diff options
Diffstat (limited to 'Common/BinaryHeap.lean')
| -rw-r--r-- | Common/BinaryHeap.lean | 192 |
1 files changed, 174 insertions, 18 deletions
diff --git a/Common/BinaryHeap.lean b/Common/BinaryHeap.lean index e876bc0..922e9bf 100644 --- a/Common/BinaryHeap.lean +++ b/Common/BinaryHeap.lean @@ -3,7 +3,7 @@ namespace BinaryHeap /--A heap, represented as a binary indexed tree. The heap predicate is a type parameter, the index is the element count.-/ inductive BinaryHeap (α : Type u) (lt : α → α → Bool): Nat → Type u | leaf : BinaryHeap α lt 0 - | branch : (val : α) → (left : BinaryHeap α lt n) → (right : BinaryHeap α lt m) → m ≤ n → (n+1).isPowerOfTwo ∨ (m+1).isPowerOfTwo → BinaryHeap α lt (n+m+1) + | branch : (val : α) → (left : BinaryHeap α lt n) → (right : BinaryHeap α lt m) → m ≤ n → n < 2*(m+1) → (n+1).isPowerOfTwo ∨ (m+1).isPowerOfTwo → BinaryHeap α lt (n+m+1) /--Please do not use this for anything meaningful. It's a debug function, with horrible performance.-/ instance {α : Type u} {lt : α → α → Bool} [ToString α] : ToString (BinaryHeap α lt n) where @@ -11,13 +11,13 @@ instance {α : Type u} {lt : α → α → Bool} [ToString α] : ToString (Binar --not very fast, doesn't matter, is for debugging let rec max_width := λ {m : Nat} (t : (BinaryHeap α lt m)) ↦ match m, t with | 0, .leaf => 0 - | (_+_+1), BinaryHeap.branch a left right _ _ => max (ToString.toString a).length $ max (max_width left) (max_width right) + | (_+_+1), BinaryHeap.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 : (BinaryHeap α lt m)) (lines : Nat) ↦ match m, t with | 0, _ => "" - | (_+_+1), BinaryHeap.branch a left right _ _ => + | (_+_+1), BinaryHeap.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) @@ -98,14 +98,150 @@ private theorem power_of_two_eq_power_of_two (n : Nat) : n.isPowerOfTwo → (n.n case h₂ => exact Nat.one_isPowerOfTwo case h₃ => exact (Nat.pos_of_isPowerOfTwo h₁) -private theorem power_of_two_iff_next_power_eq (n : Nat) : n.isPowerOfTwo ↔ (n.nextPowerOfTwo = n) := +theorem power_of_two_iff_next_power_eq (n : Nat) : n.isPowerOfTwo ↔ (n.nextPowerOfTwo = n) := Iff.intro (power_of_two_eq_power_of_two n) (eq_power_of_two_power_of_two n) +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 + have h₂ := Nat.succ_le_of_lt h₂ + rewrite[←Nat.not_ge_eq] at h₂ + simp_arith at h₄ + contradiction + else if h₅ : n+1 = 2*m then by + have h₆ := Nat.mul2_isPowerOfTwo_of_isPowerOfTwo h₁ + rewrite[Nat.mul_comm 2 m] at h₅ + rewrite[←h₅] at h₆ + contradiction + else by + simp_arith at h₄ + exact Nat.lt_of_le_of_ne h₄ h₅ + +theorem Nat.mul2_ispowerOfTwo_iff_isPowerOfTwo (n : Nat) : n.isPowerOfTwo ↔ (2*n).isPowerOfTwo := + have h₁ : n.isPowerOfTwo → (2*n).isPowerOfTwo := by + simp[Nat.mul_comm] + apply Nat.mul2_isPowerOfTwo_of_isPowerOfTwo + have h₂ : (2*n).isPowerOfTwo → n.isPowerOfTwo := + if h₂ : n.nextPowerOfTwo = n then by + simp[power_of_two_iff_next_power_eq,h₂] + else by + intro h₃ + simp[←power_of_two_iff_next_power_eq] at h₂ + have h₅ := h₃ + unfold Nat.isPowerOfTwo at h₃ + let ⟨k,h₄⟩ := h₃ + cases n with + | zero => contradiction + | succ m => cases k with + | zero => simp_arith at h₄ + have h₆ (m : Nat) : 2*m+2 > 2^0 := by + induction m with + | zero => decide + | succ o o_ih => simp_arith at * + have h₆ := Nat.le_step $ Nat.le_step o_ih + simp_arith at h₆ + assumption + have h₆ := Nat.ne_of_lt (h₆ m) + simp_arith at h₆ + rewrite[h₄] at h₆ --why is this needed?!? + contradiction + | succ l => rewrite[Nat.pow_succ 2 l] at h₄ + rewrite[Nat.mul_comm (2^l) 2] at h₄ + have h₄ := Nat.eq_of_mul_eq_mul_left (by decide : 0<2) h₄ + exists l + Iff.intro h₁ h₂ + +mutual + def Nat.isEven : Nat → Bool + | .zero => True + | .succ n => Nat.isOdd n + def Nat.isOdd : Nat → Bool + | .zero => False + | .succ n => Nat.isEven n +end + +theorem Nat.mul_2_is_even {n m : Nat} (h₁ : n = 2 * m) : Nat.isEven n := by + cases m with + | zero => simp[Nat.isEven, h₁] + | succ o => simp_arith at h₁ + simp[Nat.isEven, Nat.isOdd, h₁] + exact Nat.mul_2_is_even (rfl) + +theorem Nat.isPowerOfTwo_even_or_one {n : Nat} (h₁ : n.isPowerOfTwo) : (n = 1 ∨ (Nat.isEven n)) := by + simp[Nat.isPowerOfTwo] at h₁ + let ⟨k,h₂⟩ := h₁ + cases k with + | zero => simp_arith[h₂] + | succ l => rewrite[Nat.pow_succ] at h₂ + rewrite[Nat.mul_comm (2^l) 2] at h₂ + exact Or.inr $ Nat.mul_2_is_even h₂ + +mutual + theorem Nat.not_even_odd {n : Nat} (h₁ : ¬Nat.isEven n) : (Nat.isOdd n) := by + simp[Nat.isEven] at h₁ + cases n with + | zero => contradiction + | succ o => simp[Nat.isEven, Nat.isOdd] at * + exact (Nat.not_odd_even (by simp[h₁])) + theorem Nat.not_odd_even {n : Nat} (h₁ : ¬Nat.isOdd n) : (Nat.isEven n) := by + simp[Nat.isOdd] at h₁ + cases n with + | zero => trivial + | succ o => simp[Nat.isEven, Nat.isOdd] at * + exact (Nat.not_even_odd (by simp[h₁])) +end + +mutual + theorem Nat.even_not_odd {n : Nat} (h₁ : Nat.isEven n) : ¬(Nat.isOdd n ):= by + cases n with + | zero => simp + | succ o => simp[Nat.isEven, Nat.isOdd] at * + simp[Nat.odd_not_even h₁] + theorem Nat.odd_not_even {n : Nat} (h₁ : Nat.isOdd n) : ¬(Nat.isEven n ):= by + cases n with + | zero => contradiction + | succ o => simp[Nat.isEven, Nat.isOdd] at * + simp[Nat.even_not_odd h₁] +end + + +theorem Nat.pred_even_odd {n : Nat} (h₁ : Nat.isEven n) (h₂ : n > 0) : Nat.isOdd n.pred := by + cases n with + | zero => contradiction + | succ o => simp[Nat.isEven] at h₁ + assumption + +theorem power_of_two_mul_two_le {n m : Nat} (h₁ : (n+1).isPowerOfTwo) (h₂ : n < 2*(m+1)) (h₃ : ¬(m+1).isPowerOfTwo) (h₄ : m > 0): n < 2*m := + if h₅ : n > 2*m then by + simp_arith at h₂ + simp_arith at h₅ + have h₆ : n+1 = 2*(m+1) := by simp_arith[Nat.le_antisymm h₂ h₅] + rewrite[h₆] at h₁ + rewrite[←(Nat.mul2_ispowerOfTwo_iff_isPowerOfTwo (m+1))] at h₁ + contradiction + else if h₆ : n = 2*m then by + -- since (n+1) is a power of 2, n cannot be an even number, but n = 2*m means it's even + -- ha, thought I wouldn't see that, didn't you? Thought you're smarter than I, computer? + have h₇ : n > 0 := by rewrite[h₆] + apply Nat.mul_lt_mul_of_pos_left h₄ (by decide :2 > 0) + have h₈ : n ≠ 0 := Ne.symm $ Nat.ne_of_lt h₇ + have h₉ := Nat.isPowerOfTwo_even_or_one h₁ + simp[h₈] at h₉ + have h₉ := Nat.pred_even_odd h₉ (by simp_arith[h₇]) + simp at h₉ + have h₁₀ := Nat.mul_2_is_even h₆ + have h₁₀ := Nat.even_not_odd h₁₀ + contradiction + else by + simp[Nat.not_gt_eq] at h₅ + have h₅ := Nat.eq_or_lt_of_le h₅ + simp[h₆] at h₅ + assumption + /--Adds a new element to a given BinaryHeap.-/ def BinaryHeap.insert (elem : α) (heap : BinaryHeap α lt o) : BinaryHeap α lt (o+1) := match o, heap with - | 0, .leaf => BinaryHeap.branch elem (BinaryHeap.leaf) (BinaryHeap.leaf) (by simp) (by simp[Nat.one_isPowerOfTwo]) - | (n+m+1), .branch a left right p subtree_complete => + | 0, .leaf => BinaryHeap.branch elem (BinaryHeap.leaf) (BinaryHeap.