| import Mathlib |
| import OAI.Computability.VertexCover.Machines.FiniteFunction |
| import OAI.Computability.VertexCover.Machines.GraphConstraint |
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| namespace OAI |
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| namespace VertexCover.Machine.GraphMachine |
| open LabelCover |
| variable {a b d : ℕ} |
| attribute [local instance 0] Classical.propDecidable |
| noncomputable instance descriptorFintype (a b d : ℕ) : Fintype (Descriptor a b d) := by |
| classical |
| letI : Fintype (ProfileRep a b d → Bool) := inferInstance |
| letI : Fintype (TracePos d × TracePos d → Bool) := inferInstance |
| letI : Fintype (ProfileRep a b d × ProfileRep a b d → Bool) := inferInstance |
| letI : Fintype (TraceWeights a b d) := inferInstance |
| letI : Fintype ((ProfileRep a b d × ProfileRep a b d → Bool) × TraceWeights a b d) := inferInstance |
| letI : Fintype ((TracePos d × TracePos d → Bool) × |
| ((ProfileRep a b d × ProfileRep a b d → Bool) × TraceWeights a b d)) := inferInstance |
| exact inferInstanceAs (Fintype ((_ → Bool) × _)) |
| noncomputable def validPoly (ha : 0<a) (hb : 0<b) (x : TracePos d) (A : Query.Profile a b d) : |
| Poly (frameCode (a := a) (b := b) (d := d)) boolBits |
| (fun p => decide ((Q p x).ProfileValid A)) := by |
| let c₁ := Poly.const (frameCode (a := a) (b := b) (d := d)) boolBits |
| (decide (∀ k, ¬x.2 < k → A.1 k=⟨0,ha⟩)) |
| let c₂ := Poly.const (frameCode (a := a) (b := b) (d := d)) boolBits |
| (decide (∀ k, ¬k < x.2 → A.2 k=⟨0,hb⟩)) |
| exact (Poly.decideAnd frameCode _ _ c₁ (Poly.decideAnd frameCode _ _ c₂ (pairPoly ha hb x x A A))).congr |
| (fun _ => by apply Bool.eq_iff_iff.mpr; simp only [decide_eq_true_iff] |
| noncomputable def queryEqualPoly (hb : 0<b) (x y : TracePos d) : |
| Poly (frameCode (a := a) (b := b) (d := d)) boolBits |
| (fun p => decide (Q p x=Q p y)) := by |
| rcases x with ⟨s,j⟩ |
| rcases y with ⟨t,k⟩ |
| by_cases h : j=k |
| · subst k |
| let cu := Poly.decideForall (frameCode (a := a) (b := b) (d := d)) |
| (fun p (i : {k : Fin d // j<k}) => (Q p (s,j)).2.1 i=(Q p (t,j)).2.1 i) |
| (fun i => (eqUPoly hb (s,j) (t,j) i i).congr (fun _ => by simp)) |
| let cv := Poly.decideForall (frameCode (a := a) (b := b) (d := d)) |
| (fun p (i : {k : Fin d // k<j}) => (Q p (s,j)).2.2 i=(Q p (t,j)).2.2 i) |
| (fun i => (eqVPoly hb (s,j) (t,j) i i).congr (fun _ => by simp)) |
| exact (Poly.decideAnd frameCode _ _ cu cv).congr (fun p => by |
| apply Bool.decide_congr |
| simp only [Q,LabelCover.query,Sigma.mk.inj_iff,heq_eq_eq, |
| Prod.mk.injEq,funext_iff,true_and]) |
| · exact (Poly.const frameCode boolBits false).congr (fun p => by |
| symm |
| apply decide_eq_false |
| intro hh |
| exact h (congrArg Sigma.fst hh)) |
| noncomputable def boolFinite : Poly boolBits (finiteCode Bool) id := |
| Poly.finite boolBits (finiteCode Bool) (by intro a b h |
| noncomputable def descriptorPoly (ha : 0<a) (hb : 0<b) : |
| Poly (frameCode (a := a) (b := b) (d := d)) (finiteCode (Descriptor a b d)) descriptor := by |
| let cv := Poly.finiteFunction (frameCode (a := a) (b := b) (d := d)) |
