diff options
Diffstat (limited to 'Common')
| -rw-r--r-- | Common/BinaryHeap.lean | 104 |
1 files changed, 52 insertions, 52 deletions
diff --git a/Common/BinaryHeap.lean b/Common/BinaryHeap.lean index c2f1901..583e0f5 100644 --- a/Common/BinaryHeap.lean +++ b/Common/BinaryHeap.lean @@ -2,51 +2,51 @@ import Common.Nat namespace BinaryHeap -inductive BalancedTree (α : Type u) : Nat → Type u - | leaf : BalancedTree α 0 +inductive CompleteTree (α : Type u) : Nat → Type u + | leaf : CompleteTree α 0 | branch : (val : α) - → (left : BalancedTree α n) - → (right : BalancedTree α m) + → (left : CompleteTree α n) + → (right : CompleteTree α m) → m ≤ n → n < 2*(m+1) → (n+1).isPowerOfTwo ∨ (m+1).isPowerOfTwo - → BalancedTree α (n+m+1) + → CompleteTree α (n+m+1) -def BalancedTree.root (tree : BalancedTree α n) (_ : 0 < n) : α := match tree with +def CompleteTree.root (tree : CompleteTree α n) (_ : 0 < n) : α := match tree with | .branch a _ _ _ _ _ => a structure WellDefinedLt (lt : α → α → Bool) : Prop where not_equal : ∀ (a b : α), lt a b ↔ ¬lt b a transitive : ∀ (a b c : α), lt a b ∧ lt b c → lt a c -def HeapPredicate {α : Type u} {n : Nat} (heap : BalancedTree α n) (lt : α → α → Bool) : Prop := +def HeapPredicate {α : Type u} {n : Nat} (heap : CompleteTree α n) (lt : α → α → Bool) : Prop := match heap with | .leaf => True | .branch a left right _ _ _ => HeapPredicate left lt ∧ HeapPredicate right lt ∧ notSmallerOrLeaf a left ∧ notSmallerOrLeaf a right - where notSmallerOrLeaf := λ {m : Nat} (v : α) (h : BalancedTree α m) ↦ match m with + where notSmallerOrLeaf := λ {m : Nat} (v : α) (h : CompleteTree α m) ↦ match m with | .zero => true | .succ o => !lt (h.root (by simp_arith)) v structure BinaryHeap (α : Type u) (lt : α → α → Bool) (n : Nat) where - tree : BalancedTree α n + tree : CompleteTree α n valid : HeapPredicate tree lt wellDefinedLt : WellDefinedLt lt /--Please do not use this for anything meaningful. It's a debug function, with horrible performance.-/ -instance {α : Type u} [ToString α] : ToString (BalancedTree α n) where +instance {α : Type u} [ToString α] : ToString (CompleteTree α n) where toString := λt ↦ --not very fast, doesn't matter, is for debugging - let rec max_width := λ {m : Nat} (t : (BalancedTree α m)) ↦ match m, t with + let rec max_width := λ {m : Nat} (t : (CompleteTree α m)) ↦ match m, t with | 0, .leaf => 0 - | (_+_+1), BalancedTree.branch a left right _ _ _ => max (ToString.toString a).length $ max (max_width left) (max_width right) + | (_+_+1), CompleteTree.branch a left right _ _ _ => max (ToString.toString a).length $ max (max_width left) (max_width right) let max_width := max_width t let lines_left := Nat.log2 (n+1).nextPowerOfTwo - let rec print_line := λ (mw : Nat) {m : Nat} (t : (BalancedTree α m)) (lines : Nat) ↦ + let rec print_line := λ (mw : Nat) {m : Nat} (t : (CompleteTree α m)) (lines : Nat) ↦ match m, t with | 0, _ => "" - | (_+_+1), BalancedTree.branch a left right _ _ _ => + | (_+_+1), CompleteTree.branch a left right _ _ _ => let thisElem := ToString.toString a let thisElem := (List.replicate (mw - thisElem.length) ' ').asString ++ thisElem let elems_in_last_line := if lines == 0 then 0 else 2^(lines-1) @@ -63,12 +63,12 @@ instance {α : Type u} [ToString α] : ToString (BalancedTree α n) where print_line max_width t lines_left /-- Extracts the element count. For when pattern matching is too much work. -/ -def BalancedTree.length : BalancedTree α n → Nat := λ_ ↦ n +def CompleteTree.length : CompleteTree α n → Nat := λ_ ↦ n -/--Creates an empty BalancedTree. Needs the heap predicate as parameter.-/ -abbrev BalancedTree.empty {α : Type u} := BalancedTree.leaf (α := α) +/--Creates an empty CompleteTree. Needs the heap predicate as parameter.-/ +abbrev CompleteTree.empty {α : Type u} := CompleteTree.leaf (α := α) -theorem BalancedTree.emptyIsHeap {α : Type u} (lt : α → α → Bool) : HeapPredicate BalancedTree.empty lt := by trivial +theorem CompleteTree.emptyIsHeap {α : Type u} (lt : α → α → Bool) : HeapPredicate CompleteTree.empty lt := by trivial theorem power_of_two_mul_two_lt {n m : Nat} (h₁ : m.isPowerOfTwo) (h₂ : n < 2*m) (h₃ : ¬(n+1).isPowerOfTwo) : n+1 < 2*m := if h₄ : n+1 > 2*m then by @@ -112,10 +112,10 @@ theorem power_of_two_mul_two_le {n m : Nat} (h₁ : (n+1).isPowerOfTwo) (h₂ : simp[h₆] at h₅ assumption -/--Adds a new element to a given BalancedTree.-/ -private def BalancedTree.heapInsert (lt : α → α → Bool) (elem : α) (heap : BalancedTree α o) : BalancedTree α (o+1) := +/--Adds a new element to a given CompleteTree.-/ +private def CompleteTree.heapInsert (lt : α → α → Bool) (elem : α) (heap : CompleteTree α o) : CompleteTree α (o+1) := match o, heap with - | 0, .leaf => BalancedTree.branch elem (BalancedTree.leaf) (BalancedTree.leaf) (by simp) (by simp) (by simp[Nat.one_isPowerOfTwo]) + | 0, .leaf => CompleteTree.branch elem (CompleteTree.leaf) (CompleteTree.leaf) (by simp) (by simp) (by simp[Nat.one_isPowerOfTwo]) | (n+m+1), .branch a left right p max_height_difference subtree_complete => let (elem, a) := if lt elem a then (a, elem) else (elem, a) -- okay, based on n and m we know if we want to add left or right. @@ -165,12 +165,12 @@ private def BalancedTree.heapInsert (lt : α → α → Bool) (elem : α) (heap have letMeSpellItOutForYou : n + 1 + m + 1 = n + m + 1 + 1 := by simp_arith letMeSpellItOutForYou ▸ result -private theorem BalancedTree.rootSeesThroughCast +private theorem CompleteTree.rootSeesThroughCast (n m : Nat) (h₁ : n+1+m+1=n+m+1+1) (h₂ : 0<n+1+m+1) (h₃ : 0<n+m+1+1) - (heap : BalancedTree α (n+1+m+1)) : (h₁▸heap).root h₃ = heap.root h₂ := by + (heap : CompleteTree α (n+1+m+1)) : (h₁▸heap).root h₃ = heap.root h₂ := by induction m generalizing n case zero => simp case succ o ho => @@ -182,24 +182,24 @@ private theorem BalancedTree.rootSeesThroughCast revert heap h₁ h₂ h₃ assumption -theorem BalancedTree.heapInsertRootSameOrElem (elem : α) (heap : BalancedTree α o) (lt : α → α → Bool) (h₂ : 0 < o): (BalancedTree.root (heap.heapInsert lt elem) (by simp_arith) = elem) ∨ (BalancedTree.root (heap.heapInsert lt elem) (by simp_arith) = BalancedTree.root heap h₂) := +theorem CompleteTree.heapInsertRootSameOrElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : 