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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.CanonicalNames
import OAI.GroupTheory.SimpleOvergroups.PrefixEffective
import OAI.GroupTheory.SimpleOvergroups.PrefixProduct
namespace OAI
namespace BinaryPrefixOperators.Arithmetic
theorem primrec_cancel : Primrec₂ cancel := by
unfold cancel
exact Primrec.ite
(Primrec.eq.comp (Primrec.list_take.comp (Primrec.list_length.comp Primrec.fst) Primrec.snd)
Primrec.fst)
(Primrec.option_some.comp ((Primrec.list_drop.comp
(Primrec.list_length.comp Primrec.fst) Primrec.snd).pair (Primrec.const [])))
(Primrec.ite
(Primrec.eq.comp (Primrec.list_take.comp (Primrec.list_length.comp Primrec.snd) Primrec.fst)
Primrec.snd)
(Primrec.option_some.comp ((Primrec.const []).pair
(Primrec.list_drop.comp (Primrec.list_length.comp Primrec.snd) Primrec.fst)))
(Primrec.const none))
theorem primrec_multiplyPair : Primrec₂ multiplyPair := by
unfold multiplyPair
exact Primrec.option_map
(primrec_cancel.comp (Primrec.snd.comp Primrec.fst) (Primrec.fst.comp Primrec.snd))
((Primrec.list_append.comp
(Primrec.fst.comp (Primrec.fst.comp Primrec.fst)) (Primrec.fst.comp Primrec.snd)).pair
(Primrec.list_append.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.fst))
(Primrec.snd.comp Primrec.snd))).to₂
namespace Lists
variable {K : Type*} [CommRing K]
def negate (p : Poly K) : Poly K := p.map (fun a => (-a.1,a.2))
@[simp] theorem value_negate (p : Poly K) : value (negate p) = -value p := by
induction p with
| nil => simp [negate]
| cons a p ih =>
rcases a with ⟨a,v⟩
simp only [negate] at ih ⊢
simp [ih, neg_smul, neg_add_rev, add_comm]
def oneName : Poly K := [(1,([],[]))]
@[simp] theorem value_oneName : value (oneName (K:=K)) = 1 := by
simp [oneName, replacement]
variable [Primcodable K] [Finite K]
theorem primrec_negate : Primrec (negate (K:=K)) :=
Primrec.list_map Primrec.id (((Primrec.dom_finite (fun a : K => -a)).comp
(Primrec.fst.comp Primrec.snd)).pair (Primrec.snd.comp Primrec.snd)).to₂
theorem primrec_mulTerm : Primrec₂ (mulTerm (K:=K)) := by
have h : Primrec₂ (fun a b : K × Pair =>
(multiplyPair a.2 b.2).map (fun p => (a.1*b.1,p))) := by
exact Primrec.option_map
(primrec_multiplyPair.comp (Primrec.snd.comp Primrec.fst) (Primrec.snd.comp Primrec.snd))
((((Primrec.dom_finite (fun a : K × K => a.1*a.2)).comp
((Primrec.fst.comp (Primrec.fst.comp Primrec.fst)).pair
(Primrec.fst.comp (Primrec.snd.comp Primrec.fst)))).pair Primrec.snd)).to₂
have ht : Primrec (fun x : Option (K × Pair) => x.toList) := by
refine (Primrec.option_casesOn Primrec.id (Primrec.const [])
(Primrec.list_cons.comp Primrec.snd (Primrec.const [])).to₂).of_eq ?_
intro x; cases x <;> rfl
exact ht.comp h
theorem primrec_product : Primrec₂ (product (K:=K)) := by
unfold product
apply Primrec.list_flatMap Primrec.fst
change Primrec (fun p : (Poly K × Poly K) × (K × Pair) => p.1.2.flatMap (mulTerm p.2))
apply Primrec.list_flatMap (Primrec.snd.comp Primrec.fst)
exact (primrec_mulTerm.comp (Primrec.snd.comp Primrec.fst) Primrec.snd).to₂
variable [DecidableEq K]
noncomputable section
open scoped Classical
theorem computable_value_eq : ComputablePred
(fun p : Poly K × Poly K => value p.1 = value p.2) := by
have hc : ComputablePred (fun p : Poly K × Poly K =>
zeroTest (p.1 ++ negate p.2) = true) :=
(Primrec.eq.comp (primrec_zeroTest.comp
(Primrec.list_append.comp Primrec.fst (primrec_negate.comp Primrec.snd)))
(Primrec.const true)).computablePred
apply hc.of_eq
intro p
rw [zeroTest_iff]
change value (p.1 ++ negate p.2) = 0 ↔ _
rw [value_append, value_negate, ← sub_eq_add_neg, sub_eq_zero]
def coefficientKey : Poly K → ℕ := SimpleFInftyOvergroups.EffectiveNames.key value
omit [Finite K] [DecidableEq K] in
theorem coefficientKey_eq_iff (p q : Poly K) :
coefficientKey p = coefficientKey q ↔ value p = value q :=
SimpleFInftyOvergroups.EffectiveNames.key_eq_iff value p q
theorem computable_coefficientKey : Computable (coefficientKey (K:=K)) :=
SimpleFInftyOvergroups.EffectiveNames.computable_key value computable_value_eq
end
end Lists
end BinaryPrefixOperators.Arithmetic
end OAI
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