| import Mathlib |
| import OAI.Computability.VertexCover.Machines.ListFilter |
| import OAI.Computability.VertexCover.Machines.GraphDescriptor |
| import OAI.Computability.VertexCover.Machines.NumberedGraph |
|
|
| namespace OAI |
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| namespace VertexCover.Machine.GraphMachine |
| open LabelCover |
| noncomputable section |
| variable {a b d : ℕ} |
| def weightCount (a b d : ℕ) : ℕ := Fintype.card (Weight a b d) |
| theorem weightCount_pos : 0<weightCount a b d := |
| Fintype.card_pos_iff.mpr ⟨defaultWeight⟩ |
| def count (I : FixedLC a b) (d : ℕ) : ℕ := I.M ^ Fintype.card (PositionPair d) * weightCount a b d |
| theorem count_pos (I : FixedLC a b) (d : ℕ) : 0<count I d := |
| Nat.mul_pos (pow_pos I.Mpos _) weightCount_pos |
|
|
| def seedEquiv (I : FixedLC a b) (d : ℕ) : Fin (I.M ^ Fintype.card (PositionPair d)) ≃ I.toLC.Seeds d := |
| finFunctionFinEquiv.symm.trans (Equiv.arrowCongr (Fintype.equivFin (PositionPair d)).symm (Equiv.refl _)) |
| theorem seedEquiv_apply (I : FixedLC a b) (n : Fin (I.M ^ Fintype.card (PositionPair d))) : |
| seedEquiv I d n=seed I n.val := by |
| rfl |
|
|
| def numberEquiv (I : FixedLC a b) (d : ℕ) : Fin (count I d) ≃ I.toLC.Vertex d := |
| finProdFinEquiv.symm.trans (Equiv.prodCongr (seedEquiv I d) (Fintype.equivFin (Weight a b d)).symm) |
| def decode (n : ℕ) : RawVertex a b d := |
| (n / weightCount a b d, (Fintype.equivFin (Weight a b d)).symm ⟨n % weightCount a b d,Nat.mod_lt _ weightCount_pos⟩) |
| theorem numberEquiv_apply (I : FixedLC a b) (n : Fin (count I d)) : |
| numberEquiv I d n=rawVertex I (decode n.val) := rfl |
|
|
| def fullEquiv (I : FixedLC a b) (d : ℕ) : Fin (count I d) ≃ I.toLC.Vertex d := |
| (numberEquiv I d).trans (I.toLC.profileVertexEquiv d) |
| def graph (I : FixedLC a b) (d : ℕ) (t : ℝ) : SimpleGraph (Fin (count I d)) := |
| (I.toLC.graph d t).comap (fullEquiv I d) |
| def explicitGraph (I : FixedLC a b) (d : ℕ) (t : ℝ) : ExplicitGraph := |
| ExplicitGraph.numbered (graph I d t) |
|
|
| def countPoly (hb : 0<b) (d : ℕ) : Poly (FixedLC.code (a := a) (b := b)) natBits (fun I => count I d) := |
| ((((FixedLC.MPoly hb).comp (Poly.natPow (Fintype.card (PositionPair d)))).pair |
| (Poly.const FixedLC.code natBits (weightCount a b d))).comp Poly.natMul).congr (fun _ => rfl) |
| def decodePoly : Poly natBits (vertexCode (a := a) (b := b) (d := d)) decode := by |
| let q := ((Poly.identity natBits).pair (Poly.const natBits natBits (weightCount a b d))).comp Poly.natDiv |
| let r := (Poly.finMod (weightCount a b d) weightCount_pos).comp |
| (Poly.finite finCode weightCode (finCode_injective _) (Fintype.equivFin (Weight a b d)).symm) |
| exact (q.pair r).congr (fun _ => rfl) |
|
|
| def numberFrame (p : FixedLC a b × (ℕ × ℕ)) : Frame a b d := |
| (p.1,(decode p.2.1,decode p.2.2)) |
| def numberFramePoly : Poly (prodBits (FixedLC.code (a := a) (b := b)) (prodBits natBits natBits)) |
| (frameCode (a := a) (b := b) (d := d)) numberFrame := by |
| let I := Poly.fst (FixedLC.code (a := a) (b := b)) (prodBits natBits natBits) |
| let p := Poly.snd (FixedLC.code (a := a) (b := b)) (prodBits natBits natBits) |
| exact I.pair (((p.comp (Poly.fst natBits natBits)).comp decodePoly).pair |
| ((p.comp (Poly.snd natBits natBits)).comp decodePoly)) |
|
|
| def test (t : ℝ) (p : FixedLC a b × (ℕ × ℕ)) : Bool := |
| decide (p.2.1<p.2.2) && (descriptor (numberFrame (d := d) p)).test t |
|
|
| attribute [local irreducible] Descriptor.test |
|
|
| def normNumberPoly (ha : 0<a) (hb : 0<b) (t : ℝ) := |
| (numberFramePoly (a := a) (b := b) (d := d)).comp |
| (normTestPoly (a := a) (b := b) (d := d) ha hb t) |
| def lessPoly (a b : ℕ) := |
| (Poly.snd (FixedLC.code (a := a) (b := b)) (prodBits natBits natBits)).comp Poly.natLt |
| def testPrePoly (ha : 0<a) (hb : 0<b) (t : ℝ) := |
| ((lessPoly a b).pair (normNumberPoly (d := d) ha hb t)).comp |
| (Poly.bool₂ (fun p => p.1 && p.2)) |
| def testPoly (ha : 0<a) (hb : 0<b) (t : ℝ) : |
| Poly (prodBits (FixedLC.code (a := a) (b := b)) (prodBits natBits natBits)) boolBits (test (d := d) t) := |
| (testPrePoly (d := d) ha hb t).congr (fun _ => rfl) |
|
|
| theorem test_correct (I : FixedLC a b) (t : ℝ) (u v : Fin (count I d)) : |
| test (d := d) t (I,(u.val,v.val))=true ↔ u<v ∧ (graph I d t).Adj u v := by |
| classical |
| simp only [test,Bool.and_eq_true,decide_eq_true_iff] |
| rw [normTest_correct] |
| change (u.val<v.val ∧ _) ↔ (u<v ∧ fullEquiv I d u ≠ fullEquiv I d v ∧ _) |
| have he (n : Fin (count I d)) : fullEquiv I d n=I.toLC.profileVertexEquiv d (rawVertex I (decode n.val)) := rfl |
| simp only [he,numberFrame] |
| constructor |
| · rintro ⟨hl,hn⟩ |
| exact ⟨hl,fun h => (ne_of_lt hl) (congrArg Fin.val ((fullEquiv I d).injective h)),hn⟩ |
| · rintro ⟨hl,_,hn⟩; exact ⟨hl,hn⟩ |
| end |
| end VertexCover.Machine.GraphMachine |
| end |
| |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
