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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.MathematicalPhysics.RapidForcing.RatComputable
import OAI.MathematicalPhysics.RapidForcing.FormulaCoding
import OAI.MathematicalPhysics.RapidForcing.RatOperations
namespace OAI
section
open Encodable
open scoped BigOperators
namespace RapidForcing.EffectiveProfile
open EffectiveArithmetic
private def ballEquiv : Ball ≃ ℚ × ℚ :=
⟨fun b => (b.center, b.radius), fun p => ⟨p.1, p.2⟩, fun _ => rfl, fun _ => rfl⟩
instance : Primcodable Ball := Primcodable.ofEquiv (ℚ × ℚ) ballEquiv
@[fun_prop] lemma primrec_ball_center : Primrec Ball.center := by
exact Primrec.fst.comp (Primrec.of_equiv (e := ballEquiv))
@[fun_prop] lemma primrec_ball_radius : Primrec Ball.radius := by
exact Primrec.snd.comp (Primrec.of_equiv (e := ballEquiv))
@[fun_prop] lemma primrec_ball_mk : Primrec (fun p : ℚ × ℚ => Ball.mk p.1 p.2) := by
apply Primrec.encode_iff.mp
exact (Primrec.encode : Primrec (@encode (ℚ × ℚ) _))
@[fun_prop] lemma computable_ball_exact : Computable Ball.exact := by
unfold Ball.exact; fun_prop
@[fun_prop] lemma computable_ball_add : Computable (fun p : Ball × Ball => p.1.add p.2) := by
unfold Ball.add; fun_prop
@[fun_prop] lemma computable_ball_mul : Computable (fun p : Ball × Ball => p.1.mul p.2) := by
unfold Ball.mul; fun_prop
@[fun_prop] lemma computable_ball_denInv : Computable Ball.denInv := by
unfold Ball.denInv; fun_prop
@[fun_prop] lemma computable_ball_pow : Computable (fun p : Ball × ℕ => p.1.pow p.2) := by
have h := Computable.nat_rec (f := fun p : Ball × ℕ => p.2)
(g := fun _ => Ball.exact 1) (h := fun p r => r.2.mul p.1)
(by fun_prop) (by fun_prop) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p.2 <;> simp [Ball.pow, *])
@[fun_prop] lemma computable_glueDiff : Computable glueDiff := by
have h := computable_list_foldr (f := id)
(h := fun (_ : GluePoly) (a : (ℚ × ℕ) × GluePoly) =>
(a.1.1, a.1.2 + 2) :: (-(a.1.2 : ℚ) * a.1.1, a.1.2 + 1) :: a.2)
(g := fun _ => ([] : GluePoly))
(by fun_prop) (by fun_prop) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p <;> simp_all [glueDiff])
@[fun_prop] lemma computable_glueBound : Computable glueBound := by
have h := computable_list_foldr (f := id)
(h := fun (_ : GluePoly) (a : (ℚ × ℕ) × ℚ) => |a.1.1| * (a.1.2.factorial : ℚ) + a.2)
(g := fun _ => (0 : ℚ))
(by fun_prop) (by fun_prop) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p <;> simp_all [glueBound])
lemma computable_sum_range {A S : Type} [Primcodable A] [Primcodable S] [AddCommMonoid S]
(hadd : Computable (fun p : S × S => p.1 + p.2))
{n : A → ℕ} (hn : Computable n) {f : A → ℕ → S} (hf : Computable₂ f) :
Computable (fun a => ∑ j ∈ Finset.range (n a), f a j) := by
have h := Computable.nat_rec hn (Computable.const (0 : S))
(h := fun a r => r.2 + f a r.1)
(hadd.comp ((Computable.snd.comp Computable.snd).pair (hf.comp Computable.fst (Computable.fst.comp Computable.snd)))).to₂
apply h.of_eq
intro a
induction n a <;> simp [Finset.sum_range_succ, *]
@[fun_prop] lemma computable_expTerms : Computable (fun p : ℚ × ℕ => expTerms p.1 p.2) := by
unfold expTerms; fun_prop
private lemma computable_taylor_term : Computable (fun p : (ℚ × ℕ) × ℕ =>
p.1.1 ^ p.2 / (p.2.factorial : ℚ)) := by fun_prop
private lemma computable_taylor_sum : Computable (fun p : ℚ × ℕ =>
∑ j ∈ Finset.range (expTerms p.1 p.2), p.1 ^ j / (j.factorial : ℚ)) :=
computable_sum_range computable_rat_add computable_expTerms computable_taylor_term.to₂
private lemma computable_taylor_error_fn : Computable (fun p : ℚ × ℕ =>
|p.1| ^ p.2 / (p.2.factorial : ℚ) * 2) := by fun_prop
