File size: 6,401 Bytes
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 | import OAI.MathematicalPhysics.AlternatingFlow.BaseNames
namespace OAI
section VelocityNamesDevelopment
open scoped BigOperators Topology ContDiff
namespace AlternatingNS.Effective
attribute [local instance] Arithmetic.rationalCoding
variable {A : Type*} [Primcodable A]
lemma JetBound.curl_named {f : A → Space → ℝ} (hf : JetBound f)
(hn : Named (fun z : A × RationalPoint => f z.1 (rationalPoint z.2).2))
(i j : A → Fin 3) (hi : Computable i) (hj : Computable j) :
Named (fun z : (A × Fin 3) × RationalPoint =>
Spatial.curl (i z.1.1) (j z.1.1) (f z.1.1) (rationalPoint z.2).2 z.1.2) := by
have hp : Computable (fun z : (A × Fin 3) × RationalPoint => (z.1.1,z.2)) :=
(Computable.fst.comp Computable.fst).pair Computable.snd
have hdj := (hf.spatial_d_named hn j hj).comp hp
have hdi := (hf.spatial_d_named hn i hi).comp hp
have hqi : Computable (fun z : (A × Fin 3) × RationalPoint => decide (z.1.2 = i z.1.1)) := Primrec.eq.decide.to_comp.comp (Computable.snd.comp Computable.fst)
(hi.comp (Computable.fst.comp Computable.fst))
have hqj : Computable (fun z : (A × Fin 3) × RationalPoint => decide (z.1.2 = j z.1.1)) := Primrec.eq.decide.to_comp.comp (Computable.snd.comp Computable.fst)
(hj.comp (Computable.fst.comp Computable.fst))
exact ((Named.ite _ hqi hdj (Named.const 0)).sub
(Named.ite _ hqj hdi (Named.const 0))).congr (by
intro z; simp [Spatial.curl, Spatial.e_apply])
lemma loading_named : Named (fun z : ((Machine × List ℕ) × Fin 3) × RationalPoint =>
Construction.loading z.1.1.1 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by
exact loadPotential_bound.curl_named loadPotential_named (fun _ => 1) (fun _ => 0)
(Computable.const 1) (Computable.const 0)
lemma slot_named : Named (fun z : (((Machine × ℕ) × ℕ) × Fin 3) × RationalPoint =>
Construction.slot z.1.1.1.1 z.1.1.1.2 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by
exact slotPotential_bound.curl_named slotPotential_named
(fun p => Construction.receiverIndex p.2) (fun _ => 0)
(receiver_computable.comp Computable.snd) (Computable.const 0)
lemma piece_named : Named (fun z : (((Machine × List ℕ) × ℕ) × Fin 3) × RationalPoint =>
Construction.piece z.1.1.1.1 z.1.1.1.2 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by
let P := (((Machine × List ℕ) × ℕ) × Fin 3) × RationalPoint
have hM : Computable (fun z : P => z.1.1.1.1) :=
Computable.fst.comp (Computable.fst.comp (Computable.fst.comp Computable.fst))
have hw : Computable (fun z : P => z.1.1.1.2) :=
Computable.snd.comp (Computable.fst.comp (Computable.fst.comp Computable.fst))
have hn : Computable (fun z : P => z.1.1.2) :=
Computable.snd.comp (Computable.fst.comp Computable.fst)
have hi : Computable (fun z : P => z.1.2) := Computable.snd.comp Computable.fst
have hload := loading_named.comp (((hM.pair hw).pair hi).pair Computable.snd)
have hslot := slot_named.comp ((((hM.pair (Computable.list_length.comp hw)).pair
(Primrec.nat_sub.to_comp.comp hn (Computable.const 1))).pair hi).pair Computable.snd)
exact (Named.ite (fun z : P => z.1.1.2 = 0)
