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0dd3af4 | 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 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.PrefixLists
namespace OAI
namespace BinaryPrefixOperators.Arithmetic.Lists
variable {K : Type*} [CommRing K] [Primcodable K] [Finite K] [DecidableEq K]
private theorem prim_add {K : Type*} [CommRing K] [Primcodable K] [Finite K]
[DecidableEq K] : Primrec₂ ((· + ·) : K → K → K) :=
(Primrec.dom_finite (fun p : K × K => p.1+p.2)).to₂
theorem primrec_coefficient : Primrec₂ (coefficient (K:=K)) := by
change Primrec (fun p : Poly K × Pair =>
p.1.foldr (fun x a => (if x.2 = p.2 then x.1 else 0) + a) 0)
apply Primrec.list_foldr (h := fun (p : Poly K × Pair) (a : (K × Pair) × K) =>
(if a.1.2 = p.2 then a.1.1 else 0) + a.2) Primrec.fst (Primrec.const 0)
exact (prim_add.comp
(Primrec.ite (Primrec.eq.comp
(Primrec.snd.comp (Primrec.fst.comp Primrec.snd)) (Primrec.snd.comp Primrec.fst))
(Primrec.fst.comp (Primrec.fst.comp Primrec.snd)) (Primrec.const 0))
(Primrec.snd.comp Primrec.snd)).to₂
theorem primrec_isZero : Primrec (isZero (K:=K)) := by
have hr : PrimrecRel (fun (a : K × Pair) (p : Poly K) => coefficient p a.2 = 0) :=
Primrec.eq.comp (primrec_coefficient.comp Primrec.snd (Primrec.snd.comp Primrec.fst))
(Primrec.const 0)
have hp : PrimrecPred (fun p : Poly K => ∀ a ∈ p, coefficient p a.2 = 0) :=
hr.forall_mem_list.comp Primrec.id Primrec.id
exact hp.decide.of_eq fun p => by
apply Bool.eq_iff_iff.mpr
simp [isZero]
theorem primrec_suffixes : Primrec suffixes := by
have h0 : Primrec (fun _ : Unit => ([[]] : List Word)) := Primrec.const _
have hs : Primrec₂ (fun (_ : Unit) (p : ℕ × List Word) =>
p.2.map (0::·) ++ p.2.map (1::·)) := by
exact (Primrec.list_append.comp
(Primrec.list_map (Primrec.snd.comp Primrec.snd)
(Primrec.list_cons.comp (Primrec.const 0) Primrec.snd).to₂)
(Primrec.list_map (Primrec.snd.comp Primrec.snd)
(Primrec.list_cons.comp (Primrec.const 1) Primrec.snd).to₂)).to₂
refine ((Primrec.nat_rec h0 hs).comp (Primrec.const ()) Primrec.id).of_eq ?_
intro n
dsimp only [id, Prod.snd]
induction n with
| zero => rfl
| succ n ih => simp [suffixes, ih]
theorem primrec_expandTerm {K : Type*} [CommRing K] [Primcodable K] [Finite K]
[DecidableEq K] : Primrec₂ (expandTerm (K:=K)) := by
unfold expandTerm
apply Primrec.list_map (primrec_suffixes.comp Primrec.fst)
exact ((Primrec.fst.comp (Primrec.snd.comp Primrec.fst)).pair
((Primrec.list_append.comp
(Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.fst))) Primrec.snd).pair
(Primrec.list_append.comp
(Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.fst))) Primrec.snd))).to₂
theorem primrec_depth {K : Type*} [CommRing K] [Primcodable K] [Finite K]
[DecidableEq K] : Primrec (depth (K:=K)) := by
have h : Primrec (fun p : Poly K =>
p.foldr (fun a b => max a.2.2.length b) 0) :=
Primrec.list_foldr Primrec.id (Primrec.const 0)
(Primrec.nat_max.comp
(Primrec.list_length.comp (Primrec.snd.comp (Primrec.snd.comp
(Primrec.fst.comp Primrec.snd)))) (Primrec.snd.comp Primrec.snd)).to₂
refine h.of_eq ?_
intro p
induction p with
| nil => rfl
| cons a p ih => simp [depth, ih]
theorem primrec_spread : Primrec₂ (spread (K:=K)) := by
unfold spread
apply Primrec.list_flatMap Primrec.snd
exact (primrec_expandTerm.comp
(Primrec.nat_sub.comp (Primrec.fst.comp Primrec.fst)
(Primrec.list_length.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))))
Primrec.snd).to₂
theorem primrec_zeroTest : Primrec (zeroTest (K:=K)) :=
primrec_isZero.comp (primrec_spread.comp primrec_depth Primrec.id)
theorem computable_zero : ComputablePred
(fun p : Poly K => evaluate K (toArray p) = 0) :=
(show ComputablePred (fun p : Poly K => zeroTest p = true) from
⟨inferInstance, (Primrec.eq.comp primrec_zeroTest (Primrec.const true)).decide.to_comp⟩).of_eq
(fun p => zeroTest_iff p)
end BinaryPrefixOperators.Arithmetic.Lists
end OAI
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