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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 | import OAI.MathematicalPhysics.RapidForcing.NamedForce
import OAI.MathematicalPhysics.RapidForcing.FormulaPrograms
namespace OAI
section
open Encodable Nat.Partrec
open scoped BigOperators
namespace RapidForcing.EffectiveProfile.Formula
open EffectiveArithmetic
attribute [local irreducible] runTyped bound Expr.bound Expr.diff diff
variable {d : ℕ}
private lemma computable_lipMul : Computable (fun p : (Formula d × ℚ) ×
(Formula d × Formula d × ℚ × ℚ) =>
p.2.1.bound p.1.2 * p.2.2.2.2 + p.2.2.1.bound p.1.2 * p.2.2.2.1) := by
have ha := (@computable_bound d).comp (show Computable (fun p : (Formula d × ℚ) ×
(Formula d × Formula d × ℚ × ℚ) => (p.2.1, p.1.2)) by fun_prop)
have hb := (@computable_bound d).comp (show Computable (fun p : (Formula d × ℚ) ×
(Formula d × Formula d × ℚ × ℚ) => (p.2.2.1, p.1.2)) by fun_prop)
have hm₁ := computable_rat_mul.comp (ha.pair (show Computable (fun p : (Formula d × ℚ) ×
(Formula d × Formula d × ℚ × ℚ) => p.2.2.2.2) by fun_prop))
have hm₂ := computable_rat_mul.comp (hb.pair (show Computable (fun p : (Formula d × ℚ) ×
(Formula d × Formula d × ℚ × ℚ) => p.2.2.2.1) by fun_prop))
exact (computable_rat_add.comp (hm₁.pair hm₂)).of_eq (fun _ => rfl)
@[fun_prop] lemma computable_lipBound : Computable (fun p : Formula d × ℚ => p.1.lipBound p.2) := by
have h := computable_recOn (e := fun p : Formula d × ℚ => p.1) (by fun_prop)
(c := fun _ _ => (0 : ℚ)) (by unfold Computable₂; fun_prop)
(v := fun _ _ => (1 : ℚ)) (by unfold Computable₂; fun_prop)
(a := fun _ p => p.2.2.1 + p.2.2.2) (by unfold Computable₂; fun_prop)
(m := fun a p => p.1.bound a.2 * p.2.2.2 + p.2.1.bound a.2 * p.2.2.1) computable_lipMul.to₂
(p := fun _ p => p.1.diff.bound * p.2.2) (by unfold Computable₂; fun_prop)
exact h.of_eq (fun p => by induction p.1 <;> simp [lipBound, *])
lemma partrec_choosePrecision : Partrec (fun p : ℚ × ℚ => choosePrecision p.1 p.2) := by
apply Partrec.rfindOpt
unfold Computable₂
fun_prop
abbrev NameArgs (d : ℕ) := Formula d × Code × ℚ
private def nameLip (p : NameArgs d × (Fin d → ℚ)) : ℚ :=
p.1.1.lipBound (2 + ∑ i, |p.2 i|)
private lemma computable_nameLip : Computable (@nameLip d) := by
have h := computable_fin_sum computable_rat_add (f := fun q : Fin d → ℚ => fun i => |q i|)
(fun i => computable_rat_abs.comp (Computable.fin_app.comp Computable.id (Computable.const i)))
have ht := computable_rat_add.comp ((Computable.const 2).pair h)
have hR := ht.comp (show Computable (fun p : NameArgs d × (Fin d → ℚ) => p.2) by fun_prop)
exact ((@computable_lipBound d).comp ((show Computable (fun p : NameArgs d × (Fin d → ℚ) => p.1.1) by fun_prop).pair hR)).of_eq (fun _ => rfl)
lemma partrec_evalName : Partrec (fun p : NameArgs d =>
p.1.evalName (runTyped p.2.1) p.2.2) := by
have h₀ := (partrec_runTyped (B := Fin d → ℚ)).comp
(show Computable (fun p : NameArgs d => (p.2.1, (0 : ℕ))) by fun_prop)
have hn := partrec_choosePrecision.comp
(computable_nameLip.pair (show Computable (fun p : NameArgs d × (Fin d → ℚ) => p.1.2.2) by fun_prop))
have hq := (partrec_runTyped (B := Fin d → ℚ)).comp
(show Computable (fun p : (NameArgs d × (Fin d → ℚ)) × ℕ => (p.1.1.2.1, p.2)) by fun_prop)
have hb := partrec_evalAt.comp (show Computable
(fun p : ((NameArgs d × (Fin d → ℚ)) × ℕ) × (Fin d → ℚ) =>
(p.1.1.1.1, p.2, p.1.1.1.2.2 / 2)) by fun_prop)
have hf : Computable (fun p : (((NameArgs d × (Fin d → ℚ)) × ℕ) × (Fin d → ℚ)) × Ball =>
(⟨p.2.center, p.2.radius + (1 / 2 : ℚ) ^ p.1.1.2 * nameLip p.1.1.1⟩ : Ball)) := by
have hL := computable_nameLip.comp (show Computable (fun p :
(((NameArgs d × (Fin d → ℚ)) × ℕ) × (Fin d → ℚ)) × Ball => p.1.1.1) by fun_prop)
fun_prop (disch := assumption)
exact (h₀.bind (hn.bind (hq.bind (hb.bind hf.partrec.to₂).to₂).to₂).to₂).of_eq (fun _ => rfl)
lemma partrec_readReal : Partrec (fun p : Program × ℕ => readReal p.1 p.2) := by
exact (partrec_runTyped (B := ℚ)).of_eq (fun _ => by simp [runTyped, readReal])
lemma partrec_readPoint : Partrec (fun p : Program × ℕ => readPoint p.1 p.2) := by
exact (partrec_runTyped (B := RationalPoint)).of_eq (fun _ => by simp [runTyped, readPoint])
lemma partrec_coordinateName : Partrec (fun p : (Program × Program) × ℕ =>
coordinateName p.1.1 p.1.2 p.2) := by
have hv := partrec_readReal.comp (show Computable (fun p : (Program × Program) × ℕ =>
(p.1.1, p.2)) by fun_prop)
have hq := partrec_readPoint.comp (show Computable (fun p : ((Program × Program) × ℕ) × ℚ =>
(p.1.1.2, p.1.2)) by fun_prop)
have h : Computable (fun p : (((Program × Program) × ℕ) × ℚ) × RationalPoint =>
![p.2.1, p.2.2.1, p.2.2.2.1, p.2.2.2.2, p.1.2]) := by
apply computable_fin_lambda
intro i
fin_cases i <;> dsimp <;> fun_prop
exact (hv.bind (hq.bind h.partrec.to₂).to₂).of_eq (fun _ => rfl)
noncomputable def coordinateProgram (p : Program × Program) : Program :=
(typedCode _ partrec_coordinateName).curry (encode p)
@[fun_prop] lemma computable_coordinateProgram : Computable coordinateProgram :=
computable_specialize.comp ((Computable.const _).pair Computable.id)
lemma runTyped_coordinateProgram (p : Program × Program) (n : ℕ) :
runTyped (coordinateProgram p) n = coordinateName p.1 p.2 n := by
rw [coordinateProgram, runTyped_curry, runTyped_typedCode]
end RapidForcing.EffectiveProfile.Formula
end
end OAI
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