| import OAI.Combinatorics.MatrixRemoval.Model |
|
|
| namespace OAI |
|
|
| /-! |
| The anchor/body extension step for arbitrary repairs. |
| The finite-order gluing lemma is independent of the host construction. |
| -/ |
|
|
| universe uAlpha |
|
|
| noncomputable section |
|
|
| namespace Problem348 |
|
|
| / |
| def appendTwo {α : Type uAlpha} {k : ℕ} (f : Fin k → α) (a b : α) : |
| Fin (k + 2) → α := fun i => |
| if hi : i.val < k then f ⟨i.val, hi⟩ else if i.val = k then a else b |
|
|
| @[simp] theorem appendTwo_initial {α : Type uAlpha} {k : ℕ} |
| (f : Fin k → α) (a b : α) (i : Fin k) : |
| appendTwo f a b ⟨i.val, by omega⟩ = f i := by |
| simp [appendTwo, i.isLt] |
|
|
| @[simp] theorem appendTwo_first {α : Type uAlpha} {k : ℕ} |
| (f : Fin k → α) (a b : α) : |
| appendTwo f a b ⟨k, by omega⟩ = a := by |
| simp [appendTwo] |
|
|
| @[simp] theorem appendTwo_second {α : Type uAlpha} {k : ℕ} |
| (f : Fin k → α) (a b : α) : |
| appendTwo f a b ⟨k + 1, by omega⟩ = b := by |
| simp [appendTwo] |
|
|
| theorem strictMono_appendTwo {α : Type uAlpha} [Preorder α] {k : ℕ} |
| (f : Fin k → α) (a b : α) (hf : StrictMono f) |
| (ha : ∀ i, f i < a) (hab : a < b) : |
| StrictMono (appendTwo f a b) := by |
| intro i j hij |
| have hij' : i.val < j.val := hij |
| by_cases hi : i.val < k |
| · by_cases hj : j.val < k |
| · simpa [appendTwo, hi, hj] using |
| hf (show (⟨i.val, hi⟩ : Fin k) < ⟨j.val, hj⟩ from hij') |
| · by_cases hjk : j.val = k |
| · simpa [appendTwo, hi, hj, hjk] using ha ⟨i.val, hi⟩ |
| · simpa [appendTwo, hi, hj, hjk] using lt_trans (ha ⟨i.val, hi⟩) hab |
| · have hik : i.val = k := by omega |
| have hjk : j.val = k + 1 := by omega |
| simpa [appendTwo, hik, hjk] using hab |
|
|
| / |
| def appendTwoIncreasing {k n : ℕ} (r : IncreasingMap k n) |
| (a b : Fin n) (ha : ∀ i, r.val i < a) (hab : a < b) : |
| IncreasingMap (k + 2) n := |
| ⟨appendTwo r.val a b, strictMono_appendTwo r.val a b r.property ha hab⟩ |
|
|
| / |
| theorem appendTwo_matches_fixedH {n : ℕ} (B : BinaryMatrix n) |
| (r c : IncreasingMap 64 n) (r₀ r₁ c₀ c₁ : Fin n) |
| (hanchor : ∀ u v, B (r.val u) (c.val v) = anchor64 u v) |
| (hrow₀ : ∀ v, B r₀ (c.val v) = decide (v.val = 0)) |
| (hrow₁ : ∀ v, B r₁ (c.val v) = decide (v.val = 1)) |
| (hcol₀ : ∀ u, B (r.val u) c₀ = decide (u.val = 0)) |
| (hcol₁ : ∀ u, B (r.val u) c₁ = decide (u.val = 1)) |
| (h₀₀ : B r₀ c₀ = true) (h₀₁ : B r₀ c₁ = false) |
| (h₁₀ : B r₁ c₀ = true) (h₁₁ : B r₁ c₁ = true) : |
| ∀ i j : Fin 66, |
| B (appendTwo r.val r₀ r₁ i) (appendTwo c.val c₀ c₁ j) = fixedH i j := by |
| intro i j |
| by_cases hi : i.val < 64 |
| · by_cases hj : j.val < 64 |
| · simpa [appendTwo, fixedH, hi, hj] using |
| hanchor ⟨i.val, hi⟩ ⟨j.val, hj⟩ |
| · have hjcases : j.val = 64 ∨ j.val = 65 := by omega |
| rcases hjcases with hj₀ | hj₁ |
| · simpa [appendTwo, fixedH, hi, hj, hj₀] using hcol₀ ⟨i.val, hi⟩ |
| · simpa [appendTwo, fixedH, hi, hj, hj₁] using hcol₁ ⟨i.val, hi⟩ |
| · have hicases : i.val = 64 ∨ i.val = 65 := by omega |
| by_cases hj : j.val < 64 |
| · rcases hicases with hi₀ | hi₁ |
| · simpa [appendTwo, fixedH, hi, hj, hi₀] using hrow₀ ⟨j.val, hj⟩ |
| · simpa [appendTwo, fixedH, hi, hj, hi₁] using hrow₁ ⟨j.val, hj⟩ |
| · have hjcases : j.val = 64 ∨ j.val = 65 := by omega |
| rcases hicases with hi₀ | hi₁ <;> rcases hjcases with hj₀ | hj₁ <;> |
| simp [appendTwo, fixedH, *] |
|
|
| / |
| theorem not_HFree_of_anchor_body {n : ℕ} (B : BinaryMatrix n) |
| (r c : IncreasingMap 64 n) (r₀ r₁ c₀ c₁ : Fin n) |
| (hr : ∀ i, r.val i < r₀) (hr₀₁ : r₀ < r₁) |
| (hc : ∀ j, c.val j < c₀) (hc₀₁ : c₀ < c₁) |
| (hanchor : ∀ u v, B (r.val u) (c.val v) = anchor64 u v) |
| (hrow₀ : ∀ v, B r₀ (c.val v) = decide (v.val = 0)) |
| (hrow₁ : ∀ v, B r₁ (c.val v) = decide (v.val = 1)) |
| (hcol₀ : ∀ u, B (r.val u) c₀ = decide (u.val = 0)) |
| (hcol₁ : ∀ u, B (r.val u) c₁ = decide (u.val = 1)) |
| (h₀₀ : B r₀ c₀ = true) (h₀₁ : B r₀ c₁ = false) |
| (h₁₀ : B r₁ c₀ = true) (h₁₁ : B r₁ c₁ = true) : |
| ¬ HFree fixedH B := by |
| classical |
| let rr := appendTwoIncreasing r r₀ r₁ hr hr₀₁ |
| let cc := appendTwoIncreasing c c₀ c₁ hc hc₀₁ |
| have hmem : (rr, cc) ∈ orderedCopies fixedH B := by |
| simp only [orderedCopies, Finset.mem_filter, Finset.mem_univ, true_and] |
| exact appendTwo_matches_fixedH B r c r₀ r₁ c₀ c₁ hanchor hrow₀ hrow₁ |
| hcol₀ hcol₁ h₀₀ h₀₁ h₁₀ h₁₁ |
| have hpos : 0 < copyCount fixedH B := |
| Finset.card_pos.mpr ⟨(rr, cc), hmem⟩ |
| exact Nat.ne_of_gt hpos |
|
|
| end Problem348 |
|
|
| end |
|
|
| end OAI |
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|
