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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.NormalizedBranchTrace
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarClockMargins
namespace OAI
/-! Clock and observation margins for the entire filled branch trajectory. -/
noncomputable section
namespace ForcedComputation.Recorder.Planar
open ShearFlows PlanarHamiltonian PlanarRouting Set
/-- The parking rectangles have room for the rational initial translation. -/
theorem shifted_parking_bounds {n : ℕ} {κ : ℚ} (hκ : 0 ≤ κ) (hκ' : κ ≤ 1 / 64)
(i : Fin n) (δ : Fin 2 → ℚ)
(hδ₀ : |(δ 0 : ℝ)| ≤ 1 / 512) (hδ₁ : |(δ 1 : ℝ)| ≤ 3 / 1024)
{x : Plane} (hx : x ∈ (parkingBox n κ i).carrier) :
translatedPoint δ x ∈ clockRectangle.carrier ∧ 3 / 16 ≤ translatedPoint δ x 1 := by
have hb := parkingBox_bounds hκ hκ' i hx
have hs : (parkingScale n : ℝ) ≤ 1 / 64 := by
have hh := (Rat.cast_le (K := ℝ)).mpr (parkingScale_le n)
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] at hh
exact hh
have hk : (0 : ℝ) ≤ κ := by exact_mod_cast hκ
have hk' : (κ : ℝ) ≤ 1 / 64 := by
have hh := (Rat.cast_le (K := ℝ)).mpr hκ'
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] at hh
exact hh
have hc : (parkingCenter n i 0 : ℝ) < 3 / 4 := by
have hh := (Rat.cast_lt (K := ℝ)).mpr (parkingCenter_bounds n i).2
norm_num only [Rat.cast_div, Rat.cast_ofNat] at hh
exact hh
have hxu := (hx 0).2
change x 0 ≤ ((parkingCenter n i 0 + parkingScale n * κ : ℚ) : ℝ) at hxu
push_cast at hxu
have hm := mul_le_mul hs hk' hk (by norm_num : (0 : ℝ) ≤ 1 / 64)
obtain ⟨hd₀, hd₀'⟩ := abs_le.mp hδ₀
obtain ⟨hd₁, hd₁'⟩ := abs_le.mp hδ₁
constructor
· intro j
fin_cases j
· change ((1 / 8 : ℚ) : ℝ) ≤ x 0 + (δ 0 : ℝ) ∧
x 0 + (δ 0 : ℝ) ≤ ((7 / 8 : ℚ) : ℝ)
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat]
constructor <;> linarith [hb.1]
· change ((1 / 32 : ℚ) : ℝ) ≤ x 1 + (δ 1 : ℝ) ∧
x 1 + (δ 1 : ℝ) ≤ ((7 / 8 : ℚ) : ℝ)
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat]
constructor <;> linarith [hb.2.2.1, hb.2.2.2]
· change 3 / 16 ≤ x 1 + (δ 1 : ℝ)
linarith [hb.2.2.1]
theorem horizontalPoint_vertical (R : RationalBox 2) (p : Fin 2 → ℚ) (x : Plane) :
horizontalPoint R p x 1 = x 1 := by
simp [horizontalPoint, horizontalCenter]
theorem shifted_horizontalPoint_clock (M : Alternating.Machine) (hM : M.WellFormed)
(b : Branch (finiteMachine M hM)) (R : RationalBox 2)
(hwidth : R.halfWidth 0 ≤ 1 / 128)
(δ : Fin 2 → ℚ) (hδ₀ : |(δ 0 : ℝ)| ≤ 1 / 512)
(hδ₁ : |(δ 1 : ℝ)| ≤ 3 / 1024)
{x : Plane} (hx : x ∈ R.carrier)
(hy : (5 / 128 : ℝ) ≤ x 1 ∧ x 1 ≤ 19 / 64) :
translatedPoint δ (horizontalPoint R (parkingFor M hM b) x) ∈ clockRectangle.carrier := by
have hc := parkingCenter_bounds (geometricBranches M hM).length (branchIndex M hM b)
have hcl : (1 / 2 : ℝ) < (parkingFor M hM b 0 : ℝ) := by
have hh := (Rat.cast_lt (K := ℝ)).mpr hc.1
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] at hh
exact hh
have hcu : (parkingFor M hM b 0 : ℝ) < 3 / 4 := by
have hh := (Rat.cast_lt (K := ℝ)).mpr hc.2
norm_num only [Rat.cast_div, Rat.cast_ofNat] at hh
exact hh
have hw := RationalBox.mem_carrier_iff.mp hx 0
obtain ⟨hwl, hwu⟩ := abs_le.mp hw
obtain ⟨hd₀, hd₀'⟩ := abs_le.mp hδ₀
obtain ⟨hd₁, hd₁'⟩ := abs_le.mp hδ₁
intro j
fin_cases j
· change ((1 / 8 : ℚ) : ℝ) ≤
horizontalPoint R (parkingFor M hM b) x 0 + (δ 0 : ℝ) ∧
horizontalPoint R (parkingFor M hM b) x 0 + (δ 0 : ℝ) ≤ ((7 / 8 : ℚ) : ℝ)
have he : horizontalPoint R (parkingFor M hM b) x 0 =
x 0 + (parkingFor M hM b 0 : ℝ) - R.center 0 := by
simp [horizontalPoint, horizontalCenter]
ring