0 < o): (CompleteTree.root (heap.heapInsert lt elem) (by simp_arith) = elem) ∨ (CompleteTree.root (heap.heapInsert lt elem) (by simp_arith) = CompleteTree.root heap h₂) := match o, heap with | (n+m+1), .branch v l r _ _ _ => if h : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then by - unfold BalancedTree.heapInsert + unfold CompleteTree.heapInsert simp[h] - cases (lt elem v) <;> simp[instDecidableEqBool, Bool.decEq, BalancedTree.root] + cases (lt elem v) <;> simp[instDecidableEqBool, Bool.decEq, CompleteTree.root] else by - unfold BalancedTree.heapInsert + unfold CompleteTree.heapInsert simp[h] rw[rootSeesThroughCast n m (by simp_arith) (by simp_arith) (by simp_arith)] cases (lt elem v) - <;> simp[instDecidableEqBool, Bool.decEq, BalancedTree.root] + <;> simp[instDecidableEqBool, Bool.decEq, CompleteTree.root] -theorem BalancedTree.heapInsertEmptyElem (elem : α) (heap : BalancedTree α o) (lt : α → α → Bool) (h₂ : ¬0 < o) : (BalancedTree.root (heap.heapInsert lt elem) (by simp_arith) = elem) := +theorem CompleteTree.heapInsertEmptyElem (elem : α) (heap : CompleteTree α o) (lt : α → α → Bool) (h₂ : ¬0 < o) : (CompleteTree.root (heap.heapInsert lt elem) (by simp_arith) = 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[BalancedTree.heapInsert, root] + | 0, .leaf => by simp[CompleteTree.heapInsert, root] private theorem HeapPredicate.notSmallerOrLeaf_transitive (h₁ : WellDefinedLt lt) : lt a b → HeapPredicate.notSmallerOrLeaf lt b h → HeapPredicate.notSmallerOrLeaf lt a h := by have h₂ := h₁.not_equal @@ -220,7 +220,7 @@ private theorem HeapPredicate.seesThroughCast (h₁ : n+1+m+1=n+m+1+1) (h₂ : 0<n+1+m+1) (h₃ : 0<n+m+1+1) - (heap : BalancedTree α (n+1+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by + (heap : CompleteTree α (n+1+m+1)) : HeapPredicate heap lt → HeapPredicate (h₁▸heap) lt := by unfold HeapPredicate intro h₄ induction m generalizing n @@ -234,18 +234,18 @@ private theorem HeapPredicate.seesThroughCast revert heap h₁ h₂ h₃ assumption -theorem BalancedTree.heapInsertIsHeap {elem : α} {heap : BalancedTree α o} {lt : α → α → Bool} (h₁ : HeapPredicate heap lt) (h₂ : WellDefinedLt lt) : HeapPredicate (heap.heapInsert lt elem) lt := +theorem CompleteTree.heapInsertIsHeap {elem : α} {heap : CompleteTree α o} {lt : α → α → Bool} (h₁ : HeapPredicate heap lt) (h₂ : WellDefinedLt lt) : HeapPredicate (heap.heapInsert lt elem) lt := match o, heap with | 0, .leaf => by trivial | (n+m+1), .branch v l r m_le_n _ _ => if h₃ : m < n ∧ Nat.nextPowerOfTwo (n + 1) = n + 1 then by - unfold BalancedTree.heapInsert + unfold CompleteTree.heapInsert simp[h₃] cases h₄ : (lt elem v) <;> simp[instDecidableEqBool, Bool.decEq] <;> unfold HeapPredicate <;> unfold HeapPredicate at h₁ case true => - have h₅ : (HeapPredicate (BalancedTree.heapInsert lt v r) lt) := BalancedTree.heapInsertIsHeap h₁.right.left h₂ + have h₅ : (HeapPredicate (CompleteTree.heapInsert lt v r) lt) := CompleteTree.heapInsertIsHeap h₁.right.left h₂ simp[h₁, h₅] simp[HeapPredicate.notSmallerOrLeaf_transitive h₂ h₄ h₁.right.right.left] have h₇ := h₂.not_equal @@ -270,7 +270,7 @@ theorem BalancedTree.heapInsertIsHeap {elem : α} {heap : BalancedTree α o} {lt simp at h₉ assumption case false => - have h₅ : (HeapPredicate (BalancedTree.heapInsert lt elem r) lt) := BalancedTree.heapInsertIsHeap h₁.right.left h₂ + have h₅ : (HeapPredicate (CompleteTree.heapInsert lt elem r) lt) := CompleteTree.heapInsertIsHeap h₁.right.left h₂ simp[h₁, h₅] unfold HeapPredicate.notSmallerOrLeaf simp_arith @@ -290,7 +290,7 @@ theorem BalancedTree.heapInsertIsHeap {elem : α} {heap : BalancedTree α o} {lt case inr.zero => contradiction case inr.succ => simp[h₈] else by - unfold BalancedTree.heapInsert + unfold CompleteTree.heapInsert simp[h₃] apply HeapPredicate.seesThroughCast <;> try simp_arith[h₂] --gets rid of annoying cast. -- this should be more or less identical to the other branch, just with l r m n swapped. @@ -299,7 +299,7 @@ theorem BalancedTree.heapInsertIsHeap {elem : α} {heap : BalancedTree α o} {lt <;> unfold HeapPredicate <;> unfold HeapPredicate at h₁ case a.true => - have h₅ : (HeapPredicate (BalancedTree.heapInsert lt v l) lt) := BalancedTree.heapInsertIsHeap h₁.left h₂ + have h₅ : (HeapPredicate (CompleteTree.heapInsert lt v l) lt) := CompleteTree.heapInsertIsHeap h₁.left h₂ simp[h₁, h₅] simp[HeapPredicate.notSmallerOrLeaf_transitive h₂ h₄ h₁.right.right.right] have h₇ := h₂.not_equal @@ -324,7 +324,7 @@ theorem BalancedTree.heapInsertIsHeap {elem : α} {heap : BalancedTree α o} {lt simp at h₉ assumption case a.false => - have h₅ : (HeapPredicate (BalancedTree.heapInsert lt elem l) lt) := BalancedTree.heapInsertIsHeap h₁.left h₂ + have h₅ : (HeapPredicate (CompleteTree.heapInsert lt elem l) lt) := CompleteTree.heapInsertIsHeap h₁.left h₂ simp[h₁, h₅] unfold HeapPredicate.notSmallerOrLeaf simp_arith @@ -350,8 +350,8 @@ def BinaryHeap.insert {α : Type u} {lt : α → α → Bool} {n : Nat} : α → let tree := tree.heapInsert lt elem {tree, valid, wellDefinedLt} -/--Helper function for BalancedTree.indexOf.-/ -def BalancedTree.indexOfAux {α : Type u} [BEq α] (elem : α) (heap : BalancedTree α o) (currentIndex : Nat) : Option (Fin (o+currentIndex)) := +/--Helper function for CompleteTree.indexOf.-/ +def CompleteTree.indexOfAux {α : Type u} [BEq α] (elem : α) (heap : CompleteTree α o) (currentIndex : Nat) : Option (Fin (o+currentIndex)) := match o, heap with | 0, .leaf => none | (n+m+1), .branch a left right _ _ _ => @@ -368,7 +368,7 @@ def BalancedTree.indexOfAux {α : Type u} [BEq α] (elem : α) (heap : BalancedT found_right /--Finds the first occurance of a given element in the heap and returns its index.-/ -def BalancedTree.indexOf {α : Type u} [BEq α] (elem : α) (heap : BalancedTree α o) : Option (Fin o) := +def CompleteTree.indexOf {α : Type u} [BEq α] (elem : α) (heap : CompleteTree α o) : Option (Fin o) := indexOfAux elem heap 0 private inductive Direction @@ -381,11 +381,11 @@ theorem two_n_not_zero_n_not_zero (n : Nat) (p : ¬2*n = 0) : (¬n = 0) := by case zero => contradiction case succ => simp -def BalancedTree.popLast {α : Type u} (heap : BalancedTree α (o+1)) : (α × BalancedTree α o) := +def CompleteTree.popLast {α : Type u} (heap : CompleteTree α (o+1)) : (α × CompleteTree α o) := match o, heap with - | (n+m), .branch a (left : BalancedTree α n) (right : BalancedTree α m) m_le_n max_height_difference subtree_complete => + | (n+m), .branch a (left : CompleteTree α n) (right : CompleteTree α m) m_le_n max_height_difference subtree_complete => if p : 0 = (n+m) then - (a, p▸BalancedTree.leaf) + (a, p▸CompleteTree.leaf) else --let leftIsFull : Bool := (n+1).nextPowerOfTwo = n+1 let rightIsFull : Bool := (m+1).nextPowerOfTwo = m+1 @@ -394,7 +394,7 @@ def BalancedTree.popLast {α : Type u} (heap : BalancedTree α (o+1)) : (α × B --remove left match n, left with | (l+1), left => - let (res, (newLeft : BalancedTree α l)) := left.popLast + let (res, (newLeft : CompleteTree α l)) := left.popLast have q : l + m + 1 = l + 1 +m := by simp_arith have s : m ≤ l := match r with | .intro a _ => by apply Nat.le_of_lt_succ @@ -404,7 +404,7 @@ def BalancedTree.popLast {α : Type u} (heap : BalancedTree α (o+1)) : (α × B simp[←Nat.power_of_two_iff_next_power_eq] at r assumption have l_lt_2_m_succ : l < 2 * (m+1) := by apply Nat.lt_of_succ_lt; assumption - (res, q▸BalancedTree.branch a newLeft right s l_lt_2_m_succ (Or.inr rightIsFull)) + (res, q▸CompleteTree.branch a newLeft right s l_lt_2_m_succ (Or.inr rightIsFull)) else --remove right have : m > 0 := by @@ -427,7 +427,7 @@ def BalancedTree.popLast {α : Type u} (heap : BalancedTree α (o+1)) : (α × B assumption match h₂ : m, right with | (l+1), right => - let (res, (newRight : BalancedTree α l)) := right.popLast + let (res, (newRight : CompleteTree α l)) := right.popLast have s : l ≤ n := by have x := (Nat.add_le_add_left (Nat.zero_le 1) l) have x := Nat.le_trans x m_le_n assumption @@ -455,18 +455,18 @@ def BalancedTree.popLast {α : Type u} (heap : BalancedTree α (o+1)) : (α × B | inr h₁ => simp[←Nat.power_of_two_iff_next_power_eq, h₂] at h₁ apply power_of_two_mul_two_le <;> assumption - (res, BalancedTree.branch a left newRight s still_in_range (Or.inl leftIsFull)) + (res, CompleteTree.branch a left newRight s still_in_range (Or.inl leftIsFull)) -/--Removes the element at a given index. Use `BalancedTree.indexOf` to find the respective index.-/ -def BalancedTree.heapRemoveAt {α : Type u} (lt : α → α → Bool) {o : Nat} (index : Fin (o+1)) (heap : BalancedTree α (o+1)) : BalancedTree α o := +/--Removes the element at a given index. Use `CompleteTree.indexOf` to find the respective index.-/ +def CompleteTree.heapRemoveAt {α : Type u} (lt : α → α → Bool) {o : Nat} (index : Fin (o+1)) (heap : CompleteTree α (o+1)) : CompleteTree α o := -- first remove the last element and remember its value sorry ------------------------------------------------------------------------------------------------------- private def TestHeap := - let ins : {n: Nat} → Nat → BalancedTree Nat n → BalancedTree Nat (n+1) := λ x y ↦ BalancedTree.heapInsert (.<.) x y - ins 5 BalancedTree.empty + let ins : {n: Nat} → Nat → CompleteTree Nat n → CompleteTree Nat (n+1) := λ x y ↦ CompleteTree.heapInsert (.<.) x y + ins 5 CompleteTree.empty |> ins 3 |> ins 7 |> ins 12 |