reviewing section headings

src/plfa/DeBruijn.lagda

@@ -214,7 +214,7 @@ For each new section, we show the corresponding old section or
 sections.
 
   * DeBruijn: Inherently typed de Bruijn representation
-    + Inherently typed terms (1.1, 1.5)
+    + Syntax (1.1, 1.5)
     + Values (1.2, 2.1)
     + Renaming (2.3)
     + Substitution (1.3, 2.3)
@@ -222,16 +222,14 @@ sections.
     + Progress (2.2)
     + Evaluation (2.6)
 
-The definition of terms now combines both the syntax of terms
+The syntax of terms now incorporates their typing rules, and the
-and their typing rules, and the definition of values now
+definition of values now incorporates the Canonical Forms lemma.  The
-incorporates the Canonical Forms lemma.  The definition of
+definition of substitution is somewhat more involved, but incorporates
-substitution is somewhat more involved, but incorporates the
+the trickiest part of the previous proof, the lemma establishing that
-trickiest part of the previous proof, the lemma establishing
+substitution preserves types.  The definition of reduction
-that substitution preserves types.  The definition of
+incorporates preservation, which no longer requires a separate proof.
-reduction incorporates preservation, which no longer requires
-a separate proof.
 
-## Inherently typed terms
+## Syntax
 
 We now begin our formal development.
 
@@ -423,7 +421,7 @@ _ = ƛ ƛ (` S Z · (` S Z · ` Z))
 The final inherently-typed term represents the Church numeral
 two.
 
-### A convenient abbreviation
+### Abbreviating de Bruijn indices
 
 We can use a natural number to select a type from a context.
 \begin{code}
@@ -775,7 +773,7 @@ And combining definition with proof makes it harder for errors
 to sneak in.
 
 
-## Reductions
+## Reduction
 
 With substitution out of the way, it is straightforward to
 define values and reduction.
@@ -869,7 +867,7 @@ preserves types, which we previously is built-in to our
 definition of substitution.
 
 
-## Reflexive and transitive closure
+### Reflexive and transitive closure
 
 The reflexive and transitive closure is exactly as before.
 We simply cut-and-paste the previous definition.
@@ -899,7 +897,7 @@ begin M—↠N = M—↠N
 \end{code}
 
 
-### Examples
+## Examples
 
 We reiterate each of our previous examples.  First, the Church
 numeral two applied to the successor function and zero yields

src/plfa/Lambda.lagda

@@ -10,13 +10,6 @@ permalink : /Lambda/
 module plfa.Lambda where
 \end{code}
 
-<!--
-[This chapter was originally based on Chapter _Stlc_
-of _Software Foundations_ (_Programming Language Foundations_).
-It has now been updated, but if you spot any plagiarism
-please let me know. – P]
--->
-
 The _lambda-calculus_, first published by the logician Alonzo Church in
 1932, is a core calculus with only three syntactic constructs:
 variables, abstraction, and application.  It captures the key concept of
@@ -675,6 +668,13 @@ The three constructors specify, respectively, that `—↠` includes `—→`
 and is reflexive and transitive.  A good exercise is to show that
 the two definitions are equivalent (indeed, isomoprhic).
 
+#### Exercise (`—↠≃—↠′`)
+
+Show that the two notions of reflexive and transitive closure
+above are isomorphic.
+
+## Confluence
+
 One important property a reduction relation might satisfy is
 to be _confluent_.  If term `L` reduces to two other terms,
 `M` and `N`, then both of these reduce to a common term `P`.
@@ -721,11 +721,6 @@ the diamond property, and that every relation that satisfies
 the diamond property is confluent.  Hence, all the reduction
 systems studied in this text are trivially confluent.
 
-#### Exercise (`—↠≃—↠′`)
-
-Show that the two notions of reflexive and transitive closure
-above are isomorphic.
-
 
 ## Examples
 
@@ -1302,3 +1297,5 @@ This chapter uses the following unicode
     ⊢    U+22A2:  RIGHT TACK (\vdash or \|-)
     ⦂    U+2982:  Z NOTATION TYPE COLON (\:)
 
+We compose reduction —→ from an em dash — and an arrow →.
+Similarly for reflexive and transitive closure —↠.

src/plfa/More.lagda

@@ -10,14 +10,22 @@ module plfa.More where
 
 So far, we have focussed on a relatively minimal language, based on
 Plotkin's PCF, which supports functions, naturals, and fixpoints.  In
-this chapter we extend our calculus to support `let` bindings and more
+this chapter we extend our calculus to support the following:
-datatypes, including products, sums, unit type, empty type, and lists,
+
-all of which are familiar from Part I of this textbook.  We also give
+  * _let_ bindings
-alternative formulations of products and unit type.  For `let` and the
+  * products
-alternative formulations we show how they translate to other constructs
+  * an alternative formulation of products
-in the calculus.  Most of the description will be informal. We show
+  * sums
-how to formalise a few of the constructs and leave the rest as an
+  * unit type
-exercise for the reader.
+  * and an alternative formulation of unit type
+  * empty type
+  * lists
+
+All of the data types should be familiar from Part I of this textbook.
+For _let_ and the alternative formulations we show how they translate
+to other constructs in the calculus.  Most of the description will be
+informal. We show how to formalise the first three constructs and leave
+the rest as an exercise for the reader.
 
