| import Mathlib |
| import OAI.Computability.VertexCover.Machines.Codec |
| import OAI.Computability.VertexCover.Machines.LCData |
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| namespace OAI |
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| namespace VertexCover.Machine |
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|
| noncomputable def Poly.finFunction {α β : Type} [Fintype β] |
| (ea : α → List Bool) : (n : ℕ) → (f : α → Fin n → β) → |
| (∀ i, Poly ea (finiteCode β) (fun a => f a i)) → |
| Poly ea (finiteCode (Fin n → β)) f |
| | 0,f,_ => (Poly.const ea (finiteCode (Fin 0 → β)) (fun i => Fin.elim0 i)).congr |
| (fun a => by funext i; exact Fin.elim0 i) |
| | n+1,f,cf => |
| (((cf 0).pair (Poly.finFunction ea n (fun a i => f a i.succ) (fun i => cf i.succ))).comp |
| (Poly.finite _ _ (prodBits_injective (finiteCode_injective β) (finiteCode_injective (Fin n → β))) |
| (fun p : β × (Fin n → β) => Fin.cons p.1 p.2))).congr |
| (fun a => by funext i; refine Fin.cases ?_ (fun j => ?_) i <;> rfl) |
|
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| noncomputable def Poly.finiteFunction {α β ι : Type} [Fintype β] [Fintype ι] [DecidableEq ι] |
| (ea : α → List Bool) (f : α → ι → β) |
| (cf : ∀ i, Poly ea (finiteCode β) (fun a => f a i)) : |
| Poly ea (finiteCode (ι → β)) f := by |
| let E := Fintype.equivFin ι |
| exact ((Poly.finFunction ea (Fintype.card ι) (fun a i => f a (E.symm i)) |
| (fun i => cf (E.symm i))).comp |
| (Poly.finite _ _ (finiteCode_injective _) (fun g : Fin (Fintype.card ι) → β => fun i => g (E i)))).congr |
| (fun a => by funext i; exact congrArg (f a) (E.symm_apply_apply i)) |
|
|
| noncomputable def Poly.finiteRecode {α : Type} [Fintype α] (ea : α → List Bool) |
| (ha : Function.Injective ea) : Poly ea (finiteCode α) id := |
| Poly.finite ea (finiteCode α) ha id |
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| noncomputable def Poly.fixedSum {α ι : Type} (ea : α → List Bool) |
| (f : α → ι → ℕ) (cf : ∀ i, Poly ea natBits (fun a => f a i)) : |
| (xs : List ι) → Poly ea natBits (fun a => (xs.map (f a)).sum) |
| | [] => (Poly.const ea natBits 0).congr (fun _ => rfl) |
| | i::xs => (((cf i).pair (Poly.fixedSum ea f cf xs)).comp Poly.natAdd).congr |
| (fun a => by simp only [Function.comp_apply,List.map_cons,List.sum_cons]) |
|
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| noncomputable def Poly.finiteSum {α ι : Type} [Fintype ι] (ea : α → List Bool) |
| (f : α → ι → ℕ) (cf : ∀ i, Poly ea natBits (fun a => f a i)) : |
| Poly ea natBits (fun a => ∑ i, f a i) := by |
| classical |
| exact (Poly.fixedSum ea f cf Finset.univ.toList).congr (fun a => by simp) |
| end VertexCover.Machine |
| end |
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| end |
| end |
| end |
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| end |
| end |
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| end |
| end |
| end |
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| end OAI |
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