| import OAI.MathematicalPhysics.AlternatingFlow.Machines |
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|
| namespace OAI |
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| section EffectiveDevelopment |
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| 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 |
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|
| 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) |
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|
| 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) |
|
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| lemma alphabet : Primrec Machine.alphabet := |
| Primrec.nat_add.comp (Primrec.fst.comp Primrec.snd) (Primrec.const 1) |
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| lemma initialState : Primrec Machine.initialState := |
| Primrec.fst.comp (Primrec.snd.comp Primrec.snd) |
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| lemma haltingStates : Primrec Machine.haltingStates := |
| Primrec.fst.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd)) |
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| lemma table : Primrec Machine.table := |
| Primrec.snd.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.snd)) |
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| 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) |
|
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| lemma member {α : Type*} [Primcodable α] : PrimrecRel (fun l : List α => fun a => a ∈ l) := |
| (Primrec.eq.exists_mem_list).of_eq (by simp) |
|
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| 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) |
|
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| 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)⟩ |
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|
| end Effective |
| end AlternatingNS |
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| end EffectiveDevelopment |
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| end OAI |
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|
