diff options
Diffstat (limited to 'BinaryHeap')
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs.lean | 693 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/EmptyIsHeap.lean | 6 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapPop.lean | 9 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapPush.lean | 122 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapRemoveAt.lean | 15 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapRemoveLast.lean | 169 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapUpdateAt.lean | 128 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/HeapProofs/HeapUpdateRoot.lean | 239 | ||||
| -rw-r--r-- | BinaryHeap/CompleteTree/Lemmas.lean | 8 |
9 files changed, 702 insertions, 687 deletions
diff --git a/BinaryHeap/CompleteTree/HeapProofs.lean b/BinaryHeap/CompleteTree/HeapProofs.lean index 2946c30..6b5840c 100644 --- a/BinaryHeap/CompleteTree/HeapProofs.lean +++ b/BinaryHeap/CompleteTree/HeapProofs.lean @@ -1,687 +1,6 @@ -import BinaryHeap.CompleteTree.HeapOperations -import BinaryHeap.CompleteTree.Lemmas - -namespace BinaryHeap - -theorem CompleteTree.emptyIsHeap {α : Type u} (le : α → α → Bool) : HeapPredicate CompleteTree.empty le := True.intro - ----------------------------------------------------------------------------------------------- --- heapPush - -private theorem CompleteTree.rootSeesThroughCast - (n m : Nat) - (h₁ : n + 1 + m = n + m + 1) - (h₂ : 0 < n + 1 + m) - (h₃ : 0 < n + m + 1) - (heap : CompleteTree α (n+1+m)) : (h₁▸heap).root h₃ = heap.root h₂ := by - induction m generalizing n - case zero => simp - case succ o ho => - have h₄ := ho (n+1) - have h₅ : n + 1 + 1 + o = n + 1 + (Nat.succ o) := by simp_arith - have h₆ : n + 1 + o + 1 = n + (Nat.succ o + 1) := by simp_arith - rewrite[h₅, h₆] at h₄ - revert heap h₁ h₂ h₃ - assumption - -private theorem HeapPredicate.seesThroughCast - (n m : Nat) - (lt : α → α → Bool) - (h₁ : n+1+m+1=n+m+1+1) - (h₂ : 0<n+1+m+1) - (h₃ : 0<n+m+1+1) - (heap : CompleteTree α (n+1+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by - unfold HeapPredicate - intro h₄ - induction m generalizing n - case zero => simp[h₄] - case succ o ho => - have h₅ := ho (n+1) - have h₆ : n+1+1+o+1 = n+1+(Nat.succ o)+1 := by simp_arith - rw[h₆] at h₅ - have h₆ : n + 1 + o + 1 + 1 = n + (Nat.succ o + 1 + 1) := by simp_arith - rewrite[h₆] at h₅ - revert heap h₁ h₂ h₃ - assumption - -theorem CompleteTree.heapPushRootSameOrElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : 0 < o): (CompleteTree.root (heap.heapPush lt elem) (Nat.succ_pos o) = elem) ∨ (CompleteTree.root (heap.heapPush lt elem) (Nat.succ_pos o) = CompleteTree.root heap h₂) := by - unfold heapPush - split --match o, heap - contradiction - simp - rename_i n m v l r _ _ _ - split -- if h : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then - case h_2.isTrue h => - cases (lt elem v) <;> simp[instDecidableEqBool, Bool.decEq, CompleteTree.root] - case h_2.isFalse h => - rw[rootSeesThroughCast n (m+1) (by simp_arith) (by simp_arith) (by simp_arith)] - cases (lt elem v) - <;> simp[instDecidableEqBool, Bool.decEq, CompleteTree.root] - -theorem CompleteTree.heapPushEmptyElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : ¬0 < o) : (CompleteTree.root (heap.heapPush lt elem) (Nat.succ_pos o) = elem) := - have : o = 0 := Nat.eq_zero_of_le_zero $ (Nat.not_lt_eq 0 o).mp h₂ - match o, heap with - | 0, .leaf => by simp[CompleteTree.heapPush, root] - -theorem HeapPredicate.leOrLeaf_transitive (h₁ : transitive_le le) : le a b → HeapPredicate.leOrLeaf le b h → HeapPredicate.leOrLeaf le a h := by - unfold leOrLeaf - intros h₂ h₃ - rename_i n - cases n <;> simp - apply h₁ a b _ - exact ⟨h₂, h₃⟩ - -mutual -/-- - Helper for CompleteTree.heapPushIsHeap, to make one function out of both branches. - Sorry for the ugly signature, but this was the easiest thing to extract. - -/ -private theorem CompleteTree.heapPushIsHeapAux {α : Type u} (le : α → α → Bool) (n m : Nat) (v elem : α) (l : CompleteTree α n) (r : CompleteTree α m) (h₁ : HeapPredicate l le ∧ HeapPredicate r le ∧ HeapPredicate.leOrLeaf le v l ∧ HeapPredicate.leOrLeaf le v r) (h₂ : transitive_le le) (h₃ : total_le le): HeapPredicate l le ∧ - let smaller := (if le elem v then elem else v) - let larger := (if le elem v then v else elem) - HeapPredicate (heapPush le larger r) le - ∧ HeapPredicate.leOrLeaf le smaller l - ∧ HeapPredicate.leOrLeaf le smaller (heapPush le larger r) - := by - cases h₆ : (le elem v) <;> simp only [Bool.false_eq_true, reduceIte] - case true => - have h₇ : (HeapPredicate (CompleteTree.heapPush le v r) le) := CompleteTree.heapPushIsHeap h₁.right.left h₂ h₃ - simp only [true_and, h₁, h₇, HeapPredicate.leOrLeaf_transitive h₂ h₆ h₁.right.right.left] - cases m - case zero => - have h₈ := heapPushEmptyElem v r le (Nat.not_lt_zero 0) - simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, Nat.reduceAdd, h₈] - assumption - case succ _ => - simp only [HeapPredicate.leOrLeaf] - cases heapPushRootSameOrElem v r le (Nat.succ_pos _) - <;> rename_i h₇ - <;> rw[h₇] - . assumption - apply h₂ elem v - exact ⟨h₆, h₁.right.right.right⟩ - case false => - have h₇ : (HeapPredicate (CompleteTree.heapPush le elem r) le) := CompleteTree.heapPushIsHeap h₁.right.left h₂ h₃ - simp only [true_and, h₁, h₇] - have h₈ : le v elem := not_le_imp_le h₃ elem v (by simp only [h₆, Bool.false_eq_true, not_false_eq_true]) - cases m - case zero => - have h₇ := heapPushEmptyElem elem r le (Nat.not_lt_zero 0) - simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, Nat.reduceAdd, h₇] - assumption - case succ _ => - cases heapPushRootSameOrElem elem r le (Nat.succ_pos _) - <;> rename_i h₉ - <;> simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, h₈, h₉] - exact h₁.right.right.right - -theorem CompleteTree.heapPushIsHeap {α : Type u} {elem : α} {heap : CompleteTree α o} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate (heap.heapPush le elem) le := by - unfold heapPush - split -- match o, heap with - trivial - case h_2 n m v l r m_le_n _ _ => - simp - split -- if h₅ : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then - case' isTrue => - have h := heapPushIsHeapAux le n m v elem l r h₁ h₂ h₃ - case' isFalse => - apply HeapPredicate.seesThroughCast <;> try simp_arith[h₂] --gets rid of annoying cast. - have h := heapPushIsHeapAux le m n v elem r l (And.intro h₁.right.left $ And.intro h₁.left $ And.intro h₁.right.right.right h₁.right.right.left) h₂ h₃ - all_goals - unfold HeapPredicate - cases h₆ : (le elem v) - <;> simp only [h₆, Bool.false_eq_true, reduceIte] at h - <;> simp only [instDecidableEqBool, Bool.decEq, h, and_self] -end - ----------------------------------------------------------------------------------------------- --- heapRemoveLast - ---- Same as rootSeesThroughCast, but in the other direction. -private theorem CompleteTree.rootSeesThroughCast2 - (n m : Nat) - (h₁ : n + m + 1 = n + 1 + m) - (h₂ : 0 < n + m + 1) - (h₃ : 0 < n + 1 + m) - (heap : CompleteTree α (n+m+1)) : (h₁▸heap).root h₃ = heap.root h₂ := by - induction m generalizing n - case zero => simp - case succ mm mh => - have h₄ := mh (n+1) - have h₅ : n + 1 + mm + 1 = n + Nat.succ mm + 1 := by simp_arith - have h₆ : n + 1 + 1 + mm = n + 1 + Nat.succ mm := by simp_arith - rw[h₅, h₆] at h₄ - revert heap h₁ h₂ h₃ - assumption - -private theorem HeapPredicate.seesThroughCast2 - (n m : Nat) - (lt : α → α → Bool) - (h₁ : n+m+1=n+1+m) - (h₂ : 0<n+1+m) - (h₃ : 0<n+m+1) - (heap : CompleteTree α (n+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by - unfold HeapPredicate - intro h₄ - induction m generalizing n - case zero => simp[h₄] - case succ o ho => - have h₅ := ho (n+1) - have h₆ : n+1+o+1 = n+(Nat.succ o)+1 := by simp_arith - rw[h₆] at h₅ - have h₆ : n + 1 + 1 + o = n + 1 + Nat.succ o := by simp_arith - rewrite[h₆] at h₅ - revert heap