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caec230 | 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 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.MathematicalPhysics.RapidForcing.RationalProfiles
import OAI.MathematicalPhysics.RapidForcing.ComputableAutomation
import OAI.MathematicalPhysics.RapidForcing.RatPrimrec
namespace OAI
open Encodable Denumerable Nat.Partrec
namespace RapidForcing.EffectiveArithmetic
@[fun_prop] lemma primrec_code_pair : Primrec (fun p : Code × Code => Code.pair p.1 p.2) :=
Code.primrec₂_pair
@[fun_prop] lemma primrec_code_comp : Primrec (fun p : Code × Code => Code.comp p.1 p.2) :=
Code.primrec₂_comp
@[fun_prop] lemma primrec_code_prec : Primrec (fun p : Code × Code => Code.prec p.1 p.2) :=
Code.primrec₂_prec
attribute [fun_prop] Nat.Partrec.Code.primrec_rfind'
@[fun_prop] lemma primrec_code_ofNat : Primrec Code.ofNatCode := Primrec.ofNat Code
@[instance_reducible] def retractEncodable {A B : Type} [Encodable B] (f : A → B) (g : B → A)
(h : Function.LeftInverse g f) : Encodable A := Encodable.ofLeftInverse f g h
@[instance_reducible] def primcodable_retract {A B : Type} [Primcodable B] (f : A → B) (g : B → A)
(h : Function.LeftInverse g f) (hr : Primrec (fun b => f (g b))) :
Primcodable A :=
{ toEncodable := retractEncodable f g h
prim := by
let := retractEncodable f g h
have hp := Primrec.nat_iff.mp <| Primrec.encode.comp <|
Primrec.option_map Primrec.decode (hr.comp Primrec.snd).to₂
apply hp.of_eq
intro n
change encode ((decode (α := B) n).map (fun b => f (g b))) =
@encode (Option A) _ ((decode (α := B) n).bind (some ∘ g))
cases decode (α := B) n <;> rfl }
@[simp] lemma code_encode_ofNat (n : ℕ) : encode (Code.ofNatCode n) = n :=
Denumerable.encode_ofNat (α := Code) n
end RapidForcing.EffectiveArithmetic
namespace RapidForcing.EffectiveProfile
open EffectiveArithmetic
namespace Expr
local instance : DecidableEq Code := Encodable.decidableEqOfEncodable Code
def toCode : Expr → Code
| .const q => .pair .zero (.ofNatCode (encode q))
| .left p => .pair .succ (.ofNatCode (encode p))
| .right p => .pair .left (.ofNatCode (encode p))
| .invDen n => .pair .right (.ofNatCode n)
| .add a b => .comp a.toCode b.toCode
| .mul a b => .prec a.toCode b.toCode
def fromCode : Code → Expr
| .pair t v => if t = .zero then .const ((decode (encode v)).getD 0)
else if t = .succ then .left ((decode (encode v)).getD [])
else if t = .left then .right ((decode (encode v)).getD [])
else .invDen (encode v)
| .comp a b => .add (fromCode a) (fromCode b)
| .prec a b => .mul (fromCode a) (fromCode b)
| _ => .const 0
@[simp] lemma fromCode_toCode (e : Expr) : fromCode e.toCode = e := by
induction e <;> simp [toCode, fromCode, *]
lemma primrec_normalize : Primrec (fun c => toCode (fromCode c)) := by
let z : Code := toCode (.const 0)
have h := Code.primrec_recOn (σ := Code) Primrec.id
(Primrec.const z) (Primrec.const z) (Primrec.const z) (Primrec.const z)
(pr := fun _ t v _ _ => if t = .zero then toCode (.const ((decode (encode v)).getD 0))
else if t = .succ then toCode (.left ((decode (encode v)).getD []))
else if t = .left then toCode (.right ((decode (encode v)).getD []))
else toCode (.invDen (encode v)))
(by
dsimp [toCode]
apply Primrec.ite (by fun_prop) _ _
· apply primrec_code_pair.comp ((Primrec.const _).pair ?_)
exact primrec_code_ofNat.comp <| Primrec.encode.comp <|
Primrec.option_getD.comp (Primrec.decode.comp (Primrec.encode.comp (by fun_prop))) (Primrec.const 0)
· apply Primrec.ite (by fun_prop) _ _
· apply primrec_code_pair.comp ((Primrec.const _).pair ?_)
exact primrec_code_ofNat.comp <| Primrec.encode.comp <|
Primrec.option_getD.comp (Primrec.decode.comp (Primrec.encode.comp (by fun_prop))) (Primrec.const [])
· apply Primrec.ite (by fun_prop) _ _
· apply primrec_code_pair.comp ((Primrec.const _).pair ?_)
exact primrec_code_ofNat.comp <| Primrec.encode.comp <|
Primrec.option_getD.comp (Primrec.decode.comp (Primrec.encode.comp (by fun_prop))) (Primrec.const [])
· fun_prop)
(co := fun _ _ _ a b => .comp a b) (by fun_prop)
(pc := fun _ _ _ a b => .prec a b) (by fun_prop)
(rf := fun _ _ _ => z) (by fun_prop)
apply h.of_eq
intro c
induction c <;> simp_all only [id_eq, fromCode, toCode]
all_goals first | rfl | (split_ifs <;> rfl)
instance : Primcodable Expr := primcodable_retract toCode fromCode fromCode_toCode primrec_normalize
@[fun_prop] lemma primrec_toCode : Primrec toCode := by
apply Primrec.encode_iff.mp
exact (Primrec.encode : Primrec (@encode Expr _))
@[fun_prop] lemma primrec_fromCode : Primrec fromCode := by
apply Primrec.encode_iff.mp
exact Primrec.encode.comp primrec_normalize
@[simp] lemma encode_toCode (e : Expr) : encode (toCode e) = encode e := rfl
@[fun_prop] lemma primrec_const : Primrec Expr.const := by
apply Primrec.encode_iff.mp
change Primrec (fun q => encode (toCode (.const q)))
unfold toCode
fun_prop
@[fun_prop] lemma primrec_left : Primrec Expr.left := by
apply Primrec.encode_iff.mp
change Primrec (fun q => encode (toCode (.left q)))
unfold toCode
fun_prop
@[fun_prop] lemma primrec_right : Primrec Expr.right := by
apply Primrec.encode_iff.mp
change Primrec (fun q => encode (toCode (.right q)))
unfold toCode
fun_prop
@[fun_prop] lemma primrec_invDen : Primrec Expr.invDen := by
apply Primrec.encode_iff.mp
change Primrec (fun q => encode (toCode (.invDen q)))
unfold toCode
fun_prop
@[fun_prop] lemma primrec_add : Primrec (fun p : Expr × Expr => Expr.add p.1 p.2) := by
apply Primrec.encode_iff.mp
change Primrec (fun p : Expr × Expr => encode (toCode (.add p.1 p.2)))
unfold toCode
fun_prop
@[fun_prop] lemma primrec_mul : Primrec (fun p : Expr × Expr => Expr.mul p.1 p.2) := by
apply Primrec.encode_iff.mp
change Primrec (fun p : Expr × Expr => encode (toCode (.mul p.1 p.2)))
unfold toCode
fun_prop
end Expr
end RapidForcing.EffectiveProfile
end OAI
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