refactor: pass at removing double spaces

HepLean/StandardModel/HiggsBoson/Basic.lean

@@ -72,7 +72,7 @@ We also define the Higgs bundle.
 /-- The trivial vector bundle 𝓡² × ℂ². -/
 abbrev HiggsBundle := Bundle.Trivial SpaceTime HiggsVec
 
-instance : SmoothVectorBundle HiggsVec HiggsBundle SpaceTime.asSmoothManifold  :=
+instance : SmoothVectorBundle HiggsVec HiggsBundle SpaceTime.asSmoothManifold :=
   Bundle.Trivial.smoothVectorBundle HiggsVec 𝓘(ℝ, SpaceTime)
 
 /-- A Higgs field is a smooth section of the Higgs bundle. -/
@@ -133,11 +133,11 @@ lemma apply_smooth (φ : HiggsField) :
   (smooth_pi_space).mp (φ.toVec_smooth)
 
 lemma apply_re_smooth (φ : HiggsField) (i : Fin 2) :
-    Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : SpaceTime) =>  (φ x i))) :=
+    Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
   (ContinuousLinearMap.smooth reCLM).comp (φ.apply_smooth i)
 
 lemma apply_im_smooth (φ : HiggsField) (i : Fin 2) :
-    Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : SpaceTime) =>  (φ x i))) :=
+    Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
   (ContinuousLinearMap.smooth imCLM).comp (φ.apply_smooth i)
 
 /-!

HepLean/StandardModel/HiggsBoson/GaugeAction.lean

@@ -34,9 +34,9 @@ open ComplexConjugate
 @[simps!]
 noncomputable def higgsRepUnitary : GaugeGroup →* unitaryGroup (Fin 2) ℂ where
   toFun g := repU1 g.2.2 * fundamentalSU2 g.2.1
-  map_mul'  := by
+  map_mul' := by
     intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
-    change repU1 (a3 * b3) *  fundamentalSU2 (a2 * b2) = _
+    change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
     rw [repU1.map_mul, fundamentalSU2.map_mul, mul_assoc, mul_assoc,
       ← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
     repeat rw [mul_assoc]
@@ -97,31 +97,31 @@ def rotateMatrix (φ : HiggsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
 
 lemma rotateMatrix_star (φ : HiggsVec) :
     star φ.rotateMatrix =
-    ![![conj (φ 1) /‖φ‖ ,  φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
+    ![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
   simp_rw [star, rotateMatrix, conjTranspose]
   ext i j
   fin_cases i <;> fin_cases j <;> simp [conj_ofReal]
 
 lemma rotateMatrix_det {φ : HiggsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
-  have h1 : (‖φ‖ : ℂ)  ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
+  have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
   field_simp [rotateMatrix, det_fin_two]
   rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
-  simp [PiLp.inner_apply, Complex.inner,  neg_mul, sub_neg_eq_add,
+  simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
     Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
 
 lemma rotateMatrix_unitary {φ : HiggsVec} (hφ : φ ≠ 0) :
     (rotateMatrix φ) ∈ unitaryGroup (Fin 2) ℂ := by
   rw [mem_unitaryGroup_iff', rotateMatrix_star, rotateMatrix]
   erw [mul_fin_two, one_fin_two]
-  have : (‖φ‖ : ℂ)  ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
+  have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
   ext i j
   fin_cases i <;> fin_cases j <;> field_simp
   <;> rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
-  · simp [PiLp.inner_apply, Complex.inner,  neg_mul, sub_neg_eq_add,
+  · simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
       Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
   · ring_nf
   · ring_nf
-  · simp [PiLp.inner_apply, Complex.inner,  neg_mul, sub_neg_eq_add,
+  · simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
       Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
 
 lemma rotateMatrix_specialUnitary {φ : HiggsVec} (hφ : φ ≠ 0) :
@@ -147,10 +147,10 @@ lemma rotateGuageGroup_apply {φ : HiggsVec} (hφ : φ ≠ 0) :
   · simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons, mulVec, Fin.isValue,
     cons_val', empty_val', cons_val_fin_one, vecHead, cons_dotProduct, vecTail, Nat.succ_eq_add_one,
     Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, dotProduct_empty, add_zero]
-    have : (‖φ‖ : ℂ)  ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
+    have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
     field_simp
     rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
-    simp [PiLp.inner_apply, Complex.inner,  neg_mul, sub_neg_eq_add,
+    simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
       Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
 
 theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
@@ -166,7 +166,7 @@ theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
 end HiggsVec
 
