File size: 2,836 Bytes
caec230 | 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 | import OAI.MathematicalPhysics.RapidForcing.Model
namespace OAI
section
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
end
end OAI
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