| (fun p (r : ProfileRep a b d) => decide ((Q p r.1).ProfileValid r.2)) |
| (fun r => ((validPoly ha hb r.1 r.2).comp boolFinite).congr (fun _ => rfl)) |
| let ce := Poly.finiteFunction (frameCode (a := a) (b := b) (d := d)) |
| (fun p (xy : TracePos d × TracePos d) => decide (Q p xy.1=Q p xy.2)) |
| (fun xy => ((queryEqualPoly hb xy.1 xy.2).comp boolFinite).congr (fun _ => rfl)) |
| let cp := Poly.finiteFunction (frameCode (a := a) (b := b) (d := d)) |
| (fun p (rs : ProfileRep a b d × ProfileRep a b d) => decide (Query.ProfilePair (Q p rs.1.1) (Q p rs.2.1) rs.1.2 rs.2.2)) |
| (fun rs => ((pairPoly ha hb rs.1.1 rs.2.1 rs.1.2 rs.2.2).comp boolFinite).congr (fun _ => rfl)) |
| let cw := weightPoly (a := a) (b := b) (d := d) |
| let ec : Descriptor a b d → List Bool := prodBits (finiteCode (ProfileRep a b d → Bool)) |
| (prodBits (finiteCode (TracePos d × TracePos d → Bool)) |
| (prodBits (finiteCode (ProfileRep a b d × ProfileRep a b d → Bool)) |
| (finiteCode (TraceWeights a b d)))) |
| let cc : Poly ec (finiteCode (Descriptor a b d)) id := Poly.finite _ _ |
| (prodBits_injective (finiteCode_injective _) (prodBits_injective (finiteCode_injective _) |
| (prodBits_injective (finiteCode_injective _) (finiteCode_injective _)))) id |
| exact ((cv.pair (ce.pair (cp.pair cw))).comp cc).congr (fun p => by |
| apply Prod.ext |
| · funext r |
| · apply Prod.ext |
| · funext xy |
| · apply Prod.ext |
| · funext rs |
| · rfl) |
| noncomputable def testFinitePoly (t : ℝ) : |
| Poly (finiteCode (Descriptor a b d)) boolBits (Descriptor.test (a := a) (b := b) (d := d) t) := |
| Poly.finite (finiteCode (Descriptor a b d)) boolBits |
| (finiteCode_injective (Descriptor a b d)) (Descriptor.test t) |
| noncomputable def normTestPoly (ha : 0<a) (hb : 0<b) (t : ℝ) := |
| @Poly.comp (Frame a b d) (Descriptor a b d) Bool |
| (frameCode (a := a) (b := b) (d := d)) (finiteCode (Descriptor a b d)) boolBits |
| (descriptor (a := a) (b := b) (d := d)) (Descriptor.test (a := a) (b := b) (d := d) t) |
| (descriptorPoly (a := a) (b := b) (d := d) ha hb) (testFinitePoly (a := a) (b := b) (d := d) t) |
| theorem diagramSum_frame (p : Frame a b d) : |
| diagramSum (Q p) (W p)=p.1.toLC.sumVector (p.1.toLC.profileVertexEquiv d (rawVertex p.1 p.2.1)) + |
| p.1.toLC.sumVector (p.1.toLC.profileVertexEquiv d (rawVertex p.1 p.2.2)) := by |
| classical |
| simp only [diagramSum,Fintype.sum_prod_type,Fintype.sum_bool] |
| exact add_comm _ _ |
| theorem normTest_correct (p : Frame a b d) (t : ℝ) : |
| (descriptor p).test t=true ↔ p.1.toLC.compatibilityNorm |
| (p.1.toLC.sumVector (p.1.toLC.profileVertexEquiv d (rawVertex p.1 p.2.1))+ |
| p.1.toLC.sumVector (p.1.toLC.profileVertexEquiv d (rawVertex p.1 p.2.2))) ≤ 2*t := by |
| have h := diagram_test (Φ := p.1.toLC) (Q p) (W (a := a) (b := b) p) t |
| exact h.trans (Iff.of_eq (congrArg |
| (fun f : p.1.toLC.Coordinate d → ℝ => p.1.toLC.compatibilityNorm f ≤ 2*t) |
| (diagramSum_frame p))) |
| end VertexCover.Machine.GraphMachine |
| end |
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| end |
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| end OAI |
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