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refactor: Last batch of multi-goal proofs

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jstoobysmith 2024-08-21 06:40:58 -04:00
parent b9479c904d
commit c0499483a8
43 changed files with 910 additions and 955 deletions
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12
HepLean/StandardModel/HiggsBoson/PointwiseInnerProd.lean
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@ -139,12 +139,12 @@ lemma normSq_zero (φ : HiggsField) (x : SpaceTime) : φ.normSq x = 0 ↔ φ x =
lemma normSq_smooth (φ : HiggsField) : Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
rw [normSq_expand]
refine Smooth.add ?_ ?_
simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
exact ((φ.apply_re_smooth 0).smul (φ.apply_re_smooth 0)).add $
(φ.apply_im_smooth 0).smul (φ.apply_im_smooth 0)
simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
exact ((φ.apply_re_smooth 1).smul (φ.apply_re_smooth 1)).add $
(φ.apply_im_smooth 1).smul (φ.apply_im_smooth 1)
· simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
exact ((φ.apply_re_smooth 0).smul (φ.apply_re_smooth 0)).add $
(φ.apply_im_smooth 0).smul (φ.apply_im_smooth 0)
· simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
exact ((φ.apply_re_smooth 1).smul (φ.apply_re_smooth 1)).add $
(φ.apply_im_smooth 1).smul (φ.apply_im_smooth 1)
end HiggsField

32
HepLean/StandardModel/HiggsBoson/Potential.lean
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@ -115,24 +115,24 @@ lemma eq_zero_at (φ : HiggsField) (x : SpaceTime)
linear_combination h1
simp at h2
cases' h2 with h2 h2
simp_all
apply Or.inr
field_simp at h2 ⊢
ring_nf
linear_combination h2
· simp_all
· apply Or.inr
field_simp at h2 ⊢
ring_nf
linear_combination h2
lemma eq_zero_at_of_μSq_nonpos {μ2 : ℝ} (hμ2 : μ2 ≤ 0)
(φ : HiggsField) (x : SpaceTime) (hV : potential μ2 𝓵 φ x = 0) : φ x = 0 := by
cases' (eq_zero_at μ2 h𝓵 φ x hV) with h1 h1
exact h1
by_cases hμSqZ : μ2 = 0
simpa [hμSqZ] using h1
refine ((?_ : ¬ 0 ≤ μ2 / 𝓵) (?_)).elim
· simp_all [div_nonneg_iff]
intro h
exact lt_imp_lt_of_le_imp_le (fun _ => h) (lt_of_le_of_ne hμ2 hμSqZ)
· rw [← h1]
exact normSq_nonneg φ x
· exact h1
· by_cases hμSqZ : μ2 = 0
· simpa [hμSqZ] using h1
· refine ((?_ : ¬ 0 ≤ μ2 / 𝓵) (?_)).elim
· simp_all [div_nonneg_iff]
intro h
exact lt_imp_lt_of_le_imp_le (fun _ => h) (lt_of_le_of_ne hμ2 hμSqZ)
· rw [← h1]
exact normSq_nonneg φ x
lemma bounded_below (φ : HiggsField) (x : SpaceTime) :
- μ2 ^ 2 / (4 * 𝓵) ≤ potential μ2 𝓵 φ x := by
@ -176,8 +176,8 @@ lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
have h1 := as_quad μ2 𝓵 φ x
rw [quadratic_eq_zero_iff_of_discrim_eq_zero _
(discrim_eq_zero_of_eq_bound μ2 h𝓵 φ x hV)] at h1
simp_rw [h1, neg_neg]
exact ne_of_gt h𝓵
· simp_rw [h1, neg_neg]
· exact ne_of_gt h𝓵
lemma eq_bound_iff (φ : HiggsField) (x : SpaceTime) :
potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) ↔ ‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) :=