docs: Fix typos in docs

HepLean/PerturbationTheory/FieldOpAlgebra/Basic.lean

@@ -42,7 +42,7 @@ def fieldOpIdealSet : Set (FieldOpFreeAlgebra 𝓕) :=
   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,
-  when combined with the conditions above, that the super-commutator is in the center of the
+  when combined with the conditions above, that the super-commutator is in the center
   of the algebra.
 -/
 abbrev FieldOpAlgebra : Type := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.Quotient

HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Basic.lean

@@ -223,9 +223,9 @@ lemma ι_normalOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b)
 
   `FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
 
-  defined as the decent of `ι ∘ₗ normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
+  defined as the descent of `ι ∘ₗ normalOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
   from `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
-  This decent exists because `ι ∘ₗ normalOrderF` is well-defined on equivalence classes.
+  This descent exists because `ι ∘ₗ normalOrderF` is well-defined on equivalence classes.
 
   The notation `𝓝(a)` is used for `normalOrder a` for `a` an element of `FieldOpAlgebra 𝓕`. -/
 noncomputable def normalOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where

HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/Lemmas.lean

@@ -350,7 +350,7 @@ then `φ * 𝓝(φ₀φ₁…φₙ)` is equal to
 
 `𝓝(φφ₀φ₁…φₙ) + ∑ i, (𝓢(φ,φ₀φ₁…φᵢ₋₁) • [anPart φ, φᵢ]ₛ) * 𝓝(φ₀…φᵢ₋₁φᵢ₊₁…φₙ)`.
 
-The proof of ultimately goes as follows:
+The proof ultimately goes as follows:
 - `ofFieldOp_eq_crPart_add_anPart` is used to split `φ` into its creation and annihilation parts.
 - The following relation is then used
 

HepLean/PerturbationTheory/FieldOpAlgebra/NormalOrder/WickContractions.lean

@@ -32,7 +32,7 @@ variable {𝓕 : FieldSpecification}
 
   where `s` is the exchange sign for `φ` and the uncontracted fields in `φ₀…φᵢ₋₁`.
 
-  The prove of this result ultimately a consequence of `normalOrder_superCommute_eq_zero`.
+  The proof of this result ultimately is a consequence of `normalOrder_superCommute_eq_zero`.
 -/
 lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     (i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) :
@@ -104,7 +104,7 @@ lemma normalOrder_uncontracted_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp
   `𝓝([φsΛ ↩Λ φ i (some k)]ᵘᶜ)` is equal to the normal ordering of `[φsΛ]ᵘᶜ` with the `𝓕.FieldOp`
   corresponding to `k` removed.
 
-  The proof of this result ultimately a consequence of definitions.
+  The proof of this result ultimately is a consequence of definitions.
 -/
 lemma normalOrder_uncontracted_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     (i : Fin φs.length.succ) (φsΛ : WickContraction φs.length) (k : φsΛ.uncontracted) :

HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTerm.lean

@@ -27,7 +27,7 @@ noncomputable section
 
   `φsΛ.sign • φsΛ.staticContract * 𝓝([φsΛ]ᵘᶜ)`.
 
-  This is term which appears in the static version Wick's theorem. -/
+  This is a term which appears in the static version Wick's theorem. -/
 def staticWickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
   φsΛ.sign • φsΛ.staticContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
 
@@ -120,7 +120,7 @@ holds
 
 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:
+The proof 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`.
 - Then `staticWickTerm_insert_zero_none` and `staticWickTerm_insert_zero_some` are

HepLean/PerturbationTheory/FieldOpAlgebra/StaticWickTheorem.lean

@@ -32,7 +32,7 @@ The proof is via induction on `φs`.
 The inductive step works as follows:
 
 For the LHS:
-1. The proof considers `φ₀…φₙ` as `φ₀(φ₁…φₙ)` and use the induction hypothesis on `φ₁…φₙ`.
+1. The proof considers `φ₀…φₙ` as `φ₀(φ₁…φₙ)` and uses the induction hypothesis on `φ₁…φₙ`.
 2. This gives terms of the form `φ * φsΛ.staticWickTerm` on which
   `mul_staticWickTerm_eq_sum` is used where `φsΛ` is a Wick contraction of `φ₁…φₙ`,
   to rewrite terms as a sum over optional uncontracted elements of `φsΛ`
@@ -45,7 +45,7 @@ For the RHS:
   is split via `insertLift_sum` into a sum over Wick contractions `φsΛ` of `φ₁…φₙ` and
   sum over optional uncontracted elements of `φsΛ`.
 
