chore: Bump to 4.11.0
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48 changed files with 404 additions and 137 deletions
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@ -85,6 +85,7 @@ variable {𝓵 : ℝ}
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variable (μ2 : ℝ)
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variable (h𝓵 : 0 < 𝓵)
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include h𝓵
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/-- The second term of the potential is non-negative. -/
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lemma snd_term_nonneg (φ : HiggsField) (x : SpaceTime) :
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0 ≤ 𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x := by
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@ -93,11 +94,13 @@ lemma snd_term_nonneg (φ : HiggsField) (x : SpaceTime) :
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simp_all only [normSq, gt_iff_lt, mul_nonneg_iff_of_pos_left, ge_iff_le, norm_nonneg, pow_nonneg,
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and_self]
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omit h𝓵
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lemma as_quad (μ2 𝓵 : ℝ) (φ : HiggsField) (x : SpaceTime) :
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𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- μ2) * ‖φ‖_H ^ 2 x + (- potential μ2 𝓵 φ x) = 0 := by
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simp only [normSq, neg_mul, potential, neg_add_rev, neg_neg]
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ring
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include h𝓵
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/-- The discriminant of the quadratic formed by the potential is non-negative. -/
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lemma discrim_nonneg (φ : HiggsField) (x : SpaceTime) :
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0 ≤ discrim 𝓵 (- μ2) (- potential μ2 𝓵 φ x) := by
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@ -141,7 +144,7 @@ lemma bounded_below (φ : HiggsField) (x : SpaceTime) :
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ring_nf at h1
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rw [← neg_le_iff_add_nonneg'] at h1
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rw [show 𝓵 * potential μ2 𝓵 φ x * 4 = (4 * 𝓵) * potential μ2 𝓵 φ x by ring] at h1
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have h2 := (div_le_iff' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
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have h2 := (div_le_iff₀' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
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ring_nf at h2 ⊢
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exact h2
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@ -164,12 +167,14 @@ variable (h𝓵 : 0 < 𝓵)
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-/
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include h𝓵
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lemma discrim_eq_zero_of_eq_bound (φ : HiggsField) (x : SpaceTime)
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(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
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discrim 𝓵 (- μ2) (- potential μ2 𝓵 φ x) = 0 := by
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rw [discrim, hV]
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field_simp
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include h𝓵
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lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
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(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
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‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) := by
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