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5bdeca1 | 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 | import OAI.NumberTheory.SingleFold.RecursiveEnumeration
namespace OAI
namespace SingleFold
open Compiler
theorem recursively_enumerable_sf {n : ℕ} {p : (Fin n → ℕ) → Prop} (hp : REPred p) : SF p := by
obtain ⟨Q,hQ,hD,M,hM⟩:=TM2Macro.exists_counter hp
let := hQ
let := hD
have hc := CounterCompilation.halts_sf M
have hi : MapSF (TM2Macro.counterInput (n:=n)) := by
apply MapSF.vector
intro j
cases hs : TM2Macro.inputSlot j with
| none => simpa only [TM2Macro.counterInput,hs,Option.elim_none] using (ScalarSF.const (α:=Fin n) 0)
| some i => simpa only [TM2Macro.counterInput,hs,Option.elim_some] using ScalarSF.proj i
exact (hi.pullback hc).congr (fun a=>(hM a).symm)
theorem main : MainStatement := by
intro n _ S hS
obtain ⟨r⟩:=recursively_enumerable_sf hS
obtain ⟨m,hm,P,hP,hU⟩:=export_polynomial r
exact ⟨m,hm,P,fun a=>⟨hP a,hU a⟩⟩
end SingleFold
end OAI
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