 Our informal descriptions will be in the style of
 Chapter [Lambda]({{ site.baseurl }}{% link out/plfa/Lambda.md %}),
@@ -29,8 +37,8 @@ using de Bruijn indices and inherently typed terms.
 By now, explaining with symbols should be more concise, more precise,
 and easier to follow than explaining in prose.
 For each construct, we give syntax, typing, reductions, and an example.
-We also give translations where relevant.  Formally establishing the
+We also give translations where relevant; formally establishing the
-correctness of such translations will be the subject of the next chapter.
+correctness of translations will be the subject of the next chapter.
 
 ## Let bindings
 
@@ -484,6 +492,17 @@ Here is the map function for lists.
                | x ∷ xs ⇒ f · x `∷ ml · f · xs ]
 
 
+## Formalisation
+
+We now show how to formalise
+
+  * _let_ bindings
+  * products
+  * an alternative formulation of products
+
+and leave formalisation of the remaining constructs as an exercise.
+
+
 ## Imports
 
 \begin{code}
@@ -521,7 +540,11 @@ infix  9 #_
 
 infixr 6 _V-∷_
 infix  8 V-suc_
+\end{code}
 
+### Types
+
+\begin{code}
 data Type : Set where
   `ℕ    : Type
   _⇒_   : Type → Type → Type
@@ -530,11 +553,19 @@ data Type : Set where
   `⊤    : Type
   `⊥    : Type
   `List : Type → Type
+\end{code}
 
+### Contexts
+
+\begin{code}
 data Context : Set where
   ∅   : Context
   _,_ : Context → Type → Context
+\end{code}
 
+### Variables and the lookup judgement
+
+\begin{code}
 data _∋_ : Context → Type → Set where
 
   Z : ∀ {Γ A}
@@ -545,7 +576,11 @@ data _∋_ : Context → Type → Set where
     → Γ ∋ B
       ---------
     → Γ , A ∋ B
+\end{code}   
 
+### Terms and the typing judgment
+
+\begin{code}
 data _⊢_ : Context → Type → Set where
 
   -- variables
@@ -695,22 +730,17 @@ lookup (Γ , A) zero     =  A
 lookup (Γ , _) (suc n)  =  lookup Γ n
 lookup ∅       _        =  ⊥-elim impossible
   where postulate impossible : ⊥
-\end{code}
 
-\begin{code}
 count : ∀ {Γ} → (n : ℕ) → Γ ∋ lookup Γ n
 count {Γ , _} zero     =  Z
 count {Γ , _} (suc n)  =  S (count n)
 count {∅}     _        =  ⊥-elim impossible
   where postulate impossible : ⊥
-\end{code}
 
-\begin{code}
 #_ : ∀ {Γ} → (n : ℕ) → Γ ⊢ lookup Γ n
 # n  =  ` count n
 \end{code}
 
-
 ## Substitution
 
 ### Renaming
@@ -803,7 +833,7 @@ _[_][_] {Γ} {A} {B} N V W =  subst {Γ , A , B} {Γ} σ N
   σ (S (S x))  =  ` x
 \end{code}
 
-## Reductions
+## Reduction
 
 ### Value
 
@@ -869,7 +899,7 @@ data Value : ∀ {Γ A} → Γ ⊢ A → Set where
 Implicit arguments need to be supplied when they are
 not fixed by the given arguments.
 
-## Reduction
+### Reduction step
 
 \begin{code}
 infix 2 _—→_
@@ -1056,7 +1086,7 @@ data _—→_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where
     → caseL (V `∷ W) M N —→ N [ V ][ W ]
 \end{code}
 
-## Reflexive and transitive closure
+### Reflexive and transitive closure
 
 \begin{code}
 infix  2 _—↠_

src/plfa/Properties.lagda

@@ -8,10 +8,6 @@ permalink : /Properties/
 module plfa.Properties where
 \end{code}
 
-[Parts of this chapter take their text from chapter _StlcProp_
-of _Software Foundations_ (_Programming Language Foundations_).
-Those parts will be revised.]
-
 This chapter covers properties of the simply-typed lambda calculus, as
 introduced in the previous chapter.  The most important of these
 properties are progress and preservation.  We introduce these below,
@@ -1431,9 +1427,6 @@ checker complains, because the arguments have merely switched order
 and neither is smaller.
 
 
-
-
-
 #### Exercise: `progress-preservation`
 
 Without peeking at their statements above, write down the progress