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. -- the left tree is full, if (n+1) is a power of two AND n != m @@ -116,7 +252,8 @@ def BinaryHeap.insert (elem : α) (heap : BinaryHeap α lt o) : BinaryHeap α lt rewrite[Nat.succ_eq_add_one] rewrite[s] simp[r] - let result := branch a left (right.insert elem) (q) (by simp[(eq_power_of_two_power_of_two (n+1)), r]) + have difference_decreased := Nat.le_succ_of_le $ Nat.le_succ_of_le max_height_difference + let result := branch a left (right.insert elem) (q) difference_decreased (by simp[(eq_power_of_two_power_of_two (n+1)), r]) result else have q : m ≤ n+1 := by apply (Nat.le_of_succ_le) @@ -141,7 +278,15 @@ def BinaryHeap.insert (elem : α) (heap : BinaryHeap α lt o) : BinaryHeap α lt cases subtree_complete case inl => contradiction case inr => trivial - let result := branch a (left.insert elem) right q (Or.inr right_is_power_of_two) + have still_in_range : n + 1 < 2 * (m + 1) := by + rewrite[Decidable.not_and_iff_or_not (m<n) leftIsFull] at r + cases r + case inl h₁ => rewrite[Nat.not_gt_eq n m] at h₁ + simp_arith[Nat.le_antisymm h₁ p] + case inr h₁ => simp[←power_of_two_iff_next_power_eq] at h₁ + simp[h₁] at subtree_complete + exact power_of_two_mul_two_lt subtree_complete max_height_difference h₁ + let result := branch a (left.insert elem) right q still_in_range (Or.inr right_is_power_of_two) have letMeSpellItOutForYou : n + 1 + m + 1 = n + m + 1 + 1 := by simp_arith letMeSpellItOutForYou ▸ result @@ -150,7 +295,7 @@ def BinaryHeap.insert (elem : α) (heap : BinaryHeap α lt o) : BinaryHeap α lt def BinaryHeap.indexOfAux {α : Type u} {lt : α → α → Bool} [BEq α] (elem : α) (heap : BinaryHeap α lt o) (currentIndex : Nat) : Option (Fin (o+currentIndex)) := match o, heap with | 0, .leaf => none - | (n+m+1), .branch a left right _ _ => + | (n+m+1), .branch a left right _ _ _ => if a == elem then let result := Fin.ofNat' currentIndex (by simp_arith) some result @@ -179,7 +324,7 @@ theorem two_n_not_zero_n_not_zero (n : Nat) (p : ¬2*n = 0) : (¬n = 0) := by def BinaryHeap.popLast {α : Type u} {lt : α → α → Bool} (heap : BinaryHeap α lt (o+1)) : (α × BinaryHeap α lt o) := match o, heap with - | (n+m), .branch a (left : BinaryHeap α lt n) (right : BinaryHeap α lt m) m_le_n subtree_complete => + | (n+m), .branch a (left : BinaryHeap α lt n) (right : BinaryHeap α lt m) m_le_n max_height_difference subtree_complete => if p : 0 = (n+m) then (a, p▸BinaryHeap.leaf) else @@ -199,7 +344,8 @@ def BinaryHeap.popLast {α : Type u} {lt : α → α → Bool} (heap : BinaryHea have r := And.right r simp[←power_of_two_iff_next_power_eq] at r assumption - (res, q▸BinaryHeap.branch a newLeft right s (Or.inr rightIsFull)) + have l_lt_2_m_succ : l < 2 * (m+1) := by apply Nat.lt_of_succ_lt; assumption + (res, q▸BinaryHeap.branch a newLeft right s l_lt_2_m_succ (Or.inr rightIsFull)) else --remove right have : m > 0 := by @@ -234,13 +380,23 @@ def BinaryHeap.popLast {α : Type u} {lt : α → α → Bool} (heap : BinaryHea rewrite[←h₂] at subtree_complete simp at subtree_complete assumption - case inr h₁ => simp at h₁ - rewrite[←power_of_two_iff_next_power_eq] at h₁ - subst m - cases subtree_complete - case inl => assumption - case inr => contradiction - (res, BinaryHeap.branch a left newRight s (Or.inl leftIsFull)) + case inr h₁ => simp_arith[←power_of_two_iff_next_power_eq] at h₁ + rewrite[h₂] at h₁ + simp[h₁] at subtree_complete + assumption + have still_in_range : n < 2*(l+1) := by + rewrite[Decidable.not_and_iff_or_not (l+1<n) rightIsFull] at r + cases r with + | inl h₁ => simp_arith at h₁ + have h₃ := Nat.le_antisymm m_le_n h₁ + subst n + have h₄ := Nat.mul_lt_mul_of_pos_right (by decide : 1 < 2) this + simp at h₄ + assumption + | inr h₁ => simp[←power_of_two_iff_next_power_eq, h₂] at h₁ + apply power_of_two_mul_two_le <;> assumption + + (res, BinaryHeap.branch a left newRight s still_in_range (Or.inl leftIsFull)) /--Removes the element at a given index. Use `BinaryHeap.indexOf` to find the respective index.-/ def BinaryHeap.removeAt {α : Type u} {lt : α → α → Bool} {o : Nat} (index : Fin (o+1)) (heap : BinaryHeap α lt (o+1)) : BinaryHeap α lt o := |