File size: 2,840 Bytes
2f887db | 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 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackRelabel
import OAI.Computability.QuantumFactoring.BitStackProcedures
namespace OAI
section
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
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])
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
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
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
end
end OAI
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