-Both side now are sums over the same thing and their terms equate by the nature of the
+Both sides are now sums over the same thing and their terms equate by the nature of the
 lemmas used.
 
 -/

HepLean/PerturbationTheory/FieldOpAlgebra/SuperCommute.lean

@@ -92,7 +92,7 @@ lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.FieldOpFreeAlgebra) (h : a1
 
   `FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
 
-  defined as the decent of `ι ∘ superCommuteF` in both arguments.
+  defined as the descent of `ι ∘ superCommuteF` in both arguments.
   In particular for `φs` and `φs'` lists of `𝓕.CrAnFieldOp` in `FieldOpAlgebra 𝓕` the following
   relation holds:
 

HepLean/PerturbationTheory/FieldOpAlgebra/TimeOrder.lean

@@ -371,9 +371,9 @@ lemma ι_timeOrderF_eq_of_equiv (a b : 𝓕.FieldOpFreeAlgebra) (h : a ≈ b) :
 
 `FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕`
 
-defined as the decent of `ι ∘ₗ timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕` from
+defined as the descent of `ι ∘ₗ timeOrderF : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕` from
 `FieldOpFreeAlgebra 𝓕` to `FieldOpAlgebra 𝓕`.
-This decent exists because `ι ∘ₗ timeOrderF` is well-defined on equivalence classes.
+This descent exists because `ι ∘ₗ timeOrderF` is well-defined on equivalence classes.
 
 The notation `𝓣(a)` is used for `timeOrder a`. -/
 noncomputable def timeOrder : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where

HepLean/PerturbationTheory/FieldOpAlgebra/WickTerm.lean

@@ -29,7 +29,7 @@ noncomputable section
 
   `φsΛ.sign • φsΛ.timeContract * 𝓝([φsΛ]ᵘᶜ)`.
 
-  This is term which appears in the Wick's theorem. -/
+  This is a term which appears in the Wick's theorem. -/
 def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 𝓕.FieldOpAlgebra :=
   φsΛ.sign • φsΛ.timeContract * 𝓝(ofFieldOpList [φsΛ]ᵘᶜ)
 
@@ -95,10 +95,10 @@ 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 `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
+  such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less than `φ` and
+  `φ` has a time greater than or equal to all `FieldOp` in `φ₀…φₙ`, then
   `(φsΛ ↩Λ φ i (some k)).staticWickTerm`
-is equal the product of
+is equal to the product of
 - the sign `𝓢(φ, φ₀…φᵢ₋₁) `
 - the sign `φsΛ.sign`
 - `φsΛ.timeContract`
@@ -177,14 +177,14 @@ lemma wickTerm_insert_some (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`, and `i ≤ φs.length`
-  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
+  such that all `𝓕.FieldOp` in `φ₀…φᵢ₋₁` have time strictly less than `φ` and
+  `φ` has a time greater than 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 `k` is either `none` or `some k`.
 
-The proof of proceeds as follows:
+The proof 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`.
 - Then `wickTerm_insert_none` and `wickTerm_insert_some` are used to equate terms.

HepLean/PerturbationTheory/FieldOpAlgebra/WicksTheorem.lean

@@ -91,7 +91,7 @@ For the RHS:
   is split via `insertLift_sum` into a sum over Wick contractions `φsΛ` of `φ₀…φᵢ₋₁φᵢ₊₁φ` and
   sum over optional uncontracted elements of `φsΛ`.
 
-Both side now are sums over the same thing and their terms equate by the nature of the
+Both sides are now sums over the same thing and their terms equate by the nature of the
 lemmas used.
 -/
 theorem wicks_theorem : (φs : List 𝓕.FieldOp) → 𝓣(ofFieldOpList φs) =

HepLean/PerturbationTheory/FieldOpAlgebra/WicksTheoremNormal.lean

@@ -117,9 +117,9 @@ For a list `φs` of `𝓕.FieldOp`, then `𝓣(φs)` is equal to the sum of
   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 proceeds as follows
+The proof proceeds as follows
 - `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
+- The sum over all Wick contractions is then split additively into two parts based on having
   or not having an equal time contractions.
 - Using `join`, the sum `∑ φsΛ, _` over Wick contractions which do have equal time contractions
   is split into two sums `∑ φsΛ, ∑ φsucΛ, _`, the first over non-zero elements