Theorem: project.deleteMin_non_empty
Proof state before:
case a.a
α : Type u_1
le : α → α → Bool
le_trans : ∀ (x y z : α), le x y = true → le y z = true → le x z = true
le_rfl : ∀ (z : α), le z z = true
xs : Heap α
y : α
ys : Heap α
h : HeapNodeAux α (Heap α)
hs : List (HeapNodeAux α (Heap α))
h₁ : IsHeap (Heap.heap (h :: hs))
h₂ : IsMinHeap le (Heap.heap (h :: hs))
xs₂ : Heap α
t₁ : HeapNodeAux α (Heap α)
ts : List (BinTree α)
n : ℕ
btxn : BinTree α × ℕ
bt : BinTree α
ts₂ : List (BinTree α)
bt₂ : BinTree α
n₂ : ℕ
lebt : le bt₂.val bt.val = true
not_le_le : ∀ (x y : α), le x y = false → le y x = true
eqeq : t₁ = bt₂ ∧ 0 = n₂
hne₂ : True
heqh : h = t₁ ∧ hs = bt :: ts₂
dmeq :
  (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.val = y ∧
    merge le (Heap.heap (List.eraseIdx (t₁ :: bt :: ts₂) (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd))
        (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children =
      ys
ys₂ xs₃ : Heap α
ts₃ ts₄ : List (HeapNodeAux α (Heap α))
heq₂ : (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children = Heap.heap ts₄
heq :
  (match t₁ :: bt :: ts₂, (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd with
    | [], x => []
    | head :: as, 0 => as
    | a :: as, Nat.succ n => a :: List.eraseIdx as n) =
    ts₃
h₁₂ : IsMinHeap le (Heap.heap ts₃)
⊢ IsSearchTree le (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst

case a.a
α : Type u_1
le : α → α → Bool
le_trans : ∀ (x y z : α), le x y = true → le y z = true → le x z = true
le_rfl : ∀ (z : α), le z z = true
xs : Heap α
y : α
ys : Heap α
h : HeapNodeAux α (Heap α)
hs : List (HeapNodeAux α (Heap α))
h₁ : IsHeap (Heap.heap (h :: hs))
h₂ : IsMinHeap le (Heap.heap (h :: hs))
xs₂ : Heap α
t₁ : HeapNodeAux α (Heap α)
ts : List (BinTree α)
n : ℕ
btxn : BinTree α × ℕ
bt : BinTree α
ts₂ : List (BinTree α)
bt₂ : BinTree α
n₂ : ℕ
lebt : le bt₂.val bt.val = true
not_le_le : ∀ (x y : α), le x y = false → le y x = true
eqeq : t₁ = bt₂ ∧ 0 = n₂
hne₂ : True
heqh : h = t₁ ∧ hs = bt :: ts₂
dmeq :
  (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.val = y ∧
    merge le (Heap.heap (List.eraseIdx (t₁ :: bt :: ts₂) (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd))
        (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children =
      ys
ys₂ xs₃ : Heap α
ts₃ ts₄ : List (HeapNodeAux α (Heap α))
heq₂ : (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children = Heap.heap ts₄
heq :
  (match t₁ :: bt :: ts₂, (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd with
    | [], x => []
    | head :: as, 0 => as
    | a :: as, Nat.succ n => a :: List.eraseIdx as n) =
    ts₃
h₁₂ : IsMinHeap le (Heap.heap ts₃)
⊢ IsMinHeap le (Heap.heap [])
Tactic:
. apply IsSearchTree_findMin
  . rw[heqh.right] at h₂
    cases h₂ with | cons _ mh => 
    cases mh
    assumption
  . rw[← eqeq.left, ← heqh.left]
    cases h₂
    assumption
Proof state after:
case a.a
α : Type u_1
le : α → α → Bool
le_trans : ∀ (x y z : α), le x y = true → le y z = true → le x z = true
le_rfl : ∀ (z : α), le z z = true
xs : Heap α
y : α
ys : Heap α
h : HeapNodeAux α (Heap α)
hs : List (HeapNodeAux α (Heap α))
h₁ : IsHeap (Heap.heap (h :: hs))
h₂ : IsMinHeap le (Heap.heap (h :: hs))
xs₂ : Heap α
t₁ : HeapNodeAux α (Heap α)
ts : List (BinTree α)
n : ℕ
btxn : BinTree α × ℕ
bt : BinTree α
ts₂ : List (BinTree α)
bt₂ : BinTree α
n₂ : ℕ
lebt : le bt₂.val bt.val = true
not_le_le : ∀ (x y : α), le x y = false → le y x = true
eqeq : t₁ = bt₂ ∧ 0 = n₂
hne₂ : True
heqh : h = t₁ ∧ hs = bt :: ts₂
dmeq :
  (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.val = y ∧
    merge le (Heap.heap (List.eraseIdx (t₁ :: bt :: ts₂) (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd))
        (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children =
      ys
ys₂ xs₃ : Heap α
ts₃ ts₄ : List (HeapNodeAux α (Heap α))
heq₂ : (findMin le ts₂ (1 + 1) (bt₂, n₂)).fst.children = Heap.heap ts₄
heq :
  (match t₁ :: bt :: ts₂, (findMin le ts₂ (1 + 1) (bt₂, n₂)).snd with
    | [], x => []
    | head :: as, 0 => as
    | a :: as, Nat.succ n => a :: List.eraseIdx as n) =
    ts₃
h₁₂ : IsMinHeap le (Heap.heap ts₃)
⊢ IsMinHeap le (Heap.heap [])