| import Mathlib |
| import OAI.Computability.QuantumFactoring.BitStackBasicProcedures |
| import OAI.Computability.QuantumFactoring.BitStackSums |
| import OAI.Computability.QuantumFactoring.BitStackWords |
|
|
| namespace OAI |
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| section |
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| namespace ExactQuantumFactoring.BitStackProgram |
| variable {α β : Type} |
| def listCode (ea : α→List Bool) : List α→List Bool |
| | []=>[false] |
| | a::as=>true::pairBits (ea a) (listCode ea as) |
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| def emptyCode (_ : Unit) : List Bool := [] |
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| def listView : List α→Unit⊕(α×List α) |
| | []=>.inl () |
| | a::as=>.inr (a,as) |
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| lemma listCode_length_cons (ea : α→List Bool) (a : α) (as : List α) : |
| (listCode ea (a::as)).length=2*(ea a).length+2+(listCode ea as).length := by |
| simp [listCode,pairBits_length];omega |
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| lemma listCode_length_append (ea : α→List Bool) (xs ys : List α) : |
| (listCode ea (xs++ys)).length+1=(listCode ea xs).length+(listCode ea ys).length := by |
| induction xs with |
| | nil=>simp [listCode,Nat.add_comm] |
| | cons a as ih=>simp only [List.cons_append,listCode_length_cons];omega |
|
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| lemma listCode_length_rev (ea : α→List Bool) (xs : List α) : |
| (listCode ea xs.reverse).length=(listCode ea xs).length := by |
| induction xs with |
| | nil=>rfl |
| | cons a as ih=> |
| rw [List.reverse_cons] |
| have hh:=listCode_length_append ea as.reverse [a] |
| simp only [listCode_length_cons,ih] at hh ⊢ |
| have hz : (listCode ea []).length=1:=rfl |
| omega |
|
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| lemma listCode_length_pos (ea : α→List Bool) (xs : List α) : 0<(listCode ea xs).length := by |
| cases xs <;> simp [listCode] |
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| lemma list_length_le_code (ea : α→List Bool) (xs : List α) : xs.length≤(listCode ea xs).length := by |
| induction xs with |
| | nil=>simp |
| | cons a as ih=>simp only [List.length_cons,listCode_length_cons];omega |
|
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| lemma listCode_tail_le (ea : α→List Bool) (xs : List α) : |
| (listCode ea xs.tail).length≤(listCode ea xs).length := by |
| cases xs with |
| | nil=>rfl |
| | cons a as=>simp only [List.tail_cons,listCode_length_cons];omega |
|
|
| lemma code_le_of_mem (ea : α→List Bool) {a : α} {xs : List α} (h : a∈xs) : |
| (ea a).length≤(listCode ea xs).length := by |
| induction xs with |
| | nil=>simp at h |
| | cons b bs ih=> |
| rw [List.mem_cons] at h |
| rw [listCode_length_cons] |
| rcases h with h|h |
| · subst b;omega |
| · have hh:=ih h;omega |
|
|
| namespace Procedure |
| variable (ea : α→List Bool) |
| noncomputable def listUncons : Procedure (listCode ea) |
| (sumCode emptyCode (prodCode ea (listCode ea))) listView := |
| (identity (listCode ea)).result (by intro xs;cases xs <;> rfl) |
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| noncomputable def listCons : Procedure (prodCode ea (listCode ea)) (listCode ea) |
| (fun x=>x.1::x.2) := |
| (prepend (prodCode ea (listCode ea)) [true]).result (by intro x;rfl) |
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| noncomputable def listHead (d : α) : Procedure (listCode ea) ea (fun xs=>xs.headD d) := |
| ((casesSum (constant emptyCode ea d) (first ea (listCode ea))).comp (listUncons ea)).congrFun |
| (by intro xs;cases xs <;> rfl) |
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| noncomputable def listTail : Procedure (listCode ea) (listCode ea) List.tail := |
| ((casesSum (constant emptyCode (listCode ea) []) (second ea (listCode ea))).comp |
| (listUncons ea)).congrFun (by intro xs;cases xs <;> rfl) |
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| noncomputable def listEmpty : Procedure (listCode ea) boolCode List.isEmpty := |
| ((casesSum (constant emptyCode boolCode true) (constant (prodCode ea (listCode ea)) boolCode false)).comp |
| (listUncons ea)).congrFun (by intro xs;cases xs <;> rfl) |
| end Procedure |
| end ExactQuantumFactoring.BitStackProgram |
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| end |
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| end OAI |
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