| import OAI.MathematicalPhysics.RapidForcing.Model |
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|
| namespace OAI |
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| section |
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|
| open Encodable Denumerable |
| namespace RapidForcing |
| namespace EffectiveArithmetic |
|
|
| lemma encode_int_ofNat (n : ℕ) : Encodable.encode (Int.ofNat n) = 2 * n := by |
| simp [encode, Equiv.intEquivNat, Equiv.intEquivNatSumNat, |
| Equiv.natSumNatEquivNat] |
| change Nat.bit false n = _ |
| simp [Nat.bit_val] |
|
|
| lemma encode_int_negSucc (n : ℕ) : Encodable.encode (Int.negSucc n) = 2 * n + 1 := by |
| simp [encode, Equiv.intEquivNat, Equiv.intEquivNatSumNat, |
| Equiv.natSumNatEquivNat] |
|
|
| lemma ofNat_int (n : ℕ) : Denumerable.ofNat ℤ n = |
| if n % 2 = 0 then Int.ofNat (n / 2) else Int.negSucc (n / 2) := by |
| apply Encodable.encode_injective |
| rw [encode_ofNat] |
| split_ifs with hn |
| · rw [encode_int_ofNat] |
| omega |
| · rw [encode_int_negSucc] |
| omega |
|
|
| lemma primrec_int_ofNat : Primrec Int.ofNat := |
| Primrec.encode_iff.mp ((Primrec.nat_mul.comp (Primrec.const 2) Primrec.id).of_eq |
| (fun n => (encode_int_ofNat n).symm)) |
|
|
| lemma primrec_int_negSucc : Primrec Int.negSucc := |
| Primrec.encode_iff.mp ((Primrec.nat_add.comp |
| (Primrec.nat_mul.comp (Primrec.const 2) Primrec.id) (Primrec.const 1)).of_eq |
| (fun n => (encode_int_negSucc n).symm)) |
|
|
| lemma primrec_int_cases {A B : Type} [Primcodable A] [Primcodable B] |
| {f : A → ℤ} (hf : Primrec f) {p q : A → ℕ → B} |
| (hp : Primrec₂ p) (hq : Primrec₂ q) : |
| Primrec (fun a => Int.rec (motive := fun _ => B) (p a) (q a) (f a)) := by |
| have he := Primrec.encode.comp hf |
| have hd := Primrec.nat_div.comp he (Primrec.const 2) |
| have hb := Primrec.nat_mod.comp he (Primrec.const 2) |
| have h := Primrec.ite (Primrec.eq.comp hb (Primrec.const 0)) |
| (hp.comp Primrec.id hd) (hq.comp Primrec.id hd) |
| apply h.of_eq |
| intro a |
| have hh := ofNat_int (Encodable.encode (f a)) |
| rw [ofNat_encode] at hh |
| conv_rhs => rw [hh] |
| split_ifs <;> rfl |
|
|
| lemma primrec_int_natAbs : Primrec Int.natAbs := by |
| have h := primrec_int_cases Primrec.id Primrec₂.right (Primrec.succ.comp Primrec.snd).to₂ |
| exact h.of_eq (fun z => by cases z <;> rfl) |
|
|
| lemma primrec_int_toNat : Primrec Int.toNat := by |
| have h := primrec_int_cases Primrec.id Primrec₂.right (Primrec₂.const 0) |
| exact h.of_eq (fun z => by cases z <;> rfl) |
|
|
| lemma primrec_int_neg : Primrec (fun z : ℤ => -z) := by |
| have hn : Primrec (fun n : ℕ => -(n : ℤ)) := |
| (Primrec.ite (Primrec.eq.comp Primrec.id (Primrec.const 0)) (Primrec.const 0) |
| (primrec_int_negSucc.comp (Primrec.nat_sub.comp Primrec.id (Primrec.const 1)))).of_eq |
| (fun n => by cases n <;> simp [Int.negSucc_eq]) |
| have h := primrec_int_cases Primrec.id (hn.comp Primrec.snd).to₂ |
| (primrec_int_ofNat.comp (Primrec.succ.comp Primrec.snd)).to₂ |
| exact h.of_eq (fun z => by cases z <;> rfl) |
|
|
| end EffectiveArithmetic |
| end RapidForcing |
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| end |
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| end OAI |
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|
