refactor: Lint

HepLean.lean

@@ -129,6 +129,8 @@ import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.SuperCommute
 import HepLean.PerturbationTheory.Algebras.CrAnAlgebra.TimeOrder
 import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.Basic
 import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.NormalOrder
+import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.SuperCommute
+import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.TimeOrder
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.Basic
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.NormalOrder
 import HepLean.PerturbationTheory.Algebras.ProtoOperatorAlgebra.TimeContraction

HepLean/Mathematics/List/InsertIdx.lean

@@ -122,7 +122,8 @@ lemma insertIdx_length_fst_append {I : Type} (φ : I) : (φs φs' : List I) →
     List.insertIdx φs.length φ (φs ++ φs') = (φs ++ φ :: φs')
   | [], φs' => by simp
   | φ' :: φs, φs' => by
-    simp
+    simp only [List.length_cons, List.cons_append, List.insertIdx_succ_cons, List.cons.injEq,
+      true_and]
     exact insertIdx_length_fst_append φ φs φs'
 
 lemma get_eq_insertIdx_succAbove {I : Type} (i : I) (r : List I) (k : Fin r.length.succ) :

HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Grading.lean

@@ -25,7 +25,7 @@ def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
   Submodule.span ℂ {a | ∃ φs, a = ofCrAnList φs ∧ (𝓕 |>ₛ φs) = f}
 
 lemma ofCrAnList_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
-  (h : (𝓕 |>ₛ φs) = f) :
+    (h : (𝓕 |>ₛ φs) = f) :
     ofCrAnList φs ∈ statisticSubmodule f := by
   refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
 
@@ -234,6 +234,7 @@ lemma directSum_eq_bosonic_plus_fermionic
     conv_lhs => rw [hx, hy]
     abel
 
+/-- The instance of a graded algebra on `CrAnAlgebra`. -/
 instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
   one_mem := by
     simp only [statisticSubmodule]

HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean

@@ -134,7 +134,6 @@ lemma ι_superCommute_ofCrAnState_superCommute_ofCrAnState_ofCrAnState (φ1 φ2
   left
   use φ1, φ2, φ3
 
-@[simp]
 lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
     ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnState φ3]ₛca = 0 := by
   rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
@@ -147,24 +146,23 @@ lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2
     · rw [superCommute_fermionic_fermionic_symm h h']
       simp [ofCrAnList_singleton]
 
-@[simp]
 lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2 : 𝓕.CrAnStates)
     (φs : List 𝓕.CrAnStates) :
     ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnList φs]ₛca = 0 := by
   rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
   rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
   · rw [superCommute_bosonic_ofCrAnList_eq_sum _ _ h]
-    simp [ofCrAnList_singleton]
+    simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
   · rw [superCommute_fermionic_ofCrAnList_eq_sum _ _ h]
-    simp [ofCrAnList_singleton]
+    simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
 
 @[simp]
 lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
     (a : 𝓕.CrAnAlgebra) : ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, a]ₛca = 0 := by
   change (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
   have h1 : (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
     apply (ofCrAnListBasis.ext fun l ↦ ?_)
-    simp
+    simp [ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList]
   rw [h1]
   simp
 
@@ -239,7 +237,7 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
 lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
     x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
   have hx' := hx
-  simp only [fieldOpIdealSet,  exists_prop, Set.mem_setOf_eq] at hx
+  simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
   rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
     ⟨φ, φ', hdiff, rfl⟩
   · rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h

HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/SuperCommute.lean

@@ -49,8 +49,11 @@ lemma ι_superCommute_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈
   simp only [LinearMap.mem_ker, ← map_sub]
   exact ι_superCommute_right_zero_of_mem_ideal a (b1 - b2) h
 
-noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
-  toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a) (ι_superCommute_eq_of_equiv_right a)
+/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
+noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
+  FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
+  toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a)
+    (ι_superCommute_eq_of_equiv_right a)
   map_add' x y := by
     obtain ⟨x, hx⟩ := ι_surjective x
     obtain ⟨y, hy⟩ := ι_surjective y
@@ -64,11 +67,11 @@ noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) : FieldOpAlgebra 𝓕
     rw [← map_smul, ι_apply, ι_apply]
     simp
 
-lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) : superCommuteRight a (ι b) = ι [a, b]ₛca := by
-  rfl
+lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) :
+    superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
 
-lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) : superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by
-  rfl
+lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) :
+    superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
 
 lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
     superCommuteRight a1 = superCommuteRight a2 := by
@@ -85,6 +88,7 @@ lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
   simp only [add_sub_cancel]
   simp
 
+/-- The super commutor on the `FieldOpAlgebra`. -/
 noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
     FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
   toFun := Quotient.lift superCommuteRight superCommuteRight_eq_of_equiv

HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean

@@ -20,33 +20,35 @@ namespace FieldOpAlgebra
 variable {𝓕 : FieldSpecification}
 
 lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
-     (φs1 φs2 : List 𝓕.CrAnStates) (h :
+    (φs1 φs2 : List 𝓕.CrAnStates) (h :
       crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
       crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
-      crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2):
-    ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
-    = 0 := by
+      crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2) :
+    ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
+    ofCrAnList φs2) = 0 := by
   let l1 :=
-    (List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
+    (List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c)
+    ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
     ++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs1)
   let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
-    ++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
+    ++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1)
+    ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
   have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
       crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
-      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)):= by
+      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)) := by
     have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
-       (by simp_all)
-    rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2 by simp,
-     crAnTimeOrderList, h1]
+      (by simp_all)
+    rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
+      by simp, crAnTimeOrderList, h1]
     simp only [List.append_assoc, List.singleton_append, decide_not,
       Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
   have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
       crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
-      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)):= by
+      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)) := by
     have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
-       (by simp_all)
-    rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2 by simp,
-      crAnTimeOrderList, h1]
+        (by simp_all)
+    rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
+      by simp, crAnTimeOrderList, h1]
     simp only [List.singleton_append, decide_not,
       Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
     congr 1
@@ -67,11 +69,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 
       exact List.Perm.swap φ1 φ2 [φ3]
   have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
       crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
-      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)):= by
+      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)) := by
     have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
-       (by simp_all)
-    rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2 by simp,
-      crAnTimeOrderList, h1]
+        (by simp_all)
+    rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
+      by simp, crAnTimeOrderList, h1]
     simp only [List.singleton_append, decide_not,
       Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
     congr 1
@@ -85,11 +87,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 
         simp_all
   have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
       crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
-      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)):= by
+      • (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)) := by
     have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
-       (by simp_all)
-    rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2 by simp,
-      crAnTimeOrderList, h1]
+        (by simp_all)
+    rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
+      by simp, crAnTimeOrderList, h1]
     simp only [List.singleton_append, decide_not,
       Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
     congr 1
@@ -125,7 +127,8 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 
   repeat rw [mul_assoc]
   rw [← mul_sub, ← mul_sub, ← mul_sub]
   rw [← sub_mul, ← sub_mul, ← sub_mul]
-  trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ι (ofCrAnList l2)
+  trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
+    ι (ofCrAnList l2)
   rw [mul_assoc]
   congr
   rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
@@ -137,7 +140,7 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 
   simp_all
 
 lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
-     (φs1 φs2 : List 𝓕.CrAnStates):
+    (φs1 φs2 : List 𝓕.CrAnStates) :
     ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
     = 0 := by
   by_cases h :
@@ -150,7 +153,7 @@ lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAn
     simp
 
 @[simp]
-lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra):
+lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
     ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
   let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
     Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
@@ -215,20 +218,21 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
       rw [h1]
       simp only [map_smul]
       have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
-       (by simp_all)
+        (by simp_all)
       rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
       have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
-       (by simp_all)
+        (by simp_all)
       rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
       repeat rw [ofCrAnList_append]
       rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
       rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
       rw [← mul_smul_comm]
-      rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
+      rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc]
       rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
         ← smul_mul_assoc]
       rw [← sub_mul]
-      have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
+      have h1 : (ι (ofCrAnList [φ, ψ]) -
+          (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
         ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
         rw [superCommute_ofCrAnState_ofCrAnState]
         rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
@@ -237,7 +241,8 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
         rw [← ofCrAnList_append]
         simp
       rw [h1]
-      have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
+      have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈
+          Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
         apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
       rw [Subalgebra.mem_center_iff] at hc
       repeat rw [← mul_assoc]
@@ -272,26 +277,23 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
   · intro x hx hpx
     simp_all [pb, hpx]
 
-
 lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
     (hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
     ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
   rw [timeOrder_timeOrder_mid]
   have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
     exact Decidable.not_and_iff_or_not.mp hφψ
   rcases hφψ with hφψ | hφψ
-  · rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
-    have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
+  · rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
     simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
     simp_all
   · rw [superCommute_ofCrAnState_ofCrAnState_symm]
     simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
       neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
-    rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
+    rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
     simp only [mul_zero, zero_mul, map_zero, or_true]
     simp_all
 
-
 /-!
 
 ## Defining normal order for `FiedOpAlgebra`.

HepLean/PerturbationTheory/Koszul/KoszulSign.lean

@@ -280,7 +280,7 @@ lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (
     apply Wick.koszulSignInsert_eq_perm
     exact List.Perm.append_left φs (List.Perm.swap ψ φ φs')
 
-lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
+lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le]
     (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
     koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by
   intro φs
@@ -294,7 +294,7 @@ lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal
   rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
   simp
 
-lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
+lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
     (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
     koszulSign q le (φs' ++ φ :: ψ :: φs) = koszulSign q le (φs' ++ φs)
   | [], φs => by
@@ -395,7 +395,7 @@ lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le]
 
 -/
 
-lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
+lemma koszulSign_perm_eq_append [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
     (hp : φs.Perm φs') : (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
     koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2) := by
   let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
@@ -423,7 +423,7 @@ lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕)
     refine h φ' ?_
     exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ
 
-lemma koszulSign_perm_eq [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
+lemma koszulSign_perm_eq [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
     (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → (hp : φs.Perm φs') →
     koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)
   | [], φs, φs', φs2, h, hp => by

HepLean/PerturbationTheory/Koszul/KoszulSignInsert.lean

@@ -247,7 +247,7 @@ lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀ b
         exact h b (List.mem_cons_of_mem _ hb)
     · exact h φ1 (List.mem_cons_self _ _)
 
-lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
+lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
     (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
     koszulSignInsert q le φ φs = koszulSignInsert q le ψ φs
   | [] => by
@@ -268,7 +268,7 @@ lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans
       simp only [hr, ↓reduceIte, hψφ']
       rw [koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs]
 
-lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
+lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTrans 𝓕 le]
     (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
     koszulSignInsert q le φ' (φ :: ψ :: φs) = koszulSignInsert q le φ' φs := by
   intro φs