| import Mathlib |
| import OAI.Computability.MinUncut.Graphs.DemandQueryEffectivity |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Preprocess |
| open MinUncut.Inner MinUncut.Outer MinUncut.Outer.LocalTemplate MinUncut.FiniteGaussian |
| open MinUncutGames.Foundations.Hastad.SourceOccurrences UEncoding |
| open MinUncut.Costed MinUncut.Costed.SourceWords MinUncut.FiniteProof |
| lemma c_replicate {X : Type} [Primcodable X] : Computable₂ (fun n (x : X)=>List.replicate n x) := by |
| have hf : Computable₂ (fun (x : X) (_ : ℕ)=>x) := Computable.fst |
| have hh : Computable (fun p : ℕ × X=>(List.range p.1).map (fun _=>p.2)) := |
| (c_rangeMap hf).comp Computable.snd Computable.fst |
| change Computable (fun p : ℕ × X=>List.replicate p.1 p.2) |
| simpa using hh |
| attribute [local irreducible] rationalScoreTable MinUncut.FiniteGaussian.midpoint |
| namespace DemandBody |
| variable {P : Type} [Primcodable P] {p : P} |
| variable (t : P → ℕ) (ht : Computable t) (h : ∀p,Fin (t p) → Bool) |
| variable (hc : Computable (fun p=>(((Encoding.fin (t p)).function Encoding.bool).code (h p)).val)) |
| variable (m n : P → ℕ) (hm : Computable m) (hn : Computable n) (hnpos : ∀p,0<n p) |
| variable (g : P → GridData) (hg : Computable g) |
| def samples := ((Weights.localSamples t ht h hc m n hm hn hnpos g hg).prod (tests P)).prod |
| (Alphabet.positionsList t ht) |
| def bodyQuery (σ η : ℚ) (x : (LocalSample (Alphabet (h p)) (m p) (n p) (g p) × Test) × (Fin (t p) → Fin 3)) := |
| query (h p) x.1.1.1 x.1.1.2.1 x.1.1.2.2.1 |
| (rationalScoreTable σ η (fun i=>midpoint (g p).T (g p).L (x.1.1.2.2.2 i))) x.1.2 |
| lemma map_bodyScores (σ η : P → ℚ) (hσ : Computable σ) (hη : Computable η) : |
| (samples t ht h hc m n hm hn hnpos g hg).Map |
| ((Grid.indices m n hm hn hnpos).function UEncoding.bool) |
| (fun p x=>rationalScoreTable (σ p) (η p) (fun i=>midpoint (g p).T (g p).L (x.1.1.2.2.2 i))) := by |
| let s:=Weights.localSamples t ht h hc m n hm hn hnpos g hg |
| let c:=(s.prod (tests P)).prod (Alphabet.positionsList t ht) |
| have hs : c.Map s (fun _ x=>x.1.1) := (map_fst _ _).first |
| have hq : c.Map (Grid.samples m n hm hn hnpos g hg) (fun _ x=>x.1.1.2.2.2) := |
| map_comp (map_comp (map_comp hs (map_snd _ _)) (map_snd _ _)) (map_snd _ _) |
| have htable : (Grid.samples m n hm hn hnpos g hg).Map |
| ((Grid.indices m n hm hn hnpos).function UEncoding.bool) |
| (fun p q=>rationalScoreTable (σ p) (η p) (fun i=>midpoint (g p).T (g p).L (q i))) := |
| map_lambda (Grid.map_scoreTable m n hm hn hnpos g hg σ η hσ hη) |
| exact map_comp (a:=c) (b:=Grid.samples m n hm hn hnpos g hg) |
| (c:=(Grid.indices m n hm hn hnpos).function UEncoding.bool) |
| (f:=fun _ x=>x.1.1.2.2.2) |
| (g:=fun p q=>rationalScoreTable (σ p) (η p) (fun i=>midpoint (g p).T (g p).L (q i))) hq htable |
| lemma map_bodyQuery (σ η : P → ℚ) (hσ : Computable σ) (hη : Computable η) : |
| (samples t ht h hc m n hm hn hnpos g hg).Map (Queries.queries t ht h hc) |
| (fun p x=>bodyQuery t h m n g (σ p) (η p) x) := by |
| let s:=Weights.localSamples t ht h hc m n hm hn hnpos g hg |
| let c:=(s.prod (tests P)).prod (Alphabet.positionsList t ht) |
| let ar:=Alphabet.arrays t ht h hc m n hm hn hnpos |
| have hs : c.Map s (fun _ x=>x.1.1) := (map_fst _ _).first |
