| 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 |
| |
