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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 | import OAI.MathematicalPhysics.RapidForcing.RatComputable
import OAI.MathematicalPhysics.RapidForcing.ComputableAutomation
namespace OAI
section
open Encodable Denumerable
namespace RapidForcing.EffectiveArithmetic
lemma rat_divInt_graph (n d : ℤ) (q : ℚ) (hd : d ≠ 0) :
q.num * d = 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 = n * (q.den : ℤ) ↔ _)
@[fun_prop] lemma computable_rat_divInt : Computable (fun p : ℤ × ℤ => (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 = p.1 * (q.den : ℤ))
· have hi : PrimrecPred (fun p : (ℤ × ℤ) × ℚ =>
(p.1.2 = 0 ∧ p.2 = 0) ∨ (p.1.2 ≠ 0 ∧ p.2.num * p.1.2 = 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_divInt_graph _ _ _ hd
@[fun_prop] lemma computable_rat_inv : Computable (fun q : ℚ => q⁻¹) := by
have := computable_rat_divInt.comp (show Computable (fun q : ℚ => ((q.den : ℤ), q.num)) by fun_prop)
exact this.of_eq (fun q => by simpa [Int.cast_natCast, inv_div] using congrArg Inv.inv (Rat.num_div_den q))
@[fun_prop] lemma computable_rat_div : Computable (fun p : ℚ × ℚ => p.1 / p.2) := by
simpa [div_eq_mul_inv] using (show Computable (fun p : ℚ × ℚ => p.1 * p.2⁻¹) by fun_prop)
@[fun_prop] lemma computable_rat_abs : Computable (fun q : ℚ => |q|) := by
apply Computable.of_eq (show Computable (fun q : ℚ => if 0 ≤ q then q else -q) by fun_prop)
intro q
split_ifs with h
· exact (abs_of_nonneg h).symm
· exact (abs_of_neg (lt_of_not_ge h)).symm
@[fun_prop] lemma computable_rat_max : Computable (fun p : ℚ × ℚ => max p.1 p.2) := by
simpa only [max_def] using (show Computable (fun p : ℚ × ℚ => if p.1 ≤ p.2 then p.2 else p.1) by fun_prop)
@[fun_prop] lemma computable_rat_pow : Computable (fun p : ℚ × ℕ => p.1 ^ p.2) := by
have h := Computable.nat_rec (f := fun p : ℚ × ℕ => p.2)
(g := fun _ => (1 : ℚ)) (h := fun p r => r.2 * p.1)
(by fun_prop) (by fun_prop) (by unfold Computable₂; fun_prop)
apply h.of_eq
intro p
induction p.2 <;> simp [pow_succ, *]
@[fun_prop] lemma computable_rat_natCeil : Computable (fun q : ℚ => ⌈q⌉₊) := by
apply computable_of_compGraph _ (fun q n : _ =>
if n = (0 : ℕ) then q ≤ 0 else (n - 1 : ℚ) < q ∧ q ≤ (n : ℚ))
· have hi : CompPred (fun p : ℚ × ℕ =>
(p.2 = 0 ∧ p.1 ≤ 0) ∨ (p.2 ≠ 0 ∧ (p.2 - 1 : ℚ) < p.1 ∧ p.1 ≤ (p.2 : ℚ))) := by fun_prop
exact hi.of_eq (fun p => by split_ifs <;> simp_all)
· intro q n
by_cases hn : n = 0
· simp [hn, eq_comm, Nat.ceil_eq_zero]
· simp only [ite_eq_right hn]
rw [eq_comm, Nat.ceil_eq_iff hn]
simp only [Nat.cast_sub (Nat.one_le_iff_ne_zero.mpr hn), Nat.cast_one]
@[fun_prop] lemma computable_rat_natFloor : Computable (fun q : ℚ => ⌊q⌋₊) := by
apply computable_of_compGraph _ (fun q n : _ =>
if n = (0 : ℕ) then q < 1 else (n : ℚ) ≤ q ∧ q < (n : ℚ) + 1)
· have hi : CompPred (fun p : ℚ × ℕ =>
(p.2 = 0 ∧ p.1 < 1) ∨ (p.2 ≠ 0 ∧ (p.2 : ℚ) ≤ p.1 ∧ p.1 < (p.2 : ℚ) + 1)) := by fun_prop
exact hi.of_eq (fun p => by split_ifs <;> simp_all)
· intro q n
by_cases hn : n = 0
· simp [hn, eq_comm, Nat.floor_eq_zero]
· simp only [ite_eq_right hn]
rw [eq_comm, Nat.floor_eq_iff' hn]
end RapidForcing.EffectiveArithmetic
end
end OAI
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