refactor: More docs for Wick's theorems

HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTerm.lean

@@ -31,8 +31,8 @@ noncomputable section
 def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
   φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
 
-/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the empty Wick contraction
+/-- For the empty list `[]` of `𝓕.FieldOp`, the `staticWickTerm` of the Wick contraction
-  `empty` of `[]` (its only Wick contraction) is `1`. -/
+  corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
 @[simp]
 lemma staticWickTerm_empty_nil :
     staticWickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
@@ -116,9 +116,9 @@ lemma staticWickTerm_insert_zero_some (φ : 𝓕.FieldOp) (φs : List 𝓕.Field
 For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, the following relation
 holds
 
-`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
+`φ * φsΛ.staticWickTerm = ∑ k, (φsΛ ↩Λ φ 0 k).staticWickTerm`
 
-where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
+where the sum is over all `k` in `Option φsΛ.uncontracted`,  so `k` is either `none` or `some k`.
 
 The proof of proceeds as follows:
 - `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as

HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTheorem.lean

@@ -22,7 +22,7 @@ namespace FieldOpAlgebra
 /--
 For a list `φs` of `𝓕.FieldOp`, the static version of Wick's theorem states that
 
-`φs =∑ φsΛ, φsΛ.staticWickTerm`
+`φs = ∑ φsΛ, φsΛ.staticWickTerm`
 
 where the sum is over all Wick contraction `φsΛ`.
 

HepLean/PerturbationTheory/FieldOpAlgebra/TimeContraction.lean

@@ -24,7 +24,7 @@ namespace FieldOpAlgebra
 open FieldStatistic
 
 /-- For a field specification `𝓕`, and `φ` and `ψ` elements of `𝓕.FieldOp`, the element of
-  `𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ)- 𝓝(φψ)`. -/
+  `𝓕.FieldOpAlgebra`, `timeContract φ ψ` is defined to be `𝓣(φψ) - 𝓝(φψ)`. -/
 def timeContract (φ ψ : 𝓕.FieldOp) : 𝓕.FieldOpAlgebra :=
     𝓣(ofFieldOp φ * ofFieldOp ψ) - 𝓝(ofFieldOp φ * ofFieldOp ψ)
 

HepLean/PerturbationTheory/FieldOpAlgebra/WickTerm.lean

@@ -33,8 +33,8 @@ noncomputable section
 def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
   φsΛ.sign • φsΛ.timeContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
 
-/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the empty Wick contraction
+/-- For the empty list `[]` of `𝓕.FieldOp`, the `wickTerm` of the Wick contraction
-  `empty` of `[]` (its only Wick contraction) is `1`. -/
+  corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
 @[simp]
 lemma wickTerm_empty_nil :
     wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
@@ -95,9 +95,9 @@ lemma wickTerm_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
 
 /-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
   `𝓕.FieldOp`, `i ≤ φs.length` and a `k` in `φsΛ.uncontracted`,
-  such that all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
+  such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
   `φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
-  `(φsΛ ↩Λ φ i (some k)).wickTerm`
+  `(φsΛ ↩Λ φ i (some k)).staticWickTerm`
 is equal the product of
 - the sign `𝓢(φ, φ₀…φᵢ₋₁) `
 - the sign `φsΛ.sign`
@@ -175,17 +175,18 @@ lemma wickTerm_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
       exact hg'
 
 /--
-Let `φsΛ` be a Wick contraction for `φs = φ₀φ₁…φₙ`. Let `φ` be a field with time
+For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
-greater then or equal to all the `FieldOp` in `φs`. Let `i ≤ φs.length` be such that
+  `𝓕.FieldOp`, and `i ≤ φs.length`
-all `FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ`. Then
+  such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less then `φ` and
+  `φ` has a time greater then or equal to all `FieldOp` in `φ₀…φₙ`, then
 
 `φ * φsΛ.wickTerm = 𝓢(φ, φ₀…φᵢ₋₁) • ∑ k, (φsΛ ↩Λ φ i k).wickTerm`
 
-where the sum is over all `k` in `Option φsΛ.uncontracted` (so either `none` or `some k`).
+where the sum is over all `k` in `Option φsΛ.uncontracted`, so `k` is either `none` or `some k`.
 
