Fixed some typos (#519)

src/plfa/part2/BigStep.lagda.md

@@ -215,7 +215,7 @@ sub-id = plfa.part2.Substitution.sub-id
 ```
 
 
-We define an auxilliary function for extending a substitution.
+We define an auxiliary function for extending a substitution.
 
 ```
 ext-subst : ∀{Γ Δ} → Subst Γ Δ → Δ ⊢ ★ → Subst (Γ , ★) Δ
@@ -393,7 +393,7 @@ underneath lambda abstractions via the `ζ` rule. The call-by-name
 semantics does not reduce under lambda, so a straightforward proof by
 induction on the reduction sequence is impossible.  In the article
 _Call-by-name, call-by-value, and the λ-calculus_, Plotkin proves the
-theorem in two steps, using two auxilliary reduction relations. The
+theorem in two steps, using two auxiliary reduction relations. The
 first step uses a classic technique called Curry-Feys standardisation.
 It relies on the notion of _standard reduction sequence_, which acts
 as a half-way point between full beta reduction and call-by-name by

src/plfa/part2/Confluence.lagda.md

@@ -49,7 +49,7 @@ diamond property. Here is a counter example.
 Both terms can reduce to `a a`, but the second term requires two steps
 to get there, not one.
 
-To side-step this problem, we'll define an auxilliary reduction
+To side-step this problem, we'll define an auxiliary reduction
 relation, called _parallel reduction_, that can perform many
 reductions simultaneously and thereby satisfy the diamond property.
 Furthermore, we show that a parallel reduction sequence exists between
@@ -407,7 +407,7 @@ The heart of the confluence proof is made of stone, or rather, of
 diamond!  We show that parallel reduction satisfies the diamond
 property: that if `M ⇛ N` and `M ⇛ N′`, then `N ⇛ L` and `N′ ⇛ L` for
 some `L`.  The typical proof is an induction on `M ⇛ N` and `M ⇛ N′`
-so that every possible pair gives rise to a witeness `L` given by
+so that every possible pair gives rise to a witness `L` given by
 performing enough beta reductions in parallel.
 
 However, a simpler approach is to perform as many beta reductions in
@@ -611,7 +611,7 @@ confluence L↠M₁ L↠M₂
 ## Notes
 
 Broadly speaking, this proof of confluence, based on parallel
-reduction, is due to W. Tait and P. Martin-Lof (see Barendredgt 1984,
+reduction, is due to W. Tait and P. Martin-Löf (see Barendredgt 1984,
 Section 3.2).  Details of the mechanization come from several sources.
 The `subst-par` lemma is the "strong substitutivity" lemma of Shafer,
 Tebbi, and Smolka (ITP 2015). The proofs of `par-triangle`, `strip`,

src/plfa/part2/Inference.lagda.md

@@ -618,7 +618,7 @@ Consider the context:
 
   + If they differ, we recurse:
 
-    - If lookup fails, we apply `ext∋` to conver the proof
+    - If lookup fails, we apply `ext∋` to convert the proof
       there is no derivation from the contained context
       to the extended context.
 

src/plfa/part2/Subtyping.lagda.md

@@ -1235,7 +1235,7 @@ Come up with an algorithmic subtyping rule for variant types.
 
 #### Exercise `<:-alternative` (stretch)
 
-Revise this formalization of records with subtyping (including proofs
+Revise this formalisation of records with subtyping (including proofs
 of progress and preservation) to use the non-algorithmic subtyping
 rules for Chapter 15 of Types and Programming Languages, which we list here: