refactor: Lint
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jstoobysmith
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36752fa5c4
commit
73df7d24a7
4 changed files with 53 additions and 96 deletions
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@ -187,24 +187,16 @@ open Marked
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/-- Contracting the indices of `ofMatUpDown` returns the trace of the matrix. -/
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lemma contr_ofMatUpDown_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatUpDown M) (by rfl) = ofReal d M.trace := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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exact Empty.elim i
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· funext i
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simp only [Fin.isValue, contr, IndexValue, Equiv.cast_apply, trace, diag_apply, ofReal,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
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apply Finset.sum_congr rfl
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intro x _
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rfl
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/-- Contracting the indices of `ofMatDownUp` returns the trace of the matrix. -/
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lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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contr (ofMatDownUp M) (by rfl) = ofReal d M.trace := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-!
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@ -217,46 +209,38 @@ lemma contr_ofMatDownUp_eq_trace {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1
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lemma mul_ofVecUp_ofVecDown_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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mapIso d (@Equiv.equivEmpty (Empty ⊕ Empty) instIsEmptySum)
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(mul (ofVecUp v₁) (ofVecDown v₂) (by rfl)) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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/-- Multiplying `ofVecDown` with `ofVecUp` gives the dot product. -/
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@[simp]
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lemma mul_ofVecDown_ofVecUp_eq_dot_prod {d : ℕ} (v₁ v₂ : Fin 1 ⊕ Fin d → ℝ) :
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mapIso d (Equiv.equivEmpty (Empty ⊕ Empty))
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(mul (ofVecDown v₁) (ofVecUp v₂) rfl) = ofReal d (v₁ ⬝ᵥ v₂) := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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exact Empty.elim i
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· funext i
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rfl
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lemma mul_ofMatUpDown_ofVecUp_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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mapIso d ((Equiv.sumEmpty (Empty ⊕ Fin 1) Empty))
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(mul (unmarkFirst $ ofMatUpDown M) (ofVecUp v) rfl) = ofVecUp (M *ᵥ v) := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, mapIso_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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lemma mul_ofMatDownUp_ofVecDown_eq_mulVec {d : ℕ} (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)
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(v : Fin 1 ⊕ Fin d → ℝ) :
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mapIso d (Equiv.sumEmpty (Empty ⊕ Fin 1) Empty)
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(mul (unmarkFirst $ ofMatDownUp M) (ofVecDown v) rfl) = ofVecDown (M *ᵥ v) := by
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refine ext ?_ ?_
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refine ext ?_ rfl
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· funext i
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simp only [Nat.succ_eq_add_one, Nat.reduceAdd, mapIso_apply_color, mul_color, Equiv.symm_symm]
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fin_cases i
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rfl
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· funext i
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rfl
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/-!
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@ -290,7 +274,7 @@ lemma lorentzAction_ofVecUp (v : Fin 1 ⊕ Fin d → ℝ) :
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intro i
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simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
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intro j _
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erw [toTensorRepMat_apply, Finset.prod_singleton]
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simp_all only [Finset.mem_univ, Fin.isValue, Finset.prod_singleton, indexValueIso_refl]
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rfl
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/-- The action of the Lorentz group on `ofVecDown v` is
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@ -310,7 +294,7 @@ lemma lorentzAction_ofVecDown (v : Fin 1 ⊕ Fin d → ℝ) :
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intro i
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simp_all only [Finset.mem_univ, Fin.isValue, Equiv.coe_fn_mk]
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intro j _
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erw [toTensorRepMat_apply, Finset.prod_singleton]
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simp_all only [Finset.mem_univ, Fin.isValue, Finset.prod_singleton, indexValueIso_refl]
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rfl
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@[simp]
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@ -327,12 +311,8 @@ lemma lorentzAction_ofMatUpUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [toTensorRepMat_apply]
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simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
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Fin.isValue, one_mul, indexValueIso_refl, IndexValue]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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congr
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simp only [Fin.prod_univ_two, Fin.isValue, indexValueIso_refl, IndexValue]
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exact mul_right_comm _ _ _
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@[simp]
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lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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@ -348,12 +328,8 @@ lemma lorentzAction_ofMatDownDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [toTensorRepMat_apply]
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simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
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Fin.isValue, one_mul, lorentzGroupIsGroup_inv, indexValueIso_refl, IndexValue]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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congr
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simp only [Fin.prod_univ_two, Fin.isValue, indexValueIso_refl, IndexValue]
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exact mul_right_comm _ _ _
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@[simp]
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lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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@ -369,12 +345,8 @@ lemma lorentzAction_ofMatUpDown (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [toTensorRepMat_apply]
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simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
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Fin.isValue, one_mul, indexValueIso_refl, IndexValue, lorentzGroupIsGroup_inv]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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congr
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simp only [Fin.prod_univ_two, Fin.isValue, indexValueIso_refl, IndexValue]
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exact mul_right_comm _ _ _
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@[simp]
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lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
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@ -391,12 +363,8 @@ lemma lorentzAction_ofMatDownUp (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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refine Finset.sum_congr rfl (fun x _ => ?_)
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rw [Finset.sum_mul]
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refine Finset.sum_congr rfl (fun y _ => ?_)
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erw [toTensorRepMat_apply]
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simp only [Fintype.prod_sum_type, Finset.univ_eq_empty, Finset.prod_empty, Fin.prod_univ_two,
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Fin.isValue, one_mul, lorentzGroupIsGroup_inv, indexValueIso_refl, IndexValue]
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simp only [colorMatrix, Fin.isValue, MonoidHom.coe_mk, OneHom.coe_mk]
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rw [mul_assoc, mul_comm _ (M _ _), ← mul_assoc]
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congr
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simp only [Fin.prod_univ_two, Fin.isValue, indexValueIso_refl, IndexValue]
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exact mul_right_comm _ _ _
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end lorentzAction
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