From 82fae67ba3e2633a476452739afdc87a0eb87e29 Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Thu, 13 Feb 2025 10:44:15 +0000
Subject: [PATCH] refactor: Update supercommute notation

---
 .../FieldOpAlgebra/Basic.lean                 |  52 +++---
 .../FieldOpAlgebra/NormalOrder/Basic.lean     |  32 ++--
 .../FieldOpAlgebra/NormalOrder/Lemmas.lean    |   2 +-
 .../FieldOpAlgebra/SuperCommute.lean          |  14 +-
 .../FieldOpAlgebra/TimeOrder.lean             |  31 ++--
 .../FieldOpFreeAlgebra/Basic.lean             |   2 +-
 .../FieldOpFreeAlgebra/NormalOrder.lean       |  22 +--
 .../FieldOpFreeAlgebra/SuperCommute.lean      | 164 +++++++++---------
 .../FieldOpFreeAlgebra/TimeOrder.lean         |  18 +-
 .../FieldSpecification/CrAnFieldOp.lean       |  20 +--
 .../WickContraction/InsertAndContract.lean    |  19 +-
 .../WickContraction/Sign/Basic.lean           |   9 +-
 12 files changed, 197 insertions(+), 188 deletions(-)

diff --git a/HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean b/HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean
index d700622..9bfa5c9 100644
--- a/HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean
+++ b/HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean
@@ -24,24 +24,24 @@ variable (𝓕 : FieldSpecification)
 def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
   { x |
     (∃ (φ1 φ2 φ3 : 𝓕.CrAnFieldOp),
-        x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca)
+        x = [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF)
     ∨ (∃ (φc φc' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φc = .create) (_ : 𝓕 |>ᶜ φc' = .create),
-      x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛca)
+      x = [ofCrAnOpF φc, ofCrAnOpF φc']ₛF)
     ∨ (∃ (φa φa' : 𝓕.CrAnFieldOp) (_ : 𝓕 |>ᶜ φa = .annihilate) (_ : 𝓕 |>ᶜ φa' = .annihilate),
-      x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛca)
+      x = [ofCrAnOpF φa, ofCrAnOpF φa']ₛF)
     ∨ (∃ (φ φ' : 𝓕.CrAnFieldOp) (_ : ¬ (𝓕 |>ₛ φ) = (𝓕 |>ₛ φ')),
-      x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛca)}
+      x = [ofCrAnOpF φ, ofCrAnOpF φ']ₛF)}
 
 /-- For a field specification `𝓕`, the algebra `𝓕.FieldOpAlgebra` is defined as the quotient
   of the free algebra `𝓕.FieldOpFreeAlgebra` by the ideal generated by
-- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛca` for `φc` and `φc'` field creation operators.
+- `[ofCrAnOpF φc, ofCrAnOpF φc']ₛF` for `φc` and `φc'` field creation operators.
   This corresponds to the condition that two creation operators always super-commute.
-- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛca` for `φa` and `φa'` field annihilation operators.
+- `[ofCrAnOpF φa, ofCrAnOpF φa']ₛF` for `φa` and `φa'` field annihilation operators.
   This corresponds to the condition that two annihilation operators always super-commute.
-- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛca` for `φ` and `φ'` operators with different statistics.
+- `[ofCrAnOpF φ, ofCrAnOpF φ']ₛF` for `φ` and `φ'` operators with different statistics.
   This corresponds to the condition that two operators with different statistics always
   super-commute. In other words, fermions and bosons always super-commute.
-- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca`. This corresponds to the condition,
+- `[ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF`. This corresponds to the condition,
   when combined with the conditions above, that the super-commutator is in the center
   of the algebra.
 -/
@@ -70,7 +70,7 @@ lemma equiv_iff_exists_add (x y : FieldOpFreeAlgebra 𝓕) :
     rw [equiv_iff_sub_mem_ideal]
     simp [ha]
 
-/-- For a field specification `𝓕`, the projection
+/-- For a field specification `𝓕`, `ι` is defined as the projection
 
 `𝓕.FieldOpFreeAlgebra →ₐ[ℂ] FieldOpAlgebra 𝓕`
 
@@ -100,7 +100,7 @@ lemma ι_of_mem_fieldOpIdealSet (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.f
   simpa using hx
 
 lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = .create)
-    (hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛca = 0 := by
+    (hφc' : 𝓕 |>ᶜ φc' = .create) : ι [ofCrAnOpF φc, ofCrAnOpF φc']ₛF = 0 := by
   apply ι_of_mem_fieldOpIdealSet
   simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
   simp only [exists_prop]
@@ -110,7 +110,7 @@ lemma ι_superCommuteF_of_create_create (φc φc' : 𝓕.CrAnFieldOp) (hφc : 
 
 lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
     (hφa : 𝓕 |>ᶜ φa = .annihilate) (hφa' : 𝓕 |>ᶜ φa' = .annihilate) :
-    ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛca = 0 := by
+    ι [ofCrAnOpF φa, ofCrAnOpF φa']ₛF = 0 := by
   apply ι_of_mem_fieldOpIdealSet
   simp only [fieldOpIdealSet, exists_and_left, Set.mem_setOf_eq]
   simp only [exists_prop]
@@ -120,7 +120,7 @@ lemma ι_superCommuteF_of_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
   use φa, φa', hφa, hφa'
 
 lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnFieldOp}
-    (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
+    (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
   apply ι_of_mem_fieldOpIdealSet
   simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
   right
@@ -129,8 +129,8 @@ lemma ι_superCommuteF_of_diff_statistic {φ ψ : 𝓕.CrAnFieldOp}
   use φ, ψ
 
 lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnFieldOp)
-    (h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule fermionic) :
-    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
+    (h : [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ statisticSubmodule fermionic) :
+    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton] at h ⊢
   rcases statistic_neq_of_superCommuteF_fermionic h with h | h
   · simp only [ofCrAnListF_singleton]
@@ -139,8 +139,8 @@ lemma ι_superCommuteF_zero_of_fermionic (φ ψ : 𝓕.CrAnFieldOp)
   · simp [h]
 
 lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnFieldOp) :
-    [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ statisticSubmodule bosonic ∨
-    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca = 0 := by
+    [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ statisticSubmodule bosonic ∨
+    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF = 0 := by
   rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [ψ] with h | h
   · simp_all [ofCrAnListF_singleton]
   · simp_all only [ofCrAnListF_singleton]
@@ -155,14 +155,14 @@ lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero (φ ψ : 𝓕.CrAnFie
 
 @[simp]
 lemma ι_superCommuteF_ofCrAnOpF_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
-    ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca = 0 := by
+    ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF = 0 := by
   apply ι_of_mem_fieldOpIdealSet
   simp only [fieldOpIdealSet, exists_prop, exists_and_left, Set.mem_setOf_eq]
   left
   use φ1, φ2, φ3
 
 lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
-    ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnOpF φ3]ₛca = 0 := by
+    ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, ofCrAnOpF φ3]ₛF = 0 := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
   · rw [bonsonic_superCommuteF_symm h]
@@ -175,7 +175,7 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 φ3
 
 lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 : 𝓕.CrAnFieldOp)
     (φs : List 𝓕.CrAnFieldOp) :
-    ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, ofCrAnListF φs]ₛca = 0 := by
+    ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, ofCrAnListF φs]ₛF = 0 := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rcases superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ1] [φ2] with h | h
   · rw [superCommuteF_bosonic_ofCrAnListF_eq_sum _ _ h]
@@ -185,27 +185,27 @@ lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF (φ1 φ2 :
 
