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f18e7de | 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 | import OAI.MathematicalPhysics.AlternatingFlow.Machines
namespace OAI
section EffectiveDevelopment
open scoped BigOperators ENNReal Topology ContDiff
open MeasureTheory
namespace AlternatingNS
namespace Effective
lemma nat_pow : Primrec₂ (fun b n : ℕ => b ^ n) :=
Primrec₂.unpaired'.mp Nat.Primrec.pow
lemma capacity : Primrec₂ Scales.K :=
Primrec.nat_add.comp (Primrec.nat_add.comp Primrec.fst
(Primrec.nat_mul.comp (Primrec.const 2) Primrec.snd)) (Primrec.const 2)
lemma exponent : Primrec₂ Scales.s :=
Primrec.nat_mul.comp (Primrec.nat_add.comp Primrec.fst (Primrec.const 1))
(nat_pow.comp (Primrec.const 2) (nat_pow.comp
(Primrec.nat_add.comp Primrec.snd (Primrec.const 3)) (Primrec.const 2)))
lemma states : Primrec Machine.states :=
Primrec.nat_add.comp Primrec.fst (Primrec.const 1)
lemma alphabet : Primrec Machine.alphabet :=
Primrec.nat_add.comp (Primrec.fst.comp Primrec.snd) (Primrec.const 1)
lemma initialState : Primrec Machine.initialState :=
Primrec.fst.comp (Primrec.snd.comp Primrec.snd)
lemma haltingStates : Primrec Machine.haltingStates :=
Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))
lemma table : Primrec Machine.table :=
Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd))
lemma base : Primrec Machine.base :=
Primrec.nat_mul.comp (Primrec.const 2)
(Primrec.nat_add.comp (Primrec.const 1)
(Primrec.nat_max.comp (Primrec.nat_add.comp states (Primrec.const 1)) alphabet))
lemma instruction : Primrec (fun z : Machine × ℕ × ℕ => z.1.instruction z.2.1 z.2.2) := by
have h := Primrec.list_getElem?.comp (table.comp Primrec.fst)
(Primrec.nat_add.comp (Primrec.nat_mul.comp (Primrec.fst.comp Primrec.snd)
(alphabet.comp Primrec.fst)) (Primrec.snd.comp Primrec.snd))
exact (Primrec.option_getD.comp h (Primrec.const none)).of_eq (by
intro z; dsimp only [Machine.instruction];
cases (z.1.table[z.2.1 * z.1.alphabet + z.2.2]?) <;> rfl)
lemma member {α : Type*} [Primcodable α] : PrimrecRel (fun l : List α => fun a => a ∈ l) :=
(Primrec.eq.exists_mem_list).of_eq (by simp)
lemma command : Primrec (fun z : Machine × ℕ × ℕ => z.1.command z.2.1 z.2.2) := by
apply Primrec.ite
· exact (Primrec.eq.comp (Primrec.fst.comp Primrec.snd) (states.comp Primrec.fst)).or
(member.comp (haltingStates.comp Primrec.fst) (Primrec.fst.comp Primrec.snd))
· exact (Primrec.fst.comp Primrec.snd).pair
((Primrec.snd.comp Primrec.snd).pair (Primrec.const (1 : Fin 3)))
· exact Primrec.option_getD.comp instruction ((states.comp Primrec.fst).pair
((Primrec.snd.comp Primrec.snd).pair (Primrec.const (1 : Fin 3))))
lemma program_of_computable {A B : Type*} [Primcodable A] [Primcodable B]
(f : A → B) (hf : Computable f) : ∃ c : Program, ∀ a, Returns c a (f a) := by
obtain ⟨c, hc⟩ := Nat.Partrec.Code.exists_code.mp hf
refine ⟨c, fun a => ?_⟩
simp [Returns, hc, Encodable.encodek]
lemma compiler_of_computable {A B C : Type*} [Primcodable A] [Primcodable B] [Primcodable C]
(f : A → B → C) (hf : Computable₂ f) :
∃ compile : A → Program, Computable compile ∧
∀ a b, Returns (compile a) b (f a b) := by
obtain ⟨c, hc⟩ := program_of_computable (fun z : A × B => f z.1 z.2) hf
refine ⟨fun a => c.curry (Encodable.encode a),
(Nat.Partrec.Code.primrec₂_curry.comp (Primrec.const c) Primrec.encode).to_comp, ?_⟩
intro a b
simpa only [Returns, Nat.Partrec.Code.eval_curry, Encodable.encode_prod_val] using hc (a, b)
theorem command_program : ∃ compile : Machine → Program, Computable compile ∧
∀ M q a, Returns (compile M) (q, a) (M.command q a) := by
obtain ⟨c, hC, h⟩ := compiler_of_computable
(fun M : Machine => fun z : ℕ × ℕ => M.command z.1 z.2) command.to_comp
exact ⟨c, hC, fun M q a => h M (q, a)⟩
end Effective
end AlternatingNS
end EffectiveDevelopment
end OAI
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