| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.ExpressionEffectivity |
| import OAI.Computability.MinUncut.Estimates.ReversePrefixProgram |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Preprocess.Syntax |
| open MinUncut.Costed MinUncut.CodeEffective Turing.ToPartrec |
|
|
| def prefCode (cs : List Code) : Code := |
| cs.foldr (fun c b=>.cons (.comp c Blocks.extractor.code) b) (PolyProgram.drop 2).code |
| lemma p_prefCode : Primrec prefCode := |
| Primrec.list_foldr Primrec.id (Primrec.const _) |
| (primrec_cons.comp |
| (primrec_comp.comp (Primrec.fst.comp Primrec.snd) (Primrec.const _)) |
| (Primrec.snd.comp Primrec.snd)).to₂ |
|
|
| def stepCode (cs : List Code) : Code := |
| .cons PolyProgram.succ.code |
| (.cons (bin PolyProgram.add.code (PolyProgram.projection 1).code (constCode cs.length)) (prefCode cs)) |
| lemma p_stepCode : Primrec stepCode := |
| primrec_cons.comp (Primrec.const _) |
| (primrec_cons.comp |
| ((p_bin _).comp (Primrec.const _) (p_constCode.comp Primrec.list_length)) p_prefCode) |
| lemma prefCode_eq (es : List AExpr) : prefCode (es.map (fun e=>e.program.code)) = |
| (Blocks.stepProgram.go es).code := by |
| induction es with |
| | nil => rfl |
| | cons e es ih => simp only [prefCode,List.map_cons,List.foldr_cons] at *; rw [ih]; rfl |
| lemma stepCode_eq (es : List AExpr) : stepCode (es.map (fun e=>e.program.code))=(Blocks.stepProgram es).code := by |
| simp only [stepCode,List.length_map,prefCode_eq] |
| rfl |
| lemma p_iterCode : Primrec iterCode := by |
| exact primrec_fix.comp (primrec_case.comp (Primrec.const _) |
| (primrec_cons.comp (Primrec.const _) |
| (primrec_cons.comp (Primrec.const _) (primrec_comp.comp Primrec.id (Primrec.const _))))) |
| def forwardCode (cs : List Code) (N : Code) : Code := |
| .comp reversePrefixProgram.code (.comp PolyProgram.tail.code |
| (.comp (iterCode (stepCode cs.reverse)) |
| (.cons N (.cons (constCode 0) (.cons (constCode 0) Code.id))))) |
| lemma p_forwardCode : Primrec₂ forwardCode := by |
| exact primrec_comp.comp (Primrec.const _) |
| (primrec_comp.comp (Primrec.const _) |
| (primrec_comp.comp (p_iterCode.comp (p_stepCode.comp (Primrec.list_reverse.comp Primrec.fst))) |
| (primrec_cons.comp Primrec.snd (Primrec.const _)))) |
| lemma forwardCode_eq (es : List AExpr) (N : AExpr) : |
| forwardCode (es.map (fun e=>e.program.code)) N.program.code=(Blocks.forward es N).code := by |
| unfold forwardCode |
| rw [←List.map_reverse,stepCode_eq] |
| rfl |
| end MinUncut.Preprocess.Syntax |
|
|
| end |
| end |
|
|
| end OAI |
|
|
