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f18e7de | 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 | import OAI.MathematicalPhysics.AlternatingFlow.BaseNames
namespace OAI
section MixedProgramsDevelopment
open scoped BigOperators Topology ContDiff
namespace AlternatingNS.Effective
attribute [local instance] Arithmetic.rationalCoding
variable {A E F : Type*} [Primcodable A]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
lemma wordJet_linear {f : ℝ × Space → E} (hf : ContDiff ℝ ∞ f)
(L : E →L[ℝ] F) (w : List (Fin 4)) :
wordJet (L ∘ f) w = L ∘ wordJet f w := by
induction w with
| nil => rfl
| cons i w ih =>
funext z
simp only [wordJet, ih]
rw [fderiv_comp _ L.differentiableAt ((wordJet_smooth hf w).differentiable (by simp) z), L.fderiv]
rfl
lemma wordJet_component {f : ℝ × Space → Space} (hf : ContDiff ℝ ∞ f)
(i : Fin 3) (w : List (Fin 4)) :
wordJet (fun z => f z i) w = fun z => wordJet f w z i :=
wordJet_linear hf (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i) w
lemma JetBound.named_component_words {f : A → ℝ × Space → Space} (hf : JetBound f)
(hn : Named (fun z : (A × Fin 3) × RationalPoint => f z.1.1 (rationalPoint z.2) z.1.2)) :
Named (fun z : ((A × Fin 3) × List (Fin 4)) × RationalPoint =>
wordJet (f z.1.1.1) z.1.2 (rationalPoint z.2) z.1.1.2) := by
have hp := (hf.param (Computable.fst : Computable (Prod.fst : A × Fin 3 → A))).linear_unit
(fun p => PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) p.2) (fun p => norm_proj p.2)
exact (hp.named_wordJet hn).congr (fun z => congrFun (wordJet_component (hf.1 z.1.1.1) z.1.1.2 z.1.2) (rationalPoint z.2))
lemma wordJet_append (f : ℝ × Space → E) (v w : List (Fin 4)) :
wordJet f (v ++ w) = wordJet (wordJet f w) v := by
induction v with
| nil => rfl
| cons i v ih => simp only [List.cons_append, wordJet, ih]
lemma wordJet_dx (f : Field E) (hf : ContDiff ℝ ∞ (Function.uncurry f))
(i : Fin 3) (n : ℕ) :
Function.uncurry ((dx i)^[n] f) = wordJet (Function.uncurry f) (List.replicate n i.succ) := by
induction n with
| zero => rfl
| succ n ih =>
funext z
rw [Function.iterate_succ_apply']
change dx i ((dx i)^[n] f) z.1 z.2 = _
rw [Bounds.dx_eq_full_directional _ (Analytic.iterate_dx_smooth f hf i n), ih]
simp only [List.replicate_succ, wordJet, direction_succ]
lemma wordJet_dt (f : Field E) (hf : ContDiff ℝ ∞ (Function.uncurry f)) (n : ℕ) :
Bounds.HalfEq (dt^[n] f) (fun t x => wordJet (Function.uncurry f) (List.replicate n 0) (t,x)) := by
induction n with
| zero => exact fun _ _ _ => rfl
| succ n ih =>
intro t ht x
rw [Function.iterate_succ_apply']
have hh := ih.dt t ht x
have hs := wordJet_smooth hf (List.replicate n 0)
rw [Analytic.dt_eq_full (fun t x => wordJet (Function.uncurry f) (List.replicate n 0) (t,x)) hs t ht x,
Bounds.fullDt_eq_full_directional (fun t x => wordJet (Function.uncurry f) (List.replicate n 0) (t,x)) hs (t,x)] at hh
change dt (dt^[n] f) t x = fderiv ℝ (wordJet (Function.uncurry f) (List.replicate n 0)) (t,x) (1,0) at hh
simpa only [List.replicate_succ, wordJet, direction, Matrix.cons_val_zero] using hh
def mixedWord (l : ℕ) (α : IndexCode) : List (Fin 4) :=
List.replicate l 0 ++ (List.replicate α.1 1 ++
(List.replicate α.2.1 2 ++ List.replicate α.2.2 3))
lemma wordJet_spatial (f : Field E) (hf : ContDiff ℝ ∞ (Function.uncurry f)) (α : IndexCode) :
Function.uncurry (spatial (index α) f) =
wordJet (Function.uncurry f) (List.replicate α.1 1 ++
(List.replicate α.2.1 2 ++ List.replicate α.2.2 3)) := by
simp only [spatial, index, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val]
rw [wordJet_dx _ (Analytic.iterate_dx_smooth _ (Analytic.iterate_dx_smooth f hf 2 _) 1 _),
wordJet_dx _ (Analytic.iterate_dx_smooth f hf 2 _), wordJet_dx f hf]
simp only [wordJet_append]
rfl
lemma mixed_eq_wordJet (f : Field E) (hf : ContDiff ℝ ∞ (Function.uncurry f))
(l : ℕ) (α : IndexCode) (t : ℝ) (ht : 0 ≤ t) (x : Space) :
mixed l (index α) f t x = wordJet (Function.uncurry f) (mixedWord l α) (t,x) := by
have h := wordJet_dt (spatial (index α) f) (Analytic.spatial_smooth f hf _) l t ht x
rw [wordJet_spatial f hf α] at h
simpa only [mixedWord, wordJet_append, mixed] using h
lemma replicate_primrec {B : Type*} [Primcodable B] (b : B) : Primrec (fun n => List.replicate n b) := by
have h := Primrec.nat_rec₁ (f := fun _ : ℕ => fun w : List B => b :: w) ([] : List B)
(Primrec.list_cons.comp (Primrec.const b) Primrec.snd)
apply h.of_eq
intro n
induction n with
| zero => rfl
| succ n ih =>
change b :: _ = List.replicate (n+1) b
rw [ih, List.replicate_succ]
lemma mixedWord_computable : Computable₂ mixedWord := by
have hr (i : Fin 4) := (replicate_primrec i).to_comp
exact Primrec.list_append.to_comp.comp ((hr 0).comp Computable.fst)
(Primrec.list_append.to_comp.comp ((hr 1).comp (Computable.fst.comp Computable.snd))
(Primrec.list_append.to_comp.comp ((hr 2).comp (Computable.fst.comp (Computable.snd.comp Computable.snd)))
((hr 3).comp (Computable.snd.comp (Computable.snd.comp Computable.snd)))))
end AlternatingNS.Effective
end MixedProgramsDevelopment
end OAI
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