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2f887db | 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 | import Mathlib
import OAI.Computability.MinUncut.Estimates.CodeComputability
namespace OAI
section
namespace MinUncut.CodeEffective
open Turing.ToPartrec
variable {σ : Type*} [Primcodable σ]
def fold (z s t : σ) (b3 b4 b5 : σ → σ → σ) (f6 : σ → σ) : Code → σ
| .zero' => z
| .succ => s
| .tail => t
| .cons f g => b3 (fold z s t b3 b4 b5 f6 f) (fold z s t b3 b4 b5 f6 g)
| .comp f g => b4 (fold z s t b3 b4 b5 f6 f) (fold z s t b3 b4 b5 f6 g)
| .case f g => b5 (fold z s t b3 b4 b5 f6 f) (fold z s t b3 b4 b5 f6 g)
| .fix f => f6 (fold z s t b3 b4 b5 f6 f)
lemma primrec_fold (z s t : σ) {b3 b4 b5 : σ → σ → σ} {f6 : σ → σ}
(h3 : Primrec₂ b3) (h4 : Primrec₂ b4) (h5 : Primrec₂ b5) (h6 : Primrec f6) :
Primrec (fold z s t b3 b4 b5 f6) := by
have h := Nat.Partrec.Code.primrec_recOn (α:=Code) (σ:=σ) primrec_embed
(z:=fun _=>z) (Primrec.const _) (s:=fun _=>s) (Primrec.const _)
(l:=fun _=>t) (Primrec.const _) (r:=fun _=>t) (Primrec.const _)
(pr:=fun _ _ _ x y=>b3 x y)
(h3.comp (Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd)))
(Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))))
(co:=fun _ _ _ x y=>b4 x y)
(h4.comp (Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd)))
(Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))))
(pc:=fun _ _ _ x y=>b5 x y)
(h5.comp (Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd)))
(Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))))
(rf:=fun _ _ x=>f6 x) (h6.comp (Primrec.snd.comp Primrec.snd))
apply h.of_eq
intro c
induction c <;> simp_all [fold,embed]
end MinUncut.CodeEffective
end
end OAI
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