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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 | import OAI.MathematicalPhysics.AlternatingFlow.VelocityNames
import OAI.MathematicalPhysics.AlternatingFlow.MixedPrograms
namespace OAI
section ForceNamesDevelopment
open scoped BigOperators Topology ContDiff
namespace AlternatingNS.Effective
attribute [local instance] Arithmetic.rationalCoding
variable {A : Type*} [Primcodable A]
lemma JetBound.named_fullDt {f : A → Velocity}
(hf : JetBound (fun a => Function.uncurry (f a)))
(hn : Named (fun z : (A × Fin 3) × RationalPoint => f z.1.1 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2)) :
Named (fun z : (A × Fin 3) × RationalPoint => Analytic.fullDt (f z.1.1) (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
have h := hf.named_component_words hn |>.comp
((Computable.fst.pair (Computable.const [0])).pair Computable.snd)
apply h.congr
intro z
rw [Bounds.fullDt_eq_full_directional _ (hf.1 z.1.1) (rationalPoint z.2)]
simp only [wordJet, direction, Matrix.cons_val_zero, Function.comp_apply]
lemma JetBound.named_dx {f : A → Velocity}
(hf : JetBound (fun a => Function.uncurry (f a)))
(hn : Named (fun z : (A × Fin 3) × RationalPoint => f z.1.1 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2)) (i : Fin 3) :
Named (fun z : (A × Fin 3) × RationalPoint => dx i (f z.1.1) (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
have h := hf.named_component_words hn |>.comp
((Computable.fst.pair (Computable.const [i.succ])).pair Computable.snd)
apply h.congr
intro z
rw [Bounds.dx_eq_full_directional _ (hf.1 z.1.1) i (rationalPoint z.2)]
simp only [wordJet, direction_succ, Function.comp_apply]
lemma Named.sum3 {f : Fin 3 → A → ℝ} (hf : ∀ i, Named (f i)) :
Named (fun z => ∑ i : Fin 3, f i z) :=
((hf 0).add ((hf 1).add (hf 2))).congr (fun _ => by simp [Fin.sum_univ_succ])
lemma Certified.named_advection {f : A → Velocity}
(hf : Certified (fun a => Function.uncurry (f a)))
(hn : Named (fun z : (A × Fin 3) × RationalPoint => f z.1.1 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2)) :
Named (fun z : (A × Fin 3) × RationalPoint => AlternatingNS.advection (f z.1.1) (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
have hi (i : Fin 3) := (hn.comp
(((Computable.fst.comp Computable.fst).pair (Computable.const i)).pair Computable.snd)).mul
(hf.jetBound.named_dx hn i)
apply (Named.sum3 hi).congr
intro z
rw [Bounds.advection_eq_sum]
simp only [Function.comp_apply, WithLp.ofLp_sum, Finset.sum_apply, PiLp.smul_apply, smul_eq_mul]
lemma Certified.named_laplacian {f : A → Velocity}
(hf : Certified (fun a => Function.uncurry (f a)))
(hn : Named (fun z : (A × Fin 3) × RationalPoint => f z.1.1 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2)) :
Named (fun z : (A × Fin 3) × RationalPoint => AlternatingNS.laplacian (f z.1.1) (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
have hi (i : Fin 3) := (hf.dx i).jetBound.named_dx (hf.jetBound.named_dx hn i) i
exact (Named.sum3 hi).congr (fun z => by simp only [AlternatingNS.laplacian, WithLp.ofLp_sum, Finset.sum_apply])
lemma Named.computableReal_const {r : ℝ} (hr : ComputableReal r) : Named (fun _ : A => r) := by
obtain ⟨a,ha,h⟩ := hr
have ha' : Computable a := (Arithmetic.rat_mk.to_comp.comp
(Computable.fst.comp ha) (Computable.snd.comp ha)).of_eq (fun n => by
simp [Rat.mkRat_eq_div, Rat.num_div_den])
refine ⟨fun z => a z.2, ha'.comp Computable.snd, fun _ n => ?_⟩
simpa only [tolerance_cast] using h n
lemma force_named (ν : ℝ) (hν : ComputableReal ν) :
Named (fun z : ((Machine × List ℕ) × Fin 3) × RationalPoint =>
fullForce ν z.1.1.1 z.1.1.2 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
exact ((velocity_certified.jetBound.named_fullDt velocity_named).add
(velocity_certified.named_advection velocity_named)).sub
((Named.computableReal_const hν).mul (velocity_certified.named_laplacian velocity_named))
end AlternatingNS.Effective
end ForceNamesDevelopment
end OAI
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