 The proof of proceeds as follows:
 - `ofFieldOp_mul_normalOrder_ofFieldOpList_eq_sum` is used to expand `φ 𝓝([φsΛ]ᵘᶜ)` as
-  a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`..
+  a sum over `k` in `Option φsΛ.uncontracted` of terms involving `[anPart φ, φs[k]]ₛ`.
 - Then `wickTerm_insert_none` and `wickTerm_insert_some` are used to equate terms.
 -/
 lemma mul_wickTerm_eq_sum (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (i : Fin φs.length.succ)

HepLean/PerturbationTheory/FieldOpAlgebra/WicksTheorem.lean

@@ -75,11 +75,11 @@ The inductive step works as follows:
 
 For the LHS:
 1. `timeOrder_eq_maxTimeField_mul_finset` is used to write
-  `𝓣(φ₀…φₙ)` as ` 𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
+  `𝓣(φ₀…φₙ)` as `𝓢(φᵢ,φ₀…φᵢ₋₁) • φᵢ * 𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` where `φᵢ` is
   the maximal time field in `φ₀…φₙ`
 2. The induction hypothesis is then used on `𝓣(φ₀…φᵢ₋₁φᵢ₊₁φₙ)` to expand it as a sum over
   Wick contractions of `φ₀…φᵢ₋₁φᵢ₊₁φₙ`.
-3. This gives terms of the form `φᵢ * φsΛ.timeContract` on which
+3. This gives terms of the form `φᵢ * φsΛ.wickTerm` on which
   `mul_wickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₀…φᵢ₋₁φᵢ₊₁φ`,
   to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
 

HepLean/PerturbationTheory/FieldOpAlgebra/WicksTheoremNormal.lean

@@ -31,9 +31,10 @@ This result follows from
 - `static_wick_theorem` to rewrite `𝓣(φs)` on the left hand side as a sum of
   `𝓣(φsΛ.staticWickTerm)`.
 - `EqTimeOnly.timeOrder_staticContract_of_not_mem` and `timeOrder_timeOrder_mid` to set to
-  zero those terms in which the contracted elements do not have equal time.
+  those `𝓣(φsΛ.staticWickTerm)` for which `φsΛ` has a contracted pair which are not
-- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract as a time contract
+  equal time to zero.
-  for those terms which have equal time.
+- `staticContract_eq_timeContract_of_eqTimeOnly` to rewrite the static contract
+  in the reminaing `𝓣(φsΛ.staticWickTerm)` as a time contract.
 - `timeOrder_timeContract_mul_of_eqTimeOnly_left` to move the time contracts out of the time
   ordering.
 -/
@@ -111,20 +112,20 @@ For a list `φs` of `𝓕.FieldOp`, then `𝓣(φs)` is equal to the sum of
 
 - `∑ φsΛ, φsΛ.wickTerm` where the sum is over all Wick contraction `φsΛ` which have
   no contractions of equal time.
-- `∑ φsΛ, sign φs ↑φsΛ • (φsΛ.1).timeContract ∑ φssucΛ, φssucΛ.wickTerm`, where
+- `∑ φsΛ, φsΛ.sign • φsΛ.timeContract  * (∑ φssucΛ, φssucΛ.wickTerm)`, where
   the first sum is over all Wick contraction `φsΛ` which only have equal time contractions
   and the second sum is over all Wick contraction `φssucΛ` of the uncontracted elements of `φsΛ`
   which do not have any equal time contractions.
 
