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37bcf52 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedVelocity
namespace OAI
/-! The loading trajectory and the compact planar trajectory join with zero
velocity at time one. This identifies the fixed Euclidean particle without
assuming a computational flow or a halting correspondence. -/
noncomputable section
namespace ForcedComputation.BalancedPath
open ShearFlows Recorder.Planar PlanarHamiltonian Set Filter
open scoped ContDiff Topology
def bodyPath (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(Ψ : ℝ → ℝ → Plane → Plane) (t : ℝ) : Space :=
BalancedStretch.stretch (atHeight (Ψ 0 (t - 1) (BalancedPlanar.inputCode I hI)) 0)
def path (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(Ψ : ℝ → ℝ → Plane → Plane) (t : ℝ) : Space :=
if t ≤ 1 then BalancedLoader.path (BalancedPlanar.inputCode I hI) t else bodyPath I hI Ψ t
private theorem body_zero_phase (M : Alternating.Machine) (hM : M.WellFormed) (x : Space) :
BalancedBody.velocity M hM (0,x) = 0 := by
have hz := (periodicVelocity_zero_near_zero
(normalizedPulse_valid (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM))
(normalizedPulse_inUnit (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM))).self_of_nhds
have hv : ∀ z, BalancedPlanar.bodyVelocity M hM 0 z = 0 := hz
simp only [BalancedBody.velocity, BalancedStretch.velocity, BalancedStretch.push,
BalancedBody.raw, CompactLift.velocity, CompactLift.spatial,
VelocityDetector.triangularLift, VelocityDetector.horizontalLinear_eq, hv,
map_zero, zero_smul, add_zero, smul_zero]
theorem bodyPath_one (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (BalancedPlanar.bodyVelocity I.1 hI.1) Ψ) :
bodyPath I hI Ψ 1 = BalancedLoader.path (BalancedPlanar.inputCode I hI) 1 := by
rw [BalancedLoader.path_after _ (by norm_num)]
simp only [bodyPath, sub_self, hΨ.initial]
theorem bodyPath_deriv (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (BalancedPlanar.bodyVelocity I.1 hI.1) Ψ) (t : ℝ) :
HasDerivAt (bodyPath I hI Ψ)
(BalancedBody.velocity I.1 hI.1 (t-1, bodyPath I hI Ψ t)) t := by
have hp := hΨ.ode 0 (t-1) (BalancedPlanar.inputCode I hI)
have hl := CompactLift.trajectory_atHeight
(χ := BalancedBody.verticalCutoff) (a := 0) (z := 0)
BalancedBody.verticalCutoff_zero hp
have hs := BalancedStretch.trajectory hl
have hc := hs.scomp t ((hasDerivAt_id t).sub_const 1)
apply hasDerivAt_pi.mpr
intro j
simpa only [Function.comp_def, one_smul, zero_add, id_eq, bodyPath,
BalancedBody.velocity, BalancedBody.raw] using (hasDerivAt_pi.mp hc) j
private theorem glue_deriv {f g : ℝ → Space} {v : Space}
(hf : HasDerivAt f v 1) (hg : HasDerivAt g v 1) (he : f 1 = g 1) :
HasDerivAt (fun t => if t ≤ 1 then f t else g t) v 1 := by
have hl : HasDerivWithinAt (fun t => if t ≤ 1 then f t else g t) v (Iic 1) 1 :=
(hf.hasDerivWithinAt (s := Iic (1 : ℝ))).congr_of_mem
(fun t ht => ite_eq_left ht)
(by change (1 : ℝ) ≤ 1; exact le_rfl)
have hr : HasDerivWithinAt (fun t => if t ≤ 1 then f t else g t) v (Ici 1) 1 := by
apply (hg.hasDerivWithinAt (s := Ici (1 : ℝ))).congr_of_mem _
(by change (1 : ℝ) ≤ 1; exact le_rfl)
intro t ht
by_cases ht' : t ≤ 1
· have h : t = 1 := le_antisymm ht' ht
simpa only [h, ite_eq_left le_rfl] using he
· exact ite_eq_right ht'
simpa only [Iic_union_Ici, hasDerivWithinAt_univ] using hl.union hr
