| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackProducts |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
| namespace Procedure |
| variable |
| |
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| /-- Duplicate the input word and quote the first copy to represent a pair. -/ |
| noncomputable def duplicate (ea : α→List Bool) : |
| Procedure ea (prodCode ea ea) (fun a=>(a,a)) where |
| K:=Fin 4 |
| finiteK:=inferInstance |
| decideK:=inferInstance |
| input:=0 |
| output:=3 |
| program:=.seq (duplicateReverse 0 1 2) (.seq (reverseMove 1 3) (quoteMove 0 3 1)) |
| bound:=Polynomial.C 14*Polynomial.X+Polynomial.C 8 |
| runs a:=by |
| let s : Store (Fin 4):=singletonStore 0 (ea a) |
| let u : Store (Fin 4):=Function.update s 1 (ea a).reverse |
| let v : Store (Fin 4):=Function.update s 3 (ea a) |
| have h1 : Runs (duplicateReverse 0 1 2) s u (5*(ea a).length+2) := by |
| have hh:=duplicateReverse_runs (0 : Fin 4) 1 2 (by decide) (by decide) (by decide) |
| s (by simp [s,singletonStore]) |
| simpa only [u,s,singletonStore,Function.update_self, |
| Function.update_of_ne (by decide : (1 : Fin 4)≠0),List.append_nil] using hh |
| have h2 : Runs (reverseMove 1 3) u v (2*(ea a).length+1) := by |
| have hh:=reverseMove_runs (1 : Fin 4) 3 (by decide) u |
| convert hh using 1 |
| · funext k; fin_cases k <;> simp [u,v,s,singletonStore,Function.update] |
| · simp [u] |
| have h3 : Runs (quoteMove 0 3 1) v |
| (singletonStore 3 (prodCode ea ea (a,a))) (7*(ea a).length+5) := by |
| have hh:=quoteMove_runs (0 : Fin 4) 3 1 (by decide) (by decide) (by decide) v |
| (by simp [v,s,singletonStore]) |
| convert hh using 1 |
| · funext k; fin_cases k <;> simp [v,s,singletonStore,Function.update,prodCode,pairBits] |
| · simp [v,s,singletonStore] |
| refine ⟨(7*(ea a).length+5)+(2*(ea a).length+1)+(5*(ea a).length+2),?_, |
| Runs.seq h1 (Runs.seq h2 h3)⟩ |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_C,Polynomial.eval_X] |
| omega |
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| noncomputable def pair (p : Procedure ea eb f) (q : Procedure ea ec g) : |
| Procedure ea (prodCode eb ec) (fun a=>(f a,g a)) := |
| (p.parallel q).comp (duplicate ea) |
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| /-- Projection parses the actual prefix code and explicitly clears its tail. -/ |
| noncomputable def first (ea : α→List Bool) (eb : β→List Bool) : |
| Procedure (prodCode ea eb) ea Prod.fst where |
| K:=Fin 4 |
| finiteK:=inferInstance |
| decideK:=inferInstance |
| input:=0 |
| output:=1 |
| program:=.seq (unquoteMove 0 1 2 3) (clear 0) |
| bound:=Polynomial.C 7*Polynomial.X+Polynomial.C 7 |
| runs x:=by |
| let s : Store (Fin 4):=singletonStore 0 (prodCode ea eb x) |
| let u : Store (Fin 4):=Function.update (singletonStore 0 (eb x.2)) 1 (ea x.1) |
| have h1 : Runs (unquoteMove 0 1 2 3) s u (7*(ea x.1).length+6) := by |
| have hh:=unquoteMove_runs (0 : Fin 4) 1 2 3 (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (ea x.1) (eb x.2) s |
| (by simp [s,singletonStore,prodCode,pairBits]) |
| (by simp [s,singletonStore]) (by simp [s,singletonStore]) |
| convert hh using 1 |
| funext k; fin_cases k <;> simp [s,u,singletonStore,Function.update] |
| have h2 : Runs (clear 0) u (singletonStore 1 (ea x.1)) (2*(eb x.2).length+1) := by |
| have hh:=clear_runs (0 : Fin 4) u |
| convert hh using 1 |
| · funext k; fin_cases k <;> simp [u,singletonStore,Function.update] |
| · simp [u,singletonStore] |
| refine ⟨(2*(eb x.2).length+1)+(7*(ea x.1).length+6),?_,Runs.seq h1 h2⟩ |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_C,Polynomial.eval_X, |
| prodCode,pairBits_length] |
| omega |
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| noncomputable def second (ea : α→List Bool) (eb : β→List Bool) : |
| Procedure (prodCode ea eb) eb Prod.snd := |
| ((first eb ea).comp (swap ea eb)).congrFun (by intro x; rfl) |
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| end Procedure |
| end ExactQuantumFactoring.BitStackProgram |
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| end |
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| end OAI |
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