refactor: improve docs

HepLean/PerturbationTheory/WickContraction/InsertAndContract.lean

@@ -268,6 +268,20 @@ lemma insert_fin_eq_self (φ : 𝓕.States) {φs : List 𝓕.States}
     use z
     rfl
 
+
+lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
+    (i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
+    ∑ c, f c =
+    ∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) := by
+  rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
+    (insertIdx_length_fin φ φs i)]
+  rfl
+
+/-!
+
+## Uncontracted list
+
+-/
 lemma insertAndContract_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.States}
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
     [φsΛ ↩Λ φ i none]ᵘᶜ = List.insertIdx (φsΛ.uncontractedListOrderPos i) φ [φsΛ]ᵘᶜ := by
@@ -283,12 +297,4 @@ lemma insertAndContract_uncontractedList_none_map (φ : 𝓕.States) {φs : List
   conv_rhs => rw [get_eq_insertIdx_succAbove φ φs i]
   rfl
 
-lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
-    (i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
-    ∑ c, f c =
-    ∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted), f (φsΛ ↩Λ φ i k) := by
-  rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
-    (insertIdx_length_fin φ φs i)]
-  rfl
-
 end WickContraction

HepLean/PerturbationTheory/WickContraction/TimeContract.lean

@@ -18,7 +18,7 @@ namespace WickContraction
 variable {n : ℕ} (c : WickContraction n)
 open HepLean.List
 
-/-- Given a Wick contraction `c` associated with a list `φs`, the
+/-- Given a Wick contraction `φsΛ` associated with a list `φs`, the
   product of all time-contractions of pairs of contracted elements in `φs`,
   as a member of the center of `𝓞.A`. -/
 noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List 𝓕.States}
@@ -28,6 +28,11 @@ noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List
     (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)),
     𝓞.timeContract_mem_center _ _⟩
 
+/-- For `φsΛ` a Wick contraction for `φs`, the time contraction `(φsΛ ↩Λ φ i none).timeContract 𝓞`
+  is equal to `φsΛ.timeContract 𝓞`.
+
+This result follows from the fact that `timeContract` only depends on contracted pairs,
+and `(φsΛ ↩Λ φ i none)` has the 'same' contracted pairs as `φsΛ`. -/
 @[simp]
 lemma timeContract_insertAndContract_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :