [55, 7] | simp | C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
l : List α
⊢ (toArray l).data.get? ((toArray l).size - 1) = l.get? (l.length - 1)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_data | [62, 1] | [63, 46] | rw [List.isEmpty_eq_decide_length, isEmpty] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.data.isEmpty = a.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_toList | [65, 1] | [66, 31] | rw [isEmpty_data, toList_eq] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toList.isEmpty = a.isEmpty
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/ToList.lean | Array.isEmpty_toListRev | [68, 1] | [69, 56] | rw [toListRev_eq, List.isEmpty_reverse, isEmpty_data] | C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.isEmpty = a.isEmpty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
a : Array α
⊢ a.toListRev.isEmpty = a.isEmpty
TACTIC:
|
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