| have hB : c.Map ar (fun _ x=>x.1.1.1) := map_comp hs (map_fst _ _) |
| have hC : c.Map ar (fun _ x=>x.1.1.2.1) := map_comp (map_comp hs (map_snd _ _)) (map_fst _ _) |
| have hz : c.Map (Face.codes m n hm hn hnpos) (fun _ x=>x.1.1.2.2.1) := |
| map_comp (map_comp (map_comp hs (map_snd _ _)) (map_snd _ _)) (map_fst _ _) |
| have hj : c.Map (tests P) (fun _ x=>x.1.2) := (map_snd _ _).first |
| exact Queries.map_query t ht h hc m n hm hn hnpos hB hC hz |
| (map_bodyScores t ht h hc m n hm hn hnpos g hg σ η hσ hη) hj |
| lemma out_bodyExpressions (σ η : P → ℚ) (hσ : Computable σ) (hη : Computable η) : |
| (samples t ht h hc m n hm hn hnpos g hg).Out (fun p x=> |
| ([leftExpr (bodyQuery t h m n g (σ p) (η p) x) x.2, |
| rightExpr (bodyQuery t h m n g (σ p) (η p) x) x.2, |
| signExpr (bodyQuery t h m n g (σ p) (η p) x) x.2]).map (fun e=>e.program.code)) := |
| Syntax.out_queryExpressions t ht h hc (map_bodyQuery t ht h hc m n hm hn hnpos g hg σ η hσ hη) |
| (map_snd _ _) |
| lemma out_bodyWeight (a : P → ℚ) (ha : Computable a) |
| (b : P → Test → ℚ) (hb : (tests P).Out (fun p j=>b p j)) : |
| (samples t ht h hc m n hm hn hnpos g hg).Out |
| (fun p x=>localWeight (g p) (a p) (b p x.1.2) x.1.2 x.1.1) := by |
| have hw:=(Weights.out_localWeight t ht h hc m n hm hn hnpos g hg a (fun _=>1) ha (ca_const 1)).first (b:=Alphabet.positionsList t ht) |
| have hj : (samples t ht h hc m n hm hn hnpos g hg).Map (tests P) (fun _ x=>x.1.2) := |
| (map_snd _ _).first |
| have hbud:=hb.comp hj |
| apply (hw.ratDiv hbud).ofEq |
| intro p x |
| simp only [localWeight,div_one] |
| lemma out_bodyRepetitions (a : P → ℚ) (ha : Computable a) |
| (b : P → Test → ℚ) (hb : (tests P).Out (fun p j=>b p j)) : |
| (samples t ht h hc m n hm hn hnpos g hg).Out (fun p x=> |
| multiplicity (innerDenominator (t p) (m p) (n p) (g p) (a p) (b p)) |
| (localWeight (g p) (a p) (b p x.1.2) x.1.2 x.1.1)) := by |
| have hw := out_bodyWeight t ht h hc m n hm hn hnpos g hg a ha b hb |
| have hd := out_const (samples t ht h hc m n hm hn hnpos g hg) |
| (Denominator.c_innerDenominator t m n ht hm hn hnpos g hg a ha b hb) |
| exact ((hw.map primrec_ratNum.to_comp).map primrec_intToNat.to_comp).map₂ |
| (hd.map₂ (hw.map primrec_ratDen.to_comp) Primrec.nat_div.to_comp) Primrec.nat_mul.to_comp |
| lemma out_repeatedExpressions (a : P → ℚ) (ha : Computable a) |
| (b : P → Test → ℚ) (hb : (tests P).Out (fun p j=>b p j)) |
| (σ η : P → ℚ) (hσ : Computable σ) (hη : Computable η) : |
| (samples t ht h hc m n hm hn hnpos g hg).Out (fun p x=> |
| (List.replicate (multiplicity (innerDenominator (t p) (m p) (n p) (g p) (a p) (b p)) |
| (localWeight (g p) (a p) (b p x.1.2) x.1.2 x.1.1)) |
| (([leftExpr (bodyQuery t h m n g (σ p) (η p) x) x.2, |
| rightExpr (bodyQuery t h m n g (σ p) (η p) x) x.2, |
| signExpr (bodyQuery t h m n g (σ p) (η p) x) x.2]).map (fun e=>e.program.code))).flatten) := by |
| exact ((out_bodyRepetitions t ht h hc m n hm hn hnpos g hg a ha b hb).map₂ |
| (out_bodyExpressions t ht h hc m n hm hn hnpos g hg σ η hσ hη) |
| c_replicate).map Primrec.list_flatten.to_comp |
| end DemandBody |
| end MinUncut.Preprocess |
|
|
| end |
| end |
|
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| end OAI |
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