Merge pull request #46 from pitmonticone/golf-proofs

Golf a few proofs
HEPLean · Jun 9, 2024 · 1cb2cdf · 1cb2cdf
2 parents de720c6 + 9850a9c
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diff --git a/HepLean/SpaceTime/Metric.lean b/HepLean/SpaceTime/Metric.lean
@@ -85,40 +85,22 @@ lemma η_transpose : η.transpose = η := by
 
 @[simp]
 lemma det_η : η.det = - 1 := by
-  simp only [η_explicit, det_succ_row_zero, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue,
-    of_apply, cons_val', empty_val', cons_val_fin_one, cons_val_zero, submatrix_apply,
-    Fin.succ_zero_eq_one, cons_val_one, head_cons, submatrix_submatrix, Function.comp_apply,
-    Fin.succ_one_eq_two, cons_val_two, tail_cons, det_unique, Fin.default_eq_zero, cons_val_succ,
-    head_fin_const, Fin.sum_univ_succ, Fin.val_zero, pow_zero, one_mul, Fin.zero_succAbove,
-    Finset.univ_unique, Fin.val_succ, Fin.coe_fin_one, zero_add, pow_one, neg_mul,
-    Fin.succ_succAbove_zero, Finset.sum_neg_distrib, Finset.sum_singleton, Fin.succ_succAbove_one,
-    even_two, Even.neg_pow, one_pow, mul_one, mul_neg, neg_neg, mul_zero, neg_zero, add_zero,
-    zero_mul, Finset.sum_const_zero]
+  simp [η_explicit, det_succ_row_zero, Fin.sum_univ_succ]
 
 @[simp]
 lemma η_sq : η * η = 1 := by
   funext μ ν
-  rw [mul_apply, Fin.sum_univ_four]
   fin_cases μ <;> fin_cases ν <;>
-     simp [η_explicit, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
-      Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
-      Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
-      vecHead, vecTail, Function.comp_apply]
+     simp [η_explicit, vecHead, vecTail]
 
 lemma η_diag_mul_self (μ : Fin 4) : η μ μ * η μ μ  = 1 := by
-  fin_cases μ
-    <;> simp [η_explicit]
+  fin_cases μ <;> simp [η_explicit]
 
 lemma η_mulVec (x : spaceTime) : η *ᵥ x = ![x 0, -x 1, -x 2, -x 3] := by
-  rw [explicit x]
-  rw [η_explicit]
+  rw [explicit x, η_explicit]
   funext i
-  rw [mulVec, dotProduct, Fin.sum_univ_four]
   fin_cases i <;>
-    simp [Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one, Matrix.cons_val_one,
-      Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
-      Matrix.head_fin_const, Matrix.head_cons, Matrix.vecCons_const, Fin.mk_one, Fin.mk_one,
-      vecHead, vecTail, Function.comp_apply]
+  simp [vecHead, vecTail]
 
 /-- Given a point in spaceTime `x` the linear map `y → x ⬝ᵥ (η *ᵥ y)`. -/
 @[simps!]
@@ -128,7 +110,7 @@ def linearMapForSpaceTime (x : spaceTime) : spaceTime →ₗ[ℝ] ℝ where
     simp only
     rw [mulVec_add, dotProduct_add]
   map_smul' c y := by
-    simp only
+    simp only [RingHom.id_apply, smul_eq_mul]
     rw [mulVec_smul, dotProduct_smul]
     rfl
 
@@ -168,7 +150,7 @@ lemma time_elm_sq_of_ηLin (x : spaceTime) : x 0 ^ 2 = ηLin x x + ‖x.space‖
 
 lemma ηLin_leq_time_sq (x : spaceTime) : ηLin x x ≤ x 0 ^ 2 := by
   rw [time_elm_sq_of_ηLin]
-  apply (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖x.space‖
+  exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖x.space‖
 