rw [he]
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat]
constructor <;> linarith
· change ((1 / 32 : ℚ) : ℝ) ≤
horizontalPoint R (parkingFor M hM b) x 1 + (δ 1 : ℝ) ∧
horizontalPoint R (parkingFor M hM b) x 1 + (δ 1 : ℝ) ≤ ((7 / 8 : ℚ) : ℝ)
rw [horizontalPoint_vertical]
norm_num only [Rat.cast_div, Rat.cast_one, Rat.cast_ofNat]
constructor <;> linarith [hy.1, hy.2]
theorem branch_scaledStage_mem (M : Alternating.Machine) (hM : M.WellFormed)
(b : Branch (finiteMachine M hM)) {x : Plane}
(hx : x ∈ (instruction M hM b).source.carrier) (k : Fin 5) :
scaledStage (instruction M hM b) (parkingScale (geometricBranches M hM).length)
(3 / 8) (parkingFor M hM b) x k ∈
(parkingBox (geometricBranches M hM).length (bandScale M) (branchIndex M hM b)).carrier := by
have hw := instruction_halfWidths M hM b
exact scaledStage_mem_parking (bandScale_pos M).le (branchIndex M hM b) _
(instruction_factor_pos M hM b) (instruction_image M hM b)
hw.1 hw.2.1 hw.2.2.1 hw.2.2.2 hx k
theorem shifted_branchAnchors_clock (M : Alternating.Machine) (hM : M.WellFormed)
(b : Branch (finiteMachine M hM)) (δ : Fin 2 → ℚ)
(hδ₀ : |(δ 0 : ℝ)| ≤ 1 / 512) (hδ₁ : |(δ 1 : ℝ)| ≤ 3 / 1024)
{x : Plane} (hx : x ∈ (instruction M hM b).source.carrier) (k : Fin 9) :
translatedPoint δ (branchAnchors M hM b x k) ∈ clockRectangle.carrier := by
have ht : (instruction M hM b).affine x ∈ (instruction M hM b).target.carrier := by
rw [← instruction_image M hM b]
exact mem_image_of_mem _ hx
have hs := instruction_source_bounds M hM b hx
have ht' := instruction_target_bounds M hM b ht
have hw := instruction_halfWidthQ_bound M hM b
have hws : (instruction M hM b).source.halfWidth 0 ≤ 1 / 128 := by
have hh := (Rat.cast_le (K := ℝ)).mpr (hw.1 0)
simpa only [halfWidthQ_cast, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] using hh
have hwt : (instruction M hM b).target.halfWidth 0 ≤ 1 / 128 := by
have hh := (Rat.cast_le (K := ℝ)).mpr (hw.2 0)
simpa only [halfWidthQ_cast, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] using hh
have hp (j : Fin 5) := (shifted_parking_bounds (bandScale_pos M).le (bandScale_le M)
(branchIndex M hM b) δ hδ₀ hδ₁ (branch_scaledStage_mem M hM b hx j)).1
fin_cases k
· exact shifted_coding_rectangle_clock M hM δ hδ₀ hδ₁ hs
· exact shifted_horizontalPoint_clock M hM b _ hws δ hδ₀ hδ₁ hx hs.2
· exact hp 0
· exact hp 1
· exact hp 2
· exact hp 3
· exact hp 4
· exact shifted_horizontalPoint_clock M hM b _ hwt δ hδ₀ hδ₁ ht ht'.2
· exact shifted_coding_rectangle_clock M hM δ hδ₀ hδ₁ ht'
theorem shifted_branchAnchors_safe (M : Alternating.Machine) (hM : M.WellFormed)
(b : Branch (finiteMachine M hM)) (hn : recorderHalting M b.target = false)
(δ : Fin 2 → ℚ) (hδ₀ : |(δ 0 : ℝ)| ≤ 1 / 512) (hδ₁ : |(δ 1 : ℝ)| ≤ 3 / 1024)
{x : Plane} (hx : x ∈ (instruction M hM b).source.carrier) (k : Fin 9) :
3 / 16 ≤ translatedPoint δ (branchAnchors M hM b x k) 1 := by
have ht : (instruction M hM b).affine x ∈ (instruction M hM b).target.carrier := by
rw [← instruction_image M hM b]
exact mem_image_of_mem _ hx
have hs := (shifted_instruction_source_height M hM b δ hδ₁ hx).1
have ht' := (shifted_instruction_target_nonterminal_height M hM b hn δ hδ₁ ht).1
have hp (j : Fin 5) := (shifted_parking_bounds (bandScale_pos M).le (bandScale_le M)
(branchIndex M hM b) δ hδ₀ hδ₁ (branch_scaledStage_mem M hM b hx j)).2
fin_cases k
· exact hs
· change 3 / 16 ≤ horizontalPoint _ _ x 1 + (δ 1 : ℝ)
rw [horizontalPoint_vertical]
exact hs
· exact hp 0
· exact hp 1
· exact hp 2
· exact hp 3
· exact hp 4
