File size: 2,403 Bytes
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 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackListFold
import OAI.Computability.QuantumFactoring.BitStackDivision
namespace OAI
section
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
end
end OAI
|