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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 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackDivision
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
def unaryDecodeStep (s : ℕ×ℕ) : ℕ×ℕ := if s.1=0 then s else (s.1-1,s.2+1)
lemma unaryDecodeStep_iterate (k : ℕ) (s : ℕ×ℕ) :
(unaryDecodeStep^[k]) s=(s.1-k,s.2+min k s.1) := by
induction k generalizing s with
| zero=>simp
| succ k ih=>
rw [Function.iterate_succ_apply,ih]
unfold unaryDecodeStep
split
· rename_i hz
apply Prod.ext <;> dsimp only <;> omega
· rename_i hz
apply Prod.ext <;> dsimp only <;> omega
namespace Procedure
noncomputable def unaryDecodeStepP : Procedure (prodCode Nat.bits unaryCode) (prodCode Nat.bits unaryCode)
unaryDecodeStep := by
let x:=first Nat.bits unaryCode
let y:=second Nat.bits unaryCode
let next:=(binarySub.comp (x.pair (Procedure.constant _ Nat.bits 1))).pair (unarySuccessor.comp y)
exact (conditional (binaryZero.comp x) (identity _) next).congrFun (by
intro s;unfold unaryDecodeStep;split <;> simp_all)
noncomputable def clippedUnary : Procedure (prodCode unaryCode Nat.bits) unaryCode (fun x=>min x.1 x.2) := by
let rep:=unaryDecodeStepP.iterate Polynomial.X (by
intro n s i hi
rw [unaryDecodeStep_iterate]
have hbits:=bits_length_mono (Nat.sub_le s.1 i)
simp only [prodCode,pairBits_length,unaryCode,List.length_replicate,Polynomial.eval_X]
omega)
let inp:=(first unaryCode Nat.bits).pair
((second unaryCode Nat.bits).pair (Procedure.constant _ unaryCode 0))
exact ((second Nat.bits unaryCode).comp (rep.comp inp)).congrFun (by
intro x;dsimp only [Function.comp_def];rw [unaryDecodeStep_iterate];simp)
noncomputable def boundedUnary {α : Type} {ea : α→List Bool} {f B : α→ℕ}
(p : Procedure ea Nat.bits f) (b : Procedure ea unaryCode B) (h : ∀a,f a ≤ B a) :
Procedure ea unaryCode f :=
(clippedUnary.comp (b.pair p)).congrFun (by intro a;exact min_eq_right (h a))
end Procedure
end ExactQuantumFactoring.BitStackProgram
end
end OAI
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