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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 | import Mathlib
namespace OAI
namespace SimpleFInftyOvergroups.EffectiveNames.Finite
variable {A B P : Type*} {n : ℕ}
def traverse (f : A → Part B) (x : Fin n → A) : Part (Fin n → B) :=
(List.Vector.mOfFn (fun i => f (x i))).map List.Vector.get
theorem vector_full (f : Fin n → Part B) (g : Fin n → B) (h : ∀ i, g i ∈ f i) :
List.Vector.ofFn g ∈ List.Vector.mOfFn f := by
induction n with
| zero => exact Part.mem_some _
| succ n ih =>
simp only [List.Vector.mOfFn]
refine Part.mem_bind_iff.mpr ⟨g 0, h 0, ?_⟩
exact Part.mem_bind_iff.mpr ⟨_,ih (fun i => f i.succ) (fun i => g i.succ)
(fun i => h i.succ), Part.mem_some _⟩
theorem traverse_full (f : A → Part B) (g : A → B) (x : Fin n → A)
(h : ∀ i, g (x i) ∈ f (x i)) : (fun i => g (x i)) ∈ traverse f x := by
change _ ∈ (List.Vector.mOfFn (fun i => f (x i))).map List.Vector.get
convert Part.mem_map List.Vector.get (vector_full (fun i => f (x i)) (fun i => g (x i)) h) using 1
funext i
exact (List.Vector.get_ofFn (fun i => g (x i)) i).symm
variable [Primcodable A] [Primcodable B] [Primcodable P]
theorem partrec_traverse {f : P → A → Part B} (hf : Partrec₂ f) :
Partrec₂ (fun p : P => traverse (n:=n) (f p)) := by
have hh : ∀ i : Fin n, Partrec (fun p : P × (Fin n → A) => f p.1 (p.2 i)) :=
fun i => hf.comp Computable.fst (Primrec.fin_app.to_comp.comp Computable.snd (Computable.const i))
exact (Partrec.vector_mOfFn hh).map (Primrec.vector_get'.to_comp.comp Computable.snd).to₂
theorem computable_function {f : P → Fin n → B} (hf : ∀ i, Computable (fun p => f p i)) :
Computable f := by
exact (Primrec.vector_get'.to_comp.comp (Computable.vector_ofFn hf)).of_eq
(fun p => by funext i; simp)
theorem primrec_function {f : P → Fin n → B} (hf : ∀ i, Primrec (fun p => f p i)) :
Primrec f := by
exact (Primrec.vector_get'.comp (Primrec.vector_ofFn hf)).of_eq
(fun p => by funext i; simp)
end SimpleFInftyOvergroups.EffectiveNames.Finite
end OAI
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