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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 | import OAI.MathematicalPhysics.RapidForcing.SyntaxComputable
import OAI.MathematicalPhysics.RapidForcing.JetControl
namespace OAI
open Encodable
open scoped BigOperators
namespace RapidForcing.EffectiveArithmetic
lemma computable_list_sum {S : Type} [Primcodable S] [AddMonoid S]
(hadd : Computable (fun p : S × S => p.1 + p.2)) :
Computable (List.sum : List S → S) := by
have h := computable_list_foldr Computable.id (Computable.const (0 : S))
(hadd.comp Computable.snd).to₂
exact h.of_eq (fun _ => rfl)
lemma computable_list_flatMap {A B C : Type} [Primcodable A] [Primcodable B] [Primcodable C]
{f : A → List B} {g : A → B → List C} (hf : Computable f) (hg : Computable₂ g) :
Computable (fun a => (f a).flatMap (g a)) := by
have h := computable_list_foldr hf (Computable.const ([] : List C))
(h := fun a p => g a p.1 ++ p.2)
((Primrec.list_append.to_comp.comp (hg.comp Computable.fst (Computable.fst.comp Computable.snd))
(Computable.snd.comp Computable.snd)).to₂)
exact h.of_eq (fun a => by induction f a <;> simp [*])
end RapidForcing.EffectiveArithmetic
namespace RapidForcing
open EffectiveArithmetic EffectiveProfile
namespace EffectiveProfile.Formula
variable {d : ℕ}
attribute [local irreducible] diff Expr.diff bound Expr.bound
@[fun_prop] lemma computable_allDiff : Computable (@allDiff d) := by
unfold allDiff
apply computable_list_flatMap Computable.id
unfold Computable₂
apply computable_list_map (Computable.const (List.finRange d))
exact (computable_diff.comp
((Computable.snd.comp Computable.fst).pair Computable.snd)).to₂
@[fun_prop] lemma computable_jetBound : Computable (fun p : Formula d × ℚ × ℕ =>
p.1.jetBound p.2.1 p.2.2) := by
unfold jetBound
apply (computable_list_sum computable_rat_add).comp
apply computable_list_map
· apply computable_nat_iterate (by fun_prop)
(Computable.list_cons.comp Computable.fst (Computable.const []))
exact (computable_allDiff.comp Computable.snd).to₂
· exact (computable_bound.comp (Computable.snd.pair
((Computable.fst.comp Computable.snd).comp Computable.fst))).to₂
attribute [local irreducible] jetBound scaledFormula
@[fun_prop] lemma computable_curlJetBound : Computable curlJetBound := by
unfold curlJetBound
apply computable_fin_sum computable_rat_add
intro i
exact computable_jetBound.comp ((Computable.const (scaledFormula i)).pair
((Computable.const 3).pair Computable.id))
end EffectiveProfile.Formula
attribute [local irreducible] ScaleData.δQ ScaleData.bQ ScaleData.s ScaleData.L
ScaleData.C EffectiveProfile.Formula.curlJetBound
lemma ScaleData.computable_coefficientTerm : Computable
(fun p : (ScaleData × ℕ × ℕ) × ℕ =>
((p.2:ℚ)+4)^p.1.2.1 * (p.1.1.bQ p.2 / p.1.1.δQ p.2 ^ p.1.2.2)) := by
have hb := ScaleData.computable_bQ.comp (show Computable
(fun p : (ScaleData × ℕ × ℕ) × ℕ => (p.1.1, p.2)) by fun_prop)
have hd := ScaleData.computable_deltaQ.comp (show Computable
(fun p : (ScaleData × ℕ × ℕ) × ℕ => (p.1.1, p.2)) by fun_prop)
have he := computable_rat_pow.comp (hd.pair (show Computable
(fun p : (ScaleData × ℕ × ℕ) × ℕ => p.1.2.2) by fun_prop))
have hw : Computable (fun p : (ScaleData × ℕ × ℕ) × ℕ => ((p.2:ℚ)+4)^p.1.2.1) := by fun_prop
exact (computable_rat_mul.comp (hw.pair (computable_rat_div.comp (hb.pair he)))).of_eq (fun _ => rfl)
lemma ScaleData.computable_coefficientSum : Computable
(fun p : ScaleData × ℕ × ℕ =>
∑ n ∈ Finset.range (max p.2.1 p.2.2),
((n:ℚ)+4)^p.2.1 * (p.1.bQ n / p.1.δQ n ^ p.2.2)) := by
exact computable_sum_range computable_rat_add
(primrec_nat_max.to_comp.comp (show Computable
(fun p : ScaleData × ℕ × ℕ => (p.2.1, p.2.2)) by fun_prop))
ScaleData.computable_coefficientTerm.to₂
@[fun_prop] lemma ScaleData.computable_coefficientBound : Computable
(fun p : ScaleData × ℕ × ℕ => p.1.coefficientBound p.2.1 p.2.2) := by
unfold ScaleData.coefficientBound
exact (computable_rat_add.comp (ScaleData.computable_coefficientSum.pair (Computable.const 2))).of_eq (fun _ => rfl)
attribute [local irreducible] ScaleData.coefficientBound
@[fun_prop] lemma computable_velocityJetBound : Computable
(fun p : ScaleData × ℕ × ℕ => velocityJetBound p.1 p.2.1 p.2.2) := by
unfold velocityJetBound
fun_prop
attribute [local irreducible] velocityJetBound
@[fun_prop] lemma computable_forceJetBound : Computable
(fun p : ScaleData × ℚ × ℕ × ℕ => forceJetBound p.1 p.2.1 p.2.2.1 p.2.2.2) := by
have h : Computable (fun p : ScaleData × ℚ × ℕ × ℕ =>
∑ i ∈ Finset.range (p.2.2.2+1), velocityJetBound p.1 p.2.2.1 i *
velocityJetBound p.1 0 (p.2.2.2-i+1)) := by
apply computable_sum_range computable_rat_add (by fun_prop)
unfold Computable₂
fun_prop
unfold forceJetBound
fun_prop (disch := assumption)
end RapidForcing
end OAI
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