File size: 3,965 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackProducts
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
namespace Procedure
variable {α β γ : Type} {ea : α→List Bool} {eb : β→List Bool} {ec : γ→List Bool}
{f : α→β} {g : α→γ}
/-- 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
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)
/-- 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
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)
end Procedure
end ExactQuantumFactoring.BitStackProgram
end
end OAI
|