| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace TreeEdit.BinaryLower |
|
|
| /-- A componentwise chain in a rectangular integer grid has at most |
| `M + N - 1` points. This counts the block pairs visited by an increasing |
| subsequence matching; no alignment of block boundaries is assumed. -/ |
| theorem card_grid_chain_le (s : Finset (ℕ × ℕ)) (M N : ℕ) |
| (hbound : ∀ p ∈ s, p.1 < M ∧ p.2 < N) |
| (hchain : ∀ p ∈ s, ∀ q ∈ s, |
| (p.1 ≤ q.1 ∧ p.2 ≤ q.2) ∨ (q.1 ≤ p.1 ∧ q.2 ≤ p.2)) : |
| s.card ≤ M + N - 1 := by |
| let rank : ℕ × ℕ → ℕ := fun p => p.1 + p.2 |
| have hinj : Set.InjOn rank s := by |
| intro p hp q hq heq |
| have h := hchain p hp q hq |
| dsimp [rank] at heq |
| rcases p with ⟨i, j⟩ |
| rcases q with ⟨i', j'⟩ |
| have hi : i = i' := by rcases h with h | h <;> omega |
| have hj : j = j' := by rcases h with h | h <;> omega |
| simp [hi, hj] |
| have hsub : s.image rank ⊆ Finset.range (M + N - 1) := by |
| intro a ha |
| obtain ⟨p, hp, rfl⟩ := Finset.mem_image.mp ha |
| have hp' := hbound p hp |
| simp only [Finset.mem_range] |
| dsimp [rank] |
| omega |
| calc |
| s.card = (s.image rank).card := (Finset.card_image_iff.mpr hinj).symm |
| _ ≤ (Finset.range (M + N - 1)).card := Finset.card_le_card hsub |
| _ = M + N - 1 := Finset.card_range _ |
|
|
| /-- The distinct values attained by a componentwise monotone sequence of |
| block indices form a short grid chain. -/ |
| theorem card_monotone_pair_image_le {r M N : ℕ} |
| (f g : Fin r → ℕ) (hf : Monotone f) (hg : Monotone g) |
| (hfb : ∀ a, f a < M) (hgb : ∀ a, g a < N) : |
| (Finset.univ.image (fun a => (f a, g a))).card ≤ M + N - 1 := by |
| apply card_grid_chain_le |
| · intro p hp |
| obtain ⟨a, _, rfl⟩ := Finset.mem_image.mp hp |
| exact ⟨hfb a, hgb a⟩ |
| · intro p hp q hq |
| obtain ⟨a, _, rfl⟩ := Finset.mem_image.mp hp |
| obtain ⟨b, _, rfl⟩ := Finset.mem_image.mp hq |
| rcases le_total a b with h | h |
| · exact Or.inl ⟨hf h, hg h⟩ |
| · exact Or.inr ⟨hf h, hg h⟩ |
|
|
| /-- The number of block pairs visited by an increasing bit matching between |
| words with `M` and `N` blocks of width `P` is at most `M + N - 1`. -/ |
| theorem card_block_pair_image_le {r M N P : ℕ} (hP : 0 < P) |
| (f g : Fin r → ℕ) (hf : Monotone f) (hg : Monotone g) |
| (hfb : ∀ a, f a < M * P) (hgb : ∀ a, g a < N * P) : |
| (Finset.univ.image (fun a => (f a / P, g a / P))).card ≤ M + N - 1 := by |
| apply card_monotone_pair_image_le |
| · intro a b hab |
| exact Nat.div_le_div_right (hf hab) |
| · intro a b hab |
| exact Nat.div_le_div_right (hg hab) |
| · intro a |
| exact (Nat.div_lt_iff_lt_mul hP).mpr (hfb a) |
| · intro a |
| exact (Nat.div_lt_iff_lt_mul hP).mpr (hgb a) |
|
|
| /-- In the equal-length case, at most `2 * M - 1` block pairs occur. -/ |
| theorem card_equal_block_pair_image_le {r M P : ℕ} (hP : 0 < P) |
| (f g : Fin r → ℕ) (hf : Monotone f) (hg : Monotone g) |
| (hfb : ∀ a, f a < M * P) (hgb : ∀ a, g a < M * P) : |
| (Finset.univ.image (fun a => (f a / P, g a / P))).card ≤ 2 * M - 1 := by |
| simpa [two_mul] using card_block_pair_image_le hP f g hf hg hfb hgb |
|
|
| /-- A fixed block pair occupies a consecutive portion of the matching. |
| This is the order-convexity fact needed for its minimal spans to contain |
| no positions matched under a different pair. -/ |
| theorem monotone_pair_fiber_convex {ι : Type*} [Preorder ι] |
| (f g : ι → ℕ) (hf : Monotone f) (hg : Monotone g) |
| {a b c : ι} (hab : a ≤ b) (hbc : b ≤ c) |
| (h : (f a, g a) = (f c, g c)) : (f b, g b) = (f a, g a) := by |
| have hfEq : f a = f c := congrArg Prod.fst h |
| have hgEq : g a = g c := congrArg Prod.snd h |
| have hfL := hf hab |
| have hfR := hf hbc |
| have hgL := hg hab |
| have hgR := hg hbc |
| apply Prod.ext <;> simp only <;> omega |
|
|
| end TreeEdit.BinaryLower |
|
|
| end OAI |
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|
