| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackDivision |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
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| 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 |
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| end |
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| end OAI |
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