| import Mathlib |
| import OAI.Computability.QuantumFactoring.EmissionCombinators |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram.Procedure |
| variable {α β : Type} {ea : α→List Bool} {eb : β→List Bool} {f : α→β} |
| noncomputable def chooseBitD (c : α→Prop) [DecidablePred c] |
| (p : Procedure (fun x:{a // ¬c a}=>ea x.val) eb (fun x=>f x.val)) |
| (q : Procedure (fun x:{a // c a}=>ea x.val) eb (fun x=>f x.val)) : |
| Procedure (fun a=>decide (c a)::ea a) eb f where |
| K:=(p.K ⊕ q.K) ⊕ Fin 2 |
| finiteK:=inferInstance |
| decideK:=inferInstance |
| input:=Sum.inr 0 |
| output:=Sum.inr 0 |
| program:=.cases (Sum.inr 0) .skip |
| (p.shared.program.relabel (leftPublic p.K q.K)) |
| (q.shared.program.relabel (rightPublic p.K q.K)) |
| bound:=p.shared.bound+q.shared.bound+1 |
| runs a:=by |
| have hpmono:=eval_nat_mono p.shared.bound (Nat.le_succ (ea a).length) |
| have hqmono:=eval_nat_mono q.shared.bound (Nat.le_succ (ea a).length) |
| by_cases h:c a |
| · obtain ⟨t,ht,hd⟩:=q.shared.runs ⟨a,h⟩ |
| have hh:=hd.relabel (rightPublic p.K q.K) (fun _=>[]) |
| simp only [overlay_singleton] at hh |
| have he : rightPublic p.K q.K q.shared.input=Sum.inr 0:=rfl |
| rw [he] at hh |
| refine ⟨t+1,?_,?_⟩ |
| · simpa only [List.length_cons,Polynomial.eval_add,Polynomial.eval_one] using |
| Nat.add_le_add_right ((ht.trans hqmono).trans (Nat.le_add_left |
| (q.shared.bound.eval ((ea a).length+1)) (p.shared.bound.eval ((ea a).length+1)))) 1 |
| · simp only [h,decide_true] |
| have hu : Function.update (singletonStore (Sum.inr 0 : (p.K⊕q.K)⊕Fin 2) |
| (true::ea a)) (Sum.inr 0) (ea a)=singletonStore (Sum.inr 0) (ea a):=by |
| simp [singletonStore] |
| simpa only [hu] using Runs.cases_true (p:=Program.skip) |
| (q:=p.shared.program.relabel (leftPublic p.K q.K)) |
| (xs:=ea a) (s:=singletonStore (Sum.inr 0) (true::ea a)) (by simp [singletonStore]) |
| (by rw [hu];exact hh) |
| · obtain ⟨t,ht,hd⟩:=p.shared.runs ⟨a,h⟩ |
| have hh:=hd.relabel (leftPublic p.K q.K) (fun _=>[]) |
| simp only [overlay_singleton] at hh |
| have he : leftPublic p.K q.K p.shared.input=Sum.inr 0:=rfl |
| rw [he] at hh |
| refine ⟨t+1,?_,?_⟩ |
| · simpa only [List.length_cons,Polynomial.eval_add,Polynomial.eval_one] using |
| Nat.add_le_add_right ((ht.trans hpmono).trans (Nat.le_add_right |
| (p.shared.bound.eval ((ea a).length+1)) (q.shared.bound.eval ((ea a).length+1)))) 1 |
| · simp only [h,decide_false] |
| have hu : Function.update (singletonStore (Sum.inr 0 : (p.K⊕q.K)⊕Fin 2) |
| (false::ea a)) (Sum.inr 0) (ea a)=singletonStore (Sum.inr 0) (ea a):=by |
| simp [singletonStore] |
| simpa only [hu] using Runs.cases_false (p:=Program.skip) |
| (r:=q.shared.program.relabel (rightPublic p.K q.K)) |
| (xs:=ea a) (s:=singletonStore (Sum.inr 0) (false::ea a)) (by simp [singletonStore]) |
| (by rw [hu];exact hh) |
| noncomputable def splitOn (c : α→Prop) [DecidablePred c] |
| (test : Procedure ea boolCode (fun a=>decide (c a))) |
| (yes : Procedure (fun x:{a // c a}=>ea x.val) eb (fun x=>f x.val)) |
| (no : Procedure (fun x:{a // ¬c a}=>ea x.val) eb (fun x=>f x.val)) : Procedure ea eb f:=by |
| let p:=no.comp (unprefix (fun x:{a // ¬c a}=>ea x.val) false) |
| let q:=yes.comp (unprefix (fun x:{a // c a}=>ea x.val) false) |
| let choice:=(chooseBitD c p q).comp (unprefix (fun a=>decide (c a)::false::ea a) true) |
| let input:= (test.pair (identity ea)).result (ec:=fun a=>true::decide (c a)::false::ea a) |
| (h:=id) (by intro a;rfl) |
| exact choice.comp input |
| end ExactQuantumFactoring.BitStackProgram.Procedure |
| |
| namespace ExactQuantumFactoring.BitStackProgram.Emits |
| variable {α β : Type} {ea : α→List Bool} {eb : β→List Bool} {f : α→β} |
| lemma splitOn (c : α→Prop) [DecidablePred c] |
| (test : Emits ea Procedure.boolCode (fun a=>decide (c a))) |
| (yes : Emits (fun x:{a // c a}=>ea x.val) eb (fun x=>f x.val)) |
| (no : Emits (fun x:{a // ¬c a}=>ea x.val) eb (fun x=>f x.val)) : Emits ea eb f:=by |
| obtain ⟨p⟩:=test;obtain ⟨q⟩:=yes;obtain ⟨r⟩:=no |
| exact ⟨Procedure.splitOn c p q r⟩ |
| end ExactQuantumFactoring.BitStackProgram.Emits |
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| end |
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| end OAI |
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