· change 3 / 16 ≤ horizontalPoint _ _ ((instruction M hM b).affine x) 1 + (δ 1 : ℝ)
rw [horizontalPoint_vertical]
exact ht'
· exact ht'
theorem normalizedAnchors_clock (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(b : Branch (finiteMachine (freshMachine I.1) (freshInput_valid hI).1))
{x : Plane} (hx : x ∈ (instruction (freshMachine I.1) (freshInput_valid hI).1 b).source.carrier)
(k : Fin 9) : normalizedAnchors I hI b x k ∈ clockRectangle.carrier := by
have hδ := initialShift_small (freshInput I) (freshInput_valid hI) (fresh_initial_nonhalting I hI)
exact shifted_branchAnchors_clock _ _ b _ hδ.1 hδ.2 hx k
theorem normalizedAnchors_safe (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(b : Branch (finiteMachine (freshMachine I.1) (freshInput_valid hI).1))
(hn : recorderHalting (freshMachine I.1) b.target = false)
{x : Plane} (hx : x ∈ (instruction (freshMachine I.1) (freshInput_valid hI).1 b).source.carrier)
(k : Fin 9) : 3 / 16 ≤ normalizedAnchors I hI b x k 1 := by
have hδ := initialShift_small (freshInput I) (freshInput_valid hI) (fresh_initial_nonhalting I hI)
exact shifted_branchAnchors_safe _ _ b hn _ hδ.1 hδ.2 hx k
theorem normalizedCurve_segment (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(b : Branch (finiteMachine (freshMachine I.1) (freshInput_valid hI).1))
(x : Plane) (k : Fin 8) (t : ℝ) :
normalizedCurve I hI b x k t = normalizedAnchors I hI b x k.castSucc +
smoothRamp (compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).start
(compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).finish t •
(normalizedAnchors I hI b x k.succ - normalizedAnchors I hI b x k.castSucc) := by
unfold normalizedCurve branchCurve
rw [Pulse.curve, Primitive.path_segment, branchCurve_endpoint]
ext j
simp only [translatedPoint, normalizedAnchors, Pi.add_apply, Pi.sub_apply, Pi.smul_apply,
smul_eq_mul]
ring
theorem normalizedCurve_clock (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(b : Branch (finiteMachine (freshMachine I.1) (freshInput_valid hI).1))
{x : Plane} (hx : x ∈ (instruction (freshMachine I.1) (freshInput_valid hI).1 b).source.carrier)
(k : Fin 8) (t : ℝ) : normalizedCurve I hI b x k t ∈ clockRectangle.carrier := by
have h₀ := normalizedAnchors_clock I hI b hx k.castSucc
have h₁ := normalizedAnchors_clock I hI b hx k.succ
have hθ := smoothRamp_range
(compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).start
(compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).finish t
rw [normalizedCurve_segment]
intro j
have hj₀ := h₀ j
have hj₁ := h₁ j
simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.sub_apply]
constructor <;> nlinarith [hθ.1, hθ.2]
theorem normalizedCurve_safe (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(b : Branch (finiteMachine (freshMachine I.1) (freshInput_valid hI).1))
(hn : recorderHalting (freshMachine I.1) b.target = false)
{x : Plane} (hx : x ∈ (instruction (freshMachine I.1) (freshInput_valid hI).1 b).source.carrier)
(k : Fin 8) (t : ℝ) : 3 / 16 ≤ normalizedCurve I hI b x k t 1 := by
have h₀ := normalizedAnchors_safe I hI b hn hx k.castSucc
have h₁ := normalizedAnchors_safe I hI b hn hx k.succ
have hθ := smoothRamp_range
(compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).start
(compiledPulse (freshMachine I.1) (freshInput_valid hI).1
(branchIndices (freshMachine I.1) (freshInput_valid hI).1 b k)).finish t
rw [normalizedCurve_segment]
simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.sub_apply]
nlinarith [hθ.1, hθ.2]
end ForcedComputation.Recorder.Planar
end
end OAI
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