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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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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