 @[simp]
 lemma ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_fieldOpFreeAlgebra (φ1 φ2 : 𝓕.CrAnFieldOp)
-    (a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca = 0 := by
-  change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) a = _
-  have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca) = 0 := by
+    (a : 𝓕.FieldOpFreeAlgebra) : ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, a]ₛF = 0 := by
+  change (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF) a = _
+  have h1 : (ι.toLinearMap ∘ₗ superCommuteF [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF) = 0 := by
     apply (ofCrAnListFBasis.ext fun l ↦ ?_)
     simp [ι_superCommuteF_superCommuteF_ofCrAnOpF_ofCrAnOpF_ofCrAnListF]
   rw [h1]
   simp
 
 lemma ι_commute_fieldOpFreeAlgebra_superCommuteF_ofCrAnOpF_ofCrAnOpF (φ1 φ2 : 𝓕.CrAnFieldOp)
-    (a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca -
-    ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca * ι a = 0 := by
+    (a : 𝓕.FieldOpFreeAlgebra) : ι a * ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF -
+    ι [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF * ι a = 0 := by
   rcases ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_zero φ1 φ2 with h | h
   swap
   · simp [h]
-  trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca, a]ₛca
+  trans - ι [[ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF, a]ₛF
   · rw [bosonic_superCommuteF h]
     simp
   · simp
 
 lemma ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center (φ ψ : 𝓕.CrAnFieldOp) :
-    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
+    ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
   rw [Subalgebra.mem_center_iff]
   intro a
   obtain ⟨a, rfl⟩ := ι_surjective a
diff --git a/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean b/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean
index 075df50..0c416a6 100644
--- a/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean
+++ b/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean
@@ -32,7 +32,7 @@ is zero.
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero
     (φa φa' : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
-    ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * ofCrAnListF φs') = 0 := by
+    ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * ofCrAnListF φs') = 0 := by
   rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa) with hφa | hφa
   <;> rcases CreateAnnihilate.eq_create_or_annihilate (𝓕 |>ᶜ φa') with hφa' | hφa'
   · rw [normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF φa φa' hφa hφa' φs φs']
@@ -53,27 +53,27 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero
     (φa φa' : 𝓕.CrAnFieldOp) (φs : List 𝓕.CrAnFieldOp)
     (a : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * a) = 0 := by
+    ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * a) = 0 := by
   have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
-      mulLinearMap (ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca) = 0 := by
+      mulLinearMap (ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF) = 0 := by
     apply ofCrAnListFBasis.ext
     intro l
     simp only [FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp, Function.comp_apply,
       AlgHom.toLinearMap_apply, LinearMap.zero_apply]
     exact ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero φa φa' φs l
   change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
-    mulLinearMap ((ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca))) a = 0
+    mulLinearMap ((ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF))) a = 0
   rw [hf]
   simp
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul (φa φa' : 𝓕.CrAnFieldOp)
     (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b) = 0 := by
+    ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * b) = 0 := by
   rw [mul_assoc]
   change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
-    ([ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b)) a = 0
+    ([ofCrAnOpF φa, ofCrAnOpF φa']ₛF * b)) a = 0
   have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ mulLinearMap.flip
-      ([ofCrAnOpF φa, ofCrAnOpF φa']ₛca * b) = 0 := by
+      ([ofCrAnOpF φa, ofCrAnOpF φa']ₛF * b) = 0 := by
     apply ofCrAnListFBasis.ext
     intro l
     simp only [mulLinearMap, FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
@@ -86,7 +86,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul (φa φa' : 𝓕.CrAnF
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
     (φs : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnListF φs]ₛca * b) = 0 := by
+    ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnListF φs]ₛF * b) = 0 := by
   rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
   rw [Finset.mul_sum, Finset.sum_mul]
   rw [map_sum, map_sum]
@@ -98,7 +98,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul (φa : 
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
     (φs : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnOpF φa]ₛca * b) = 0 := by
+    ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnOpF φa]ₛF * b) = 0 := by
   rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_symm, ofCrAnListF_singleton]
   simp only [FieldStatistic.instCommGroup.eq_1, FieldStatistic.ofList_singleton, mul_neg,
     Algebra.mul_smul_comm, neg_mul, Algebra.smul_mul_assoc, map_neg, map_smul]
@@ -107,7 +107,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul (φa : 
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul
     (φs φs' : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnListF φs']ₛca * b) = 0 := by
+    ι 𝓝ᶠ(a * [ofCrAnListF φs, ofCrAnListF φs']ₛF * b) = 0 := by
   rw [superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum, Finset.mul_sum, Finset.sum_mul]
   rw [map_sum, map_sum]
   apply Fintype.sum_eq_zero
@@ -119,7 +119,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
     (φs : List 𝓕.CrAnFieldOp)
     (a b c : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [ofCrAnListF φs, c]ₛca * b) = 0 := by
+    ι 𝓝ᶠ(a * [ofCrAnListF φs, c]ₛF * b) = 0 := by
   change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
     mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnListF φs)) c = 0
   have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
@@ -135,7 +135,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul
-    (a b c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛca * b) = 0 := by
+    (a b c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(a * [d, c]ₛF * b) = 0 := by
   change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
     mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) d = 0
   have hf : (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
@@ -151,25 +151,25 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
 
 @[simp]
 lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ([d, c]ₛca * b) = 0 := by
+    ι 𝓝ᶠ([d, c]ₛF * b) = 0 := by
   rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
   simp
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [d, c]ₛca) = 0 := by
+    ι 𝓝ᶠ(a * [d, c]ₛF) = 0 := by
   rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
   simp
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓝ᶠ(a * [d, c]ₛca * b1 * b2) = 0 := by
+    ι 𝓝ᶠ(a * [d, c]ₛF * b1 * b2) = 0 := by
   rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d]
   congr 2
   noncomm_ring
 
 @[simp]
-lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ([d, c]ₛca) = 0 := by
+lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ([d, c]ₛF) = 0 := by
   rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
   simp
 
diff --git a/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean b/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean
index cee072a..00a8c4c 100644
--- a/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean
+++ b/HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean
@@ -324,7 +324,7 @@ lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute_reorder (φ : 𝓕.Fi
 