-The proof of this result relies on `wicks_theorem` to rewrite `𝓣(φs)` as a sum over
+The proof of proceeds as follows
-all Wick contractions.
+- `wicks_theorem` is used to rewrite `𝓣(φs)` as a sum over all Wick contractions.
-The sum over all Wick contractions is then split additively into two parts using based on having or
+- The sum over all Wick contractions is then split additively into two parts using based on having
-not having equal time contractions.
+  or not having an equal time contractions.
-The sum over Wick contractions which do have equal time contractions is turned into two sums
+- Using `join`, the sum `∑ φsΛ, _` over Wick contractions which do have equal time contractions
-one over the Wick contractions which only have equal time contractions and the other over the
+  is split into two sums `∑ φsΛ, ∑ φsucΛ, _`, the first over non-zero elements
-uncontracted elements of the Wick contraction which do not have equal time contractions using
+  which only have equal time contractions and the second over Wick contractions `φsucΛ` of
-`join`.
+  `[φsΛ]ᵘᶜ` which do not have equal time contractions.
-The properties of `join_sign_timeContract` is then used to equate terms.
+- `join_sign_timeContract` is then used to equate terms.
 -/
 lemma timeOrder_haveEqTime_split (φs : List 𝓕.FieldOp) :
     𝓣(ofFieldOpList φs) = (∑ (φsΛ : {φsΛ : WickContraction φs.length // ¬ HaveEqTime φsΛ}),

HepLean/PerturbationTheory/WickContraction/TimeCond.lean

@@ -97,7 +97,7 @@ lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly :
   simp [empty]
 
 /-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` with
-  in which every contraction involves two `FieldOp`s that have the same time. Then
+  in which every contraction involves two `𝓕FieldOp`s that have the same time, then
   `φsΛ.staticContract = φsΛ.timeContract`. -/
 lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
     φsΛ.staticContract = φsΛ.timeContract := by
@@ -194,7 +194,7 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
   exact timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl
 
 /-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` with
-  in which every contraction involves two `FieldOp`s that have the same time and
+  in which every contraction involves two `𝓕.FieldOp`s that have the same time and
   `b` a general element in `𝓕.FieldOpAlgebra`. Then
   `𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b)`.
 
@@ -248,7 +248,7 @@ lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
   simp_all
 
 /-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` with
-  at least one contraction between `FieldOp` that do not have the same time. Then
+  at least one contraction between `𝓕.FieldOp` that do not have the same time. Then
   `𝓣(φsΛ.staticContract.1) = 0`. -/
 lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
     (hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.staticContract.1) = 0 := by

HepLean/PerturbationTheory/WickContraction/TimeContract.lean

@@ -21,7 +21,7 @@ open FieldOpAlgebra
 
 /-- For a list `φs` of `𝓕.FieldOp` and a Wick contraction `φsΛ` the
   element of the center of `𝓕.FieldOpAlgebra`, `φsΛ.timeContract` is defined as the product
-  of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` (both indices of `φs`) in `φsΛ`
+  of `timeContract φs[j] φs[k]` over contracted pairs `{j, k}` in `φsΛ`
   with `j < k`. -/
 noncomputable def timeContract {φs : List 𝓕.FieldOp}
     (φsΛ : WickContraction φs.length) :
@@ -86,7 +86,7 @@ open FieldStatistic
   - `[anPart φ, φs[k]]ₛ`
   - `φsΛ.timeContract`
   - two copies of the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
-    These two exchange signs cancle each other out but are included for convenience.
+    These two exchange signs cancel each other out but are included for convenience.
 
   The proof of this result ultimately a consequence of definitions and
   `timeContract_of_timeOrderRel`. -/
@@ -125,15 +125,13 @@ lemma timeContract_insert_some_of_lt
 
 /-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
   `𝓕.FieldOp`, a `i ≤ φs.length` and a `k` in `φsΛ.uncontracted` such that `k < i`, with the
-  condition that `φs[k]` does not have has greater or equal time to `φ`, then
+  condition that `φs[k]` does not have time greater or equal to `φ`, then
   `(φsΛ ↩Λ φ i (some k)).timeContract` is equal to the product of
   - `[anPart φ, φs[k]]ₛ`
   - `φsΛ.timeContract`
   - the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
   - the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ`.
 
-  Most of the contributes to the exchange signs cancle.
-
   The proof of this result ultimately a consequence of definitions and
   `timeContract_of_not_timeOrderRel_expand`. -/
 lemma timeContract_insert_some_of_not_lt