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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 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib
import OAI.Computability.QuantumFactoring.EmissionCombinators
namespace OAI
section
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
end
end OAI
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