| import OAI.Computability.WLIdentification.CircuitForms |
|
|
| namespace OAI |
|
|
| namespace WLIdentification |
| namespace FinitePresentation |
| open WordExpressions |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| structure ProductRealizer {A : Word → Type} (AP : Presented A) (d : Word → ℕ) |
| (dp : NatPoly (fun X : Fin 1 → Word => d (X 0))) (D : ∀ w,A w → ProductMap (d w)) where |
| domain : Test (AP.prod ((Presented.fin d dp).prod Presented.bool)) (fun w x => decide (x.2.2∈(D w x.1).domain x.2.1)) |
| mapping : Map (AP.prod ((Presented.fin d dp).prod Presented.bool)) Presented.bool (fun w x => (D w x.1).mapping x.2.1 x.2.2) |
|
|
| namespace ProductRealizer |
| variable {A B : Word → Type} {AP : Presented A} {BP : Presented B} |
| {d e : Word → ℕ} {dp : NatPoly (fun X : Fin 1 → Word => d (X 0))} |
| {ep : NatPoly (fun X : Fin 1 → Word => e (X 0))} |
| {D : ∀ w,A w → ProductMap (d w)} |
|
|
| noncomputable def comp (F : ProductRealizer AP d dp D) {f : ∀ w,B w → A w} (G : Map BP AP f) : |
| ProductRealizer BP d dp (fun w b => D w (f w b)) where |
| domain := F.domain.comp (Map.pair (G.comp (Map.fst _ _)) (Map.snd _ _)) |
| mapping := F.mapping.comp (Map.pair (G.comp (Map.fst _ _)) (Map.snd _ _)) |
|
|
| noncomputable def sumCases (F : ProductRealizer AP d dp D) {E : ∀ w,B w → ProductMap (d w)} |
| (G : ProductRealizer BP d dp E) : ProductRealizer (AP.sum BP) d dp (fun w x => x.elim (D w) (E w)) where |
| domain := (Test.sumFst F.domain G.domain).congr (by intro w x; rcases x with ⟨x,y⟩; cases x <;> rfl) |
| mapping := (Number.sumFst F.mapping G.mapping).congr (by intro w x; rcases x with ⟨x,y⟩; cases x <;> rfl) |
|
|
| noncomputable def identity (AP : Presented A) (d : Word → ℕ) (dp : NatPoly (fun X : Fin 1 → Word => d (X 0))) : |
| ProductRealizer AP d dp (fun w _ => ProductMap.identity (d w)) where |
| domain := (Test.const _ true).congr (by intro w x; simp [ProductMap.identity]) |
| mapping := (Map.snd _ _).comp (Map.snd _ _) |
|
|
| noncomputable def concat (F : ProductRealizer AP d dp D) {E : ∀ w,A w → ProductMap (e w)} |
| (G : ProductRealizer AP e ep E) : ProductRealizer AP (fun w => d w+e w) (NatPoly.add dp ep) |
| (fun w a => (D w a).concat (E w a)) where |
| domain := { |
| eval w v := if (Nat.unpair (Nat.unpair v).2).1<d w then F.domain.eval w v |
| else G.domain.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair ((Nat.unpair (Nat.unpair v).2).1-d w) (Nat.unpair (Nat.unpair v).2).2)) |
| correct w x := by |
| rcases x with ⟨a,i,b⟩ |
| refine Fin.addCases (fun j => ?_) (fun j => ?_) i |
| · simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair,Fin.val_castAdd,j.isLt,ite_true] |
| simpa only [Presented.prod_code,Presented.fin_code,ProductMap.concat,Fin.append_left] using F.domain.correct w (a,j,b) |
| · simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair,Fin.val_natAdd,show ¬d w+j.val<d w by omega,ite_false,Nat.add_sub_cancel_left] |
| simpa only [Presented.prod_code,Presented.fin_code,ProductMap.concat,Fin.append_right] using G.domain.correct w (a,j,b) |
| poly := BoolPoly.iteProp (NatPoly.lt (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (dp.rename (fun _ => 0))) F.domain.poly |
| (G.domain.eval_poly (Poly.var 0) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.var 1)) (NatPoly.pair |
| (NatPoly.sub (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (dp.rename (fun _ => 0))) |
| (NatPoly.unpair_snd (NatPoly.unpair_snd (NatPoly.var 1)))))) } |
| mapping := { |
| eval w v := if (Nat.unpair (Nat.unpair v).2).1<d w then F.mapping.eval w v |
| else G.mapping.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair ((Nat.unpair (Nat.unpair v).2).1-d w) (Nat.unpair (Nat.unpair v).2).2)) |
| correct w x := by |
| rcases x with ⟨a,i,b⟩ |
| refine Fin.addCases (fun j => ?_) (fun j => ?_) i |
| · simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair,Fin.val_castAdd,j.isLt,ite_true] |
| simpa only [Presented.prod_code,Presented.fin_code,ProductMap.concat,Fin.append_left] using F.mapping.correct w (a,j,b) |
| · simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair,Fin.val_natAdd,show ¬d w+j.val<d w by omega,ite_false,Nat.add_sub_cancel_left] |
| simpa only [Presented.prod_code,Presented.fin_code,ProductMap.concat,Fin.append_right] using G.mapping.correct w (a,j,b) |
| poly := NatPoly.iteProp (NatPoly.lt (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (dp.rename (fun _ => 0))) F.mapping.poly |
| (G.mapping.eval_poly (Poly.var 0) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.var 1)) (NatPoly.pair |
| (NatPoly.sub (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (dp.rename (fun _ => 0))) |
| (NatPoly.unpair_snd (NatPoly.unpair_snd (NatPoly.var 1)))))) } |
|
|
| noncomputable def inverse (F : ProductRealizer AP d dp D) : ProductRealizer AP d dp (fun w a => (D w a).inverse) where |
| domain := { |
| eval w v := (F.domain.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 0)) && |
| decide (F.mapping.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 0))=(Nat.unpair (Nat.unpair v).2).2)) || |
| (F.domain.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 1)) && |
| decide (F.mapping.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 1))=(Nat.unpair (Nat.unpair v).2).2)) |
| correct w x := by |
| simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair] |
| have hdf := F.domain.correct w (x.1,x.2.1,false) |
| have hdt := F.domain.correct w (x.1,x.2.1,true) |
| have hmf := F.mapping.correct w (x.1,x.2.1,false) |
| have hmt := F.mapping.correct w (x.1,x.2.1,true) |
| simp only [Presented.prod_code,Presented.fin_code,Presented.bool_code] at hdf hdt hmf hmt |
| simp only [show NaturalCode.bool false=0 from rfl,show NaturalCode.bool true=1 from rfl] at hdf hdt hmf hmt |
| simp only [Presented.bool_code] |
| rw [hdf,hdt,hmf,hmt] |
| apply Bool.eq_iff_iff.mpr |
| simp only [Bool.or_eq_true,Bool.and_eq_true,decide_eq_true_eq,(NaturalCode.bool).injective.eq_iff, |
| ProductMap.inverse,Finset.mem_image,Bool.exists_bool] |
| poly := by |
| have hk (b : ℕ) : NatPoly (fun X : Fin 2 → Word => Nat.pair (Nat.unpair (X 1).length).1 (Nat.pair (Nat.unpair (Nat.unpair (X 1).length).2).1 b)) := |
| NatPoly.pair (NatPoly.unpair_fst (NatPoly.var 1)) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (NatPoly.const b)) |
| exact BoolPoly.or (BoolPoly.and (F.domain.eval_poly (Poly.var 0) (hk 0)) |
| (NatPoly.eqtest (F.mapping.eval_poly (Poly.var 0) (hk 0)) (NatPoly.unpair_snd (NatPoly.unpair_snd (NatPoly.var 1))))) |
| (BoolPoly.and (F.domain.eval_poly (Poly.var 0) (hk 1)) |
| (NatPoly.eqtest (F.mapping.eval_poly (Poly.var 0) (hk 1)) (NatPoly.unpair_snd (NatPoly.unpair_snd (NatPoly.var 1))))) } |
| mapping := { |
| eval w v := if (F.domain.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 0)) && |
| decide (F.mapping.eval w (Nat.pair (Nat.unpair v).1 (Nat.pair (Nat.unpair (Nat.unpair v).2).1 0))=(Nat.unpair (Nat.unpair v).2).2)) then 0 else 1 |
| correct w x := by |
| simp only [Presented.prod_code,Presented.fin_code,Nat.unpair_pair] |
| have hdf := F.domain.correct w (x.1,x.2.1,false) |
| have hmf := F.mapping.correct w (x.1,x.2.1,false) |
| simp only [Presented.prod_code,Presented.fin_code,Presented.bool_code] at hdf hmf |
| simp only [show NaturalCode.bool false=0 from rfl] at hdf hmf |
| simp only [Presented.bool_code] |
| rw [hdf,hmf] |
| simp only [Bool.and_eq_true,decide_eq_true_eq,(NaturalCode.bool).injective.eq_iff] |
| change (if _ then 0 else 1)=NaturalCode.bool (if _ then false else true) |
| split <;> rfl |
| poly := NatPoly.ite (BoolPoly.and |
| (F.domain.eval_poly (Poly.var 0) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.var 1)) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (NatPoly.const 0)))) |
| (NatPoly.eqtest (F.mapping.eval_poly (Poly.var 0) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.var 1)) (NatPoly.pair (NatPoly.unpair_fst (NatPoly.unpair_snd (NatPoly.var 1))) (NatPoly.const 0)))) |
| (NatPoly.unpair_snd (NatPoly.unpair_snd (NatPoly.var 1))))) (NatPoly.const 0) (NatPoly.const 1) } |
|
|
| end ProductRealizer |
| end FinitePresentation |
| end WLIdentification |
|
|
| end OAI |
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