| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackRelabel |
| import OAI.Computability.QuantumFactoring.BitStackProcedures |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
| variable {K L : Type} |
| def sumStore (s : Store K) (t : Store L) : Store (K ⊕ L) := Sum.elim s t |
| @[simp] lemma sumStore_left (s : Store K) (t : Store L) (k : K) : |
| sumStore s t (Sum.inl k)=s k := rfl |
| @[simp] lemma sumStore_right (s : Store K) (t : Store L) (l : L) : |
| sumStore s t (Sum.inr l)=t l := rfl |
| lemma sumStore_empty : sumStore (fun _ : K=>[]) (fun _ : L=>[])=fun _=>[] := by |
| funext j; cases j <;> rfl |
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| def leftEmbed (K L : Type) : K ↪ K ⊕ L := ⟨Sum.inl,Sum.inl_injective⟩ |
| lemma overlay_left (s : Store K) (t : Store L) : |
| overlay (leftEmbed K L) (sumStore (fun _=>[]) t) s=sumStore s t := by |
| funext j |
| cases j with |
| | inl j => exact overlay_at _ _ _ j |
| | inr j => exact overlay_outside _ _ _ _ (by intro i; simp [leftEmbed]) |
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| variable [DecidableEq K] [DecidableEq L] |
| @[simp] lemma sumStore_update_left (s : Store K) (t : Store L) (k : K) (xs : List Bool) : |
| Function.update (sumStore s t) (Sum.inl k) xs=sumStore (Function.update s k xs) t := by |
| funext j; cases j <;> simp [Function.update,sumStore] |
| @[simp] lemma sumStore_update_right (s : Store K) (t : Store L) (k : L) (xs : List Bool) : |
| Function.update (sumStore s t) (Sum.inr k) xs=sumStore s (Function.update t k xs) := by |
| funext j; cases j <;> simp [Function.update,sumStore] |
| lemma sumStore_singleton_left (k : K) (xs : List Bool) : |
| singletonStore (Sum.inl k) xs=sumStore (singletonStore k xs) (fun _ : L=>[]) := by |
| rw [singletonStore,←sumStore_empty,sumStore_update_left]; rfl |
| lemma sumStore_singleton_right (k : L) (xs : List Bool) : |
| singletonStore (Sum.inr k) xs=sumStore (fun _ : K=>[]) (singletonStore k xs) := by |
| rw [singletonStore,←sumStore_empty,sumStore_update_right]; rfl |
| lemma Runs.left {p : Program K} {s u : Store K} {c : ℕ} (h : Runs p s u c) (t : Store L) : |
| Runs (p.relabel Sum.inl) (sumStore s t) (sumStore u t) c := by |
| have hh := h.relabel (leftEmbed K L) (sumStore (fun _=>[]) t) |
| rw [overlay_left,overlay_left] at hh |
| exact hh |
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| namespace Procedure |
| variable {α β : Type} {ea : α→List Bool} {eb : β→List Bool} {f g : α→β} |
| def congrFun (p : Procedure ea eb f) (h : ∀ a,f a=g a) : Procedure ea eb g where |
| K:=p.K |
| finiteK:=p.finiteK |
| decideK:=p.decideK |
| input:=p.input |
| output:=p.output |
| program:=p.program |
| bound:=p.bound |
| runs a:=by simpa only [h a] using p.runs a |
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| noncomputable def identity (ea : α→List Bool) : Procedure ea ea id where |
| K:=Unit |
| finiteK:=inferInstance |
| decideK:=inferInstance |
| input:=() |
| output:=() |
| program:=.skip |
| bound:=1 |
| runs a:=⟨1,by simp,Runs.skip _⟩ |
| end Procedure |
| end ExactQuantumFactoring.BitStackProgram |
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| end |
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| end OAI |
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