diff options
| -rw-r--r-- | BinaryHeap/CompleteTree/AdditionalProofs.lean | 114 |
1 files changed, 109 insertions, 5 deletions
diff --git a/BinaryHeap/CompleteTree/AdditionalProofs.lean b/BinaryHeap/CompleteTree/AdditionalProofs.lean index 442c2d3..949854d 100644 --- a/BinaryHeap/CompleteTree/AdditionalProofs.lean +++ b/BinaryHeap/CompleteTree/AdditionalProofs.lean @@ -301,7 +301,7 @@ protected theorem heapRemoveLastWithIndexReturnsItemAtIndex {α : Type u} {o : N private theorem heapRemoveLastWithIndexLeavesRoot {α : Type u} {n: Nat} (heap : CompleteTree α (n+1)) (h₁ : n > 0) : heap.root (Nat.succ_pos n) = (CompleteTree.Internal.heapRemoveLastWithIndex heap).fst.root h₁ := CompleteTree.heapRemoveLastAuxLeavesRoot heap _ _ _ h₁ -private theorem leftLen_sees_through_cast {α : Type u} {p q : Nat} (h₁ : q+p+1 = q+1+p) (heap : CompleteTree α (q+p+1)) (ha : q+p+1 > 0) (hb : q+1+p > 0): heap.leftLen ha = (h₁▸heap).leftLen hb:= by +private theorem lens_see_through_cast {α : Type u} {p q : Nat} (h₁ : q+p+1 = q+1+p) (len : {n : Nat} → CompleteTree α n → (n > 0) → Nat) (heap : CompleteTree α (q+p+1)) (ha : q+p+1 > 0) (hb : q+1+p > 0): len heap ha = len (h₁▸heap) hb:= by induction p generalizing q case zero => simp only [Nat.add_zero] case succ pp hp => @@ -323,6 +323,17 @@ private theorem left_sees_through_cast {α : Type u} {p q : Nat} (h₁ : q+p+1 = revert hb ha heap h₁ assumption +private theorem right_sees_through_cast {α : Type u} {p q : Nat} (h₁ : q+p+1 = q+1+p) (heap : CompleteTree α (q+p+1)) (ha : q+p+1 > 0) (hb : q+1+p > 0): HEq (heap.right ha) ((h₁▸heap).right hb) := by + induction p generalizing q + case zero => simp only [Nat.add_zero, heq_eq_eq] + case succ pp hp => + have h₂ := hp (q := q+1) + have h₃ : (q + 1 + pp + 1) = q + (pp + 1) + 1 := by simp_arith + have h₄ : (q + 1 + 1 + pp) = q + 1 + (pp + 1) := by simp_arith + rw[h₃, h₄] at h₂ + revert hb ha heap h₁ + assumption + /--Helper for heapRemoveLastWithIndexOnlyRemovesOneElement_Auxllength to allow splitting the goal-/ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLengthAux {α : Type u} {n : Nat} (tree : CompleteTree α (n+1)) (h₁ : n > 0) : @@ -343,7 +354,7 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLengthAux {α : match o, l, o_le_p, max_height_difference, subtree_complete, this, h₂ with | (q+1), l, o_le_p, max_height_difference, subtree_complete, this, h₂ => simp only [Nat.add_eq, Fin.zero_eta, Fin.isValue, Nat.succ_eq_add_one, Nat.pred_eq_sub_one] - rw[←leftLen_sees_through_cast _ _ (Nat.succ_pos (q+p))] + rw[←lens_see_through_cast _ leftLen _ (Nat.succ_pos (q+p))] unfold leftLen' rw[leftLen_unfold] rw[leftLen_unfold] @@ -510,6 +521,89 @@ private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxL2 {α : Type u} := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxL2Aux (branch v l r h₁ h₂ h₃) (by omega) (Nat.succ_pos _) h₄ +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength2Aux {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 +:= 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 h₄ : 0 ≠ o + p := Nat.ne_of_lt h₁ + simp only [h₄, ↓reduceDite, h₂, decide_True, and_self] + cases o + case zero => exact absurd h₂.left $ Nat.not_lt_zero p + case succ oo => + simp at h₂ ⊢ + have : (p+1).isPowerOfTwo := by simp[Nat.power_of_two_iff_next_power_eq, h₂.right] + have := lens_see_through_cast ( by simp_arith : oo + p + 1 = oo + 1 + p) rightLen (branch v + (Internal.heapRemoveLastAux (β := λn ↦ α × Fin n) l (fun a => (a, 0)) + (fun {prev_size curr_size} x h₁ => (x.fst, Fin.castLE (by omega) x.snd.succ)) + fun {prev_size curr_size} x left_size h₁ => (x.fst, Fin.castLE (by omega) (x.snd.addNat left_size).succ)).fst + r (Nat.le_of_lt_succ h₂.left) (Nat.lt_of_succ_lt max_height_difference) (Or.inr this)) (Nat.succ_pos _) h₁ + rw[←this] + simp[rightLen', rightLen_unfold] + done + +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength2 {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 + 1 +:= by + apply heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength2Aux (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 theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR2Aux {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) tree.right' +:= by + simp only + intro h₂ h₃ + generalize hi : (Internal.heapRemoveLastWithIndex tree).fst = input + unfold Internal.heapRemoveLastWithIndex Internal.heapRemoveLastAux at hi + split at hi + rename_i d1 d2 o2 p vv ll rr m_le_n max_height_difference subtree_complete + clear d1 d2 + unfold leftLen' rightLen' at* + rw[leftLen_unfold, rightLen_unfold] at * + have h₄ : 0 ≠ o2+p := Nat.ne_of_lt h₁ + simp only [h₄, h₃, and_true, reduceDite] at hi + -- okay, dealing with p.pred.succ is a guarantee for pain. "Stuck at solving universe constraint" + cases o2 + case zero => + exact absurd h₂ $ Nat.not_lt_of_le m_le_n + case succ oo => + simp at hi --whoa, how easy this gets if one just does cases o2... + unfold right' + rewrite[right_unfold] + simp at h₃ + have : (p+1).isPowerOfTwo := by simp[Nat.power_of_two_iff_next_power_eq, h₃.right] + have h₅ := right_sees_through_cast (by simp_arith: oo + p + 1 = oo + 1 + p) (branch vv + (Internal.heapRemoveLastAux (β := λn ↦ α × Fin n) ll (fun a => (a, 0)) + (fun {prev_size curr_size} x h₁ => (x.fst, Fin.castLE (by omega) x.snd.succ)) + fun {prev_size curr_size} x left_size h₁ => (x.fst, Fin.castLE (by omega) (x.snd.addNat left_size).succ)).fst + rr (Nat.le_of_lt_succ h₃.left) (Nat.lt_of_succ_lt max_height_difference) (Or.inr this)) (Nat.succ_pos _) h₁ + rw[right_unfold, hi] at h₅ + exact h₅.symm + +private theorem heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR2 {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)) r +:= + heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR2Aux (branch v l r h₁ h₂ h₃) (by omega) (Nat.succ_pos _) h₄ + + /--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 have h₆ := heapRemoveLastWithIndex'_unfold tree.left' $ Nat.zero_lt_of_lt h₂ @@ -566,6 +660,7 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n -- this should be solvable by recursion clear del simp + let rightIsFull : Bool := ((p + 1).nextPowerOfTwo = p + 1) split case isTrue j_lt_o => split @@ -576,7 +671,6 @@ protected theorem heapRemoveLastWithIndexOnlyRemovesOneElement {α : Type u} {n left --unfold Internal.heapRemoveLastWithIndex Internal.heapRemoveLastAux --unfolding fails. Need a helper, it seems. - let rightIsFull : Bool := ((p + 1).nextPowerOfTwo = p + 1) if h₅ : p < oo + 1 ∧ rightIsFull then have h₆ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxL v l r ht1 ht2 ht3 h₅ have h₇ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxLLength v l r ht1 ht2 ht3 h₅ @@ -593,9 +687,19 @@ 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 _ _ _ h₄ h₅ + rename_i p d1 d2 d3 d4 d5 h₆ pp r ht1 ht2 ht3 h₄ h₅ clear d1 d2 d3 d4 d5 rw[contains_as_root_left_right _ _ (Nat.lt_add_left o $ Nat.succ_pos pp)] right right - sorry + -- this should be the same as the left side, with minor adaptations... Let's see. + if h₇ : pp + 1 < o ∧ rightIsFull then + have h₈ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxR2 v l r ht1 ht2 ht3 h₇ + have h₉ := heapRemoveLastWithIndexOnlyRemovesOneElement_AuxRLength2 v l r ht1 ht2 ht3 h₇ + have h₁₀ := heqContains h₉ h₈ + rw[h₁₀] + rw[contains_iff_index_exists _ _ (Nat.succ_pos pp)] + 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 |