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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.MathematicalPhysics.AlternatingFlow.Machines
namespace OAI
open scoped BigOperators ENNReal NNReal Topology ContDiff
open MeasureTheory
namespace AlternatingNS
namespace Profiles
noncomputable def selector (b a : ℕ) (v : ℝ) : ℝ :=
letI := twoAtLeastTwo
letI := eightAtLeastTwo
Real.smoothTransition (8 * ((b : ℝ) * v - 2 * a) + 2) *
Real.smoothTransition (8 * (2 * a + 1 - (b : ℝ) * v) + 2)
lemma selector_smooth (b a : ℕ) : ContDiff ℝ ∞ (selector b a) := by
unfold selector
fun_prop
lemma selector_nonneg (b a : ℕ) (v : ℝ) : 0 ≤ selector b a v :=
mul_nonneg (Real.smoothTransition.nonneg _) (Real.smoothTransition.nonneg _)
lemma selector_le_one (b a : ℕ) (v : ℝ) : selector b a v ≤ 1 := by
exact (mul_le_of_le_one_left (Real.smoothTransition.nonneg _)
(Real.smoothTransition.le_one _)).trans (Real.smoothTransition.le_one _)
lemma selector_one (b a : ℕ) (v : ℝ)
(h₀ : 2 * (a : ℝ) - 1 / 8 ≤ b * v) (h₁ : (b : ℝ) * v ≤ 2 * a + 9 / 8) :
selector b a v = 1 := by
rw [selector, Real.smoothTransition.one_of_one_le (by linarith),
Real.smoothTransition.one_of_one_le (by linarith)]
norm_num
lemma selector_zero_left (b a : ℕ) (v : ℝ) (h : (b : ℝ) * v ≤ 2 * a - 1 / 4) :
selector b a v = 0 := by
rw [selector, Real.smoothTransition.zero_of_nonpos (by linarith), zero_mul]
lemma selector_zero_right (b a : ℕ) (v : ℝ) (h : 2 * (a : ℝ) + 5 / 4 ≤ b * v) :
selector b a v = 0 := by
rw [selector, Real.smoothTransition.zero_of_nonpos (by linarith :
8 * (2 * (a : ℝ) + 1 - b * v) + 2 ≤ 0), mul_zero]
lemma selector_support_bound (b a : ℕ) (v : ℝ) (h : selector b a v ≠ 0) :
2 * (a : ℝ) - 1 / 4 < b * v ∧ (b : ℝ) * v < 2 * a + 5 / 4 := by
constructor
· by_contra hn
exact h (selector_zero_left b a v (le_of_not_gt hn))
· by_contra hn
exact h (selector_zero_right b a v (le_of_not_gt hn))
lemma selector_support_disjoint (b a q : ℕ) (haq : a ≠ q) :
Disjoint (Function.support (selector b a)) (Function.support (selector b q)) := by
rw [Set.disjoint_left]
intro v ha hq
obtain ⟨ha₀, ha₁⟩ := selector_support_bound b a v ha
obtain ⟨hq₀, hq₁⟩ := selector_support_bound b q v hq
rcases lt_or_gt_of_ne haq with h | h
· have h' : (a : ℝ) + 1 ≤ q := by exact_mod_cast h
linarith
· have h' : (q : ℝ) + 1 ≤ a := by exact_mod_cast h
linarith
lemma selector_on_cell (b a q : ℕ) (v : ℝ)
(h₀ : 2 * (q : ℝ) ≤ b * v) (h₁ : (b : ℝ) * v ≤ 2 * q + 1) :
selector b a v = if a = q then 1 else 0 := by
split_ifs with h
· subst a
apply selector_one <;> linarith
· rcases lt_or_gt_of_ne h with ha | ha
· have h' : (a : ℝ) + 1 ≤ q := by exact_mod_cast ha
apply selector_zero_right
linarith
· have h' : (q : ℝ) + 1 ≤ a := by exact_mod_cast ha
apply selector_zero_left
linarith
lemma selector_code (b a q : ℕ) (hb : 2 ≤ b) (L : List ℕ)
(hL : ∀ d ∈ L, d + 2 ≤ b) :
selector b a (Encoding.code b (2 * q :: L)) = if a = q then 1 else 0 := by
have hb' : (0 : ℝ) < b := by exact_mod_cast (show 0 < b by omega)
have h := Encoding.digit_interval b (2 * q) hb L hL
have h₀ := (div_le_iff₀ hb').1 h.1
have h₁ := (le_div_iff₀ hb').1 h.2
push_cast at h₀ h₁
apply selector_on_cell <;> nlinarith
lemma selector_streamCode (b a q K : ℕ) (hb : 2 ≤ b) (hK : 0 < K) (f : ℕ → ℕ)
(hf : ∀ j, f j + 2 ≤ b) (h₀ : f 0 = 2 * q) :
selector b a (Encoding.streamCode b K f) = if a = q then 1 else 0 := by
obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hK)
rw [Encoding.streamCode, Encoding.digits_succ, h₀]
apply selector_code b a q hb
intro d hd
obtain ⟨i, rfl⟩ := List.mem_ofFn.1 hd
exact hf _
end Profiles
namespace Machine
lemma selector_recordedBlock (M : Machine) (hM : M.WellFormed) (w : List ℕ)
(hw : M.ValidInput w) (n q : ℕ) :
Profiles.selector M.base q (M.recordedBlock w n) =
if q = (M.run w n).state then 1 else 0 := by
apply Profiles.selector_code _ _ _ (by have := base_ge_four M; omega)
intro d hd
exact recordedDigits_bound M hM w hw n d (List.mem_cons_of_mem _ hd)
lemma selector_leftCode (M : Machine) (c : Configuration) (hc : M.Admissible c) (N n a : ℕ) :
Profiles.selector M.base a (leftCode M.base N n c) =
if a = c.tape (c.head - 1) then 1 else 0 := by
apply Profiles.selector_streamCode _ _ _ _ (by have := base_ge_four M; omega)
(by dsimp [Scales.K]; omega)
· intro j
exact symbol_digit_bound M _ (hc.2 _)
· simp [leftDigits]
lemma selector_rightCode (M : Machine) (c : Configuration) (hc : M.Admissible c) (N n a : ℕ) :
Profiles.selector M.base a (rightCode M.base N n c) =
if a = c.tape c.head then 1 else 0 := by
apply Profiles.selector_streamCode _ _ _ _ (by have := base_ge_four M; omega)
(by dsimp [Scales.K]; omega)
· intro j
exact symbol_digit_bound M _ (hc.2 _)
· simp [rightDigits]
end Machine
end AlternatingNS
end OAI
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