h₁ h₂ h₃ - assumption - -private theorem CompleteTree.castZeroHeap (n m : Nat) (heap : CompleteTree α 0) (h₁ : 0=n+m) {le : α → α → Bool} : HeapPredicate (h₁ ▸ heap) le := by - have h₂ : heap = (CompleteTree.empty : CompleteTree α 0) := by - simp[empty] - match heap with - | .leaf => trivial - have h₂ : HeapPredicate heap le := by simp[h₂, empty, emptyIsHeap] - cases m - case succ => contradiction - case zero => - cases n - case succ => contradiction - case zero => - simp[h₁, h₂] - --- If there is only one element left, the result is a leaf. -private theorem CompleteTree.heapRemoveLastAuxLeaf -{α : Type u} -{β : Nat → Type u} -(heap : CompleteTree α 1) -(aux0 : α → (β 1)) -(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) -(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) -: (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst = CompleteTree.leaf := by - let l := (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst - have h₁ : l = CompleteTree.leaf := match l with - | .leaf => rfl - exact h₁ - -protected theorem CompleteTree.heapRemoveLastAuxLeavesRoot -{α : Type u} -{β : Nat → Type u} -(heap : CompleteTree α (n+1)) -(aux0 : α → (β 1)) -(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) -(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) -(h₁ : n > 0) -: heap.root (Nat.zero_lt_of_ne_zero $ Nat.succ_ne_zero n) = (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst.root (h₁) := by - unfold CompleteTree.Internal.heapRemoveLastAux - split - rename_i o p v l r _ _ _ - have h₃ : (0 ≠ o + p) := Ne.symm $ Nat.not_eq_zero_of_lt h₁ - simp[h₃] - exact - if h₄ : p < o ∧ Nat.nextPowerOfTwo (p + 1) = p + 1 then by - simp[h₄] - cases o - case zero => exact absurd h₄.left $ Nat.not_lt_zero p - case succ oo _ _ _ => - simp -- redundant, but makes goal easier to read - rw[rootSeesThroughCast2 oo p _ (by simp_arith) _] - apply root_unfold - else by - simp[h₄] - cases p - case zero => - simp_arith at h₁ -- basically o ≠ 0 - simp_arith (config := {ground := true})[h₁] at h₄ -- the second term in h₄ is decidable and True. What remains is ¬(0 < o), or o = 0 - case succ pp hp => - simp_arith - apply root_unfold - -private theorem CompleteTree.heapRemoveLastAuxIsHeap -{α : Type u} -{β : Nat → Type u} -{heap : CompleteTree α (o+1)} -{le : α → α → Bool} -(aux0 : α → (β 1)) -(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) -(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) -(h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate ((Internal.heapRemoveLastAux heap aux0 auxl auxr).fst) le := by - unfold CompleteTree.Internal.heapRemoveLastAux - split - rename_i n m v l r _ _ _ - exact - if h₄ : 0 = (n+m) then by - simp only [h₄, reduceDite, castZeroHeap] - else by - simp[h₄] - exact - if h₅ : (m<n ∧ Nat.nextPowerOfTwo (m+1) = m+1) then by - simp only [h₅, and_self, ↓reduceDite] - cases n - case zero => - exact absurd h₅.left $ Nat.not_lt_zero m - case succ ll h₆ h₇ h₈ => - simp only - apply HeapPredicate.seesThroughCast2 <;> try simp_arith - cases ll - case a.zero => -- if ll is zero, then (heapRemoveLast l).snd is a leaf. - have h₆ := heapRemoveLastAuxLeaf l aux0 auxl auxr - rw[h₆] - unfold HeapPredicate at * - have h₇ : HeapPredicate .leaf le := by trivial - have h₈ : HeapPredicate.leOrLeaf le v .leaf := by trivial - exact ⟨h₇,h₁.right.left, h₈, h₁.right.right.right⟩ - case a.succ nn => -- if ll is not zero, then the root element before and after heapRemoveLast is the same. - unfold HeapPredicate at * - simp only [heapRemoveLastAuxIsHeap aux0 auxl auxr h₁.left h₂ h₃, h₁.right.left, h₁.right.right.right, and_true, true_and] - unfold HeapPredicate.leOrLeaf - simp only - rw[←CompleteTree.heapRemoveLastAuxLeavesRoot] - exact h₁.right.right.left - else by - simp[h₅] - cases m - case zero => - simp only [Nat.add_zero] at h₄ -- n ≠ 0 - simp (config := { ground := true }) only [Nat.zero_add, and_true, Nat.not_lt, Nat.le_zero_eq, Ne.symm h₄] at h₅ -- the second term in h₅ is decidable and True. What remains is ¬(0 < n), or n = 0 - case succ mm h₆ h₇ h₈ => - unfold HeapPredicate at * - simp only [h₁, heapRemoveLastAuxIsHeap aux0 auxl auxr h₁.right.left h₂ h₃, true_and] - unfold HeapPredicate.leOrLeaf - exact match mm with - | 0 => rfl - | o+1 => - have h₉ : le v ((Internal.heapRemoveLastAux r _ _ _).fst.root (Nat.zero_lt_succ o)) := by - rw[←CompleteTree.heapRemoveLastAuxLeavesRoot] - exact h₁.right.right.right - h₉ - -private theorem CompleteTree.heapRemoveLastIsHeap {α : Type u} {heap : CompleteTree α (o+1)} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate (Internal.heapRemoveLast heap).fst le := - heapRemoveLastAuxIsHeap _ _ _ h₁ h₂ h₃ - -private theorem CompleteTree.heapRemoveLastWithIndexIsHeap {α : Type u} {heap : CompleteTree α (o+1)} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate ((Internal.heapRemoveLastWithIndex heap).fst) le := - heapRemoveLastAuxIsHeap _ _ _ h₁ h₂ h₃ - ----------------------------------------------------------------------------------------------- --- heapUpdateRoot - -private theorem CompleteTree.heapUpdateRootReturnsRoot {α : Type u} {n : Nat} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) : (heap.heapUpdateRoot le value h₁).snd = heap.root h₁ := by - unfold heapUpdateRoot - split - rename_i o p v l r h₃ h₄ h₅ h₁ - simp - cases o <;> simp - case zero => - exact root_unfold v l r h₃ h₄ h₅ h₁ - case succ => - cases p <;> simp - case zero => - cases le value (root l _) - <;> simp only [Bool.false_eq_true, ↓reduceIte, root_unfold] - case succ => - cases le value (root l _) <;> cases le value (root r _) - <;> cases le (root l _) (root r _) - <;> simp only [Bool.false_eq_true, and_self, and_true, and_false, ↓reduceIte, root_unfold] - -private theorem CompleteTree.heapUpdateRootPossibleRootValuesAuxL {α : Type u} (heap : CompleteTree α n) (h₁ : n > 1) : 0 < heap.leftLen (Nat.lt_trans (Nat.lt_succ_self 0) h₁) := - match h₅: n, heap with - | (o+p+1), .branch v l r h₂ h₃ h₄ => by - simp[leftLen, length] - cases o - case zero => rewrite[(Nat.le_zero_eq p).mp h₂] at h₁; contradiction - case succ q => exact Nat.zero_lt_succ q - -private theorem CompleteTree.heapUpdateRootPossibleRootValuesAuxR {α : Type u} (heap : CompleteTree α n) (h₁ : n > 2) : 0 < heap.rightLen (Nat.lt_trans (Nat.lt_succ_self 0) $ Nat.lt_trans (Nat.lt_succ_self 1) h₁) := - match h₅: n, heap with - | (o+p+1), .branch v l r h₂ h₃ h₄ => by - simp[rightLen, length] - cases p - case zero => simp_arith at h₁; simp at h₃; exact absurd h₁ (Nat.not_le_of_gt h₃) - case succ q => exact Nat.zero_lt_succ q - -private theorem CompleteTree.heapUpdateRootPossibleRootValues1 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n = 1) : (heap.heapUpdateRoot le value (h₁.substr (Nat.lt_succ_self 0))).fst.root (h₁.substr (Nat.lt_succ_self 0)) = value := by - unfold heapUpdateRoot - generalize (h₁.substr (Nat.lt_succ_self 0) : n > 0) = hx - split - rename_i o p v l r _ _ _ h₁ - have h₃ : o = 0 := (Nat.add_eq_zero.mp $ Nat.succ.inj h₁).left - simp[h₃, root_unfold] - -private theorem CompleteTree.heapUpdateRootPossibleRootValues2 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n = 2) : -have h₂ : 0 < n := Nat.lt_trans (Nat.lt_succ_self 0) $ h₁.substr (Nat.lt_succ_self 1) -have h₃ : 0 < leftLen heap h₂ := heapUpdateRootPossibleRootValuesAuxL heap (h₁.substr (Nat.lt_succ_self 1)) -(heap.heapUpdateRoot le value h₂).fst.root h₂ = value -∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.left h₂).root h₃ -:= by - simp - unfold heapUpdateRoot - generalize (Nat.lt_trans (Nat.lt_succ_self 0) (Eq.substr h₁ (Nat.lt_succ_self 1)) : 0 < n) = h₂ - split - rename_i o p v l r h₃ h₄ h₅ h₂ - cases o <;> simp - case zero => simp only[root, true_or] - case succ oo => - have h₆ : p = 0 := by simp at h₁; omega - simp only [h₆, ↓reduceDite] - cases le value (l.root _) - <;> simp[heapUpdateRootReturnsRoot, root_unfold, left_unfold] - -private