import Mathlib
import OAI.Computability.VertexCover.Machines.FiniteFunction
import OAI.Computability.VertexCover.Machines.GraphConstraint

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]; rfl)
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; simpa [boolBits] using h) id
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; exact Bool.decide_congr Iff.rfl
    · apply Prod.ext
      · funext xy; exact Bool.decide_congr Iff.rfl
      · apply Prod.ext
        · funext rs; exact Bool.decide_congr Iff.rfl
        · 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 OAI