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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.MathematicalPhysics.RapidForcing.RatPrimrec
namespace OAI
section
open Encodable Denumerable
namespace RapidForcing.EffectiveArithmetic
lemma computable_of_graph {A B : Type} [Primcodable A] [Primcodable B]
(f : A → B) (R : A → B → Prop) [DecidableRel R]
(hR : PrimrecPred (fun p : A × B => R p.1 p.2))
(h : ∀ a b, R a b ↔ b = f a) : Computable f := by
let test : A → ℕ → Option B := fun a n =>
(decode n).bind fun b => if R a b then some b else none
have ht : Primrec₂ test :=
Primrec.option_bind (Primrec.decode.comp Primrec.snd)
(Primrec.ite (hR.comp ((Primrec.fst.comp Primrec.fst).pair Primrec.snd))
(Primrec.option_some.comp Primrec.snd) (Primrec.const none)).to₂
apply (Partrec.rfindOpt ht.to_comp).of_eq_tot
intro a
have hd : (Nat.rfindOpt (test a)).Dom := Nat.rfindOpt_dom.mpr
⟨encode (f a), f a, by simp [test, Encodable.encodek, (h a (f a)).mpr rfl]⟩
have hm : (Nat.rfindOpt (test a)).get hd ∈ Nat.rfindOpt (test a) := ⟨hd, rfl⟩
obtain ⟨n, hn⟩ := Nat.rfindOpt_spec hm
have he : (Nat.rfindOpt (test a)).get hd = f a := by
simp only [test, Option.mem_bind_iff] at hn
obtain ⟨b, _, hb⟩ := hn
split_ifs at hb with hab
· obtain rfl := Option.mem_some_iff.mp hb
exact (h a _).mp hab
· contradiction
exact he ▸ hm
@[fun_prop] lemma primrec_rat_le : PrimrecPred (fun p : ℚ × ℚ => p.1 ≤ p.2) := by
exact (show PrimrecPred (fun p : ℚ × ℚ => p.1.num * (p.2.den : ℤ) ≤
p.2.num * (p.1.den : ℤ)) by fun_prop).of_eq (fun p => (Rat.le_iff _ _).symm)
@[fun_prop] lemma primrec_rat_lt : PrimrecPred (fun p : ℚ × ℚ => p.1 < p.2) := by
exact (PrimrecPred.not (primrec_rat_le.comp (Primrec.snd.pair Primrec.fst))).of_eq
(fun p => not_le)
lemma rat_div_graph (n : ℤ) (d : ℕ) (q : ℚ) (hd : d ≠ 0) :
q.num * (d : ℤ) = n * (q.den : ℤ) ↔ q = (n : ℚ) / d := by
have hq : (q.den : ℚ) ≠ 0 := by exact_mod_cast q.den_nz
have hd' : (d : ℚ) ≠ 0 := by exact_mod_cast hd
conv_rhs => rw [← Rat.num_div_den q, div_eq_div_iff hq hd']
exact_mod_cast (Iff.rfl : q.num * (d : ℤ) = n * (q.den : ℤ) ↔ _)
@[fun_prop] lemma computable_rat_divNat : Computable (fun p : ℤ × ℕ => (p.1 : ℚ) / p.2) := by
apply computable_of_graph _ (fun p q =>
if p.2 = 0 then q = 0 else q.num * (p.2 : ℤ) = p.1 * (q.den : ℤ))
· have hi : PrimrecPred (fun p : (ℤ × ℕ) × ℚ =>
(p.1.2 = 0 ∧ p.2 = 0) ∨ (p.1.2 ≠ 0 ∧ p.2.num * (p.1.2 : ℤ) = p.1.1 * (p.2.den : ℤ))) := by fun_prop
exact hi.of_eq (fun p => by split_ifs <;> simp_all)
· intro p q
split_ifs with hd
· simp [hd]
· exact rat_div_graph _ _ _ hd
@[fun_prop] lemma computable_rat_castInt : Computable (fun z : ℤ => (z : ℚ)) := by
simpa using computable_rat_divNat.comp (Computable.id.pair (Computable.const 1))
@[fun_prop] lemma computable_rat_castNat : Computable (fun n : ℕ => (n : ℚ)) := by
simpa only [Int.cast_natCast] using computable_rat_castInt.comp primrec_int_cast.to_comp
@[fun_prop] lemma computable_rat_add : Computable (fun p : ℚ × ℚ => p.1 + p.2) := by
have hh : Primrec (fun p : ℚ × ℚ =>
(p.1.num * (p.2.den : ℤ) + p.2.num * (p.1.den : ℤ), p.1.den * p.2.den)) := by fun_prop
apply (computable_rat_divNat.comp hh.to_comp).of_eq
intro p
dsimp; push_cast
conv_rhs => rw [← Rat.num_div_den p.1, ← Rat.num_div_den p.2]
have h1 : (p.1.den : ℚ) ≠ 0 := by exact_mod_cast p.1.den_nz
have h2 : (p.2.den : ℚ) ≠ 0 := by exact_mod_cast p.2.den_nz
field_simp
@[fun_prop] lemma computable_rat_mul : Computable (fun p : ℚ × ℚ => p.1 * p.2) := by
have hh : Primrec (fun p : ℚ × ℚ => (p.1.num * p.2.num, p.1.den * p.2.den)) := by fun_prop
apply (computable_rat_divNat.comp hh.to_comp).of_eq
intro p
dsimp; push_cast
rw [mul_div_mul_comm, Rat.num_div_den, Rat.num_div_den]
@[fun_prop] lemma computable_rat_neg : Computable (fun q : ℚ => -q) := by
have hh : Primrec (fun q : ℚ => (-q.num, q.den)) := by fun_prop
apply (computable_rat_divNat.comp hh.to_comp).of_eq
intro q
simp only [Int.cast_neg, neg_div, Rat.num_div_den]
@[fun_prop] lemma computable_rat_sub : Computable (fun p : ℚ × ℚ => p.1 - p.2) := by
simpa only [sub_eq_add_neg] using (show Computable (fun p : ℚ × ℚ => p.1 + -p.2) by fun_prop)
end RapidForcing.EffectiveArithmetic
end
end OAI
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