theorem CompleteTree.heapUpdateRootPossibleRootValues3 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 2) : -have h₂ : 0 < n := Nat.lt_trans (Nat.lt_succ_self 0) $ Nat.lt_trans (Nat.lt_succ_self 1) h₁ -have h₃ : 0 < leftLen heap h₂ := heapUpdateRootPossibleRootValuesAuxL heap $ Nat.lt_trans (Nat.lt_succ_self 1) h₁ -have h₄ : 0 < rightLen heap h₂ := heapUpdateRootPossibleRootValuesAuxR heap h₁ -(heap.heapUpdateRoot le value h₂).fst.root h₂ = value -∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.left h₂).root h₃ -∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.right h₂).root h₄ -:= by - simp only - unfold heapUpdateRoot - generalize (Nat.lt_trans (Nat.lt_succ_self 0) (Nat.lt_trans (Nat.lt_succ_self 1) h₁) : 0 < n) = h₂ - split - rename_i o p v l r h₃ h₄ h₅ h₂ - cases o - case zero => simp only[root, true_or] - case succ oo => - have h₆ : p ≠ 0 := by simp at h₁; omega - simp only [Nat.add_one_ne_zero, ↓reduceDite, h₆] - cases le value (l.root _) <;> simp - rotate_right - cases le value (r.root _) <;> simp - case true.true => simp[root] - case false | true.false => - cases le (l.root _) (r.root _) - <;> simp only [Bool.false_eq_true, ↓reduceIte, heapUpdateRootReturnsRoot, root_unfold, left_unfold, right_unfold, true_or, or_true] - -private theorem CompleteTree.heapUpdateRootIsHeapLeRootAux {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le (root heap h₂) value) : HeapPredicate.leOrLeaf le (root heap h₂) (heapUpdateRoot le value heap h₂).fst := - if h₄ : n = 1 then by - have h₅ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only[h₃, h₄, heapUpdateRootPossibleRootValues1] - unfold HeapPredicate.leOrLeaf - split - · rfl - · exact h₅ - else if h₅ : n = 2 then by - have h₆ := heapUpdateRootPossibleRootValues2 le value heap h₅ - cases h₆ - case inl h₆ => - have h₇ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only [h₆, h₃] - unfold HeapPredicate.leOrLeaf - split - · rfl - · exact h₇ - case inr h₆ => - unfold HeapPredicate.leOrLeaf - unfold HeapPredicate at h₁ - split at h₁ - case h_1 => contradiction - case h_2 o p v l r h₇ h₈ h₉ => - have h₁₁ : p = 0 := by - simp at h₅ - cases o; simp only [Nat.le_zero_eq] at h₇; exact h₇; simp_arith[Nat.add_eq_zero] at h₅; exact h₅.right - have h₁₀ : o = 1 := by simp_arith[h₁₁] at h₅; assumption - simp only - rw[h₆] - have h₁₂ := h₁.right.right.left - unfold HeapPredicate.leOrLeaf at h₁₂ - cases o ; contradiction; - case succ => - exact h₁₂ - else by - have h₆ : n > 2 := by omega - have h₇ := heapUpdateRootPossibleRootValues3 le value heap h₆ - simp at h₇ - unfold HeapPredicate at h₁ - cases h₇ - case inl h₇ => - have h₈ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only [h₇, h₃] - unfold HeapPredicate.leOrLeaf - split - · rfl - · exact h₈ - case inr h₇ => - cases h₇ - case inl h₇ | inr h₇ => - unfold HeapPredicate.leOrLeaf - split at h₁ - contradiction - simp_all - case h_2 o p v l r _ _ _ => - cases o - omega - cases p - omega - have h₈ := h₁.right.right.left - have h₉ := h₁.right.right.right - assumption - -private theorem CompleteTree.heapUpdateRootIsHeapLeRootAuxLe {α : Type u} (le : α → α → Bool) {a b c : α} (h₁ : transitive_le le) (h₂ : total_le le) (h₃ : le b c) : ¬le a c ∨ ¬ le a b → le b a -| .inr h₅ => not_le_imp_le h₂ _ _ h₅ -| .inl h₅ => h₁ b c a ⟨h₃,not_le_imp_le h₂ _ _ h₅⟩ - -theorem CompleteTree.heapUpdateRootIsHeap {α : Type u} {n: Nat} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) (h₂ : HeapPredicate heap le) (h₃ : transitive_le le) (h₄ : total_le le) : HeapPredicate (heap.heapUpdateRoot le value h₁).fst le := by - unfold heapUpdateRoot - split - rename_i o p v l r h₇ h₈ h₉ heq - exact - if h₁₀ : o = 0 then by - simp only [Nat.add_eq, Nat.succ_eq_add_one, h₁₀, ↓reduceDite] - unfold HeapPredicate at h₂ ⊢ - simp only [h₂, true_and] - unfold HeapPredicate.leOrLeaf - have : p = 0 := by rw[h₁₀, Nat.le_zero_eq] at h₇; assumption - apply And.intro - case left => match o, l with - | Nat.zero, _ => trivial - case right => match p, r with - | Nat.zero, _ => trivial - else if h₁₁ : p = 0 then by - simp only [↓reduceDite, h₁₀, h₁₁] - cases h₉ : le value (root l (_ : 0 < o)) <;> simp - case true => - unfold HeapPredicate at * - simp only [h₂, true_and] - unfold HeapPredicate.leOrLeaf - apply And.intro - case right => match p, r with - | Nat.zero, _ => trivial - case left => match o, l with - | Nat.succ _, _ => assumption - case false => - rw[heapUpdateRootReturnsRoot] - have h₁₂ : le (l.root (Nat.zero_lt_of_ne_zero h₁₀)) value := by simp[h₉, h₄, not_le_imp_le] - have h₁₃ : o = 1 := Nat.le_antisymm (by simp_arith[h₁₁] at h₈; exact h₈) (Nat.succ_le_of_lt (Nat.zero_lt_of_ne_zero h₁₀)) - unfold HeapPredicate at * - simp only [h₂, true_and] --closes one sub-goal - apply And.intro <;> try apply And.intro - case right.right => match p, r with - | 0, .leaf => simp[HeapPredicate.leOrLeaf] - case right.left => - simp only [HeapPredicate.leOrLeaf] - cases o <;> simp only [Nat.succ_eq_add_one, heapUpdateRootPossibleRootValues1, h₁₃, h₁₂] - case left => - apply heapUpdateRootIsHeap - exact h₂.left - exact h₃ - exact h₄ - else if h₁₂ : le value (root l (Nat.zero_lt_of_ne_zero h₁₀)) ∧ le value (root r (Nat.zero_lt_of_ne_zero h₁₁)) then by - unfold HeapPredicate at * - simp only [↓reduceDite, and_self, ↓reduceIte, true_and, h₁₀, h₁₁, h₁₂, h₂] - unfold HeapPredicate.leOrLeaf - cases o - · contradiction - · cases p - · contradiction - · assumption - else by - simp only [↓reduceDite, ↓reduceIte, h₁₀, h₁₁, h₁₂] - have h₁₃ : ¬le value (root l _) ∨ ¬le value (root r _) := (Decidable.not_and_iff_or_not (le value (root l (Nat.zero_lt_of_ne_zero h₁₀)) = true) (le value (root r (Nat.zero_lt_of_ne_zero h₁₁)) = true)).mp h₁₂ - cases h₁₄ : le (root l (_ : 0 < o)) (root r (_ : 0 < p)) - <;> simp only [Bool.false_eq_true, ↓reduceIte] - <;> unfold HeapPredicate at * - <;> simp only [true_and, h₂] - <;> apply And.intro - <;> try apply And.intro - case false.left | true.left => - apply heapUpdateRootIsHeap - <;> simp only [h₂, h₃, h₄] - case false.right.left => - unfold HeapPredicate.leOrLeaf - have h₁₅ : le (r.root _) (l.root _) = true := not_le_imp_le h₄ (l.root _) (r.root _) $ (Bool.not_eq_true $ le (root l (_ : 0 < o)) (root r (_ : 0 < p))).substr h₁₄ - simp only[heapUpdateRootReturnsRoot] - cases o <;> simp only[h₁₅] - case true.right.right => - unfold HeapPredicate.leOrLeaf - simp only[heapUpdateRootReturnsRoot] - cases p <;> simp only[h₁₄] - case false.right.right => - have h₁₅ : le (r.root _) (l.root _) = true := not_le_imp_le h₄ (l.root _) (r.root _) $ (Bool.not_eq_true $ le (root l (_ : 0 < o)) (root r (_ : 0 < p))).substr h₁₄ - have h₁₆ : le (r.root _) value := heapUpdateRootIsHeapLeRootAuxLe le h₃ h₄ h₁₅ h₁₃ - simp only [heapUpdateRootReturnsRoot, heapUpdateRootIsHeapLeRootAux, h₂, h₁₆] - case true.right.left => - have h₁₆ : le (l.root _) value := heapUpdateRootIsHeapLeRootAuxLe le h₃ h₄ h₁₄ h₁₃.symm - simp only [heapUpdateRootReturnsRoot, heapUpdateRootIsHeapLeRootAux, h₂, h₁₆] - ----------------------------------------------------------------------------------------------- --- heapUpdateAt - -private theorem CompleteTree.heapUpdateAtIsHeapLeRootAux_RootLeValue {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le (root heap h₂) value) (h₄ : total_le le) : HeapPredicate.leOrLeaf le (root heap h₂) (heapUpdateAt le index value heap h₂).fst := by - unfold heapUpdateAt - split - case isTrue => exact heapUpdateRootIsHeapLeRootAux le value heap h₁ h₂ h₃ - case isFalse hi => - split - rename_i o p v l r h₆ h₇ h₈ index h₁ h₅ - cases h₉ : le v value <;> simp (config := { ground := true }) only - case false => rw[root_unfold] at h₃; exact absurd h₃ ((Bool.not_eq_true (le v value)).substr h₉) - case true => - rw[root_unfold] - split - <;> simp![reflexive_le, h₄] - -private theorem CompleteTree.heapUpdateAtIsHeapLeRootAux_ValueLeRoot {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le value (root heap h₂)) (h₄ : total_le le) (h₅ : transitive_le le) : HeapPredicate.leOrLeaf le value (heapUpdateAt le index value heap h₂).fst := by - unfold heapUpdateAt - split - <;> rename_i h₉ - case isTrue => - unfold heapUpdateRoot - split - rename_i o p v l r h₆ h₇ h₈ h₂ - cases o <;> cases p <;> simp only [↓reduceDite,HeapPredicate.leOrLeaf, root_unfold, h₄, reflexive_le] - <;> unfold HeapPredicate at h₁ - <;> have h₁₀ : le value $ l.root (by omega) := h₅ value v (l.root _) ⟨h₃, h₁.right.right.left⟩ - simp only [↓reduceIte, Nat.add_zero, h₁₀, root_unfold, h₄, reflexive_le] - have h₁₁ : le value $ r.root (by omega) := h₅ value v (r.root _) ⟨h₃, h₁.right.right.right⟩ - simp only [↓reduceIte, h₁₀, h₁₁, and_self, root_unfold, h₄, reflexive_le] - case isFalse => - split - rename_i o p v l r h₆ h₇ h₈ index h₂ hi - cases le v value - <;> simp (config := { ground := true }) only [root_unfold, Nat.pred_eq_sub_one] at h₃ ⊢ - <;> split - <;> unfold HeapPredicate.leOrLeaf - <;> simp only [root_unfold, h₃, h₄, reflexive_le] - -theorem CompleteTree.heapUpdateAtIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) (h₂ : HeapPredicate heap le) (h₃ : transitive_le le) (h₄ : total_le le) : HeapPredicate (heap.heapUpdateAt le index value h₁).fst le := by - unfold heapUpdateAt - split - case isTrue h₅ => - exact heapUpdateRootIsHeap le value heap h₁ h₂ h₃ h₄ - case isFalse h₅ => - split - rename_i o p v l r h₆ h₇ h₈ index h₁ h₅ - cases h₁₀ : le v value <;> simp (config := {ground := true}) -- this could probably be solved without this split, but readability... - <;> split - <;> rename_i h -- h is the same name as used in the function - <;> unfold HeapPredicate at h₂ ⊢ - <;> simp only [h₂, and_true, true_and] - case false.isFalse => - have h₁₀ := not_le_imp_le h₄ v value (Bool.eq_false_iff.mp h₁₀) - have h₁₄ : p > 0 := by cases p; exact absurd (Nat.lt_succ.mp index.isLt) h; exact Nat.zero_lt_succ _ - apply And.intro <;> try apply And.intro - case left => exact heapUpdateAtIsHeap le ⟨index.val - o - 1, _⟩ v r h₁₄ h₂.right.left h₃ h₄ - case right.left => exact HeapPredicate.leOrLeaf_transitive h₃ h₁₀ h₂.right.right.left - case right.right => - have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - o - 1, (by omega)⟩ v r h₁₄).fst le := - (heapUpdateAtIsHeap le ⟨index.val - o - 1, (by omega)⟩ v r _ h₂.right.left h₃ h₄) - cases h₁₂ : le v (r.root h₁₄) - case false => - cases p - exact absurd (Nat.lt_succ.mp index.isLt) h - exact absurd h₂.right.right.right ((Bool.eq_false_iff).mp h₁₂) - case true => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - o - 1, (by omega)⟩ v r h₂.right.left (by omega) h₁₂ h₄ h₃ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - exact h₁₀ - case false.isTrue => - have h₁₀ := not_le_imp_le h₄ v value (Bool.eq_false_iff.mp h₁₀) - have h₁₄ : o > 0 := by cases o; simp at h₅ h; exact absurd (Fin.val_inj.mp h : index = 0) h₅; exact Nat.zero_lt_succ _ - apply And.intro <;> try apply And.intro - case left => exact heapUpdateAtIsHeap le ⟨index.val - 1, _⟩ v l h₁₄ h₂.left h₃ h₄ - case right.right => exact HeapPredicate.leOrLeaf_transitive h₃ h₁₀ h₂.right.right.right - case right.left => - have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - 1, (_)⟩ v l h₁₄).fst le := - (heapUpdateAtIsHeap le ⟨index.val - 1, (by omega)⟩ v l _ h₂.left h₃ h₄) - cases h₁₂ : le v (l.root h₁₄) - case false => - cases o - contradiction -- h₁₄ is False - exact absurd h₂.right.right.left ((Bool.eq_false_iff).mp h₁₂) - case true => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - 1, (by omega)⟩ v l h₂.left (by omega) h₁₂ h₄ h₃ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - exact h₁₀ - case true.isFalse => - have h₁₄ : p > 0 := by cases p; exact absurd (Nat.lt_succ.mp index.isLt) h; exact Nat.zero_lt_succ _ - apply And.intro - case left => exact heapUpdateAtIsHeap le ⟨index.val - o - 1, _⟩ value r h₁₄ h₂.right.left h₃ h₄ - case right => - have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - o - 1, (by omega)⟩ v r (h₁₄)).fst le := - (heapUpdateAtIsHeap le ⟨index.val - o - 1, (by omega)⟩ v r _ h₂.right.left h₃ h₄) - cases h₁₂ : le value (r.root h₁₄) - case false => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_RootLeValue le ⟨index.val - o - 1, (by omega)⟩ value r h₂.right.left (by omega) (not_le_imp_le h₄ value (r.root h₁₄) (Bool.eq_false_iff.mp h₁₂)) h₄ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - cases p - contradiction -- h₁₄ is False - exact h₂.right.right.right - case true => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - o - 1, (by omega)⟩ value r h₂.right.left (by omega) h₁₂ h₄ h₃ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - exact h₁₀ - case true.isTrue => - have h₁₄ : o > 0 := by cases o; simp at h₅ h; exact absurd (Fin.val_inj.mp h : index = 0) h₅; exact Nat.zero_lt_succ _ - apply And.intro - case left => exact heapUpdateAtIsHeap le ⟨index.val - 1, _⟩ value l h₁₄ h₂.left h₃ h₄ - case right => - have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - 1, (by omega)⟩ v l h₁₄).fst le := - (heapUpdateAtIsHeap le ⟨index.val - 1, (by omega)⟩ v l _ h₂.left h₃ h₄) - cases h₁₂ : le value (l.root h₁₄) - case false => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_RootLeValue le ⟨index.val - 1, (by omega)⟩ value l h₂.left (by omega) (not_le_imp_le h₄ value (l.root h₁₄) (Bool.eq_false_iff.mp h₁₂)) h₄ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - cases o - contradiction -- h₁₄ is False - exact h₂.right.right.left - case true => - have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - 1, (by omega)⟩ value l h₂.left (by omega) h₁₂ h₄ h₃ - apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ - exact h₁₀ - ----------------------------------------------------------------------------------------------- --- heapPop - -theorem CompleteTree.heapPopIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (heap : CompleteTree α (n+1)) (h₁ : HeapPredicate heap le) (wellDefinedLe : transitive_le le ∧ total_le le) : HeapPredicate (heap.heapPop le).fst le := by - have h₂ : HeapPredicate (Internal.heapRemoveLast heap).fst le := heapRemoveLastIsHeap h₁ wellDefinedLe.left wellDefinedLe.right - unfold heapPop - cases n <;> simp[h₂, heapUpdateRootIsHeap, wellDefinedLe] - ----------------------------------------------------------------------------------------------- --- heapRemoveAt - -theorem CompleteTree.heapRemoveAtIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin (n+1)) (heap : CompleteTree α (n+1)) (h₁ : HeapPredicate heap le) (wellDefinedLe : transitive_le le ∧ total_le le) : HeapPredicate (heap.heapRemoveAt le index).fst le := by - have h₂ : HeapPredicate (Internal.heapRemoveLastWithIndex heap).fst le := heapRemoveLastWithIndexIsHeap h₁ wellDefinedLe.left wellDefinedLe.right - unfold heapRemoveAt - split - case isTrue => exact heapPopIsHeap le heap h₁ wellDefinedLe - case isFalse h₃ => - cases h: (index = (Internal.heapRemoveLastWithIndex heap).snd.snd : Bool) - <;> simp_all - split - <;> apply heapUpdateAtIsHeap <;> simp_all +import BinaryHeap.CompleteTree.HeapProofs.EmptyIsHeap +import BinaryHeap.CompleteTree.HeapProofs.HeapPush +import BinaryHeap.CompleteTree.HeapProofs.HeapUpdateRoot +import BinaryHeap.CompleteTree.HeapProofs.HeapUpdateAt +import BinaryHeap.CompleteTree.HeapProofs.HeapPop +import BinaryHeap.CompleteTree.HeapProofs.HeapRemoveAt diff --git a/BinaryHeap/CompleteTree/HeapProofs/EmptyIsHeap.lean b/BinaryHeap/CompleteTree/HeapProofs/EmptyIsHeap.lean new file mode 100644 index 0000000..ab95b80 --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/EmptyIsHeap.lean @@ -0,0 +1,6 @@ +import BinaryHeap.CompleteTree.Basic +import BinaryHeap.HeapPredicate + +namespace BinaryHeap.CompleteTree + +theorem emptyIsHeap {α : Type u} (le : α → α → Bool) : HeapPredicate CompleteTree.empty le := True.intro diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapPop.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapPop.lean new file mode 100644 index 0000000..df4f9aa --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapPop.lean @@ -0,0 +1,9 @@ +import BinaryHeap.CompleteTree.HeapProofs.HeapRemoveLast +import BinaryHeap.CompleteTree.HeapProofs.HeapUpdateRoot + +namespace BinaryHeap.CompleteTree + +theorem heapPopIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (heap : CompleteTree α (n+1)) (h₁ : HeapPredicate heap le) (wellDefinedLe : transitive_le le ∧ total_le le) : HeapPredicate (heap.heapPop le).fst le := by + have h₂ : HeapPredicate (Internal.heapRemoveLast heap).fst le := CompleteTree.heapRemoveLastIsHeap h₁ wellDefinedLe.left wellDefinedLe.right + unfold heapPop + cases n <;> simp[h₂, heapUpdateRootIsHeap, wellDefinedLe] diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapPush.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapPush.lean new file mode 100644 index 0000000..8a6e48c --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapPush.lean @@ -0,0 +1,122 @@ +import BinaryHeap.CompleteTree.HeapOperations +import BinaryHeap.CompleteTree.Lemmas + +namespace BinaryHeap.CompleteTree + +private theorem rootSeesThroughCast + (n m : Nat) + (h₁ : n + 1 + m = n + m + 1) + (h₂ : 0 < n + 1 + m) + (h₃ : 0 < n + m + 1) + (heap : CompleteTree α (n+1+m)) : (h₁▸heap).root h₃ = heap.root h₂ := by + induction m generalizing n + case zero => simp + case succ o ho => + have h₄ := ho (n+1) + have h₅ : n + 1 + 1 + o = n + 1 + (Nat.succ o) := by simp_arith + have h₆ : n + 1 + o + 1 = n + (Nat.succ o + 1) := by simp_arith + rewrite[h₅, h₆] at h₄ + revert heap h₁ h₂ h₃ + assumption + +private theorem HeapPredicate.seesThroughCast + (n m : Nat) + (lt : α → α → Bool) + (h₁ : n+1+m+1=n+m+1+1) + (h₂ : 0<n+1+m+1) + (h₃ : 0<n+m+1+1) + (heap : CompleteTree α (n+1+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by + unfold HeapPredicate + intro h₄ + induction m generalizing n + case zero => simp[h₄] + case succ o ho => + have h₅ := ho (n+1) + have h₆ : n+1+1+o+1 = n+1+(Nat.succ o)+1 := by simp_arith + rw[h₆] at h₅ + have h₆ : n + 1 + o + 1 + 1 = n + (Nat.succ o + 1 + 1) := by simp_arith + rewrite[h₆] at h₅ + revert heap h₁ h₂ h₃ + assumption + +theorem heapPushRootSameOrElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : 0 < o): (root (heap.heapPush lt elem) (Nat.succ_pos o) = elem) ∨ (root (heap.heapPush lt elem) (Nat.succ_pos o) = root heap h₂) := by + unfold heapPush + split --match o, heap + contradiction + simp + rename_i n m v l r _ _ _ + split -- if h : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then + case h_2.isTrue h => + cases (lt elem v) <;> simp[instDecidableEqBool, Bool.decEq, root] + case h_2.isFalse h => + rw[rootSeesThroughCast n (m+1) (by simp_arith) (by simp_arith) (by simp_arith)] + cases (lt elem v) + <;> simp[instDecidableEqBool, Bool.decEq, root] + +theorem heapPushEmptyElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : ¬0 < o) : (root (heap.heapPush lt elem) (Nat.succ_pos o) = elem) := + have : o = 0 := Nat.eq_zero_of_le_zero $ (Nat.not_lt_eq 0 o).mp h₂ + match o, heap with + | 0, .leaf => by simp[heapPush, root] + +mutual +/-- + Helper for heapPushIsHeap, to make one function out of both branches. + Sorry for the ugly signature, but this was the easiest thing to extract. + -/ +private theorem heapPushIsHeapAux {α : Type u} (le : α → α → Bool) (n m : Nat) (v elem : α) (l : CompleteTree α n) (r : CompleteTree α m) (h₁ : HeapPredicate l le ∧ HeapPredicate r le ∧ HeapPredicate.leOrLeaf le v l ∧ HeapPredicate.leOrLeaf le v r) (h₂ : transitive_le le) (h₃ : total_le le): HeapPredicate l le ∧ + let smaller := (if le elem v then elem else v) + let larger := (if le elem v then v else elem) + HeapPredicate (heapPush le larger r) le + ∧ HeapPredicate.leOrLeaf le smaller l + ∧ HeapPredicate.leOrLeaf le smaller (heapPush le larger r) + := by + cases h₆ : (le elem v) <;> simp only [Bool.false_eq_true, reduceIte] + case true => + have h₇ : (HeapPredicate (heapPush le v r) le) := heapPushIsHeap h₁.right.left h₂ h₃ + simp only [true_and, h₁, h₇, HeapPredicate.leOrLeaf_transitive h₂ h₆ h₁.right.right.left] + cases m + case zero => + have h₈ := heapPushEmptyElem v r le (Nat.not_lt_zero 0) + simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, Nat.reduceAdd, h₈] + assumption + case succ _ => + simp only [HeapPredicate.leOrLeaf] + cases heapPushRootSameOrElem v r le (Nat.succ_pos _) + <;> rename_i h₇ + <;> rw[h₇] + . assumption + apply h₂ elem v + exact ⟨h₆, h₁.right.right.right⟩ + case false => + have h₇ : (HeapPredicate (heapPush le elem r) le) := heapPushIsHeap h₁.right.left h₂ h₃ + simp only [true_and, h₁, h₇] + have h₈ : le v elem := not_le_imp_le h₃ elem v (by simp only [h₆, Bool.false_eq_true, not_false_eq_true]) + cases m + case zero => + have h₇ := heapPushEmptyElem elem r le (Nat.not_lt_zero 0) + simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, Nat.reduceAdd, h₇] + assumption + case succ _ => + cases heapPushRootSameOrElem elem r le (Nat.succ_pos _) + <;> rename_i h₉ + <;> simp only [HeapPredicate.leOrLeaf, Nat.succ_eq_add_one, h₈, h₉] + exact h₁.right.right.right + +theorem heapPushIsHeap {α : Type u} {elem : α} {heap : CompleteTree α o} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate (heap.heapPush le elem) le := by + unfold heapPush + split -- match o, heap with + trivial + case h_2 n m v l r m_le_n _ _ => + simp + split -- if h₅ : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then + case' isTrue => + have h := heapPushIsHeapAux le n m v elem l r h₁ h₂ h₃ + case' isFalse => + apply HeapPredicate.seesThroughCast <;> try simp_arith[h₂] --gets rid of annoying cast. + have h := heapPushIsHeapAux le m n v elem r l (And.intro h₁.right.left $ And.intro h₁.left $ And.intro h₁.right.right.right h₁.right.right.left) h₂ h₃ + all_goals + unfold HeapPredicate + cases h₆ : (le elem v) + <;> simp only [h₆, Bool.false_eq_true, reduceIte] at h + <;> simp only [instDecidableEqBool, Bool.decEq, h, and_self] +end diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveAt.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveAt.lean new file mode 100644 index 0000000..3d7975c --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveAt.lean @@ -0,0 +1,15 @@ +import BinaryHeap.CompleteTree.HeapProofs.HeapPop +import BinaryHeap.CompleteTree.HeapProofs.HeapUpdateAt + +namespace BinaryHeap.CompleteTree + +theorem heapRemoveAtIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin (n+1)) (heap : CompleteTree α (n+1)) (h₁ : HeapPredicate heap le) (wellDefinedLe : transitive_le le ∧ total_le le) : HeapPredicate (heap.heapRemoveAt le index).fst le := by + have h₂ : HeapPredicate (Internal.heapRemoveLastWithIndex heap).fst le := CompleteTree.heapRemoveLastWithIndexIsHeap h₁ wellDefinedLe.left wellDefinedLe.right + unfold heapRemoveAt + split + case isTrue => exact heapPopIsHeap le heap h₁ wellDefinedLe + case isFalse h₃ => + cases h: (index = (Internal.heapRemoveLastWithIndex heap).snd.snd : Bool) + <;> simp_all + split + <;> apply heapUpdateAtIsHeap <;> simp_all diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveLast.