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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 | import Mathlib
import OAI.Computability.FiniteCompiler.Effectivity
import OAI.Computability.FiniteCompiler.Selectors
import OAI.GroupTheory.BooneHigman.Basic
import OAI.GroupTheory.SimpleOvergroups.ArrayNames
import OAI.GroupTheory.SimpleOvergroups.CanonicalNames
namespace OAI
namespace SimpleFInftyOvergroups.EffectiveNames.Array
open FiniteAlgebraicEnvelopes
noncomputable section
open scoped Classical
variable {A X J D : Type*} [AddCommMonoid D] [Primcodable A] [Primcodable X] [Primcodable J]
(coeff : A → D)
def collect (f : X → Part J) (p : Name A X) : Part (Name A J) :=
Effect.traverse (fun a => (f a.2).map (fun k => (a.1,k))) p
def collectedKey (f : X → Part J) (p : Name A X) : Part ℕ :=
(collect f p).map (key (value coeff id))
omit [Primcodable A] [Primcodable X] [Primcodable J] in
theorem collect_full (f : X → Part J) (g : X → J) (p : Name A X)
(h : ∀ a ∈ p, g a.2 ∈ f a.2) :
p.map (fun a => (a.1,g a.2)) ∈ collect f p := by
apply FiniteAlgebraicEnvelopes.Bootstrap.Search.traverse_map
intro a ha
exact Part.mem_map _ (h a ha)
omit [Primcodable X] in
theorem collectedKey_full (hc : Function.Surjective coeff) (f : X → Part J)
{K : Type*} (index : X → K) (g : K → J) (p : Name A X)
(h : ∀ a ∈ p, g (index a.2) ∈ f a.2) :
semanticKey (value coeff id) (value_surjective coeff id hc Function.surjective_id)
(Finsupp.mapDomain g (value coeff index p)) ∈ collectedKey coeff f p := by
rw [← value_map_index coeff index (g ∘ index) id g (fun _ => rfl) p,
semanticKey_value]
exact Part.mem_map _ (collect_full f (g ∘ index) p h)
variable {P : Type*} [Primcodable P] {f : P → X → Part J}
theorem partrec_collect (hf : Partrec₂ f) :
Partrec₂ (fun p : P => (collect (f p) : Name A X → Part (Name A J))) := by
have hc : Partrec₂ (fun (p : P) (a : A × X) =>
(f p a.2).map (fun k => (a.1,k))) := by
exact ((hf.comp Computable.fst (Computable.snd.comp Computable.snd)).map
((Computable.fst.comp (Computable.snd.comp Computable.fst)).pair Computable.snd).to₂).to₂
exact Effect.partrec_traverse_parameter hc
theorem partrec_collectedKey (heq : ComputablePred
(fun p : Name A J × Name A J => value coeff id p.1 = value coeff id p.2))
(hf : Partrec₂ f) : Partrec₂ (fun p : P => collectedKey coeff (f p)) := by
exact (partrec_collect hf).map (((computable_key (value coeff id) heq).comp Computable.snd).to₂)
end
end SimpleFInftyOvergroups.EffectiveNames.Array
end OAI
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