theorem path_deriv (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (BalancedPlanar.bodyVelocity I.1 hI.1) Ψ) (t : ℝ) :
HasDerivAt (path I hI Ψ) (BalancedVelocity.velocity I hI (t,path I hI Ψ t)) t := by
have hq : ∀ j, 0 ≤ BalancedPlanar.inputCode I hI j ∧ BalancedPlanar.inputCode I hI j ≤ 1 :=
fun j => ⟨(BalancedPlanar.inputCode_in_unit I hI j).1.le,
(BalancedPlanar.inputCode_in_unit I hI j).2.le⟩
rcases lt_trichotomy t 1 with ht | rfl | ht
· have he : path I hI Ψ =ᶠ[𝓝 t] BalancedLoader.path (BalancedPlanar.inputCode I hI) := by
filter_upwards [eventually_lt_nhds ht] with s hs
exact ite_eq_left hs.le
rw [he.self_of_nhds, BalancedVelocity.before I hI ht.le]
exact (BalancedLoader.path_deriv hq t).congr_of_eventuallyEq he
· have hl := BalancedLoader.path_deriv hq 1
rw [BalancedLoader.velocity_after _ (by norm_num)] at hl
have hr := bodyPath_deriv I hI hΨ 1
rw [sub_self, body_zero_phase] at hr
have hh := glue_deriv hl hr (bodyPath_one I hI hΨ).symm
have hz : BalancedVelocity.velocity I hI (1,path I hI Ψ 1) = 0 := by
rw [BalancedVelocity.before I hI le_rfl, BalancedLoader.velocity_after _ (by norm_num)]
rw [hz]
exact hh
· have he : path I hI Ψ =ᶠ[𝓝 t] bodyPath I hI Ψ := by
filter_upwards [eventually_gt_nhds ht] with s hs
exact ite_eq_right (not_le.mpr hs)
rw [he.self_of_nhds, BalancedVelocity.tail I hI ht.le]
have hp := BalancedBody.velocity_periodic I.1 hI.1 (t-1) (bodyPath I hI Ψ t)
rw [sub_add_cancel] at hp
rw [hp]
exact (bodyPath_deriv I hI hΨ t).congr_of_eventuallyEq he
theorem path_initial (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(Ψ : ℝ → ℝ → Plane → Plane) : path I hI Ψ 0 = ![4,0,0] := by
rw [path, ite_eq_left (by norm_num), BalancedLoader.path_before _ (by norm_num)]
theorem path_after (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (BalancedPlanar.bodyVelocity I.1 hI.1) Ψ)
{t : ℝ} (ht : 1 ≤ t) : path I hI Ψ t = bodyPath I hI Ψ t := by
by_cases h : t ≤ 1
· have he : t = 1 := le_antisymm h ht
rw [he, path, ite_eq_left le_rfl, bodyPath_one I hI hΨ]
· exact ite_eq_right h
theorem path_observation (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{Ψ Ω : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (BalancedPlanar.bodyVelocity I.1 hI.1) Ψ)
(hΩ : IsPlanarTransition (planarSlice (BalancedPlanar.bodyHamiltonian I.1 hI.1)) Ω) :
(∃ t, 0 ≤ t ∧ path I hI Ψ t ∈ BalancedStretch.observer) ↔ Alternating.Halts I := by
have hsame (t : ℝ) : Ψ 0 t (BalancedPlanar.inputCode I hI) =
Ω 0 t (BalancedPlanar.inputCode I hI) :=
compactTransition_eq_slice (BalancedPlanar.referenceInput I.1)
(BalancedPlanar.referenceValid hI.1) hΨ hΩ
(BalancedPlanar.inputCode_in_unit I hI) 0 t
constructor
· rintro ⟨t, ht, he⟩
by_contra hn
by_cases h : t ≤ 1
· rw [path, ite_eq_left h] at he
exact BalancedLoader.path_safe (BalancedPlanar.inputCode_loader_guard I hI) t he
· rw [path_after I hI hΨ (le_of_not_ge h), bodyPath, hsame] at he
exact BalancedStretch.nonhalting_outside
(BalancedPlanar.body_nonhalting_corridor I hI hΩ hn (by linarith)).2 he
· intro hh
obtain ⟨n, hx, hy⟩ := BalancedPlanar.body_halting_clearance I hI hΩ hh
refine ⟨(n : ℝ)+1, by positivity, ?_⟩
rw [path_after I hI hΨ (by have hn : (0 : ℝ) ≤ n := Nat.cast_nonneg n; linarith),
bodyPath, add_sub_cancel_right, hsame]
exact BalancedStretch.halting_inside hx hy
end ForcedComputation.BalancedPath
end
end OAI
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