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveLast.lean new file mode 100644 index 0000000..e9c357d --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapRemoveLast.lean @@ -0,0 +1,169 @@ +import BinaryHeap.CompleteTree.HeapOperations +import BinaryHeap.CompleteTree.Lemmas +import BinaryHeap.CompleteTree.HeapProofs.EmptyIsHeap + +namespace BinaryHeap.CompleteTree + + +--- Same as rootSeesThroughCast, but in the other direction. +private theorem rootSeesThroughCast2 + (n m : Nat) + (h₁ : n + m + 1 = n + 1 + m) + (h₂ : 0 < n + m + 1) + (h₃ : 0 < n + 1 + m) + (heap : CompleteTree α (n+m+1)) : (h₁▸heap).root h₃ = heap.root h₂ := by + induction m generalizing n + case zero => simp + case succ mm mh => + have h₄ := mh (n+1) + have h₅ : n + 1 + mm + 1 = n + Nat.succ mm + 1 := by simp_arith + have h₆ : n + 1 + 1 + mm = n + 1 + Nat.succ mm := by simp_arith + rw[h₅, h₆] at h₄ + revert heap h₁ h₂ h₃ + assumption + +private theorem HeapPredicate.seesThroughCast2 + (n m : Nat) + (lt : α → α → Bool) + (h₁ : n+m+1=n+1+m) + (h₂ : 0<n+1+m) + (h₃ : 0<n+m+1) + (heap : CompleteTree α (n+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by + unfold HeapPredicate + intro h₄ + induction m generalizing n + case zero => simp[h₄] + case succ o ho => + have h₅ := ho (n+1) + have h₆ : n+1+o+1 = n+(Nat.succ o)+1 := by simp_arith + rw[h₆] at h₅ + have h₆ : n + 1 + 1 + o = n + 1 + Nat.succ o := by simp_arith + rewrite[h₆] at h₅ + revert heap h₁ h₂ h₃ + assumption + +private theorem castZeroHeap (n m : Nat) (heap : CompleteTree α 0) (h₁ : 0=n+m) {le : α → α → Bool} : HeapPredicate (h₁ ▸ heap) le := by + have h₂ : heap = (empty : CompleteTree α 0) := by + simp[empty] + match heap with + | .leaf => trivial + have h₂ : HeapPredicate heap le := by simp[h₂, empty, emptyIsHeap] + cases m + case succ => contradiction + case zero => + cases n + case succ => contradiction + case zero => + simp[h₁, h₂] + +-- If there is only one element left, the result is a leaf. +private theorem heapRemoveLastAuxLeaf +{α : Type u} +{β : Nat → Type u} +(heap : CompleteTree α 1) +(aux0 : α → (β 1)) +(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) +(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) +: (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst = leaf := by + let l := (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst + have h₁ : l = leaf := match l with + | .leaf => rfl + exact h₁ + +protected theorem heapRemoveLastAuxLeavesRoot +{α : Type u} +{β : Nat → Type u} +(heap : CompleteTree α (n+1)) +(aux0 : α → (β 1)) +(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) +(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) +(h₁ : n > 0) +: heap.root (Nat.zero_lt_of_ne_zero $ Nat.succ_ne_zero n) = (Internal.heapRemoveLastAux heap aux0 auxl auxr).fst.root (h₁) := by + unfold CompleteTree.Internal.heapRemoveLastAux + split + rename_i o p v l r _ _ _ + have h₃ : (0 ≠ o + p) := Ne.symm $ Nat.not_eq_zero_of_lt h₁ + simp[h₃] + exact + if h₄ : p < o ∧ Nat.nextPowerOfTwo (p + 1) = p + 1 then by + simp[h₄] + cases o + case zero => exact absurd h₄.left $ Nat.not_lt_zero p + case succ oo _ _ _ => + simp -- redundant, but makes goal easier to read + rw[rootSeesThroughCast2 oo p _ (by simp_arith) _] + apply root_unfold + else by + simp[h₄] + cases p + case zero => + simp_arith at h₁ -- basically o ≠ 0 + simp_arith (config := {ground := true})[h₁] at h₄ -- the second term in h₄ is decidable and True. What remains is ¬(0 < o), or o = 0 + case succ pp hp => + simp_arith + apply root_unfold + +private theorem heapRemoveLastAuxIsHeap +{α : Type u} +{β : Nat → Type u} +{heap : CompleteTree α (o+1)} +{le : α → α → Bool} +(aux0 : α → (β 1)) +(auxl : {prev_size curr_size : Nat} → β prev_size → (h₁ : prev_size < curr_size) → β curr_size) +(auxr : {prev_size curr_size : Nat} → β prev_size → (left_size : Nat) → (h₁ : prev_size + left_size < curr_size) → β curr_size) +(h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate ((Internal.heapRemoveLastAux heap aux0 auxl auxr).fst) le := by + unfold CompleteTree.Internal.heapRemoveLastAux + split + rename_i n m v l r _ _ _ + exact + if h₄ : 0 = (n+m) then by + simp only [h₄, reduceDite, castZeroHeap] + else by + simp[h₄] + exact + if h₅ : (m<n ∧ Nat.nextPowerOfTwo (m+1) = m+1) then by + simp only [h₅, and_self, ↓reduceDite] + cases n + case zero => + exact absurd h₅.left $ Nat.not_lt_zero m + case succ ll h₆ h₇ h₈ => + simp only + apply HeapPredicate.seesThroughCast2 <;> try simp_arith + cases ll + case a.zero => -- if ll is zero, then (heapRemoveLast l).snd is a leaf. + have h₆ := heapRemoveLastAuxLeaf l aux0 auxl auxr + rw[h₆] + unfold HeapPredicate at * + have h₇ : HeapPredicate .leaf le := by trivial + have h₈ : HeapPredicate.leOrLeaf le v .leaf := by trivial + exact ⟨h₇,h₁.right.left, h₈, h₁.right.right.right⟩ + case a.succ nn => -- if ll is not zero, then the root element before and after heapRemoveLast is the same. + unfold HeapPredicate at * + simp only [heapRemoveLastAuxIsHeap aux0 auxl auxr h₁.left h₂ h₃, h₁.right.left, h₁.right.right.right, and_true, true_and] + unfold HeapPredicate.leOrLeaf + simp only + rw[←CompleteTree.heapRemoveLastAuxLeavesRoot] + exact h₁.right.right.left + else by + simp[h₅] + cases m + case zero => + simp only [Nat.add_zero] at h₄ -- n ≠ 0 + simp (config := { ground := true }) only [Nat.zero_add, and_true, Nat.not_lt, Nat.le_zero_eq, Ne.symm h₄] at h₅ -- the second term in h₅ is decidable and True. What remains is ¬(0 < n), or n = 0 + case succ mm h₆ h₇ h₈ => + unfold HeapPredicate at * + simp only [h₁, heapRemoveLastAuxIsHeap aux0 auxl auxr h₁.right.left h₂ h₃, true_and] + unfold HeapPredicate.leOrLeaf + exact match mm with + | 0 => rfl + | o+1 => + have h₉ : le v ((Internal.heapRemoveLastAux r _ _ _).fst.root (Nat.zero_lt_succ o)) := by + rw[←CompleteTree.heapRemoveLastAuxLeavesRoot] + exact h₁.right.right.right + h₉ + +protected theorem heapRemoveLastIsHeap {α : Type u} {heap : CompleteTree α (o+1)} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate (Internal.heapRemoveLast heap).fst le := + heapRemoveLastAuxIsHeap _ _ _ h₁ h₂ h₃ + +protected theorem heapRemoveLastWithIndexIsHeap {α : Type u} {heap : CompleteTree α (o+1)} {le : α → α → Bool} (h₁ : HeapPredicate heap le) (h₂ : transitive_le le) (h₃ : total_le le) : HeapPredicate ((Internal.heapRemoveLastWithIndex heap).fst) le := + heapRemoveLastAuxIsHeap _ _ _ h₁ h₂ h₃ diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateAt.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateAt.lean new file mode 100644 index 0000000..90991d8 --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateAt.lean @@ -0,0 +1,128 @@ +import BinaryHeap.CompleteTree.HeapOperations +import BinaryHeap.CompleteTree.Lemmas +import BinaryHeap.CompleteTree.HeapProofs.HeapUpdateRoot + +namespace BinaryHeap.CompleteTree + +private theorem heapUpdateAtIsHeapLeRootAux_RootLeValue {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le (root heap h₂) value) (h₄ : total_le le) : HeapPredicate.leOrLeaf le (root heap h₂) (heapUpdateAt le index value heap h₂).fst := by + unfold heapUpdateAt + split + case isTrue => exact CompleteTree.heapUpdateRootIsHeapLeRootAux le value heap h₁ h₂ h₃ + case isFalse hi => + split + rename_i o p v l r h₆ h₇ h₈ index h₁ h₅ + cases h₉ : le v value <;> simp (config := { ground := true }) only + case false => rw[root_unfold] at h₃; exact absurd h₃ ((Bool.not_eq_true (le v value)).substr h₉) + case true => + rw[root_unfold] + split + <;> simp![reflexive_le, h₄] + +private theorem heapUpdateAtIsHeapLeRootAux_ValueLeRoot {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le value (root heap h₂)) (h₄ : total_le le) (h₅ : transitive_le le) : HeapPredicate.leOrLeaf le value (heapUpdateAt le index value heap h₂).fst := by + unfold heapUpdateAt + split + <;> rename_i h₉ + case isTrue => + unfold heapUpdateRoot + split + rename_i o p v l r h₆ h₇ h₈ h₂ + cases o <;> cases p <;> simp only [↓reduceDite,HeapPredicate.leOrLeaf, root_unfold, h₄, reflexive_le] + <;> unfold HeapPredicate at h₁ + <;> have h₁₀ : le value $ l.root (by omega) := h₅ value v (l.root _) ⟨h₃, h₁.right.right.left⟩ + simp only [↓reduceIte, Nat.add_zero, h₁₀, root_unfold, h₄, reflexive_le] + have h₁₁ : le value $ r.root (by omega) := h₅ value v (r.root _) ⟨h₃, h₁.right.right.right⟩ + simp only [↓reduceIte, h₁₀, h₁₁, and_self, root_unfold, h₄, reflexive_le] + case isFalse => + split + rename_i o p v l r h₆ h₇ h₈ index h₂ hi + cases le v value + <;> simp (config := { ground := true }) only [root_unfold, Nat.pred_eq_sub_one] at h₃ ⊢ + <;> split + <;> unfold HeapPredicate.leOrLeaf + <;> simp only [root_unfold, h₃, h₄, reflexive_le] + +theorem heapUpdateAtIsHeap {α : Type u} {n : Nat} (le : α → α → Bool) (index : Fin n) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) (h₂ : HeapPredicate heap le) (h₃ : transitive_le le) (h₄ : total_le le) : HeapPredicate (heap.heapUpdateAt le index value h₁).fst le := by + unfold heapUpdateAt + split + case isTrue h₅ => + exact heapUpdateRootIsHeap le value heap h₁ h₂ h₃ h₄ + case isFalse h₅ => + split + rename_i o p v l r h₆ h₇ h₈ index h₁ h₅ + cases h₁₀ : le v value <;> simp (config := {ground := true}) -- this could probably be solved without this split, but readability... + <;> split + <;> rename_i h -- h is the same name as used in the function + <;> unfold HeapPredicate at h₂ ⊢ + <;> simp only [h₂, and_true, true_and] + case false.isFalse => + have h₁₀ := not_le_imp_le h₄ v value (Bool.eq_false_iff.mp h₁₀) + have h₁₄ : p > 0 := by cases p; exact absurd (Nat.lt_succ.mp index.isLt) h; exact Nat.zero_lt_succ _ + apply And.intro <;> try apply And.intro + case left => exact heapUpdateAtIsHeap le ⟨index.val - o - 1, _⟩ v r h₁₄ h₂.right.left h₃ h₄ + case right.left => exact HeapPredicate.leOrLeaf_transitive h₃ h₁₀ h₂.right.right.left + case right.right => + have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - o - 1, (by omega)⟩ v r h₁₄).fst le := + (heapUpdateAtIsHeap le ⟨index.val - o - 1, (by omega)⟩ v r _ h₂.right.left h₃ h₄) + cases h₁₂ : le v (r.root h₁₄) + case false => + cases p + exact absurd (Nat.lt_succ.mp index.isLt) h + exact absurd h₂.right.right.right ((Bool.eq_false_iff).mp h₁₂) + case true => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - o - 1, (by omega)⟩ v r h₂.right.left (by omega) h₁₂ h₄ h₃ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + exact h₁₀ + case false.isTrue => + have h₁₀ := not_le_imp_le h₄ v value (Bool.eq_false_iff.mp h₁₀) + have h₁₄ : o > 0 := by cases o; simp at h₅ h; exact absurd (Fin.val_inj.mp h : index = 0) h₅; exact Nat.zero_lt_succ _ + apply And.intro <;> try apply And.intro + case left => exact heapUpdateAtIsHeap le ⟨index.val - 1, _⟩ v l h₁₄ h₂.left h₃ h₄ + case right.right => exact HeapPredicate.leOrLeaf_transitive h₃ h₁₀ h₂.right.right.right + case right.left => + have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - 1, (_)⟩ v l h₁₄).fst le := + (heapUpdateAtIsHeap le ⟨index.val - 1, (by omega)⟩ v l _ h₂.left h₃ h₄) + cases h₁₂ : le v (l.root h₁₄) + case false => + cases o + contradiction -- h₁₄ is False + exact absurd h₂.right.right.left ((Bool.eq_false_iff).mp h₁₂) + case true => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - 1, (by omega)⟩ v l h₂.left (by omega) h₁₂ h₄ h₃ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + exact h₁₀ + case true.isFalse => + have h₁₄ : p > 0 := by cases p; exact absurd (Nat.lt_succ.mp index.isLt) h; exact Nat.zero_lt_succ _ + apply And.intro + case left => exact heapUpdateAtIsHeap le ⟨index.val - o - 1, _⟩ value r h₁₄ h₂.right.left h₃ h₄ + case right => + have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - o - 1, (by omega)⟩ v r (h₁₄)).fst le := + (heapUpdateAtIsHeap le ⟨index.val - o - 1, (by omega)⟩ v r _ h₂.right.left h₃ h₄) + cases h₁₂ : le value (r.root h₁₄) + case false => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_RootLeValue le ⟨index.val - o - 1, (by omega)⟩ value r h₂.right.left (by omega) (not_le_imp_le h₄ value (r.root h₁₄) (Bool.eq_false_iff.mp h₁₂)) h₄ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + cases p + contradiction -- h₁₄ is False + exact h₂.right.right.right + case true => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - o - 1, (by omega)⟩ value r h₂.right.left (by omega) h₁₂ h₄ h₃ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + exact h₁₀ + case true.isTrue => + have h₁₄ : o > 0 := by cases o; simp at h₅ h; exact absurd (Fin.val_inj.mp h : index = 0) h₅; exact Nat.zero_lt_succ _ + apply And.intro + case left => exact heapUpdateAtIsHeap le ⟨index.val - 1, _⟩ value l h₁₄ h₂.left h₃ h₄ + case right => + have h₁₁: HeapPredicate (heapUpdateAt le ⟨index.val - 1, (by omega)⟩ v l h₁₄).fst le := + (heapUpdateAtIsHeap le ⟨index.val - 1, (by omega)⟩ v l _ h₂.left h₃ h₄) + cases h₁₂ : le value (l.root h₁₄) + case false => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_RootLeValue le ⟨index.val - 1, (by omega)⟩ value l h₂.left (by omega) (not_le_imp_le h₄ value (l.root h₁₄) (Bool.eq_false_iff.mp h₁₂)) h₄ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + cases o + contradiction -- h₁₄ is False + exact h₂.right.right.left + case true => + have h₁₃ := heapUpdateAtIsHeapLeRootAux_ValueLeRoot le ⟨index.val - 1, (by omega)⟩ value l h₂.left (by omega) h₁₂ h₄ h₃ + apply HeapPredicate.leOrLeaf_transitive h₃ _ h₁₃ + exact h₁₀ diff --git a/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateRoot.lean b/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateRoot.lean new file mode 100644 index 0000000..c2cc072 --- /dev/null +++ b/BinaryHeap/CompleteTree/HeapProofs/HeapUpdateRoot.lean @@ -0,0 +1,239 @@ +import BinaryHeap.CompleteTree.HeapOperations +import BinaryHeap.CompleteTree.Lemmas + +namespace BinaryHeap.CompleteTree + +private theorem heapUpdateRootReturnsRoot {α : Type u} {n : Nat} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) : (heap.heapUpdateRoot le value h₁).snd = heap.root h₁ := by + unfold heapUpdateRoot + split + rename_i o p v l r h₃ h₄ h₅ h₁ + simp + cases o <;> simp + case zero => + exact root_unfold v l r h₃ h₄ h₅ h₁ + case succ => + cases p <;> simp + case zero => + cases le value (root l _) + <;> simp only [Bool.false_eq_true, ↓reduceIte, root_unfold] + case succ => + cases le value (root l _) <;> cases le value (root r _) + <;> cases le (root l _) (root r _) + <;> simp only [Bool.false_eq_true, and_self, and_true, and_false, ↓reduceIte, root_unfold] + +private theorem heapUpdateRootPossibleRootValuesAuxL {α : Type u} (heap : CompleteTree α n) (h₁ : n > 1) : 0 < heap.leftLen (Nat.lt_trans (Nat.lt_succ_self 0) h₁) := + match h₅: n, heap with + | (o+p+1), .branch v l r h₂ h₃ h₄ => by + simp[leftLen, length] + cases o + case zero => rewrite[(Nat.le_zero_eq p).mp h₂] at h₁; contradiction + case succ q => exact Nat.zero_lt_succ q + +private theorem heapUpdateRootPossibleRootValuesAuxR {α : Type u} (heap : CompleteTree α n) (h₁ : n > 2) : 0 < heap.rightLen (Nat.lt_trans (Nat.lt_succ_self 0) $ Nat.lt_trans (Nat.lt_succ_self 1) h₁) := + match h₅: n, heap with + | (o+p+1), .branch v l r h₂ h₃ h₄ => by + simp[rightLen, length] + cases p + case zero => simp_arith at h₁; simp at h₃; exact absurd h₁ (Nat.not_le_of_gt h₃) + case succ q => exact Nat.zero_lt_succ q + +private theorem heapUpdateRootPossibleRootValues1 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n = 1) : (heap.heapUpdateRoot le value (h₁.substr (Nat.lt_succ_self 0))).fst.root (h₁.substr (Nat.lt_succ_self 0)) = value := by + unfold heapUpdateRoot + generalize (h₁.substr (Nat.lt_succ_self 0) : n > 0) = hx + split + rename_i o p v l r _ _ _ h₁ + have h₃ : o = 0 := (Nat.add_eq_zero.mp $ Nat.succ.inj h₁).left + simp[h₃, root_unfold] + +private theorem heapUpdateRootPossibleRootValues2 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n = 2) : +have h₂ : 0 < n := Nat.lt_trans (Nat.lt_succ_self 0) $ h₁.substr (Nat.lt_succ_self 1) +have h₃ : 0 < leftLen heap h₂ := heapUpdateRootPossibleRootValuesAuxL heap (h₁.substr (Nat.lt_succ_self 1)) +(heap.heapUpdateRoot le value h₂).fst.root h₂ = value +∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.left h₂).root h₃ +:= by + simp + unfold heapUpdateRoot + generalize (Nat.lt_trans (Nat.lt_succ_self 0) (Eq.substr h₁ (Nat.lt_succ_self 1)) : 0 < n) = h₂ + split + rename_i o p v l r h₃ h₄ h₅ h₂ + cases o <;> simp + case zero => simp