From a0993b537d2431c0b39dca6e88cc637fb519af05 Mon Sep 17 00:00:00 2001
From: Adam Chlipala <adam@chlipala.net>
Date: Sun, 9 Feb 2020 12:54:33 -0500
Subject: [PATCH] Revising Interpreters before class

---
 Interpreters.v | 18 +++++++++---------
 frap_book.tex  |  4 ++--
 2 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/Interpreters.v b/Interpreters.v
index ee96392..50b7a85 100644
--- a/Interpreters.v
+++ b/Interpreters.v
@@ -6,7 +6,7 @@
 Require Import Frap.
 
 
-(* We begin with a return to our arithmetic language from the last chapter,
+(* We begin with a return to our arithmetic language from BasicSyntax,
  * adding subtraction*, which will come in handy later.
  * *: good pun, right? *)
 Inductive arith : Set :=
@@ -25,7 +25,7 @@ Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
  * Actually, it's not quite that simple.
  * We need to consider the meaning to be a function over a valuation
  * to the variables, which in turn is itself a finite map from variable
- * names to numbers.  We use the book library's [map] type family. *)
+ * names to numbers.  We use the book library's [fmap] type family. *)
 Definition valuation := fmap var nat.
 (* That is, the domain is [var] (a synonym for [string]) and the codomain/range
  * is [nat]. *)
@@ -97,9 +97,7 @@ Proof.
 
   equality.
 
-  rewrite IHe1.
-  rewrite IHe2.
-  ring.
+  linear_arithmetic.
 Qed.
 (* Well, that's a relief! ;-) *)
 
@@ -121,11 +119,11 @@ Proof.
 
   (* One case left after our basic heuristic:
    * the variable case, naturally! *)
-  cases (x ==v replaceThis); simplify; try equality.
+  cases (x ==v replaceThis); simplify; equality.
 Qed.
 (* Great; we seem to have gotten that one right, too. *)
 
-(* Let's also defined a pared-down version of the expression-simplificaton
+(* Let's also define a pared-down version of the expression-simplification
  * functions from last chapter. *)
 Fixpoint doSomeArithmetic (e : arith) : arith :=
   match e with
@@ -208,7 +206,8 @@ Fixpoint compile (e : arith) : list instruction :=
  * Skip down to the next theorem to see the overall correctness statement.
  * It turns out that we need to strengthen the induction hypothesis with a
  * lemma, to push the proof through. *)
-Lemma compile_ok' : forall e v is stack, run (compile e ++ is) v stack = run is v (interp e v :: stack).
+Lemma compile_ok' : forall e v is stack,
+    run (compile e ++ is) v stack = run is v (interp e v :: stack).
 Proof.
   induct e; simplify.
 
@@ -310,6 +309,7 @@ Example factorial_ugly :=
  * them from our examples. *)
 Coercion Const : nat >-> arith.
 Coercion Var : var >-> arith.
+Declare Scope arith_scope.
 Infix "+" := Plus : arith_scope.
 Infix "-" := Minus : arith_scope.
 Infix "*" := Times : arith_scope.
@@ -346,7 +346,7 @@ Definition factorial_body :=
  * Note that here we're careful to put the quantified variable [input] *first*,
  * because the variables coming after it will need to *change* in the course of
  * the induction.  Try switching the order to see what goes wrong if we put
-e * [input] later. *)
+ * [input] later. *)
 Lemma factorial_ok' : forall input output v,
   v $? "input" = Some input
   -> v $? "output" = Some output
diff --git a/frap_book.tex b/frap_book.tex
index d1da57a..a96a196 100644
--- a/frap_book.tex
+++ b/frap_book.tex
@@ -138,7 +138,7 @@ However, they go better together.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\chapter{Formalizing Program Syntax}
+\chapter{Formalizing Program Syntax}\label{syntax}
 
 \section{Concrete Syntax}
 
@@ -698,7 +698,7 @@ Let's shift our attention to what programs \emph{mean}.
 \newcommand{\msel}[2]{#1(#2)}
 \newcommand{\mupd}[3]{#1[#2 \mapsto #3]}
 
-To explain the meaning of one of last chapter's arithmetic expressions, we need a way to indicate the value of each variable.
+To explain the meaning of one of Chapter \ref{syntax}'s arithmetic expressions, we need a way to indicate the value of each variable.
 \encoding
 A theory of \emph{finite maps}\index{finite map} is helpful here.
 We apply the following notations throughout the book: \\