 lemma ηLin_space_inner_product (x y : spaceTime) :
     ηLin x y = x 0 * y 0 - ⟪x.space, y.space⟫_ℝ  := by
@@ -202,18 +184,18 @@ lemma ηLin_stdBasis_apply (μ : Fin 4) (x : spaceTime) : ηLin (stdBasis μ) x
 lemma ηLin_η_stdBasis (μ ν : Fin 4) : ηLin (stdBasis μ) (stdBasis ν) = η μ ν := by
   rw [ηLin_stdBasis_apply]
   by_cases h : μ = ν
-  rw [stdBasis_apply]
-  subst h
-  simp only [↓reduceIte, mul_one]
-  rw [stdBasis_not_eq, η_off_diagonal h]
-  simp only [mul_zero]
-  exact fun a => h (id a.symm)
+  · rw [stdBasis_apply]
+    subst h
+    simp
+  · rw [stdBasis_not_eq, η_off_diagonal h]
+    simp only [mul_zero]
+    exact fun a ↦ h (id a.symm)
 
 lemma ηLin_mulVec_left (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
     ηLin (Λ *ᵥ x) y = ηLin x ((η * Λᵀ * η) *ᵥ y) := by
-  simp only [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply, mulVec_mulVec]
-  rw [(vecMul_transpose Λ x).symm, ← dotProduct_mulVec, mulVec_mulVec]
-  rw [← mul_assoc, ← mul_assoc, η_sq, one_mul]
+  simp [ηLin, LinearMap.coe_mk, AddHom.coe_mk, linearMapForSpaceTime_apply,
+    mulVec_mulVec, (vecMul_transpose Λ x).symm, ← dotProduct_mulVec, mulVec_mulVec,
+    ← mul_assoc, ← mul_assoc, η_sq, one_mul]
 
 lemma ηLin_mulVec_right (x y : spaceTime) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
     ηLin x (Λ *ᵥ y) = ηLin ((η * Λᵀ * η) *ᵥ x) y := by
@@ -231,14 +213,14 @@ lemma ηLin_matrix_stdBasis' (μ ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
 lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
     Λ = 1 ↔ ∀ (x y : spaceTime), ηLin x y = ηLin x (Λ *ᵥ y) := by
   apply Iff.intro
-  intro h
-  subst h
-  simp only [ηLin, one_mulVec, implies_true]
-  intro h
-  funext μ ν
-  have h1 := h (stdBasis μ) (stdBasis ν)
-  rw [ηLin_matrix_stdBasis, ηLin_η_stdBasis] at h1
-  fin_cases μ <;> fin_cases ν <;>
+  · intro h
+    subst h
+    simp only [ηLin, one_mulVec, implies_true]
+  · intro h
+    funext μ ν
+    have h1 := h (stdBasis μ) (stdBasis ν)
+    rw [ηLin_matrix_stdBasis, ηLin_η_stdBasis] at h1
+    fin_cases μ <;> fin_cases ν <;>
     simp_all [η_explicit, Fin.zero_eta, Matrix.cons_val', Matrix.cons_val_fin_one,
       Matrix.cons_val_one,
       Matrix.cons_val_succ', Matrix.cons_val_zero, Matrix.empty_val', Matrix.head_cons,
@@ -248,9 +230,6 @@ lemma ηLin_matrix_eq_identity_iff (Λ : Matrix (Fin 4) (Fin 4) ℝ) :
 /-- The metric as a quadratic form on `spaceTime`. -/
 def quadraticForm : QuadraticForm ℝ spaceTime := ηLin.toQuadraticForm
 
-
-
-
 end spaceTime
 
 end
diff --git a/HepLean/StandardModel/HiggsBoson/Basic.lean b/HepLean/StandardModel/HiggsBoson/Basic.lean
@@ -51,7 +51,7 @@ instance : NormedAddCommGroup (Fin 2 → ℂ) := by
 /-- Given a vector `ℂ²` the constant higgs field with value equal to that
 section. -/
 noncomputable def higgsVec.toField (φ : higgsVec) : higgsField where
-  toFun := fun _ => φ
+  toFun := fun _ ↦ φ
   contMDiff_toFun := by
     intro x
     rw [Bundle.contMDiffAt_section]
@@ -63,7 +63,6 @@ open  Complex Real
 /-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/
 def toHiggsVec (φ : higgsField) : spaceTime → higgsVec := φ
 
-
 lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, higgsVec) φ.toHiggsVec := by
   intro x0
   have h1 := φ.contMDiff x0
@@ -76,29 +75,25 @@ lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ
   exact h1
 
 lemma toField_toHiggsVec_apply (φ : higgsField) (x : spaceTime) :
-    (φ.toHiggsVec x).toField x = φ x := by
-  rfl
+    (φ.toHiggsVec x).toField x = φ x := rfl
 
-lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) : higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := by
-  ext x
-  rfl
+lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) :
+    higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := rfl
 
-lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ := by
-  rw [← φ.higgsVecToFin2ℂ_toHiggsVec]
-  exact Smooth.comp smooth_higgsVecToFin2ℂ (φ.toHiggsVec_smooth)
+lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ :=
+  smooth_higgsVecToFin2ℂ.comp φ.toHiggsVec_smooth
 
 lemma apply_smooth (φ : higgsField) :
-    ∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) := by
-  rw [← smooth_pi_space]
-  exact φ.toVec_smooth
+    ∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) :=
+  (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.comp (ContinuousLinearMap.smooth reCLM) (φ.apply_smooth 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.comp (ContinuousLinearMap.smooth imCLM) (φ.apply_smooth i)
+  (ContinuousLinearMap.smooth imCLM).comp (φ.apply_smooth i)
 
 /-- Given two `higgsField`, the map `spaceTime → ℂ` obtained by taking their inner product. -/
 def innerProd (φ1 φ2 : higgsField) : spaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ
@@ -115,9 +110,7 @@ lemma toHiggsVec_norm (φ : higgsField) (x : spaceTime) :
 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 only [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
-  rw [@norm_sq_eq_inner ℂ]
-  simp
+  simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]
 
 lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
   rw [normSq_expand]
@@ -140,10 +133,10 @@ lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ,
   exact φ.apply_im_smooth 1
 
 lemma normSq_nonneg (φ : higgsField) (x : spaceTime) : 0 ≤ φ.normSq x := by
-  simp only [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
+  simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
 
 lemma normSq_zero (φ : higgsField) (x : spaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
-  simp only [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
+  simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
 
 /-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
 @[simp]
@@ -153,10 +146,8 @@ def potential (φ : higgsField) (μSq lambda : ℝ ) (x : spaceTime) :  ℝ :=
 lemma potential_smooth (φ : higgsField) (μSq lambda : ℝ) :
     Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
   simp only [potential, normSq, neg_mul]
-  exact Smooth.add
-    (Smooth.neg (Smooth.smul smooth_const φ.normSq_smooth))
-    (Smooth.smul (Smooth.smul smooth_const φ.normSq_smooth) φ.normSq_smooth)
-
+  exact (smooth_const.smul φ.normSq_smooth).neg.add
+    ((smooth_const.smul φ.normSq_smooth).smul φ.normSq_smooth)
 
 lemma potential_apply (φ : higgsField) (μSq lambda : ℝ) (x : spaceTime) :
     (φ.potential μSq lambda) x = higgsVec.potential μSq lambda (φ.toHiggsVec x) := by
@@ -171,35 +162,20 @@ lemma isConst_of_higgsVec (φ : higgsVec) : φ.toField.isConst := by
   intro x _
   simp [higgsVec.toField]
 
-lemma isConst_iff_of_higgsVec (Φ : higgsField) : Φ.isConst ↔ ∃ (φ : higgsVec), Φ = φ.toField := by
-  apply Iff.intro
-  intro h
-  use Φ 0
-  ext x y
-  rw [← h x 0]
-  rfl
-  intro h
-  intro x y
-  obtain ⟨φ, hφ⟩ := h
-  subst hφ
-  rfl
+lemma isConst_iff_of_higgsVec (Φ : higgsField) : Φ.isConst ↔ ∃ (φ : higgsVec), Φ = φ.toField :=
+  Iff.intro (fun h ↦ ⟨Φ 0, by ext x y; rw [← h x 0]; rfl⟩) (fun ⟨φ, hφ⟩ x y ↦ by subst hφ; rfl)
 
 lemma normSq_of_higgsVec (φ : higgsVec) : φ.toField.normSq = fun x => (norm φ) ^ 2 := by
-  simp only [normSq, higgsVec.toField]
   funext x
-  simp
+  simp [normSq, higgsVec.toField]
 
 lemma potential_of_higgsVec (φ : higgsVec) (μSq lambda : ℝ) :
     φ.toField.potential μSq lambda = fun _ => higgsVec.potential μSq lambda φ := by
   simp [higgsVec.potential]
   unfold potential
   rw [normSq_of_higgsVec]
-  funext x
-  simp only [neg_mul, add_right_inj]
   ring_nf
 
-
-
 end higgsField
 end
 end StandardModel