lemma computable_expBall : Computable (fun p : ℚ × ℕ => expBall p.1 p.2) := by
have herr : Computable (fun p : ℚ × ℕ =>
(p.1, expTerms p.1 p.2)) := Computable.fst.pair computable_expTerms
have h := primrec_ball_mk.to_comp.comp
(computable_taylor_sum.pair (computable_taylor_error_fn.comp herr))
exact h.of_eq (fun p => rfl)
@[fun_prop] lemma computable_monoBall : Computable (fun p : ℕ × ℚ × ℕ => monoBall p.1 p.2.1 p.2.2) := by
have harg : Computable (fun p : ℕ × ℚ × ℕ => (-(p.2.1)⁻¹, p.2.2)) := by fun_prop
have he := computable_expBall.comp harg
have hx : Computable (fun p : ℕ × ℚ × ℕ => Ball.exact (p.2.1⁻¹ ^ p.1)) := by fun_prop
have hm := computable_ball_mul.comp (hx.pair he)
have hi := computable_ite (p := fun p : ℕ × ℚ × ℕ => p.2.1 ≤ 0)
(by fun_prop) (Computable.const (Ball.exact 0)) hm
exact hi.of_eq (fun p => rfl)
@[fun_prop] lemma computable_glueBall : Computable (fun p : GluePoly × ℚ × ℕ => glueBall p.1 p.2.1 p.2.2) := by
have h := computable_list_foldr (f := fun p : GluePoly × ℚ × ℕ => p.1)
(h := fun p (a : (ℚ × ℕ) × Ball) => ((Ball.exact a.1.1).mul (monoBall a.1.2 p.2.1 p.2.2)).add a.2)
(g := fun _ => Ball.exact 0)
(by fun_prop) (by fun_prop) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p.1 <;> simp [glueBall, *])
@[fun_prop] lemma Expr.computable_bound : Computable Expr.bound := by
have h := Expr.computable_recOn (e := id) Computable.id
(c := fun _ q => |q|) (by unfold Computable₂; fun_prop)
(l := fun _ p => glueBound p) (by unfold Computable₂; fun_prop)
(r := fun _ p => glueBound p) (by unfold Computable₂; fun_prop)
(d := fun _ n => (16 : ℚ) ^ n) (by unfold Computable₂; fun_prop)
(a := fun _ p => p.2.2.1 + p.2.2.2) (by unfold Computable₂; fun_prop)
(m := fun _ p => p.2.2.1 * p.2.2.2) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun e => by induction e <;> simp_all only [id_eq, bound])
@[fun_prop] lemma Expr.computable_diff : Computable Expr.diff := by
have h := Expr.computable_recOn (e := id) Computable.id
(c := fun _ _ => Expr.const 0) (by unfold Computable₂; fun_prop)
(l := fun _ p => Expr.left (glueDiff p)) (by unfold Computable₂; fun_prop)
(r := fun _ p => Expr.mul (.const (-1)) (.right (glueDiff p))) (by unfold Computable₂; fun_prop)
(d := fun _ n => Expr.mul (.mul (.const (-(n : ℚ)))
(.add (.left (glueDiff [(1, 0)])) (.mul (.const (-1)) (.right (glueDiff [(1, 0)])))))
(.invDen (n + 1))) (by unfold Computable₂; fun_prop)
(a := fun _ p => Expr.add p.2.2.1 p.2.2.2) (by unfold Computable₂; fun_prop)
(m := fun _ p => Expr.add (.mul p.2.2.1 p.2.1) (.mul p.1 p.2.2.2)) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun e => by induction e <;> simp_all only [id_eq, diff])
@[fun_prop] lemma Expr.computable_numeric : Computable (fun p : Expr × ℚ × ℕ => p.1.numeric p.2.1 p.2.2) := by
have h := Expr.computable_recOn (e := fun p : Expr × ℚ × ℕ => p.1) (by fun_prop)
(c := fun _ q => Ball.exact q) (by unfold Computable₂; fun_prop)
(l := fun a p => glueBall p a.2.1 a.2.2) (by unfold Computable₂; fun_prop)
(r := fun a p => glueBall p (1 - a.2.1) a.2.2) (by unfold Computable₂; fun_prop)
(d := fun a n => ((glueBall [(1, 0)] a.2.1 a.2.2).add
(glueBall [(1, 0)] (1 - a.2.1) a.2.2)).denInv.pow n) (by unfold Computable₂; fun_prop)
(a := fun _ p => p.2.2.1.add p.2.2.2) (by unfold Computable₂; fun_prop)
(m := fun _ p => p.2.2.1.mul p.2.2.2) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p.1 <;> simp [numeric, *])
lemma Expr.partrec_approx : Partrec (fun p : Expr × ℚ × ℚ => p.1.approx p.2.1 p.2.2) := by
apply Partrec.rfindOpt
unfold Computable₂
fun_prop
end RapidForcing.EffectiveProfile
end
end OAI
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