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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 | import Mathlib
import OAI.Computability.QuantumFactoring.BitStackListOps
namespace OAI
section
namespace ExactQuantumFactoring.BitStackProgram
variable {α : Type}
lemma fold_append_flatten (xss : List (List α)) (xs : List α) :
xss.foldl (fun xs ys=>xs++ys) xs=xs++xss.flatten:=by
induction xss generalizing xs with
| nil=>simp
| cons ys yss ih=>simp [ih,List.append_assoc]
lemma listCode_length_flatten_le (ea : α→List Bool) (xss : List (List α)) :
(listCode ea xss.flatten).length≤(listCode (listCode ea) xss).length:=by
induction xss with
| nil=>simp [listCode]
| cons xs xss ih=>
simp only [List.flatten_cons,listCode_length_cons]
have hh:=listCode_length_append ea xs xss.flatten
omega
namespace Procedure
noncomputable def listFlatten (ea : α→List Bool) (d : α) :
Procedure (listCode (listCode ea)) (listCode ea) List.flatten:=by
let step:=(listAppend ea d).comp ((second (listCode ea) (listCode ea)).pair
(first (listCode ea) (listCode ea)))
let p:=foldList [] step Polynomial.X (by
intro xss xs i
change (listCode ea ((xss.take i).foldl (fun xs ys=>xs++ys) xs)).length≤_
rw [fold_append_flatten]
have hh:=listCode_length_append ea xs (xss.take i).flatten
have hl:=listCode_length_flatten_le ea (xss.take i)
have ht:=listCode_length_take_le (listCode ea) i xss
simp only [Polynomial.eval_X];omega)
exact (p.comp ((identity (listCode (listCode ea))).pair (constant _ (listCode ea) []))).congrFun (by
intro xss;exact fold_append_flatten xss [])
end Procedure
end ExactQuantumFactoring.BitStackProgram
end
end OAI
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