| import Mathlib |
| import OAI.Computability.VertexCover.Machines.Naturals |
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| namespace OAI |
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| namespace VertexCover.Machine |
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|
| noncomputable def Poly.fixedList {α ι β : Type} (ea : α → List Bool) (eb : β → List Bool) |
| (f : α → ι → β) (cf : ∀ i, Poly ea eb (fun a => f a i)) : |
| (xs : List ι) → Poly ea (listBits eb) (fun a => xs.map (f a)) |
| | [] => Poly.const ea (listBits eb) [] |
| | i::xs => ((cf i).pair (Poly.fixedList ea eb f cf xs)).comp (Poly.listCons eb) |
|
|
| noncomputable def Poly.fixedAll {α ι : Type} (ea : α → List Bool) (f : α → ι → Bool) |
| (cf : ∀ i, Poly ea boolBits (fun a => f a i)) : |
| (xs : List ι) → Poly ea boolBits (fun a => xs.all (f a)) |
| | [] => Poly.const ea boolBits true |
| | i::xs => (((cf i).pair (Poly.fixedAll ea f cf xs)).comp (Poly.bool₂ (fun x => x.1 && x.2))).congr |
| (fun _ => by simp only [Function.comp_apply,List.all_cons]) |
|
|
| noncomputable def Poly.finiteAll {α ι : Type} [Fintype ι] (ea : α → List Bool) |
| (f : α → ι → Bool) (cf : ∀ i, Poly ea boolBits (fun a => f a i)) : |
| Poly ea boolBits (fun a => decide (∀ i, f a i = true)) := by |
| classical |
| exact (Poly.fixedAll ea f cf Finset.univ.toList).congr (fun a => by |
| apply Bool.eq_iff_iff.mpr |
| simp only [List.all_eq_true,Finset.mem_toList,Finset.mem_univ,forall_const,decide_eq_true_iff]) |
|
|
| noncomputable def fixedSelect {α ι β : Type} [DecidableEq ι] |
| (f : α → ι → β) (i₀ : ι) : List ι → α × ι → β |
| | [], x => f x.1 i₀ |
| | i::xs, x => if x.2=i then f x.1 i else fixedSelect f i₀ xs x |
|
|
| theorem fixedSelect_mem {α ι β : Type} [DecidableEq ι] |
| (f : α → ι → β) (i₀ : ι) (a : α) : |
| ∀ (xs : List ι) (i : ι), i ∈ xs → fixedSelect f i₀ xs (a,i) = f a i := by |
| intro xs |
| induction xs with |
| | nil => simp |
| | cons j xs ih => |
| intro i hi |
| by_cases h : i=j |
| · subst i; simp [fixedSelect] |
| · rw [fixedSelect,ite_eq_right h] |
| exact ih i ((List.mem_cons.mp hi).resolve_left h) |
|
|
| noncomputable def Poly.fixedSelect {α ι β : Type} [Fintype ι] [DecidableEq ι] |
| (ea : α → List Bool) (ei : ι → List Bool) (eb : β → List Bool) (hi : Function.Injective ei) |
| (f : α → ι → β) (i₀ : ι) (cf : ∀ i, Poly ea eb (fun a => f a i)) : |
| (xs : List ι) → Poly (prodBits ea ei) eb (fixedSelect f i₀ xs) |
| | [] => (Poly.fst ea ei).comp (cf i₀) |
| | i::xs => (((Poly.snd ea ei).comp (Poly.finite ei boolBits hi (fun j => decide (j=i)))).ite |
| ((Poly.fst ea ei).comp (cf i)) (Poly.fixedSelect ea ei eb hi f i₀ cf xs)).congr (fun _ => by |
| simp only [Function.comp_apply,VertexCover.Machine.fixedSelect,decide_eq_true_eq]) |
|
|
| noncomputable def Poly.finiteBranch {α ι β : Type} [Fintype ι] [Nonempty ι] |
| (ea : α → List Bool) (ei : ι → List Bool) (eb : β → List Bool) (hi : Function.Injective ei) |
| (f : α → ι → β) (cf : ∀ i, Poly ea eb (fun a => f a i)) : |
| Poly (prodBits ea ei) eb (fun x => f x.1 x.2) := by |
| classical |
| let i₀ := Classical.choice (inferInstance : Nonempty ι) |
| exact (Poly.fixedSelect ea ei eb hi f i₀ cf Finset.univ.toList).congr |
| (fun x => fixedSelect_mem f i₀ x.1 _ x.2 (by simp)) |
|
|
| end VertexCover.Machine |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
