File size: 3,345 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackListCode
import OAI.Computability.QuantumFactoring.BitStackBooleans
import OAI.Computability.QuantumFactoring.BitStackIteration
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
variable {α β : Type}
def listScan (f : α×β→β) : List α×β→List α×β
| ([],b)=>([],b)
| (a::as,b)=>(as,f (a,b))
lemma listScan_iterate (f : α×β→β) (i : ℕ) (xs : List α) (b : β) :
(listScan f)^[i] (xs,b)=(xs.drop i,(xs.take i).foldl (fun b a=>f (a,b)) b) := by
induction i generalizing xs b with
| zero=>simp
| succ i ih=>
rw [Function.iterate_succ_apply]
cases xs with
| nil=>simpa [listScan] using ih [] b
| cons a as=>simpa [listScan] using ih as (f (a,b))
lemma listCode_length_drop_le (ea : α→List Bool) (i : ℕ) (xs : List α) :
(listCode ea (xs.drop i)).length≤(listCode ea xs).length := by
have hh:=listCode_length_append ea (xs.take i) (xs.drop i)
rw [List.take_append_drop] at hh
have hp:=listCode_length_pos ea (xs.take i)
omega
lemma listCode_length_take_le (ea : α→List Bool) (i : ℕ) (xs : List α) :
(listCode ea (xs.take i)).length≤(listCode ea xs).length := by
have hh:=listCode_length_append ea (xs.take i) (xs.drop i)
rw [List.take_append_drop] at hh
have hp:=listCode_length_pos ea (xs.drop i)
omega
namespace Procedure
variable {ea : α→List Bool} {eb : β→List Bool} {f : α×β→β}
noncomputable def listScanP (d : α) (p : Procedure (prodCode ea eb) eb f) :
Procedure (prodCode (listCode ea) eb) (prodCode (listCode ea) eb) (listScan f) := by
let xs:=first (listCode ea) eb
let b:=second (listCode ea) eb
let upd:=((listTail ea).comp xs).pair (p.comp (((listHead ea d).comp xs).pair b))
exact ((choose upd (identity (prodCode (listCode ea) eb))).comp
(((listEmpty ea).comp xs).pair (identity (prodCode (listCode ea) eb)))).congrFun
(by rintro ⟨xs,b⟩;cases xs <;> rfl)
noncomputable def foldList (d : α) (p : Procedure (prodCode ea eb) eb f)
(bound : Polynomial ℕ)
(hsize : ∀ (xs : List α) b i,
(eb ((xs.take i).foldl (fun b a=>f (a,b)) b)).length≤
bound.eval ((listCode ea xs).length+(eb b).length)) :
Procedure (prodCode (listCode ea) eb) eb (fun x=>x.1.foldl (fun b a=>f (a,b)) x.2) := by
let s:=listScanP d p
let rep:=s.iterate (Polynomial.C 2*Polynomial.X+1+bound) (by
intro n x i _
rw [listScan_iterate]
simp only [prodCode,pairBits_length]
have hs:=hsize x.1 x.2 i
have hl:=listCode_length_drop_le ea i x.1
have hm:=eval_nat_mono bound (by omega :
(listCode ea x.1).length+(eb x.2).length≤n+(2*(listCode ea x.1).length+(eb x.2).length+1))
dsimp only at hm
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_C,
Polynomial.eval_X,Polynomial.eval_one]
omega)
let wc:=(length.precompose (listCode ea)).comp (first (listCode ea) eb)
exact ((second (listCode ea) eb).comp (rep.comp (wc.pair (identity (prodCode (listCode ea) eb))))).congrFun
(by
rintro ⟨xs,b⟩
change ((listScan f)^[(listCode ea xs).length] (xs,b)).2=xs.foldl (fun b a=>f (a,b)) b
rw [listScan_iterate,List.take_of_length_le (list_length_le_code ea xs)])
end Procedure
end ExactQuantumFactoring.BitStackProgram
end
end OAI
|