| import Mathlib |
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| namespace OAI |
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| open scoped BigOperators Classical |
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| namespace BufferedIsing.DSS |
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| / |
| noncomputable def update (a c : ℝ) : ℝ := (a + c) / (1 + a * c) |
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| 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 |
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| 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)] |
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| 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 |
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| 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 |
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| 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 |
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| / |
| 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 |
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| end BufferedIsing.DSS |
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| end OAI |
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