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dd14c26 | 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 | import Mathlib
namespace OAI
open scoped BigOperators Classical
namespace BufferedIsing.DSS
/-- The rational single-spin field update. -/
noncomputable def update (a c : ℝ) : ℝ := (a + c) / (1 + a * c)
lemma update_den_pos {a c : ℝ} (ha : -1 < a ∧ a < 1)
(hc : -1 < c ∧ c < 1) : 0 < 1 + a * c := by
have h := abs_mul a c
have haa : |a| < 1 := abs_lt.mpr ha
have hcc : |c| < 1 := abs_lt.mpr hc
have hh : |a * c| < 1 := by
rw [abs_mul]
nlinarith [abs_nonneg a, abs_nonneg c]
have := (abs_lt.mp hh).1
linarith
lemma interpolation_positive_sum {a b c : ℝ}
(ha : -1 < a ∧ a < 1) (hb : -1 < b ∧ b < 1)
(hab : b ≤ a) (hc : 0 ≤ c ∧ c < 1) (hs : 0 ≤ a + b) :
∃ d : ℝ, -1 < d ∧ d < 1 ∧
(1 - a) * (1 - d) / (1 + a * d) ≤
(1 + (a - b) * c / 2) * (1 - a) / (1 + a * c) ∧
(1 + b) * (1 + d) / (1 + b * d) ≤
(1 + (a - b) * c / 2) * (1 + b) / (1 - b * c) := by
have ha0 : 0 ≤ a := by linarith
let k := 1 + (a - b) * c / 2
let A := 1 + a * c
have hk : 0 < k := by dsimp [k]; nlinarith [mul_nonneg (sub_nonneg.mpr hab) hc.1]
have hA : 0 < A := by dsimp [A]; nlinarith [mul_nonneg ha0 hc.1]
have hD : 0 < A + a * k := by positivity
let d := (A - k) / (A + a * k)
have hN : 0 ≤ A - k := by dsimp [A, k]; nlinarith [mul_nonneg hs hc.1]
have hd0 : 0 ≤ d := div_nonneg hN hD.le
have hd1 : d < 1 := by
apply (div_lt_one hD).mpr
nlinarith [mul_pos hk (by linarith : 0 < 1 + a)]
have hda : 0 < 1 + a * d := update_den_pos ha ⟨by linarith, hd1⟩
have hdb : 0 < 1 + b * d := update_den_pos hb ⟨by linarith, hd1⟩
have hbC : 0 < 1 - b * c := by
have := update_den_pos hb (a := b) (c := -c) (by constructor <;> linarith)
nlinarith
have heq : A * (1 - d) = k * (1 + a * d) := by
dsimp [d]
field_simp
ring
refine ⟨d, by linarith, hd1, ?_, ?_⟩
· apply (div_le_div_iff₀ hda hA).mpr
change (1 - a) * (1 - d) * A ≤ k * (1 - a) * (1 + a * d)
nlinarith [heq]
· have hsums : (1 + a * d) * (1 + d) + (1 + b * d) * (1 - d) ≥
2 * (1 - d) * (1 + d) := by
nlinarith [mul_nonneg hs hd0,
mul_nonneg (show 0 ≤ 2 + a - b by linarith) (sq_nonneg d)]
have ht : (1 - b * c) * (1 + d) ≤ k * (1 + b * d) := by
have hkA : A + (1 - b * c) = 2 * k := by dsimp [A,k]; ring
have hh := mul_nonneg hk.le (sub_nonneg.mpr hsums)
nlinarith [heq]
apply (div_le_div_iff₀ hdb hbC).mpr
change (1 + b) * (1 + d) * (1 - b * c) ≤ k * (1 + b) * (1 + b * d)
nlinarith [mul_nonneg (by linarith : 0 ≤ 1 + b) (sub_nonneg.mpr ht)]
lemma interpolation_pair {a b c : ℝ}
(ha : -1 < a ∧ a < 1) (hb : -1 < b ∧ b < 1)
(hab : b ≤ a) (hc : 0 ≤ c ∧ c < 1) :
∃ d : ℝ, -1 < d ∧ d < 1 ∧
(1 - a) * (1 - d) / (1 + a * d) ≤
(1 + (a - b) * c / 2) * (1 - a) / (1 + a * c) ∧
(1 + b) * (1 + d) / (1 + b * d) ≤
(1 + (a - b) * c / 2) * (1 + b) / (1 - b * c) := by
by_cases hs : 0 ≤ a + b
· exact interpolation_positive_sum ha hb hab hc hs
