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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackProcedures
import OAI.Computability.QuantumFactoring.BitStackQuotation
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
variable {K : Type} [DecidableEq K]
/-- A fixed finite list is part of the program, not a run-time oracle. -/
def pushWord (k : K) : List Bool→Program K
| []=>.skip
| b::bs=>.seq (pushWord k bs) (.push k b)
lemma pushWord_runs (k : K) (xs : List Bool) (s : Store K) :
Runs (pushWord k xs) s (Function.update s k (xs++s k)) (xs.length+1) := by
induction xs with
| nil=>simpa [pushWord] using Runs.skip s
| cons b bs ih=>
have hh:=Runs.seq ih (Runs.push (Function.update s k (bs++s k)) k b)
simpa only [pushWord,Function.update_self,Function.update_idem,List.cons_append,
List.length_cons,Nat.add_comm] using hh
namespace Procedure
variable {α β : Type} (ea : α→List Bool) (eb : β→List Bool)
noncomputable def constant (b : β) : Procedure ea eb (fun _=>b) where
K:=Fin 1
finiteK:=inferInstance
decideK:=inferInstance
input:=0
output:=0
program:=.seq (clear 0) (pushWord 0 (eb b))
bound:=Polynomial.C 2*Polynomial.X+Polynomial.C ((eb b).length+2)
runs a:=by
let s : Store (Fin 1):=singletonStore 0 (ea a)
let u : Store (Fin 1):=Function.update s 0 []
have hc:=clear_runs (0 : Fin 1) s
have hp:=pushWord_runs (0 : Fin 1) (eb b) u
have hh:=Runs.seq hc hp
refine ⟨((eb b).length+1)+(2*(ea a).length+1),by simp;omega,?_⟩
convert hh using 1
· funext k;fin_cases k;simp [u,s,singletonStore]
· simp [s,singletonStore]
noncomputable def quote (ea : α→List Bool) : Procedure ea (fun a=>quoteBits (ea a)) id where
K:=Fin 3
finiteK:=inferInstance
decideK:=inferInstance
input:=0
output:=1
program:=quoteMove 0 1 2
bound:=Polynomial.C 7*Polynomial.X+Polynomial.C 5
runs a:=by
have hh:=quoteMove_runs (0 : Fin 3) 1 2 (by decide) (by decide) (by decide)
(singletonStore 0 (ea a)) (by simp [singletonStore])
refine ⟨7*(ea a).length+5,by simp,?_⟩
convert hh using 1
· funext k; fin_cases k <;> simp [singletonStore,Function.update]
· simp [singletonStore]
/-- Count each actual symbol, returning a unary length. -/
noncomputable def length : Procedure (id : List Bool→List Bool)
(fun n=>List.replicate n true) List.length where
K:=Fin 2
finiteK:=inferInstance
decideK:=inferInstance
input:=0
output:=1
program:=.loop 0 (.push 1 true) (.push 1 true)
bound:=Polynomial.C 2*Polynomial.X+1
runs xs:=by
have h : ∀ xs (s : Store (Fin 2)),s 0=xs→
Runs (.loop 0 (.push 1 true) (.push 1 true)) s
(Function.update (Function.update s 0 []) 1 (List.replicate xs.length true++s 1))
(2*xs.length+1) := by
intro ys; induction ys with
| nil=>
intro s hs
have he : Function.update (Function.update s 0 []) 1
(List.replicate ([] : List Bool).length true++s 1)=s := by
simp only [List.length_nil,List.replicate_zero,List.nil_append]
rw [←hs,Function.update_eq_self,Function.update_eq_self]
rw [he];exact Runs.loop_nil hs
| cons y ys ih=>
intro s hs
let u:=Function.update s 0 ys
let v:=Function.update u 1 (true::u 1)
have hv : v 0=ys:=by simp [v,u]
have htail:=ih v hv
have hh : Runs (.loop 0 (.push 1 true) (.push 1 true)) s
(Function.update (Function.update v 0 []) 1 (List.replicate ys.length true++v 1))
(2*ys.length+1+1+1) := by
cases y
· exact Runs.loop_false (a:=1) (b:=2*ys.length+1) hs (Runs.push u 1 true) htail
· exact Runs.loop_true (a:=1) (b:=2*ys.length+1) hs (Runs.push u 1 true) htail
convert hh using 1
· funext k;fin_cases k <;> simp [u,v,Function.update,List.replicate_succ]
rw [←List.cons_append,←List.replicate_succ,List.replicate_succ',List.append_assoc]
rfl
refine ⟨2*xs.length+1,by simp,?_⟩
convert h xs (singletonStore 0 xs) (by simp [singletonStore]) using 1 <;>
(funext k;fin_cases k <;> simp [singletonStore,Function.update])
end Procedure
end ExactQuantumFactoring.BitStackProgram
end
end OAI
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