diff --git a/HepLean/SpaceTime/LorentzGroup/Basic.lean b/HepLean/SpaceTime/LorentzGroup/Basic.lean
index 30d2b82..5c737b5 100644
--- a/HepLean/SpaceTime/LorentzGroup/Basic.lean
+++ b/HepLean/SpaceTime/LorentzGroup/Basic.lean
@@ -89,7 +89,6 @@ lemma mem_iff_self_mul_dual :  Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1  := by
   rw [mem_iff_dual_mul_self]
   exact mul_eq_one_comm
 
-
 lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d := by
   apply Iff.intro
   · intro h
@@ -236,7 +235,6 @@ lemma toGL_embedding : Embedding (@toGL d).toFun where
     rw [isOpen_induced_iff, isOpen_induced_iff]
     exact exists_exists_and_eq_and
 
-
 instance : TopologicalGroup (LorentzGroup d) :=
 Inducing.topologicalGroup toGL toGL_embedding.toInducing
 
@@ -276,9 +274,5 @@ lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
   erw [Pi.basisFun_apply, mulVec_stdBasis]
   simp
 
-
-
-
-
 end
 end LorentzGroup
diff --git a/HepLean/SpaceTime/LorentzVector/Basic.lean b/HepLean/SpaceTime/LorentzVector/Basic.lean
index 45543c2..c3d2a5a 100644
--- a/HepLean/SpaceTime/LorentzVector/Basic.lean
+++ b/HepLean/SpaceTime/LorentzVector/Basic.lean
@@ -65,7 +65,7 @@ scoped[LorentzVector] notation "e" => stdBasis
 lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : e μ ν = if μ = ν then 1 else 0 := by
   rw [stdBasis]
   erw [Pi.basisFun_apply]
-  simp
+  exact LinearMap.stdBasis_apply' ℝ μ ν
 
 /-- The standard unit time vector. -/
 noncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)
@@ -91,16 +91,14 @@ def spaceReflectionLin : LorentzVector d →ₗ[ℝ] LorentzVector d where
   map_add' x y := by
     funext i
     rcases i with i | i
-    · simp only [Sum.elim_inl]
-      apply Eq.refl
+    · rfl
     · simp only [Sum.elim_inr, Pi.neg_apply]
       apply neg_add
   map_smul' c x := by
     funext i
     rcases i with i | i
-    · simp only [Sum.elim_inl, Pi.smul_apply]
-      apply smul_eq_mul
-    · simp [ HSMul.hSMul, SMul.smul]
+    · rfl
+    · simp [HSMul.hSMul, SMul.smul]
 
 /-- The reflection of space. -/
 @[simp]
diff --git a/HepLean/SpaceTime/LorentzVector/NormOne.lean b/HepLean/SpaceTime/LorentzVector/NormOne.lean
index 532d8dd..2866976 100644
--- a/HepLean/SpaceTime/LorentzVector/NormOne.lean
+++ b/HepLean/SpaceTime/LorentzVector/NormOne.lean
@@ -184,7 +184,7 @@ lemma metric_reflect_mem_not_mem (h : v ∈ FuturePointing d) (hw : w ∉ Future
 
 lemma metric_reflect_not_mem_mem (h : v ∉ FuturePointing d) (hw :  w ∈ FuturePointing d) :
     ⟪v.1, w.1.spaceReflection⟫ₘ ≤ 0 := by
-  rw [show (0 : ℝ) = - 0 by simp, le_neg]
+  rw [show (0 : ℝ) = - 0 from zero_eq_neg.mpr rfl, le_neg]
   have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h)  hw
   apply le_of_le_of_eq h1 ?_
   simp [neg]
@@ -207,7 +207,7 @@ open LorentzVector
 @[simps!]
 noncomputable def timeVecNormOneFuture : FuturePointing d := ⟨⟨timeVec, by
   rw [NormOneLorentzVector.mem_iff, on_timeVec]⟩, by
-  rw [mem_iff, timeVec_time]; simp⟩
+  rw [mem_iff, timeVec_time]; exact Real.zero_lt_one⟩
 
 
 /-- A continuous path from `timeVecNormOneFuture` to any other. -/
@@ -222,15 +222,12 @@ noncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFutur
         conj_trivial]
       rw [Real.mul_self_sqrt, ← @real_inner_self_eq_norm_sq, @PiLp.inner_apply]
       simp only [Function.comp_apply, RCLike.inner_apply, conj_trivial]
-      have h1 : ∑ x : Fin d, t.1 * u.1.1 (Sum.inr x) * (↑t.1 * u.1.1 (Sum.inr x)) =
-        t ^ 2 * ∑ x : Fin d, u.1.1 (Sum.inr x) * u.1.1 (Sum.inr x) := by
-        rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i _
-        ring
-      rw [h1]
+      refine Eq.symm (eq_sub_of_add_eq (congrArg (HAdd.hAdd 1) ?_))
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro i _
       ring
-      refine Right.add_nonneg (zero_le_one' ℝ) $ mul_nonneg (sq_nonneg _) (sq_nonneg _) ⟩,
+      exact Right.add_nonneg (zero_le_one' ℝ) $ mul_nonneg (sq_nonneg _) (sq_nonneg _) ⟩,
     by
       simp only [space, Function.comp_apply, mem_iff_time_nonneg, time, Real.sqrt_pos]
       exact Real.sqrt_nonneg _⟩