· obtain ⟨e, he0, he1, h1, h2⟩ := interpolation_positive_sum
(a := -b) (b := -a) (by constructor <;> linarith)
(by constructor <;> linarith) (by linarith) hc (by linarith)
refine ⟨-e, by linarith, by linarith, ?_, ?_⟩
· convert h2 using 1 <;> congr 1 <;> ring
· convert h1 using 1 <;> congr 1 <;> ring
lemma one_sub_update {a c : ℝ} (hd : 1 + a * c ≠ 0) :
1 - update a c = (1 - a) * (1 - c) / (1 + a * c) := by
dsimp [update]
field_simp
ring
lemma one_add_update {a c : ℝ} (hd : 1 + a * c ≠ 0) :
1 + update a c = (1 + a) * (1 + c) / (1 + a * c) := by
dsimp [update]
field_simp
ring
/-- Finite-field interpolation with a common finite field parameter for both one-sided mixtures. -/
theorem interpolation {a b c z : ℝ}
(ha : -1 < a ∧ a < 1) (hb : -1 < b ∧ b < 1)
(hab : b ≤ a) (hc : 0 ≤ c ∧ c < 1)
(hz : 0 ≤ z) (hgap : a - b ≤ 2 * z) :
∃ d : ℝ, -1 < d ∧ d < 1 ∧
update a c ≤ ((1 - c) / (1 + z * c)) * update a d +
(1 - (1 - c) / (1 + z * c)) ∧
((1 - c) / (1 + z * c)) * update b d -
(1 - (1 - c) / (1 + z * c)) ≤ update b (-c) := by
obtain ⟨d, hd0, hd1, h1, h2⟩ := interpolation_pair ha hb hab hc
let α := (1 - c) / (1 + z * c)
let k := 1 + (a - b) * c / 2
have hc' : -1 < c ∧ c < 1 := ⟨by linarith, hc.2⟩
have hmc : -1 < -c ∧ -c < 1 := by constructor <;> linarith
have hK : 0 < 1 + z * c := by nlinarith [mul_nonneg hz hc.1]
have hα : 0 ≤ α := div_nonneg (by linarith) hK.le
have hk : k ≤ 1 + z * c := by
dsimp [k]
nlinarith [mul_nonneg (sub_nonneg.mpr hgap) hc.1]
have hαk : α * k ≤ 1 - c := by
calc
α * k ≤ α * (1 + z * c) := mul_le_mul_of_nonneg_left hk hα
_ = 1 - c := by exact div_mul_cancel₀ _ hK.ne'
have hDa : 0 ≤ (1 - a) / (1 + a * c) :=
div_nonneg (by linarith) (update_den_pos ha hc').le
have hDb : 0 ≤ (1 + b) / (1 - b * c) := by
apply div_nonneg (by linarith)
have := update_den_pos hb hmc
nlinarith
have H1 : α * (1 - update a d) ≤ 1 - update a c := by
rw [one_sub_update (update_den_pos ha ⟨hd0, hd1⟩).ne',
one_sub_update (update_den_pos ha hc').ne']
calc
_ ≤ α * (k * (1 - a) / (1 + a * c)) := mul_le_mul_of_nonneg_left h1 hα
_ = (α * k) * ((1 - a) / (1 + a * c)) := by ring
_ ≤ (1 - c) * ((1 - a) / (1 + a * c)) := mul_le_mul_of_nonneg_right hαk hDa
_ = _ := by ring
have H2 : α * (1 + update b d) ≤ 1 + update b (-c) := by
rw [one_add_update (update_den_pos hb ⟨hd0, hd1⟩).ne',
one_add_update (update_den_pos hb hmc).ne']
calc
_ ≤ α * (k * (1 + b) / (1 - b * c)) := mul_le_mul_of_nonneg_left h2 hα
_ = (α * k) * ((1 + b) / (1 - b * c)) := by ring
_ ≤ (1 - c) * ((1 + b) / (1 - b * c)) := mul_le_mul_of_nonneg_right hαk hDb
_ = _ := by simp only [mul_neg, ← sub_eq_add_neg]; ring
refine ⟨d, hd0, hd1, ?_, ?_⟩
· change update a c ≤ α * update a d + (1 - α)
linarith
· change α * update b d - (1 - α) ≤ update b (-c)
linarith
end BufferedIsing.DSS
end OAI
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