From 28eaf0274c580e3d5bb479d7160a18cbe474a04c Mon Sep 17 00:00:00 2001 From: Andreas Grois Date: Tue, 30 Jul 2024 22:35:37 +0200 Subject: Finish heapRemoveLastWithIndexOnlyRemovesOneElement. It could use some cleanup though. Compiling it takes quite long already... --- BinaryHeap/CompleteTree/AdditionalProofs.lean | 138 ++++++++++++++++++++++++-- 1 file changed, 130 insertions(+), 8 deletions(-) diff --git a/BinaryHeap/CompleteTree/AdditionalProofs.lean b/BinaryHeap/CompleteTree/AdditionalProofs.lean index 949854d..7e5fbe5 100644 --- a/BinaryHeap/CompleteTree/AdditionalProofs.lean +++ b/BinaryHeap/CompleteTree/AdditionalProofs.lean @@ -370,6 +370,39 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLength {α : Ty simp only [decide_eq_true_eq] at h₄ assumption +/--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_Auxllength to allow splitting the goal-/ +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLengthAux {α : Type u} {n : Nat} (tree : CompleteTree α (n+1)) (h₁ : n > 0) +: + let o := tree.leftLen' + let p := tree.rightLen' + (h₂ : ¬(p < o ∧ ((p+1).nextPowerOfTwo = p+1 : Bool))) → (Internal.heapRemoveLastWithIndex tree).fst.rightLen h₁ = p.pred +:= by + simp only + intro h₂ + unfold Internal.heapRemoveLastWithIndex Internal.heapRemoveLastAux + split --finally + rename_i del1 del2 o p v l r o_le_p max_height_difference subtree_complete + clear del2 del1 + unfold rightLen' leftLen' at h₂ + simp only [leftLen_unfold, rightLen_unfold] at h₂ + have : 0 ≠ o + p := Nat.ne_of_lt h₁ + simp only [this, ↓reduceDite, h₂, decide_True, and_self] + cases p + case zero => + simp at h₂ h₁ + simp (config := {ground:=true})[h₁] at h₂ + case succ pp => + simp[rightLen', rightLen_unfold] + +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength {α : Type u} {o p : Nat} (v : α) (l : CompleteTree α o) (r : CompleteTree α (p+1)) (h₁ : p+1 ≤ o) (h₂ : o < 2 * (p + 1 + 1)) (h₃ : (o + 1).isPowerOfTwo ∨ (p + 1 + 1).isPowerOfTwo) (h₄ : ¬(p + 1 < o ∧ ((p+1+1).nextPowerOfTwo = p+1+1 : Bool))) +: + (Internal.heapRemoveLastWithIndex (branch v l r h₁ h₂ h₃)).fst.rightLen (Nat.lt_add_left o $ Nat.succ_pos p) = p +:= by + apply heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLengthAux (branch v l r h₁ h₂ h₃) (Nat.lt_add_left o $ Nat.succ_pos p) + unfold leftLen' rightLen' + simp only [leftLen_unfold, rightLen_unfold] + assumption + private def heapRemoveLastWithIndex' {α : Type u} {o : Nat} (heap : CompleteTree α o) (_ : o > 0) : (CompleteTree α o.pred × α × Fin o) := match o, heap with | _+1, heap => Internal.heapRemoveLastWithIndex heap @@ -445,6 +478,58 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxL {α : Type u} have h₅ : (branch v l r h₁ h₂ h₃).leftLen' > 0 := Nat.succ_pos o heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLAux _ _ h₅ h₄ +/--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR to allow splitting the goal-/ +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRAux {α : Type u} {n : Nat} (tree : CompleteTree α (n+1)) (h₁ : n > 0) +: + let o := tree.leftLen' + let p := tree.rightLen' + (h₂ : p > 0) → + (h₃ : ¬(p < o ∧ ((p+1).nextPowerOfTwo = p+1 : Bool))) → + HEq ((Internal.heapRemoveLastWithIndex tree).fst.right h₁) (heapRemoveLastWithIndex' (tree.right') h₂).fst +:= by + simp only + intro h₂ h₃ + --sorry for this wild mixture of working on the LHS and RHS of the goal. + --this function is trial and error, I'm fighting an uphill battle agains tactics mode here, + --which keeps randomly failing on me if I do steps in what the tactics + --percieve to be the "wrong" order. + have h₄ := heapRemoveLastWithIndex'_unfold tree.right' h₂ + split at h₄ + rename_i d3 d2 d1 pp l o_gt_0 he1 he2 he3 + clear d1 d2 d3 + --okay, I have no clue why generalizing here is needed. + --I mean, why does unfolding and simplifying work if it's a separate hypothesis, + --but not within the goal? + generalize hi : (Internal.heapRemoveLastWithIndex tree).fst = input + unfold Internal.heapRemoveLastWithIndex Internal.heapRemoveLastAux at hi + split at hi + rename_i o p2 vv ll rr m_le_n max_height_difference subtree_complete + unfold right' leftLen' rightLen' at * + rw[right_unfold] at * + rw[leftLen_unfold, rightLen_unfold] at * + subst p2 + simp at he2 + subst rr + rw[h₄] + have : (0 ≠ o + pp.succ) := by simp_arith + simp[this] at hi + have : ¬(pp + 1 < o ∧ (pp + 1 + 1).nextPowerOfTwo = pp + 1 + 1) := by simp; simp at h₃; assumption + simp[this] at hi + subst input + simp[right_unfold, Internal.heapRemoveLastWithIndex] + done + +/--Shows that if the removal happens in the left tree, the new left-tree is the old left-tree with the last element removed.-/ +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR {α : Type u} {o p : Nat} (v : α) (l : CompleteTree α o) (r : CompleteTree α (p+1)) (h₁ : (p+1) ≤ o) (h₂ : o < 2 * (p + 1 + 1)) (h₃ : (o + 1).isPowerOfTwo ∨ (p + 1 + 1).isPowerOfTwo) (h₄ : ¬(p + 1 < o ∧ ((p + 1 + 1).nextPowerOfTwo = p + 1 + 1 : Bool))) +: + HEq ((Internal.heapRemoveLastWithIndex (branch v l r h₁ h₂ h₃)).fst.right (Nat.lt_add_left o $ Nat.succ_pos p)) (Internal.heapRemoveLastWithIndex r).fst +:= + --okay, let me be frank here: I have absolutely no clue why I need heapRemoveLastWithIndexOnlyRemovesOneElement_Auxl2. + --from the looks of it, I should just be able to generalize the LHS, and unfold things and be happy. + --However, tactic unfold fails. So, let's just forward this to the helper. + have h₅ : (branch v l r h₁ h₂ h₃).rightLen' > 0 := Nat.succ_pos p + heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRAux _ _ h₅ h₄ + /--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_Auxllength to allow splitting the goal-/ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLength2Aux {α : Type u} {n : Nat} (tree : CompleteTree α (n+1)) (h₁ : n > 0) : @@ -605,7 +690,7 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR2 {α : Type u} /--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNe-/ -private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNeAux {α : Type u} {n j : Nat} (tree : CompleteTree α (n+1)) (h₁ : (n+1) > 0) (h₂ : j < tree.leftLen h₁) (h₃ : j.succ < (n+1)) (h₄ : tree.rightLen' < tree.leftLen' ∧ ((tree.rightLen'+1).nextPowerOfTwo = tree.rightLen'+1 : Bool)) (h₅ : ⟨j.succ, h₃⟩ ≠ (Internal.heapRemoveLastWithIndex tree).2.snd) : ⟨j, h₂⟩ ≠ (heapRemoveLastWithIndex' (tree.left h₁) (Nat.zero_lt_of_lt h₂)).snd.snd := by +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLIndexNeAux {α : Type u} {n j : Nat} (tree : CompleteTree α (n+1)) (h₁ : (n+1) > 0) (h₂ : j < tree.leftLen