| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Complexity.TreePolyTactic |
| import OAI.Combinatorics.MatchingCount.Complexity.TreeCostMeasure |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped |
| open Polynomial |
|
|
| variable {A B : Type} [Codec A] [Codec B] |
| lemma weight_fst (a : A×B) : weight a.1≤weight a := by rcases a with ⟨a,b⟩; simp only [weight_prod]; omega |
| lemma weight_snd (a : A×B) : weight a.2≤weight a := by rcases a with ⟨a,b⟩; simp only [weight_prod]; omega |
|
|
| variable {m n : A → ℕ} {f : A×(ℕ×ℕ) → B} |
| def grid (M : Realizer m) (N : Realizer n) (F : Realizer f) : |
| Realizer (fun a=>ExecutableDP.tabulate f (a,(m a,n a))) := by |
| let rowF : Realizer (fun x : (A×ℕ)×ℕ=>f (x.1.1,(x.1.2,x.2))) := |
| composition (pair first.fst (pair first.snd second)) F |
| let row : Realizer (fun x : A×ℕ=>(List.range (n x.1+1)).map fun r=>f (x.1,(x.2,r))) := |
| composition (pair identity (composition (composition (composition first N) successor) range)) (mapWith rowF) |
| exact composition (pair identity (composition M range)) (mapWith row) |
| lemma polynomial_grid {M : Realizer m} {N : Realizer n} {F : Realizer f} |
| (hM : PolynomialTime M) (hN : PolynomialTime N) (hF : PolynomialTime F) |
| (p q : Polynomial ℕ) (hm : ∀ a,m a≤p.eval (weight a)) (hn : ∀ a,n a≤q.eval (weight a)) : |
| PolynomialTime (grid M N F) := by |
| have hrows : PolynomialTime (composition (composition (composition (first (A:=A) (B:=ℕ)) N) successor) range) := by |
| apply polynomial_boundedRange (polynomial_composition (polynomial_composition polynomial_first hN) polynomial_successor) (q+1) |
| intro x |
| have hb:=(hn x.1).trans (polynomial_eval_monotone q (weight_fst x)) |
| simpa only [Function.comp_apply,Polynomial.eval_add,Polynomial.eval_one] using Nat.add_le_add_right hb 1 |
| unfold grid |
| dsimp only |
| apply polynomial_composition (polynomial_pair polynomial_identity (polynomial_boundedRange hM p hm)) |
| apply polynomial_mapWith |
| apply polynomial_composition (polynomial_pair polynomial_identity hrows) |
| apply polynomial_mapWith |
| apply polynomial_composition _ hF |
| exact polynomial_pair (polynomial_fst polynomial_first) |
| (polynomial_pair (polynomial_snd polynomial_first) polynomial_second) |
|
|
| def clip (x : ℕ×ℕ) : ℕ := if x.2.size≤x.1 then x.2 else 0 |
| def clipRealizer : Realizer clip := |
| (choose (composition (pair (composition second bitSize) first) leNat) second (constant 0)).congr (by |
| intro x; simp only [clip,Function.comp_apply,decide_eq_true_eq]) |
| omit [Codec A] [Codec B] in |
| lemma polynomial_clip : PolynomialTime clipRealizer := by |
| unfold clipRealizer |
| apply polynomial_congr |
| apply polynomial_choose _ polynomial_second (polynomial_constant 0) |
| apply polynomial_composition _ polynomial_leNat |
| apply polynomial_pair (polynomial_composition polynomial_second polynomial_bitSize) polynomial_first |
| omit [Codec A] [Codec B] in |
| lemma clip_size (K n : ℕ) : (clip (K,n)).size≤K := by |
| unfold clip |
| split_ifs with h |
| · exact h |
| · simp |
| omit [Codec A] [Codec B] in |
| lemma clip_eq (K n : ℕ) (h : n.size≤K) : clip (K,n)=n := by simp [clip,h] |
| end MatchingFPRAS.TreeTyped |
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| end OAI |
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