(Primrec.eq.decide.to_comp.comp hn (Computable.const 0)) hload hslot).congr
(by rintro ⟨⟨⟨⟨M,w⟩,n⟩,i⟩,q⟩; cases n <;> rfl)
lemma clock_bound : JetBound (fun _ : Unit => Profiles.clock) := by
have hclock := transition_scaled.affine (fun _ _ => zero_le_one) (fun _ _ => 2)
(fun _ _ => -1/2) (fun _ => 2) (Computable.const 2) (by intros; norm_num)
obtain ⟨hs,B,hB,h⟩ := hclock
refine ⟨fun _ => Profiles.clock_smooth, B, hB, fun p r x => ?_⟩
have he : (fun t => Real.smoothTransition (2 * t + -1/2)) = Profiles.clock := by
funext t; unfold Profiles.clock; congr 1; ring
simpa only [one_pow, mul_one, he] using h p r 0 x
lemma pulse_rational_named : Named (fun q : ℚ => Profiles.pulse (q : ℝ)) := by
have hf := clock_bound.comp_unit (fun _ => ContinuousLinearMap.fst ℝ ℝ Space) (fun _ => norm_fst)
have hn : Named (fun z : Unit × RationalPoint => Profiles.clock (rationalPoint z.2).1) := by
have ht := Named.rational (Computable.fst.comp (Computable.snd : Computable (Prod.snd : Unit × RationalPoint → _)))
exact (((Named.const 2).mul ht).sub (Named.const (1/2))).transition.congr (by
intro z; simp [Profiles.clock, rationalPoint])
have hd := hf.named_directional hn (fun _ => 0) (Computable.const 0)
have hdc := hd.comp ((Computable.const ()).pair (Computable.id.pair
((Computable.const (0:ℚ)).pair ((Computable.const (0:ℚ)).pair (Computable.const (0:ℚ))))))
apply hdc.congr
intro q
change fderiv ℝ (Profiles.clock ∘ Prod.fst) ((q:ℝ),_) (1,0) = deriv Profiles.clock (q:ℝ)
rw [fderiv_comp _ (Profiles.clock_smooth.differentiable (by simp) _) differentiableAt_fst, fderiv_fst]
rfl
lemma velocity_named : Named (fun z : ((Machine × List ℕ) × Fin 3) × RationalPoint =>
Construction.velocity z.1.1.1 z.1.1.2 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by
let P := ((Machine × List ℕ) × Fin 3) × RationalPoint
have hq : Computable (fun z : P => z.2.1) := Computable.fst.comp Computable.snd
let N : P → ℕ := fun z => z.2.1.num.natAbs+1
have hN : Computable N := Computable.succ.comp (Arithmetic.int_abs.to_comp.comp (Arithmetic.rat_num.to_comp.comp hq))
have hterm : Named (fun z : P × ℕ => Profiles.pulse ((rationalPoint z.1.2).1 - z.2) *
Construction.piece z.1.1.1.1 z.1.1.1.2 z.2 (rationalPoint z.1.2).2 z.1.1.2) := by
have hh := pulse_rational_named.comp (Arithmetic.rat_sub.to_comp.comp
(hq.comp Computable.fst) (Arithmetic.rat_natCast.to_comp.comp Computable.snd))
have hp := piece_named.comp
((((Computable.fst.comp (Computable.fst.comp Computable.fst)).pair Computable.snd).pair
(Computable.snd.comp (Computable.fst.comp Computable.fst))).pair
(Computable.snd.comp Computable.fst))
exact (hh.mul hp).congr (by intro z; simp [rationalPoint])
apply (hterm.sum_range N hN).congr
intro z
have ht : (rationalPoint z.2).1 ≤ (N z : ℝ) := by
have hr := le_trans (le_abs_self (z.2.1 : ℝ)) (abs_rat_cast_le_num z.2.1)
dsimp [N, rationalPoint]; push_cast; linarith
rw [Construction.velocity, TimeGlue.eq_sum _ Profiles.pulse_zero_left _ _ _ (N z) ht]
simp [WithLp.ofLp_sum, Finset.sum_apply]
end AlternatingNS.Effective
end VelocityNamesDevelopment
end OAI
|