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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Computability.WLIdentification.CarryPresentation
namespace OAI
namespace WLIdentification
namespace FinitePresentation
open WordExpressions
attribute [local instance] Classical.propDecidable Classical.decEq
namespace CellularRealizer
open CellularCircuit
variable (A : Type) [Fintype A] (d : Word → ℕ) (dp : NatPoly (fun X : Fin 1 → Word => d (X 0)))
noncomputable abbrev pointP := Presented.fixed (Gate A)
noncomputable abbrev tripleP := Presented.fixed (A × A × A)
abbrev digitP := Presented.fin d dp
abbrev kindP := ((digitP d dp).prod (digitP d dp).option).sum ((digitP d dp).sum
(((digitP d dp).prod (digitP d dp).option).sum (Presented.unit.sum Presented.unit)))
noncomputable abbrev edgeBase := (kindP d dp).prod (tripleP A)
def edgeEquiv (w : Word) : Connection (d w) A ≃
(((Fin (d w) × Option (Fin (d w))) ⊕ (Fin (d w) ⊕ ((Fin (d w) × Option (Fin (d w))) ⊕ (Unit ⊕ Unit)))) × (A × A × A)) :=
Connection.equiv.trans (Equiv.prodComm _ _)
noncomputable def edgeP : Presented (fun w => Connection (d w) A) := (edgeBase A d dp).retype (edgeEquiv A d)
noncomputable def casesNumber {f : ∀ w,Connection (d w) A → ℕ}
(l : Number (((digitP d dp).prod (digitP d dp).option).prod (tripleP A))
(fun w x => f w (.left x.2.1 x.2.2.1 x.2.2.2 x.1.1 x.1.2)))
(c : Number ((digitP d dp).prod (tripleP A))
(fun w x => f w (.center x.2.1 x.2.2.1 x.2.2.2 x.1)))
(r : Number (((digitP d dp).prod (digitP d dp).option).prod (tripleP A))
(fun w x => f w (.right x.2.1 x.2.2.1 x.2.2.2 x.1.1 x.1.2)))
(j : Number (Presented.unit.prod (tripleP A))
(fun w x => f w (.join x.2.1 x.2.2.1 x.2.2.2)))
(o : Number (Presented.unit.prod (tripleP A))
(fun w x => f w (.output x.2.1 x.2.2.1 x.2.2.2))) : Number (edgeP A d dp) f :=
((Number.sumFst l (Number.sumFst c (Number.sumFst r (Number.sumFst j o)))).retype (edgeEquiv A d)).congr
(by intro w x; cases x <;> rfl)
noncomputable def pointIndex : Number (pointP A) (fun _ p => gateIndex p) := Number.fixed _ gateIndex
noncomputable def dimension : Number (pointP A) (fun _ p => dimension p) := Number.fixed (Gate A) CellularCircuit.dimension
noncomputable def edgeIndex : Number (edgeP A d dp) (fun _ e => connectionIndex e) :=
((Number.pair ((Number.fixed (A × A × A) (fun a => Nat.pair (NaturalCode.fixed A a.1)
(Nat.pair (NaturalCode.fixed A a.2.1) (NaturalCode.fixed A a.2.2)))).comp (Map.snd _ _))
(Number.code (kindP d dp) |>.comp (Map.fst _ _))).retype (edgeEquiv A d)).congr
(by intro w x; cases x <;> rfl)
noncomputable def source : Map (edgeP A d dp) (pointP A) (fun _ => CellularCircuit.source) := by
apply casesNumber
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.scalar a.1))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.scalar a.2.1))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.scalar a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.first a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.second a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
noncomputable def target (F : A → A → A → A) : Map (edgeP A d dp) (pointP A) (fun _ => CellularCircuit.target F) := by
apply casesNumber
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.first a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.first a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.second a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.second a.1 a.2.1 a.2.2))).comp (Map.snd _ _)
· exact (Number.fixed _ (fun a : A × A × A => NaturalCode.fixed (Gate A) (.scalar (F a.1 a.2.1 a.2.2)))).comp (Map.snd _ _)
noncomputable def port (F : A → A → A → A) : Number (edgeP A d dp) (fun _ e => (CellularCircuit.port F e).val) := by
apply casesNumber
· exact Number.const _ 0
· exact Number.const _ 1
· exact Number.const _ 1
· exact Number.const _ 0
· exact Number.const _ 0
noncomputable def edgeMap : ProductRealizer (edgeP A d dp) (fun w => d w+d w) (NatPoly.add dp dp)
(fun _ => CellularCircuit.edgeMap) := by
let C := ProductRealizer.carry d dp
let R := ProductRealizer.cycle d dp
let I := ProductRealizer.identity (digitP d dp) d dp
let L := (C.comp (Map.fst _ _)).concat (R.comp (Map.snd _ _))
let H := C.concat I
let B := (C.comp (Map.fst _ _)).concat (R.inverse.comp (Map.snd _ _))
let J := ProductRealizer.identity Presented.unit (fun w => d w+d w) (NatPoly.add dp dp)
exact (((L.sumCases (H.sumCases (B.sumCases (J.sumCases J)))).comp (Map.fst _ _)).comp
(Map.retype (edgeBase A d dp) (edgeEquiv A d))).congr (by intro w x; cases x <;> rfl)
end CellularRealizer
end FinitePresentation
end WLIdentification
end OAI
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