only[root, true_or] + case succ oo => + have h₆ : p = 0 := by simp at h₁; omega + simp only [h₆, ↓reduceDite] + cases le value (l.root _) + <;> simp[heapUpdateRootReturnsRoot, root_unfold, left_unfold] + +private theorem heapUpdateRootPossibleRootValues3 {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 2) : +have h₂ : 0 < n := Nat.lt_trans (Nat.lt_succ_self 0) $ Nat.lt_trans (Nat.lt_succ_self 1) h₁ +have h₃ : 0 < leftLen heap h₂ := heapUpdateRootPossibleRootValuesAuxL heap $ Nat.lt_trans (Nat.lt_succ_self 1) h₁ +have h₄ : 0 < rightLen heap h₂ := heapUpdateRootPossibleRootValuesAuxR heap h₁ +(heap.heapUpdateRoot le value h₂).fst.root h₂ = value +∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.left h₂).root h₃ +∨ (heap.heapUpdateRoot le value h₂).fst.root h₂ = (heap.right h₂).root h₄ +:= by + simp only + unfold heapUpdateRoot + generalize (Nat.lt_trans (Nat.lt_succ_self 0) (Nat.lt_trans (Nat.lt_succ_self 1) h₁) : 0 < n) = h₂ + split + rename_i o p v l r h₃ h₄ h₅ h₂ + cases o + case zero => simp only[root, true_or] + case succ oo => + have h₆ : p ≠ 0 := by simp at h₁; omega + simp only [Nat.add_one_ne_zero, ↓reduceDite, h₆] + cases le value (l.root _) <;> simp + rotate_right + cases le value (r.root _) <;> simp + case true.true => simp[root] + case false | true.false => + cases le (l.root _) (r.root _) + <;> simp only [Bool.false_eq_true, ↓reduceIte, heapUpdateRootReturnsRoot, root_unfold, left_unfold, right_unfold, true_or, or_true] + +protected theorem heapUpdateRootIsHeapLeRootAux {α : Type u} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : HeapPredicate heap le) (h₂ : n > 0) (h₃ : le (root heap h₂) value) : HeapPredicate.leOrLeaf le (root heap h₂) (heapUpdateRoot le value heap h₂).fst := + if h₄ : n = 1 then by + have h₅ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only[h₃, h₄, heapUpdateRootPossibleRootValues1] + unfold HeapPredicate.leOrLeaf + split + · rfl + · exact h₅ + else if h₅ : n = 2 then by + have h₆ := heapUpdateRootPossibleRootValues2 le value heap h₅ + cases h₆ + case inl h₆ => + have h₇ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only [h₆, h₃] + unfold HeapPredicate.leOrLeaf + split + · rfl + · exact h₇ + case inr h₆ => + unfold HeapPredicate.leOrLeaf + unfold HeapPredicate at h₁ + split at h₁ + case h_1 => contradiction + case h_2 o p v l r h₇ h₈ h₉ => + have h₁₁ : p = 0 := by + simp at h₅ + cases o; simp only [Nat.le_zero_eq] at h₇; exact h₇; simp_arith[Nat.add_eq_zero] at h₅; exact h₅.right + have h₁₀ : o = 1 := by simp_arith[h₁₁] at h₅; assumption + simp only + rw[h₆] + have h₁₂ := h₁.right.right.left + unfold HeapPredicate.leOrLeaf at h₁₂ + cases o ; contradiction; + case succ => + exact h₁₂ + else by + have h₆ : n > 2 := by omega + have h₇ := heapUpdateRootPossibleRootValues3 le value heap h₆ + simp at h₇ + unfold HeapPredicate at h₁ + cases h₇ + case inl h₇ => + have h₈ : le (heap.root h₂) ( (heapUpdateRoot le value heap h₂).fst.root h₂) := by simp only [h₇, h₃] + unfold HeapPredicate.leOrLeaf + split + · rfl + · exact h₈ + case inr h₇ => + cases h₇ + case inl h₇ | inr h₇ => + unfold HeapPredicate.leOrLeaf + split at h₁ + contradiction + simp_all + case h_2 o p v l r _ _ _ => + cases o + omega + cases p + omega + have h₈ := h₁.right.right.left + have h₉ := h₁.right.right.right + assumption + +private theorem heapUpdateRootIsHeapLeRootAuxLe {α : Type u} (le : α → α → Bool) {a b c : α} (h₁ : transitive_le le) (h₂ : total_le le) (h₃ : le b c) : ¬le a c ∨ ¬ le a b → le b a +| .inr h₅ => not_le_imp_le h₂ _ _ h₅ +| .inl h₅ => h₁ b c a ⟨h₃,not_le_imp_le h₂ _ _ h₅⟩ + +theorem heapUpdateRootIsHeap {α : Type u} {n: Nat} (le : α → α → Bool) (value : α) (heap : CompleteTree α n) (h₁ : n > 0) (h₂ : HeapPredicate heap le) (h₃ : transitive_le le) (h₄ : total_le le) : HeapPredicate (heap.heapUpdateRoot le value h₁).fst le := by + unfold heapUpdateRoot + split + rename_i o p v l r h₇ h₈ h₉ heq + exact + if h₁₀ : o = 0 then by + simp only [Nat.add_eq, Nat.succ_eq_add_one, h₁₀, ↓reduceDite] + unfold HeapPredicate at h₂ ⊢ + simp only [h₂, true_and] + unfold HeapPredicate.leOrLeaf + have : p = 0 := by rw[h₁₀, Nat.le_zero_eq] at h₇; assumption + apply And.intro + case left => match o, l with + | Nat.zero, _ => trivial + case right => match p, r with + | Nat.zero, _ => trivial + else if h₁₁ : p = 0 then by + simp only [↓reduceDite, h₁₀, h₁₁] + cases h₉ : le value (root l (_ : 0 < o)) <;> simp + case true => + unfold HeapPredicate at * + simp only [h₂, true_and] + unfold HeapPredicate.leOrLeaf + apply And.intro + case right => match p, r with + | Nat.zero, _ => trivial + case left => match o, l with + | Nat.succ _, _ => assumption + case false => + rw[heapUpdateRootReturnsRoot] + have h₁₂ : le (l.root (Nat.zero_lt_of_ne_zero h₁₀)) value := by simp[h₉, h₄, not_le_imp_le] + have h₁₃ : o = 1 := Nat.le_antisymm (by simp_arith[h₁₁] at h₈; exact h₈) (Nat.succ_le_of_lt (Nat.zero_lt_of_ne_zero h₁₀)) + unfold HeapPredicate at * + simp only [h₂, true_and] --closes one sub-goal + apply And.intro <;> try apply And.intro + case right.right => match p, r with + | 0, .leaf => simp[HeapPredicate.leOrLeaf] + case right.left => + simp only [HeapPredicate.leOrLeaf] + cases o <;> simp only [Nat.succ_eq_add_one, heapUpdateRootPossibleRootValues1, h₁₃, h₁₂] + case left => + apply heapUpdateRootIsHeap + exact h₂.left + exact h₃ + exact h₄ + else if h₁₂ : le value (root l (Nat.zero_lt_of_ne_zero h₁₀)) ∧ le value (root r (Nat.zero_lt_of_ne_zero h₁₁)) then by + unfold HeapPredicate at * + simp only [↓reduceDite, and_self, ↓reduceIte, true_and, h₁₀, h₁₁, h₁₂, h₂] + unfold HeapPredicate.leOrLeaf + cases o + · contradiction + · cases p + · contradiction + · assumption + else by + simp only [↓reduceDite, ↓reduceIte, h₁₀, h₁₁, h₁₂] + have h₁₃ : ¬le value (root l _) ∨ ¬le value (root r _) := (Decidable.not_and_iff_or_not (le value (root l (Nat.zero_lt_of_ne_zero h₁₀)) = true) (le value (root r (Nat.zero_lt_of_ne_zero h₁₁)) = true)).mp h₁₂ + cases h₁₄ : le (root l (_ : 0 < o)) (root r (_ : 0 < p)) + <;> simp only [Bool.false_eq_true, ↓reduceIte] + <;> unfold HeapPredicate at * + <;> simp only [true_and, h₂] + <;> apply And.intro + <;> try apply And.intro + case false.left | true.left => + apply heapUpdateRootIsHeap + <;> simp only [h₂, h₃, h₄] + case false.right.left => + unfold HeapPredicate.leOrLeaf + have h₁₅ : le (r.root _) (l.root _) = true := not_le_imp_le h₄ (l.root _) (r.root _) $ (Bool.not_eq_true $ le (root l (_ : 0 < o)) (root r (_ : 0 < p))).substr h₁₄ + simp only[heapUpdateRootReturnsRoot] + cases o <;> simp only[h₁₅] + case true.right.right => + unfold HeapPredicate.leOrLeaf + simp only[heapUpdateRootReturnsRoot] + cases p <;> simp only[h₁₄] + case false.right.right => + have h₁₅ : le (r.root _) (l.root _) = true := not_le_imp_le h₄ (l.root _) (r.root _) $ (Bool.not_eq_true $ le (root l (_ : 0 < o)) (root r (_ : 0 < p))).substr h₁₄ + have h₁₆ : le (r.root _) value := heapUpdateRootIsHeapLeRootAuxLe le h₃ h₄ h₁₅ h₁₃ + simp only [heapUpdateRootReturnsRoot, CompleteTree.heapUpdateRootIsHeapLeRootAux, h₂, h₁₆] + case true.right.left => + have h₁₆ : le (l.root _) value := heapUpdateRootIsHeapLeRootAuxLe le h₃ h₄ h₁₄ h₁₃.symm + simp only [heapUpdateRootReturnsRoot, CompleteTree.heapUpdateRootIsHeapLeRootAux, h₂, h₁₆] diff --git a/BinaryHeap/CompleteTree/Lemmas.lean b/BinaryHeap/CompleteTree/Lemmas.lean index 24bea5b..e7a7e7b 100644 --- a/BinaryHeap/CompleteTree/Lemmas.lean +++ b/BinaryHeap/CompleteTree/Lemmas.lean @@ -81,3 +81,11 @@ theorem CompleteTree.lengthEqLeftRightLenSucc {α : Type u} {n : Nat} (tree : Co split unfold length rfl + +theorem HeapPredicate.leOrLeaf_transitive (h₁ : transitive_le le) : le a b → HeapPredicate.leOrLeaf le b h → HeapPredicate.leOrLeaf le a h := by + unfold HeapPredicate.leOrLeaf + intros h₂ h₃ + rename_i n + cases n <;> simp + apply h₁ a b _ + exact ⟨h₂, h₃⟩ |