h₁) (h₃ : j.succ < (n+1)) (h₄ : tree.rightLen' < tree.leftLen' ∧ ((tree.rightLen'+1).nextPowerOfTwo = tree.rightLen'+1 : Bool)) (h₅ : ⟨j.succ, h₃⟩ ≠ (Internal.heapRemoveLastWithIndex tree).2.snd) : ⟨j, h₂⟩ ≠ (heapRemoveLastWithIndex' (tree.left h₁) (Nat.zero_lt_of_lt h₂)).snd.snd := by have h₆ := heapRemoveLastWithIndex'_unfold tree.left' $ Nat.zero_lt_of_lt h₂ split at h₆ rename_i d3 d2 d1 oo l o_gt_0 he1 he2 he3 @@ -627,14 +712,45 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNeAux {α unfold Internal.heapRemoveLastWithIndex assumption -private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNe {α : Type u} {o p j : Nat} (v : α) (l : CompleteTree α (o+1)) (r : CompleteTree α p) (h₁ : p ≤ (o+1)) (h₂ : (o+1) < 2 * (p + 1)) (h₃ : (o + 1 + 1).isPowerOfTwo ∨ (p + 1).isPowerOfTwo) (h₄ : j < o+1) (h₅ : p < (o+1) ∧ ((p+1).nextPowerOfTwo = p+1 : Bool)) (h₆ : ⟨j.succ, (by omega)⟩ ≠ (Internal.heapRemoveLastWithIndex (branch v l r h₁ h₂ h₃)).2.snd) : ⟨j,h₄⟩ ≠ (Internal.heapRemoveLastWithIndex l).snd.snd := +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLIndexNe {α : Type u} {o p j : Nat} (v : α) (l : CompleteTree α (o+1)) (r : CompleteTree α p) (h₁ : p ≤ (o+1)) (h₂ : (o+1) < 2 * (p + 1)) (h₃ : (o + 1 + 1).isPowerOfTwo ∨ (p + 1).isPowerOfTwo) (h₄ : j < o+1) (h₅ : p < (o+1) ∧ ((p+1).nextPowerOfTwo = p+1 : Bool)) (h₆ : ⟨j.succ, (by omega)⟩ ≠ (Internal.heapRemoveLastWithIndex (branch v l r h₁ h₂ h₃)).2.snd) : ⟨j,h₄⟩ ≠ (Internal.heapRemoveLastWithIndex l).snd.snd := --splitting at h₅ does not work. Probably because we have o+1... --helper function it is... - heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNeAux (branch v l r h₁ h₂ h₃) (Nat.succ_pos _) _ (by omega) h₅ h₆ + heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLIndexNeAux (branch v l r h₁ h₂ h₃) (Nat.succ_pos _) _ (by omega) h₅ h₆ + +/--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNe-/ +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRIndexNeAux {α : Type u} {n j : Nat} (tree : CompleteTree α (n+1)) (h₁ : (n+1) > 0) (h₂ : j - tree.leftLen h₁ < tree.rightLen h₁) (h₃ : j.succ < (n+1)) (h₄ : tree.leftLen' ≤ j) (h₅ : ¬(tree.rightLen' < tree.leftLen' ∧ ((tree.rightLen'+1).nextPowerOfTwo = tree.rightLen'+1 : Bool))) (h₆ : ⟨j.succ, h₃⟩ ≠ (Internal.heapRemoveLastWithIndex tree).2.snd) : ⟨j - tree.leftLen h₁, h₂⟩ ≠ (heapRemoveLastWithIndex' (tree.right h₁) (Nat.zero_lt_of_lt h₂)).snd.snd := by + have h₇ := heapRemoveLastWithIndex'_unfold tree.right' (Nat.zero_lt_of_lt h₂) + split at h₇ + rename_i d3 d2 d1 pp r o_gt_0 he1 he2 he3 + clear d1 d2 d3 + unfold Internal.heapRemoveLastWithIndex Internal.heapRemoveLastAux at h₆ + split at h₆ + rename_i o p2 vv ll rr _ _ _ + unfold right' rightLen' leftLen' at * + rw[right_unfold] at * + rw[leftLen_unfold, rightLen_unfold] at * + subst he1 + simp at he2 + subst he2 + rw[h₇] + have : ¬0 = o + pp.succ := by omega + simp only [this, h₅, and_true, reduceDite] at h₆ + rw[←Fin.val_ne_iff] at h₆ ⊢ + simp at h₆ + unfold Internal.heapRemoveLastWithIndex + simp[leftLen_unfold] + rw[Nat.sub_eq_iff_eq_add h₄] + assumption + + +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRIndexNe {α : Type u} {o p j : Nat} (v : α) (l : CompleteTree α o) (r : CompleteTree α (p+1)) (h₁ : p+1 ≤ o) (h₂ : o < 2 * (p + 1 + 1)) (h₃ : (o + 1).isPowerOfTwo ∨ (p + 1 + 1).isPowerOfTwo) (h₄ : j - o < p + 1) (h₅ : o ≤ j) (h₆ : ¬(p + 1 < o ∧ ((p+1+1).nextPowerOfTwo = p+1+1 : Bool))) (h₇ : ⟨j.succ, (by omega)⟩ ≠ (Internal.heapRemoveLastWithIndex (branch v l r h₁ h₂ h₃)).2.snd) : ⟨j-o,h₄⟩ ≠ (Internal.heapRemoveLastWithIndex r).snd.snd := + --splitting at h₅ does not work. Probably because we have o+1... + --helper function it is... + heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRIndexNeAux (branch v l r h₁ h₂ h₃) (Nat.succ_pos _) _ _ h₅ h₆ h₇ /--If the resulting tree contains all elements except the removed one, and contains one less than the original, well, you get the idea.-/ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n : Nat} (heap : CompleteTree α (n+1)) (index : Fin (n+1)) : - let (newHeap, removedValue, removedIndex) := Internal.heapRemoveLastWithIndex heap + let (newHeap, _, removedIndex) := Internal.heapRemoveLastWithIndex heap (h₁ : index ≠ removedIndex) → newHeap.contains (heap.get index (Nat.succ_pos n)) := by simp only intro h₁ @@ -664,7 +780,7 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n split case isTrue j_lt_o => split - rename_i o d1 d2 d3 d4 d5 oo l ht1 ht2 ht3 h₄ + rename_i o d1 d2 d3 d4 d5 oo l ht1 ht2 ht3 _ clear d1 d2 d3 d4 d5 rw[contains_as_root_left_right _ _ (Nat.lt_add_right p $ Nat.succ_pos oo)] right @@ -676,7 +792,7 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n have h₇ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLength v l r ht1 ht2 ht3 h₅ have h₈ := heqContains h₇ h₆ rw[h₈] - have h₉ : ⟨j,j_lt_o⟩ ≠ (Internal.heapRemoveLastWithIndex l).snd.snd := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxlIndexNe v l r ht1 ht2 ht3 j_lt_o h₅ h₁ + have h₉ : ⟨j,j_lt_o⟩ ≠ (Internal.heapRemoveLastWithIndex l).snd.snd := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLIndexNe v l r ht1 ht2 ht3 j_lt_o h₅ h₁ exact CompleteTree.AdditionalProofs.heapRemoveLastWithIndexOnlyRemovesOneElement _ _ h₉ else have h₆ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxL2 v l r ht1 ht2 ht3 h₅ @@ -687,7 +803,7 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n exists ⟨j, j_lt_o⟩ case isFalse j_ge_o => split - rename_i p d1 d2 d3 d4 d5 h₆ pp r ht1 ht2 ht3 h₄ h₅ + rename_i p d1 d2 d3 d4 d5 h₆ pp r ht1 ht2 ht3 _ h₅ clear d1 d2 d3 d4 d5 rw[contains_as_root_left_right _ _ (Nat.lt_add_left o $ Nat.succ_pos pp)] right @@ -702,4 +818,10 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n have : p = pp.succ := (Nat.add_sub_cancel pp.succ o).subst $ (Nat.add_comm o (pp.succ)).subst (motive := λx ↦ p = x-o ) h₅.symm exists ⟨j-o,this.subst h₆⟩ else - sorry + have h₈ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR v l r ht1 ht2 ht3 h₇ + have h₉ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength v l r ht1 ht2 ht3 h₇ + have h₁₀ := heqContains h₉ h₈ + rw[h₁₀] + have : p = pp.succ := (Nat.add_sub_cancel pp.succ o).subst $ (Nat.add_comm o (pp.succ)).subst (motive := λx ↦ p = x-o ) h₅.symm + have h₉ : ⟨j-o,this.subst h₆⟩ ≠ (Internal.heapRemoveLastWithIndex r).snd.snd := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRIndexNe v l r ht1 ht2 ht3 (this.subst h₆) (by omega) h₇ h₁ + exact CompleteTree.AdditionalProofs.heapRemoveLastWithIndexOnlyRemovesOneElement _ _ h₉ -- cgit v1.2.3