| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackListFold |
| import OAI.Computability.QuantumFactoring.BitStackDivision |
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| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
| variable {α : Type} |
| def dropStep : ℕ×List α→ℕ×List α |
| | (n,xs)=>if n=0 then (n,xs) else (n-1,xs.tail) |
| lemma dropStep_iterate (i n : ℕ) (xs : List α) : |
| dropStep^[i] (n,xs)=(n-i,xs.drop (min i n)) := by |
| induction i generalizing n xs with |
| | zero=>simp |
| | succ i ih=> |
| rw [Function.iterate_succ_apply] |
| by_cases hn : n=0 |
| · subst n;simp [dropStep,ih] |
| · obtain ⟨n,rfl⟩:=Nat.exists_eq_succ_of_ne_zero hn |
| simp only [dropStep,Nat.succ_ne_zero,ite_false,Nat.succ_sub_one,ih] |
| rw [←List.drop_one,List.drop_drop] |
| congr 1 |
| · omega |
| · congr 1; omega |
| |
| namespace Procedure |
| variable (ea : α→List Bool) |
| noncomputable def listDrop : Procedure (prodCode Nat.bits (listCode ea)) (listCode ea) |
| (fun x=>x.2.drop x.1) := by |
| let en:=prodCode Nat.bits (listCode ea) |
| let n:=first Nat.bits (listCode ea) |
| let xs:=second Nat.bits (listCode ea) |
| let pred:=binarySub.comp (n.pair (constant en Nat.bits 1)) |
| let step:=(conditional (binaryZero.comp n) (identity en) |
| (pred.pair ((listTail ea).comp xs))).congrFun (by intro x;rfl) |
| have stepType : Procedure en en (dropStep (α:=α)):= |
| step.congrFun (by intro x;simp only [Function.comp_apply,decide_eq_true_eq,id_eq,dropStep]) |
| let rep:=stepType.iterate Polynomial.X (by |
| intro k x i _ |
| rw [dropStep_iterate] |
| simp only [en,prodCode,pairBits_length,Polynomial.eval_X] |
| have hn:=bits_length_mono (Nat.sub_le x.1 i) |
| have hl:=listCode_length_drop_le ea (min i x.1) x.2 |
| omega) |
| let times:=(length.precompose (listCode ea)).comp xs |
| exact (xs.comp (rep.comp (times.pair (identity en)))).congrFun (by |
| intro x |
| change (dropStep^[(listCode ea x.2).length] x).2=x.2.drop x.1 |
| rw [dropStep_iterate] |
| by_cases h:x.1≤(listCode ea x.2).length |
| · rw [Nat.min_eq_right h] |
| · have hlen:=list_length_le_code ea x.2 |
| rw [Nat.min_eq_left (by omega),List.drop_eq_nil_of_le (by omega),List.drop_eq_nil_of_le (by omega)]) |
| noncomputable def listGet (d : α) : Procedure (prodCode Nat.bits (listCode ea)) ea |
| (fun x=>(x.2.drop x.1).headD d) := (listHead ea d).comp (listDrop ea) |
| end Procedure |
| end ExactQuantumFactoring.BitStackProgram |
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| end |
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| end OAI |
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