 /--
 Within a proto-operator algebra we have that
-`φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛca`.
+`φ * 𝓝ᶠ(φ₀φ₁…φₙ) = 𝓝ᶠ(φφ₀φ₁…φₙ) + [anpart φ, 𝓝ᶠ(φ₀φ₁…φₙ)]ₛF`.
 -/
 lemma ofFieldOp_mul_normalOrder_ofFieldOpList_eq_superCommute (φ : 𝓕.FieldOp)
     (φs : List 𝓕.FieldOp) : ofFieldOp φ * 𝓝(ofFieldOpList φs) =
diff --git a/HepLean/PerturbationTheory/FieldOpAlgebra/SuperCommute.lean b/HepLean/PerturbationTheory/FieldOpAlgebra/SuperCommute.lean
index 467190f..0a0ee23 100644
--- a/HepLean/PerturbationTheory/FieldOpAlgebra/SuperCommute.lean
+++ b/HepLean/PerturbationTheory/FieldOpAlgebra/SuperCommute.lean
@@ -20,13 +20,13 @@ namespace FieldOpAlgebra
 variable {𝓕 : FieldSpecification}
 
 lemma ι_superCommuteF_eq_zero_of_ι_right_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι b = 0) :
-    ι [a, b]ₛca = 0 := by
+    ι [a, b]ₛF = 0 := by
   rw [superCommuteF_expand_bosonicProjF_fermionicProjF]
   rw [ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero] at h
   simp_all
 
 lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.FieldOpFreeAlgebra) (h : ι a = 0) :
-    ι [a, b]ₛca = 0 := by
+    ι [a, b]ₛF = 0 := by
   rw [superCommuteF_expand_bosonicProjF_fermionicProjF]
   rw [ι_eq_zero_iff_ι_bosonicProjF_fermonicProj_zero] at h
   simp_all
@@ -38,12 +38,12 @@ lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.FieldOpFreeAlgebra) (
 -/
 
 lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.FieldOpFreeAlgebra)
-    (h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛca = 0 := by
+    (h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛF = 0 := by
   apply ι_superCommuteF_eq_zero_of_ι_right_zero
   exact (ι_eq_zero_iff_mem_ideal b).mpr h
 
 lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h : b1 ≈ b2) :
-    ι [a, b1]ₛca = ι [a, b2]ₛca := by
+    ι [a, b1]ₛF = ι [a, b2]ₛF := by
   rw [equiv_iff_sub_mem_ideal] at h
   rw [LinearMap.sub_mem_ker_iff.mp]
   simp only [LinearMap.mem_ker, ← map_sub]
@@ -68,10 +68,10 @@ noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
     simp
 
 lemma superCommuteRight_apply_ι (a b : 𝓕.FieldOpFreeAlgebra) :
-    superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
+    superCommuteRight a (ι b) = ι [a, b]ₛF := by rfl
 
 lemma superCommuteRight_apply_quot (a b : 𝓕.FieldOpFreeAlgebra) :
-    superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
+    superCommuteRight a ⟦b⟧= ι [a, b]ₛF := by rfl
 
 lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.FieldOpFreeAlgebra) (h : a1 ≈ a2) :
     superCommuteRight a1 = superCommuteRight a2 := by
@@ -126,7 +126,7 @@ noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
 scoped[FieldSpecification.FieldOpAlgebra] notation "[" a "," b "]ₛ" => superCommute a b
 
 lemma superCommute_eq_ι_superCommuteF (a b : 𝓕.FieldOpFreeAlgebra) :
-    [ι a, ι b]ₛ = ι [a, b]ₛca := rfl
+    [ι a, ι b]ₛ = ι [a, b]ₛF := rfl
 
 /-!
 
diff --git a/HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean b/HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean
index 85b65b1..ad52ae5 100644
--- a/HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean
+++ b/HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean
@@ -24,7 +24,7 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
       crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
       crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
       crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2) :
-    ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca *
+    ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF *
     ofCrAnListF φs2) = 0 := by
   let l1 :=
     (List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c)
@@ -127,7 +127,7 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
   repeat rw [mul_assoc]
   rw [← mul_sub, ← mul_sub, ← mul_sub]
   rw [← sub_mul, ← sub_mul, ← sub_mul]
-  trans ι (ofCrAnListF l1) * ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca *
+  trans ι (ofCrAnListF l1) * ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF *
     ι (ofCrAnListF l2)
   rw [mul_assoc]
   congr
@@ -141,7 +141,7 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
 
 lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
     (φs1 φs2 : List 𝓕.CrAnFieldOp) :
-    ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * ofCrAnListF φs2)
+    ι 𝓣ᶠ(ofCrAnListF φs1 * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF * ofCrAnListF φs2)
     = 0 := by
   by_cases h :
       crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
@@ -155,16 +155,16 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.
 @[simp]
 lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
     (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0 := by
+    ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF * b) = 0 := by
   let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
-    Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0
+    Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF * b) = 0
   change pb b (Basis.mem_span _ b)
   apply Submodule.span_induction
   · intro x hx
     obtain ⟨φs, rfl⟩ := hx
     simp only [ofListBasis_eq_ofList, pb]
     let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
-      Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * ofCrAnListF φs) = 0
+      Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF * ofCrAnListF φs) = 0
     change pa a (Basis.mem_span _ a)
     apply Submodule.span_induction
     · intro x hx
@@ -184,19 +184,19 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
 
 lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
     (hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) =
-    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a * b)) := by
+    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) =
+    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a * b)) := by
   let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
-    Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) =
-    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a * b))
+    Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) =
+    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a * b))
   change pb b (Basis.mem_span _ b)
   apply Submodule.span_induction
   · intro x hx
     obtain ⟨φs, rfl⟩ := hx
     simp only [ofListBasis_eq_ofList, map_mul, pb]
     let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
-      Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * ofCrAnListF φs) =
-      ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * 𝓣ᶠ(a* ofCrAnListF φs))
+      Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * ofCrAnListF φs) =
+      ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a* ofCrAnListF φs))
     change pa a (Basis.mem_span _ a)
     apply Submodule.span_induction
     · intro x hx
@@ -234,7 +234,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
       rw [← sub_mul]
       have h1 : (ι (ofCrAnListF [φ, ψ]) -
           (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnListF [ψ, φ])) =
-        ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca := by
+        ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF := by
         rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
         rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append]
         simp only [instCommGroup.eq_1, List.singleton_append, Algebra.smul_mul_assoc, map_sub,
@@ -280,7 +280,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
 
 lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnFieldOp}
     (hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.FieldOpFreeAlgebra) :
-    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
+    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) = 0 := by
   rw [timeOrderF_timeOrderF_mid]
   have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
     exact Decidable.not_and_iff_or_not.mp hφψ
@@ -433,7 +433,8 @@ lemma timeOrder_ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
 
 /-- For a field specification `𝓕`, the time order operator acting on a
   list of `𝓕.FieldOp`, `𝓣(φ₀…φₙ)`, is equal to
-  `𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is the maximal time field in `φ₀…φₙ`.
+  `𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is the maximal time field
+  operator in `φ₀…φₙ`.
 
   The proof of this result ultimately relies on basic properties of ordering and signs. -/
 lemma timeOrder_eq_maxTimeField_mul_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
diff --git a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
index 8e93545..6507eee 100644
--- a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
+++ b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
@@ -96,7 +96,7 @@ lemma ofCrAnListF_singleton (φ : 𝓕.CrAnFieldOp) :
   `ofCrAnOpF` of the
   creation and annihilation parts of `φ`.
 
-  For example for `φ` an incoming asymptotic field operator we get
+  For example, for `φ` an incoming asymptotic field operator we get
   `ofCrAnOpF ⟨φ, ()⟩`, and for `φ` a
   position field operator we get `ofCrAnOpF ⟨φ, .create⟩ + ofCrAnOpF ⟨φ, .annihilate⟩`. -/
 def ofFieldOpF (φ : 𝓕.FieldOp) : FieldOpFreeAlgebra 𝓕 :=
diff --git a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
index 29deeaa..0258d94 100644
--- a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
+++ b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
@@ -239,7 +239,7 @@ lemma normalOrderF_swap_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
 lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
     (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
     (a b : 𝓕.FieldOpFreeAlgebra) :
-    𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛca * b) = 0 := by
+    𝓝ᶠ(a * [ofCrAnOpF φc, ofCrAnOpF φa]ₛF * b) = 0 := by
   simp only [superCommuteF_ofCrAnOpF_ofCrAnOpF, instCommGroup.eq_1, Algebra.smul_mul_assoc]
   rw [mul_sub, sub_mul, map_sub, ← smul_mul_assoc, ← mul_assoc, ← mul_assoc,
     normalOrderF_swap_create_annihilate φc φa hφc hφa]
@@ -248,7 +248,7 @@ lemma normalOrderF_superCommuteF_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
 lemma normalOrderF_superCommuteF_annihilate_create (φc φa : 𝓕.CrAnFieldOp)
     (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
     (a b : 𝓕.FieldOpFreeAlgebra) :
-    𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛca * b) = 0 := by
+    𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnOpF φc]ₛF * b) = 0 := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF_symm]
   simp only [instCommGroup.eq_1, neg_smul, mul_neg, Algebra.mul_smul_comm, neg_mul,
     Algebra.smul_mul_assoc, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
@@ -402,9 +402,9 @@ TODO "Split the following two lemmas up into smaller parts."
 lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
     (φc φc' : 𝓕.CrAnFieldOp) (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create)
     (hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) (φs φs' : List 𝓕.CrAnFieldOp) :
-    (𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca * ofCrAnListF φs')) =
+    (𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF * ofCrAnListF φs')) =
       normalOrderSign (φs ++ φc' :: φc :: φs') •
-    (ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛca *
+    (ofCrAnListF (createFilter φs) * [ofCrAnOpF φc, ofCrAnOpF φc']ₛF *
       ofCrAnListF (createFilter φs') * ofCrAnListF (annihilateFilter (φs ++ φs'))) := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
   conv_lhs =>
@@ -463,10 +463,10 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
     (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
     (hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate)
     (φs φs' : List 𝓕.CrAnFieldOp) :
-    𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * ofCrAnListF φs') =
+    𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF * ofCrAnListF φs') =
       normalOrderSign (φs ++ φa' :: φa :: φs') •
     (ofCrAnListF (createFilter (φs ++ φs'))
-      * ofCrAnListF (annihilateFilter φs) * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca
+      * ofCrAnListF (annihilateFilter φs) * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF
       * ofCrAnListF (annihilateFilter φs')) := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
   conv_lhs =>
@@ -533,7 +533,7 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
 -/
 
 lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛca =
+    [ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛF =
     ofCrAnListF φs * 𝓝ᶠ(ofCrAnListF φs') -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs := by
   simp only [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF,
@@ -541,7 +541,7 @@ lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.C
     Algebra.mul_smul_comm, mul_comm, Algebra.smul_mul_assoc]
 
 lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnFieldOp)
-    (φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca =
+    (φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF =
     ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs := by
   rw [ofFieldOpListF_sum, map_sum, Finset.mul_sum, Finset.smul_sum, Finset.sum_mul,
@@ -561,20 +561,20 @@ lemma ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φs : List 
     (φs' : List 𝓕.FieldOp) :
     ofCrAnListF φs * 𝓝ᶠ(ofFieldOpListF φs') =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnListF φs
-    + [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+    + [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
   simp [ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF]
 
 lemma ofCrAnOpF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.FieldOp) : ofCrAnOpF φ * 𝓝ᶠ(ofFieldOpListF φs') =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs') * ofCrAnOpF φ
-    + [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+    + [ofCrAnOpF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
   simp [← ofCrAnListF_singleton, ofCrAnListF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF]
 
 lemma anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp)
     (φs' : List 𝓕.FieldOp) :
     anPartF φ * 𝓝ᶠ(ofFieldOpListF φs') =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofFieldOpListF φs' * anPartF φ)
-    + [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛca := by
+    + [anPartF φ, 𝓝ᶠ(ofFieldOpListF φs')]ₛF := by
   rw [normalOrderF_mul_anPartF]
   match φ with
   | .inAsymp φ => simp
diff --git a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
index 2e19473..ba7fa2e 100644
--- a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
+++ b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
@@ -29,7 +29,7 @@ open FieldStatistic
 
   `superCommuteF φs φs' = φs * φs' - 𝓢(φs, φs') • φs' * φs`.
 
-  The notation `[a, b]ₛca` can be used for `superCommuteF a b`. -/
+  The notation `[a, b]ₛF` can be used for `superCommuteF a b`. -/
 noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
     𝓕.FieldOpFreeAlgebra :=
   Basis.constr ofCrAnListFBasis ℂ fun φs =>
@@ -37,7 +37,7 @@ noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.Field
   ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs)
 
 @[inherit_doc superCommuteF]
-scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca" => superCommuteF φs φs'
+scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛF" => superCommuteF φs φs'
 
 /-!
 
@@ -46,13 +46,13 @@ scoped[FieldSpecification.FieldOpFreeAlgebra] notation "[" φs "," φs' "]ₛca"
 -/
 
 lemma superCommuteF_ofCrAnListF_ofCrAnListF (φs φs' : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs, ofCrAnListF φs']ₛca =
+    [ofCrAnListF φs, ofCrAnListF φs']ₛF =
     ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs) := by
   rw [← ofListBasis_eq_ofList, ← ofListBasis_eq_ofList]
   simp only [superCommuteF, Basis.constr_basis]
 
 lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnFieldOp) :
-    [ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
+    [ofCrAnOpF φ, ofCrAnOpF φ']ₛF =
     ofCrAnOpF φ * ofCrAnOpF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnOpF φ' * ofCrAnOpF φ := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rw [superCommuteF_ofCrAnListF_ofCrAnListF, ofCrAnListF_append]
@@ -61,7 +61,7 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF (φ φ' : 𝓕.CrAnFieldOp) :
   rw [FieldStatistic.ofList_singleton, FieldStatistic.ofList_singleton, smul_mul_assoc]
 
 lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnFieldOp) (φs : List 𝓕.FieldOp) :
-    [ofCrAnListF φcas, ofFieldOpListF φs]ₛca = ofCrAnListF φcas * ofFieldOpListF φs -
+    [ofCrAnListF φcas, ofFieldOpListF φs]ₛF = ofCrAnListF φcas * ofFieldOpListF φs -
     𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofCrAnListF φcas := by
   conv_lhs => rw [ofFieldOpListF_sum]
   rw [map_sum]
@@ -74,7 +74,7 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpFsList (φcas : List 𝓕.CrAnFieldOp)
   simp
 
 lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
-    [ofFieldOpListF φ, ofFieldOpListF φs]ₛca = ofFieldOpListF φ * ofFieldOpListF φs -
+    [ofFieldOpListF φ, ofFieldOpListF φs]ₛF = ofFieldOpListF φ * ofFieldOpListF φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpListF φ := by
   conv_lhs => rw [ofFieldOpListF_sum]
   simp only [map_sum, LinearMap.coeFn_sum, Finset.sum_apply, instCommGroup.eq_1,
@@ -87,21 +87,21 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs
   rw [← Finset.sum_mul, ← Finset.smul_sum, ← Finset.mul_sum, ← ofFieldOpListF_sum]
 
 lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
-    [ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
+    [ofFieldOpF φ, ofFieldOpListF φs]ₛF = ofFieldOpF φ * ofFieldOpListF φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
   rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
     ofFieldOpListF_singleton]
   simp
 
 lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
-    [ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
+    [ofFieldOpListF φs, ofFieldOpF φ]ₛF = ofFieldOpListF φs * ofFieldOpF φ -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
   rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
     ofFieldOpListF_singleton]
   simp
 
 lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
-    [anPartF φ, crPartF φ']ₛca = anPartF φ * crPartF φ' -
+    [anPartF φ, crPartF φ']ₛF = anPartF φ * crPartF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * anPartF φ := by
   match φ, φ' with
   | FieldOp.inAsymp φ, _ =>
@@ -129,7 +129,7 @@ lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
     simp [crAnStatistics, ← ofCrAnListF_append]
 
 lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) :
-    [crPartF φ, anPartF φ']ₛca = crPartF φ * anPartF φ' -
+    [crPartF φ, anPartF φ']ₛF = crPartF φ * anPartF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ := by
     match φ, φ' with
     | FieldOp.outAsymp φ, _ =>
@@ -156,7 +156,7 @@ lemma superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) :
     simp [crAnStatistics, ← ofCrAnListF_append]
 
 lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.FieldOp) :
-    [crPartF φ, crPartF φ']ₛca = crPartF φ * crPartF φ' -
+    [crPartF φ, crPartF φ']ₛF = crPartF φ * crPartF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ := by
   match φ, φ' with
   | FieldOp.outAsymp φ, _ =>
@@ -183,7 +183,7 @@ lemma superCommuteF_crPartF_crPartF (φ φ' : 𝓕.FieldOp) :
   simp [crAnStatistics, ← ofCrAnListF_append]
 
 lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.FieldOp) :
-    [anPartF φ, anPartF φ']ₛca =
+    [anPartF φ, anPartF φ']ₛF =
     anPartF φ * anPartF φ' - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ := by
   match φ, φ' with
   | FieldOp.inAsymp φ, _ =>
@@ -208,7 +208,7 @@ lemma superCommuteF_anPartF_anPartF (φ φ' : 𝓕.FieldOp) :
     simp [crAnStatistics, ← ofCrAnListF_append]
 
 lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
-    [crPartF φ, ofFieldOpListF φs]ₛca =
+    [crPartF φ, ofFieldOpListF φs]ₛF =
     crPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs *
     crPartF φ := by
   match φ with
@@ -224,7 +224,7 @@ lemma superCommuteF_crPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.
     simp
 
 lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
-    [anPartF φ, ofFieldOpListF φs]ₛca =
+    [anPartF φ, ofFieldOpListF φs]ₛF =
     anPartF φ * ofFieldOpListF φs - 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) •
     ofFieldOpListF φs * anPartF φ := by
   match φ with
@@ -240,14 +240,14 @@ lemma superCommuteF_anPartF_ofFieldOpListF (φ : 𝓕.FieldOp) (φs : List 𝓕.
     simp [crAnStatistics]
 
 lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
-    [crPartF φ, ofFieldOpF φ']ₛca =
+    [crPartF φ, ofFieldOpF φ']ₛF =
     crPartF φ * ofFieldOpF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
   rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
   simp
 
 lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
-    [anPartF φ, ofFieldOpF φ']ₛca =
+    [anPartF φ, ofFieldOpF φ']ₛF =
     anPartF φ * ofFieldOpF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
   rw [← ofFieldOpListF_singleton, superCommuteF_anPartF_ofFieldOpListF]
@@ -263,39 +263,39 @@ multiplication with a sign plus the super commutator.
 -/
 lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
     ofCrAnListF φs * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnListF φs
-    + [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
+    + [ofCrAnListF φs, ofCrAnListF φs']ₛF := by
   rw [superCommuteF_ofCrAnListF_ofCrAnListF]
   simp [ofCrAnListF_append]
 
 lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.CrAnFieldOp) :
     ofCrAnOpF φ * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnOpF φ
-    + [ofCrAnOpF φ, ofCrAnListF φs']ₛca := by
+    + [ofCrAnOpF φ, ofCrAnListF φs']ₛF := by
   rw [← ofCrAnListF_singleton, ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF]
   simp
 
 lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.FieldOp) :
     ofFieldOpListF φs * ofFieldOpListF φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
-    + [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
+    + [ofFieldOpListF φs, ofFieldOpListF φs']ₛF := by
   rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
   simp
 
 lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
     ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
-    + [ofFieldOpF φ, ofFieldOpListF φs']ₛca := by
+    + [ofFieldOpF φ, ofFieldOpListF φs']ₛF := by
   rw [superCommuteF_ofFieldOpF_ofFieldOpFsList]
   simp
 
 lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
     ofFieldOpListF φs * ofFieldOpF φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs
-    + [ofFieldOpListF φs, ofFieldOpF φ]ₛca := by
+    + [ofFieldOpListF φs, ofFieldOpF φ]ₛF := by
   rw [superCommuteF_ofFieldOpListF_ofFieldOpF]
   simp
 
 lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     crPartF φ * anPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ +
-    [crPartF φ, anPartF φ']ₛca := by
+    [crPartF φ, anPartF φ']ₛF := by
   rw [superCommuteF_crPartF_anPartF]
   simp
 
@@ -303,27 +303,27 @@ lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     anPartF φ * crPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
     crPartF φ' * anPartF φ +
-    [anPartF φ, crPartF φ']ₛca := by
+    [anPartF φ, crPartF φ']ₛF := by
   rw [superCommuteF_anPartF_crPartF]
   simp
 
 lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     crPartF φ * crPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ +
-    [crPartF φ, crPartF φ']ₛca := by
+    [crPartF φ, crPartF φ']ₛF := by
   rw [superCommuteF_crPartF_crPartF]
   simp
 
 lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     anPartF φ * anPartF φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ +
-    [anPartF φ, anPartF φ']ₛca := by
+    [anPartF φ, anPartF φ']ₛF := by
   rw [superCommuteF_anPartF_anPartF]
   simp
 
 lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.FieldOp) : ofCrAnListF φs * ofFieldOpListF φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
-    + [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
+    + [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by
   rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
   simp
 
@@ -334,8 +334,8 @@ lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnField
 -/
 
 lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs, ofCrAnListF φs']ₛca =
-    (- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛca := by
+    [ofCrAnListF φs, ofCrAnListF φs']ₛF =
+    (- 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs')) • [ofCrAnListF φs', ofCrAnListF φs]ₛF := by
   rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF, smul_sub]
   simp only [instCommGroup.eq_1, neg_smul, sub_neg_eq_add]
   rw [smul_smul]
@@ -346,8 +346,8 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_symm (φs φs' : List 𝓕.CrAnField
   abel
 
 lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
-    [ofCrAnOpF φ, ofCrAnOpF φ']ₛca =
-    (- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛca := by
+    [ofCrAnOpF φ, ofCrAnOpF φ']ₛF =
+    (- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOpF φ', ofCrAnOpF φ]ₛF := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
   rw [smul_sub]
   simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, neg_smul, sub_neg_eq_add]
@@ -365,10 +365,10 @@ lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_symm (φ φ' : 𝓕.CrAnFieldOp) :
 -/
 
 lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φs' : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛca =
-    [ofCrAnListF φs, ofCrAnOpF φ]ₛca * ofCrAnListF φs' +
+    [ofCrAnListF φs, ofCrAnListF (φ :: φs')]ₛF =
+    [ofCrAnListF φs, ofCrAnOpF φ]ₛF * ofCrAnListF φs' +
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ)
-    • ofCrAnOpF φ * [ofCrAnListF φs, ofCrAnListF φs']ₛca := by
+    • ofCrAnOpF φ * [ofCrAnListF φs, ofCrAnListF φs']ₛF := by
   rw [superCommuteF_ofCrAnListF_ofCrAnListF]
   conv_rhs =>
     lhs
@@ -386,9 +386,9 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φ
   simp only [instCommGroup, map_mul, mul_comm]
 
 lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.CrAnFieldOp)
-    (φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛca =
-    [ofCrAnListF φs, ofFieldOpF φ]ₛca * ofFieldOpListF φs' +
-    𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
+    (φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, ofFieldOpListF (φ :: φs')]ₛF =
+    [ofCrAnListF φs, ofFieldOpF φ]ₛF * ofFieldOpListF φs' +
+    𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by
   rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
   conv_rhs =>
     lhs
@@ -409,14 +409,14 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_cons (φ : 𝓕.FieldOp) (φs : L
 For a field specification `𝓕`, and two lists `φs = φ₀…φₙ` and `φs'` of `𝓕.CrAnFieldOp`
 the following super commutation relation holds:
 
-`[φs', φ₀…φₙ]ₛca = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛca * φᵢ₊₁ … φₙ`
+`[φs', φ₀…φₙ]ₛF = ∑ i, 𝓢(φs', φ₀…φᵢ₋₁) • φ₀…φᵢ₋₁ * [φs', φᵢ]ₛF * φᵢ₊₁ … φₙ`
 
 The proof of this relation is via induction on the length of `φs`.
 -/
 lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
-    (φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛca =
+    (φs' : List 𝓕.CrAnFieldOp) → [ofCrAnListF φs, ofCrAnListF φs']ₛF =
     ∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
-    ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛca *
+    ofCrAnListF (φs'.take n) * [ofCrAnListF φs, ofCrAnOpF (φs'.get n)]ₛF *
     ofCrAnListF (φs'.drop (n + 1))
   | [] => by
     simp [← ofCrAnListF_nil, superCommuteF_ofCrAnListF_ofCrAnListF]
@@ -431,9 +431,9 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp)
 
 lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
     (φs' : List 𝓕.FieldOp) →
-    [ofCrAnListF φs, ofFieldOpListF φs']ₛca =
+    [ofCrAnListF φs, ofFieldOpListF φs']ₛF =
     ∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
-      ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛca *
+      ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛF *
       ofFieldOpListF (φs'.drop (n + 1))
   | [] => by
     simp only [superCommuteF_ofCrAnListF_ofFieldOpFsList, instCommGroup, ofList_empty,
@@ -450,10 +450,10 @@ lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnField
         FieldStatistic.ofList_cons_eq_mul, mul_comm]
 
 lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛca]ₛca =
+    [ofCrAnListF φs1, [ofCrAnListF φs2, ofCrAnListF φs3]ₛF]ₛF =
     𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
-    (- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛca]ₛca -
-    𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnListF φs2, [ofCrAnListF φs3, ofCrAnListF φs1]ₛca]ₛca) := by
+    (- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnListF φs3, [ofCrAnListF φs1, ofCrAnListF φs2]ₛF]ₛF -
+    𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnListF φs2, [ofCrAnListF φs3, ofCrAnListF φs1]ₛF]ₛF) := by
   repeat rw [superCommuteF_ofCrAnListF_ofCrAnListF]
   simp only [instCommGroup, map_sub, map_smul, neg_smul]
   repeat rw [superCommuteF_ofCrAnListF_ofCrAnListF]
@@ -501,16 +501,16 @@ lemma summerCommute_jacobi_ofCrAnListF (φs1 φs2 φs3 : List 𝓕.CrAnFieldOp)
 
 lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatistic}
     (ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
-    [a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
+    [a, b]ₛF ∈ statisticSubmodule (f1 + f2) := by
   let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
-    [a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
+    [a, a2]ₛF ∈ statisticSubmodule (f1 + f2)
   change p b hb
   apply Submodule.span_induction (p := p)
   · intro x hx
     obtain ⟨φs, rfl, hφs⟩ := hx
     simp only [add_eq_mul, instCommGroup, p]
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
-        [a2, ofCrAnListF φs]ₛca ∈ statisticSubmodule (f1 + f2)
+        [a2, ofCrAnListF φs]ₛF ∈ statisticSubmodule (f1 + f2)
     change p a ha
     apply Submodule.span_induction (p := p)
     · intro x hx
@@ -543,15 +543,15 @@ lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatisti
 
 lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = a * b - b * a := by
+    [a, b]ₛF = a * b - b * a := by
   let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
-    [a, a2]ₛca = a * a2 - a2 * a
+    [a, a2]ₛF = a * a2 - a2 * a
   change p b hb
   apply Submodule.span_induction (p := p)
   · intro x hx
     obtain ⟨φs, rfl, hφs⟩ := hx
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
-        [a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
+        [a2, ofCrAnListF φs]ₛF = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
     change p a ha
     apply Submodule.span_induction (p := p)
     · intro x hx
@@ -576,15 +576,15 @@ lemma superCommuteF_bosonic_bosonic {a b : 𝓕.FieldOpFreeAlgebra}
 
 lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
-    [a, b]ₛca = a * b - b * a := by
+    [a, b]ₛF = a * b - b * a := by
   let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
-    [a, a2]ₛca = a * a2 - a2 * a
+    [a, a2]ₛF = a * a2 - a2 * a
   change p b hb
   apply Submodule.span_induction (p := p)
   · intro x hx
     obtain ⟨φs, rfl, hφs⟩ := hx
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
-        [a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
+        [a2, ofCrAnListF φs]ₛF = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
     change p a ha
     apply Submodule.span_induction (p := p)
     · intro x hx
@@ -609,15 +609,15 @@ lemma superCommuteF_bosonic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
 
 lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = a * b - b * a := by
+    [a, b]ₛF = a * b - b * a := by
   let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
-    [a, a2]ₛca = a * a2 - a2 * a
+    [a, a2]ₛF = a * a2 - a2 * a
   change p b hb
   apply Submodule.span_induction (p := p)
   · intro x hx
     obtain ⟨φs, rfl, hφs⟩ := hx
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
-        [a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
+        [a2, ofCrAnListF φs]ₛF = a2 * ofCrAnListF φs - ofCrAnListF φs * a2
     change p a ha
     apply Submodule.span_induction (p := p)
     · intro x hx
@@ -641,7 +641,7 @@ lemma superCommuteF_fermionic_bonsonic {a b : 𝓕.FieldOpFreeAlgebra}
   · exact hb
 
 lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = a * b - b * a := by
+    [a, b]ₛF = a * b - b * a := by
   rw [← bosonicProjF_add_fermionicProjF a]
   simp only [map_add, LinearMap.add_apply]
   rw [superCommuteF_bosonic_bosonic (by simp) hb, superCommuteF_fermionic_bonsonic (by simp) hb]
@@ -649,7 +649,7 @@ lemma superCommuteF_bonsonic {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statist
   abel
 
 lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = a * b - b * a := by
+    [a, b]ₛF = a * b - b * a := by
   rw [← bosonicProjF_add_fermionicProjF b]
   simp only [map_add, LinearMap.add_apply]
   rw [superCommuteF_bosonic_bosonic ha (by simp), superCommuteF_bosonic_fermionic ha (by simp)]
@@ -658,27 +658,27 @@ lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisti
 
 lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (hb : b ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = - [b, a]ₛca := by
+    [a, b]ₛF = - [b, a]ₛF := by
   rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
   simp
 
 lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule bosonic) :
-    [a, b]ₛca = - [b, a]ₛca := by
+    [a, b]ₛF = - [b, a]ₛF := by
   rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
   simp
 
 lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
-    [a, b]ₛca = a * b + b * a := by
+    [a, b]ₛF = a * b + b * a := by
   let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
-    [a, a2]ₛca = a * a2 + a2 * a
+    [a, a2]ₛF = a * a2 + a2 * a
   change p b hb
   apply Submodule.span_induction (p := p)
   · intro x hx
     obtain ⟨φs, rfl, hφs⟩ := hx
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
-        [a2, ofCrAnListF φs]ₛca = a2 * ofCrAnListF φs + ofCrAnListF φs * a2
+        [a2, ofCrAnListF φs]ₛF = a2 * ofCrAnListF φs + ofCrAnListF φs * a2
     change p a ha
     apply Submodule.span_induction (p := p)
     · intro x hx
@@ -703,13 +703,13 @@ lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
 
 lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
-    [a, b]ₛca = [b, a]ₛca := by
+    [a, b]ₛF = [b, a]ₛF := by
   rw [superCommuteF_fermionic_fermionic ha hb]
   rw [superCommuteF_fermionic_fermionic hb ha]
   abel
 
 lemma superCommuteF_expand_bosonicProjF_fermionicProjF (a b : 𝓕.FieldOpFreeAlgebra) :
-    [a, b]ₛca = bosonicProjF a * bosonicProjF b - bosonicProjF b * bosonicProjF a +
+    [a, b]ₛF = bosonicProjF a * bosonicProjF b - bosonicProjF b * bosonicProjF a +
     bosonicProjF a * fermionicProjF b - fermionicProjF b * bosonicProjF a +
     fermionicProjF a * bosonicProjF b - bosonicProjF b * fermionicProjF a +
     fermionicProjF a * fermionicProjF b + fermionicProjF b * fermionicProjF a := by
@@ -722,8 +722,8 @@ lemma superCommuteF_expand_bosonicProjF_fermionicProjF (a b : 𝓕.FieldOpFreeAl
   abel
 
 lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnFieldOp) :
-    [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule bosonic ∨
-    [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic := by
+    [ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule bosonic ∨
+    [ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule fermionic := by
   by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
   · left
     have h : bosonic = bosonic + bosonic := by
@@ -759,14 +759,14 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : Lis
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
 
 lemma superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic (φ φ' : 𝓕.CrAnFieldOp) :
-    [ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule bosonic ∨
-    [ofCrAnOpF φ, ofCrAnOpF φ']ₛca ∈ statisticSubmodule fermionic := by
+    [ofCrAnOpF φ, ofCrAnOpF φ']ₛF ∈ statisticSubmodule bosonic ∨
+    [ofCrAnOpF φ, ofCrAnOpF φ']ₛF ∈ statisticSubmodule fermionic := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   exact superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic [φ] [φ']
 
 lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 : 𝓕.CrAnFieldOp) :
-    [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule bosonic ∨
-    [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca ∈ statisticSubmodule fermionic := by
+    [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF ∈ statisticSubmodule bosonic ∨
+    [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF ∈ statisticSubmodule fermionic := by
   rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ2 φ3 with hs | hs
     <;> rcases ofCrAnOpF_bosonic_or_fermionic φ1 with h1 | h1
   · left
@@ -796,12 +796,12 @@ lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 :
 
 lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
     (ha : a ∈ statisticSubmodule bosonic) :
-    [a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
-      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
+    [a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length),
+      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
       ofCrAnListF (φs.drop (n + 1)) := by
   let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
-      [a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length),
-      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
+      [a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length),
+      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
       ofCrAnListF (φs.drop (n + 1))
   change p a ha
   apply Submodule.span_induction (p := p)
@@ -825,12 +825,12 @@ lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φ
 
 lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra)
     (φs : List 𝓕.CrAnFieldOp) (ha : a ∈ statisticSubmodule fermionic) :
-    [a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
-      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
+    [a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
+      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
       ofCrAnListF (φs.drop (n + 1)) := by
   let p (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
-      [a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
-      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
+      [a, ofCrAnListF φs]ₛF = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
+      ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛF *
       ofCrAnListF (φs.drop (n + 1))
   change p a ha
   apply Submodule.span_induction (p := p)
@@ -858,9 +858,9 @@ lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra)
   · exact ha
 
 lemma statistic_neq_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnFieldOp}
-    (h : [ofCrAnListF φs, ofCrAnListF φs']ₛca ∈ statisticSubmodule fermionic) :
-    (𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0 := by
-  by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛca = 0
+    (h : [ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule fermionic) :
+    (𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnListF φs, ofCrAnListF φs']ₛF = 0 := by
+  by_cases h0 : [ofCrAnListF φs, ofCrAnListF φs']ₛF = 0
   · simp [h0]
   simp only [ne_eq, h0, or_false]
   by_contra hn
diff --git a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
index a6daed3..f0b127a 100644
--- a/HepLean/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
+++ b/HepLean/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
@@ -161,7 +161,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.F
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
     {φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) :
-    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
+    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF) = 0 := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
   simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton,
@@ -179,28 +179,28 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_right
     {φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
-    𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = 0 := by
+    𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF) = 0 := by
   rw [timeOrderF_timeOrderF_right,
     timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
   simp
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_left
     {φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.FieldOpFreeAlgebra) :
-    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * a) = 0 := by
+    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * a) = 0 := by
   rw [timeOrderF_timeOrderF_left,
     timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
   simp
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel_mid
     {φ ψ : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.FieldOpFreeAlgebra) :
-    𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca * b) = 0 := by
+    𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) = 0 := by
   rw [timeOrderF_timeOrderF_mid,
     timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_not_crAnTimeOrderRel h]
   simp
 
 lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
     {φ1 φ2 : 𝓕.CrAnFieldOp} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.FieldOpFreeAlgebra) :
-    𝓣ᶠ([a, [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛca]ₛca) = 0 := by
+    𝓣ᶠ([a, [ofCrAnOpF φ1, ofCrAnOpF φ2]ₛF]ₛF) = 0 := by
   rw [← bosonicProjF_add_fermionicProjF a]
   simp only [map_add, LinearMap.add_apply]
   rw [bosonic_superCommuteF (Submodule.coe_mem (bosonicProjF a))]
@@ -224,7 +224,7 @@ lemma timeOrderF_superCommuteF_superCommuteF_ofCrAnOpF_not_crAnTimeOrderRel
 lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
     {φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
     (h13 : ¬ crAnTimeOrderRel φ1 φ3) :
-    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
+    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF) = 0 := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rw [summerCommute_jacobi_ofCrAnListF]
   simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnListF_singleton, neg_smul, map_smul,
@@ -240,7 +240,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel
 lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel'
     {φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
     (h13 : ¬ crAnTimeOrderRel φ3 φ1) :
-    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
+    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF) = 0 := by
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rw [summerCommute_jacobi_ofCrAnListF]
   simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnListF_singleton, neg_smul, map_smul,
@@ -258,7 +258,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
       (crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
       crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
       crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2)) :
-    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca) = 0 := by
+    𝓣ᶠ([ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF) = 0 := by
   simp only [not_and] at h
   by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
   · simp_all only [IsEmpty.forall_iff, implies_true]
@@ -293,7 +293,7 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_eq_time
     {φ ψ : 𝓕.CrAnFieldOp} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
-    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛca) = [ofCrAnOpF φ, ofCrAnOpF ψ]ₛca := by
+    𝓣ᶠ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF) = [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF := by
   rw [superCommuteF_ofCrAnOpF_ofCrAnOpF]
   simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
   rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton,
diff --git a/HepLean/PerturbationTheory/FieldSpecification/CrAnFieldOp.lean b/HepLean/PerturbationTheory/FieldSpecification/CrAnFieldOp.lean
index ab84fc1..16fc24f 100644
--- a/HepLean/PerturbationTheory/FieldSpecification/CrAnFieldOp.lean
+++ b/HepLean/PerturbationTheory/FieldSpecification/CrAnFieldOp.lean
@@ -66,29 +66,29 @@ def fieldOpToCreateAnnihilateTypeCongr : {i j : 𝓕.FieldOp} → i = j →
 For a field specification `𝓕`, the (sigma) type `𝓕.CrAnFieldOp`
 corresponds to the type of creation and annihilation parts of field operators.
 It formally defined to consist of the following elements:
-- for each incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
+- For each incoming asymptotic field operator `φ` in `𝓕.FieldOp` an element
   written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the creation part of `φ`.
   Here `φ` has no annihilation part. (Here `()` is the unique element of `Unit`.)
-- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
+- For each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
   written as `⟨φ, .create⟩`, corresponding to the creation part of `φ`.
-- for each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
+- For each position field operator `φ` in `𝓕.FieldOp` an element of `𝓕.CrAnFieldOp`
   written as `⟨φ, .annihilate⟩`, corresponding to the annihilation part of `φ`.
-- for each outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
+- For each outgoing asymptotic field operator `φ` in `𝓕.FieldOp` an element
   written as `⟨φ, ()⟩` in `𝓕.CrAnFieldOp`, corresponding to the annihilation part of `φ`.
   Here `φ` has no creation part. (Here `()` is the unique element of `Unit`.)
 
 As an example, if `f` corresponds to a Weyl-fermion field, it would contribute
   the following elements to `𝓕.CrAnFieldOp`
-- an element corresponding to incoming asymptotic operators for each spin `s`: `a(p, s)`.
-- an element corresponding to the creation parts of position operators for each each Lorentz
-  index `a`:
+- For each spin `s`, an element corresponding to an incoming asymptotic operator: `a(p, s)`.
+- For each each Lorentz
+  index `a`, an element corresponding to the creation part of a position operator:
 
   `∑ s, ∫ d³p/(…) (xₐ (p,s)  a(p, s) e ^ (-i p x))`.
-- an element corresponding to annihilation parts of position operator,
-  for each each Lorentz index `a`:
+- For each each Lorentz
+  index `a`,an element corresponding to annihilation part of a position operator:
 
   `∑ s, ∫ d³p/(…) (yₐ(p,s) a†(p, s) e ^ (-i p x))`.
-- an element corresponding to outgoing asymptotic operators for each spin `s`: `a†(p, s)`.
+- For each spin `s`, element corresponding to an outgoing asymptotic operator: `a†(p, s)`.
 
 -/
 def CrAnFieldOp : Type := Σ (s : 𝓕.FieldOp), 𝓕.fieldOpToCrAnType s
diff --git a/HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean b/HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean
index 852f1d6..9c80005 100644
--- a/HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean
+++ b/HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean
@@ -25,24 +25,25 @@ open HepLean.Fin
 -/
 
 /-- Given a Wick contraction `φsΛ` for a list `φs` of `𝓕.FieldOp`,
-  an element `φ` of `𝓕.FieldOp`, an `i ≤ φs.length` and a `j` which is either `none` or
+  an element `φ` of `𝓕.FieldOp`, an `i ≤ φs.length` and a `k`
+  in `Option φsΛ.uncontracted` i.e. is either `none` or
   some element of `φsΛ.uncontracted`, the new Wick contraction
-  `φsΛ.insertAndContract φ i j` is defined by inserting `φ` into `φs` after
+  `φsΛ.insertAndContract φ i k` is defined by inserting `φ` into `φs` after
   the first `i`-elements and moving the values representing the contracted pairs in `φsΛ`
   accordingly.
-  If `j` is not `none`, but rather `some j`, to this contraction is added the contraction
-  of `φ` (at position `i`) with the new position of `j` after `φ` is added.
+  If `k` is not `none`, but rather `some k`, to this contraction is added the contraction
+  of `φ` (at position `i`) with the new position of `k` after `φ` is added.
 
-  In other words, `φsΛ.insertAndContract φ i j` is formed by adding `φ` to `φs` at position `i`,
-  and contracting `φ` with the field originally at position `j` if `j` is not `none`.
+  In other words, `φsΛ.insertAndContract φ i k` is formed by adding `φ` to `φs` at position `i`,
+  and contracting `φ` with the field originally at position `k` if `k` is not `none`.
   It is a Wick contraction of the list `φs.insertIdx φ i` corresponding to `φs` with `φ` inserted at
   position `i`.
 
-  The notation `φsΛ ↩Λ φ i j` is used to denote `φsΛ.insertAndContract φ i j`. -/
+  The notation `φsΛ ↩Λ φ i k` is used to denote `φsΛ.insertAndContract φ i k`. -/
 def insertAndContract {φs : List 𝓕.FieldOp} (φ : 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
-    (i : Fin φs.length.succ) (j : Option φsΛ.uncontracted) :
+    (i : Fin φs.length.succ) (k : Option φsΛ.uncontracted) :
     WickContraction (φs.insertIdx i φ).length :=
-  congr (by simp) (φsΛ.insertAndContractNat i j)
+  congr (by simp) (φsΛ.insertAndContractNat i k)
 
 @[inherit_doc insertAndContract]
 scoped[WickContraction] notation φs "↩Λ" φ:max i:max j => insertAndContract φ φs i j
diff --git a/HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean b/HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
index adc4c08..ddd2f8c 100644
--- a/HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
+++ b/HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
@@ -34,7 +34,14 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
   to the number of `fermionic`-`fermionic` exchanges that must be done to put
   contracted pairs within `φsΛ` next to one another, starting recursively
   from the contracted pair
-  whose first element occurs at the left-most position. -/
+  whose first element occurs at the left-most position.
+
+  As an example, if `[φ1, φ2, φ3, φ4]` correspond to fermionic fields then the sign
+  associated with
+- `{{0, 1}}` is `1`
+- `{{0, 1}, {2, 3}}` is `1`
+- `{{0, 2}, {1, 3}}` is `-1`
+-/
 def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
   ∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],
     𝓕 |>ₛ ⟨φs.get, φsΛ.signFinset (φsΛ.fstFieldOfContract a) (φsΛ.sndFieldOfContract a)⟩)