 /-! TODO: Define the global gauge action on HiggsField. -/
-/-! TODO: Prove `⟪φ1, φ2⟫_H`  invariant under the global gauge action. (norm_map_of_mem_unitary) -/
+/-! TODO: Prove `⟪φ1, φ2⟫_H` invariant under the global gauge action. (norm_map_of_mem_unitary) -/
 /-! TODO: Prove invariance of potential under global gauge action. -/
 
 end StandardModel

HepLean/StandardModel/HiggsBoson/PointwiseInnerProd.lean

@@ -55,13 +55,13 @@ lemma innerProd_left_zero (φ : HiggsField) : ⟪0, φ⟫_H = 0 := by
 lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
   funext x
   simp [innerProd]
-example  (x : ℝ): RCLike.ofReal x = ofReal' x := by
+example (x : ℝ): RCLike.ofReal x = ofReal' x := by
   rfl
 lemma innerProd_expand (φ1 φ2 : HiggsField) :
-    ⟪φ1, φ2⟫_H = fun x =>  equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
+    ⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
     + (φ1 x 1).re * (φ2 x 1).re+ (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im),
     ((φ1 x 0).re * (φ2 x 0).im + (φ1 x 1).re * (φ2 x 1).im
-    - (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re))  := by
+    - (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)) := by
   funext x
   simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
     equivRealProdCLM_symm_apply, ofReal_add, ofReal_mul, ofReal_sub]
@@ -124,9 +124,9 @@ lemma normSq_eq_innerProd_self (φ : HiggsField) (x : SpaceTime) :
 -/
 
 lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
-    ‖φ x‖  = ‖φ.toHiggsVec x‖ := rfl
+    ‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
 
-lemma normSq_expand (φ : HiggsField)  :
+lemma normSq_expand (φ : HiggsField) :
     φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1)).re := by
   funext x
   simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]

HepLean/StandardModel/HiggsBoson/Potential.lean

@@ -34,7 +34,7 @@ open SpaceTime
 
 /-- The Higgs potential of the form `- μ² * |φ|² + 𝓵 * |φ|⁴`. -/
 @[simp]
-def potential (μ2 𝓵 : ℝ ) (φ : HiggsField) (x : SpaceTime) :  ℝ :=
+def potential (μ2 𝓵 : ℝ ) (φ : HiggsField) (x : SpaceTime) : ℝ :=
   - μ2 * ‖φ‖_H ^ 2 x + 𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x
 
 /-!
@@ -94,7 +94,7 @@ lemma snd_term_nonneg (φ : HiggsField) (x : SpaceTime) :
     and_self]
 
 lemma as_quad (μ2 𝓵 : ℝ) (φ : HiggsField) (x : SpaceTime) :
-    𝓵  * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- μ2 ) * ‖φ‖_H ^ 2 x + (- potential μ2 𝓵 φ x) = 0 := by
+    𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- μ2 ) * ‖φ‖_H ^ 2 x + (- potential μ2 𝓵 φ x) = 0 := by
   simp only [normSq, neg_mul, potential, neg_add_rev, neg_neg]
   ring
 
@@ -121,7 +121,7 @@ lemma eq_zero_at (φ : HiggsField) (x : SpaceTime)
   ring_nf
   linear_combination h2
 
-lemma eq_zero_at_of_μSq_nonpos  {μ2 : ℝ} (hμ2 : μ2 ≤ 0)
+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
@@ -141,7 +141,7 @@ lemma bounded_below (φ : HiggsField) (x : SpaceTime) :
   ring_nf at h1
   rw [← neg_le_iff_add_nonneg'] at h1
   rw [show 𝓵 * potential μ2 𝓵 φ x * 4 = (4 * 𝓵) * potential μ2 𝓵 φ x by ring] at h1
-  have h2 :=  (div_le_iff' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
+  have h2 := (div_le_iff' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
   ring_nf at h2 ⊢
   exact h2
 
@@ -165,13 +165,13 @@ variable (h𝓵 : 0 < 𝓵)
 -/
 
 lemma discrim_eq_zero_of_eq_bound (φ : HiggsField) (x : SpaceTime)
-    (hV : potential μ2 𝓵 φ x  = - μ2 ^ 2 / (4 * 𝓵)) :
+    (hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
     discrim 𝓵 (- μ2) (- potential μ2 𝓵 φ x) = 0 := by
   rw [discrim, hV]
   field_simp
 
 lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
-    (hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4  * 𝓵)) :
+    (hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
     ‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) := by
   have h1 := as_quad μ2 𝓵 φ x
   rw [quadratic_eq_zero_iff_of_discrim_eq_zero _
@@ -180,7 +180,7 @@ lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
   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) :=
+    potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) ↔ ‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) :=
   Iff.intro (normSq_of_eq_bound μ2 h𝓵 φ x)
     (fun h ↦ by
       rw [potential, h]