
diff -r a9eb69749c93 -r 1dc03eaa7cb9 ProgTutorial/Package/Ind_Prelims.thy
--- a/ProgTutorial/Package/Ind_Prelims.thy	Mon Mar 30 17:40:20 2009 +0200
+++ b/ProgTutorial/Package/Ind_Prelims.thy	Wed Apr 01 12:28:14 2009 +0100
@@ -1,52 +1,33 @@
 theory Ind_Prelims
-imports Main LaTeXsugar"../Base" Simple_Inductive_Package
+imports Main LaTeXsugar "../Base" 
 begin
 
 section{* Preliminaries *}
   
 text {*
-  The user will just give a specification of an inductive predicate and
+  The user will just give a specification of inductive predicate(s) and
   expects from the package to produce a convenient reasoning
   infrastructure. This infrastructure needs to be derived from the definition
-  that correspond to the specified predicate. This will roughly mean that the
-  package has three main parts, namely:
+  that correspond to the specified predicate(s). Before we start with
+  explaining all parts of the package, let us first give some examples 
+  showing how to define inductive predicates and then also how
+  to generate a reasoning infrastructure for them. From the examples 
+  we will figure out a general method for
+  defining inductive predicates.  The aim in this section is \emph{not} to
+  write proofs that are as beautiful as possible, but as close as possible to
+  the ML-code we will develop in later sections.
 
 
-  \begin{itemize}
-  \item parsing the specification and typing the parsed input,
-  \item making the definitions and deriving the reasoning infrastructure, and
-  \item storing the results in the theory. 
-  \end{itemize}
 
-  Before we start with explaining all parts, let us first give three examples
-  showing how to define inductive predicates by hand and then also how to
-  prove by hand important properties about them. From these examples, we will
-  figure out a general method for defining inductive predicates.  The aim in
-  this section is \emph{not} to write proofs that are as beautiful as
-  possible, but as close as possible to the ML-code we will develop in later
-  sections.
-
-
-  We first consider the transitive closure of a relation @{text R}. It is
-  an inductive predicate characterised by the two introduction rules:
+  We first consider the transitive closure of a relation @{text R}. The 
+  ``pencil-and-paper'' specification for the transitive closure is:
 
   \begin{center}\small
   @{prop[mode=Axiom] "trcl R x x"} \hspace{5mm}
   @{prop[mode=Rule] "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"}
   \end{center}
 
-  In Isabelle, the user will state for @{term trcl\<iota>} the specification:
-*}
-
-simple_inductive
-  trcl\<iota> :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
-where
-  base: "trcl\<iota> R x x"
-| step: "trcl\<iota> R x y \<Longrightarrow> R y z \<Longrightarrow> trcl\<iota> R x z"
-
-text {*
-  As said above the package has to make an appropriate definition and provide
-  lemmas to reason about the predicate @{term trcl\<iota>}. Since an inductively
+  The package has to make an appropriate definition. Since an inductively
   defined predicate is the least predicate closed under a collection of
   introduction rules, the predicate @{text "trcl R x y"} can be defined so
   that it holds if and only if @{text "P x y"} holds for every predicate
@@ -58,25 +39,25 @@
                   \<longrightarrow> (\<forall>x y z. R x y \<longrightarrow> P y z \<longrightarrow> P x z) \<longrightarrow> P x y"
 
 text {*
-  where we quantify over the predicate @{text P}. We have to use the
-  object implication @{text "\<longrightarrow>"} and object quantification @{text "\<forall>"} for
-  stating this definition (there is no other way for definitions in
-  HOL). However, the introduction rules and induction principles 
-  should use the meta-connectives since they simplify the
-  reasoning for the user.
+  We have to use the object implication @{text "\<longrightarrow>"} and object quantification
+  @{text "\<forall>"} for stating this definition (there is no other way for
+  definitions in HOL). However, the introduction rules and induction
+  principles associated with the transitive closure should use the meta-connectives, 
+  since they simplify the reasoning for the user.
+
 
   With this definition, the proof of the induction principle for @{term trcl}
   is almost immediate. It suffices to convert all the meta-level
   connectives in the lemma to object-level connectives using the
-  proof method @{text atomize} (Line 4), expand the definition of @{term trcl}
+  proof method @{text atomize} (Line 4 below), expand the definition of @{term trcl}
   (Line 5 and 6), eliminate the universal quantifier contained in it (Line~7),
-  and then solve the goal by assumption (Line 8).
+  and then solve the goal by @{text assumption} (Line 8).
 
 *}
 
 lemma %linenos trcl_induct:
-  assumes "trcl R x y"
-  shows "(\<And>x. P x x) \<Longrightarrow> (\<And>x y z. R x y \<Longrightarrow> P y z \<Longrightarrow> P x z) \<Longrightarrow> P x y"
+assumes "trcl R x y"
+shows "(\<And>x. P x x) \<Longrightarrow> (\<And>x y z. R x y \<Longrightarrow> P y z \<Longrightarrow> P x z) \<Longrightarrow> P x y"
 apply(atomize (full))
 apply(cut_tac prems)
 apply(unfold trcl_def)
@@ -90,7 +71,7 @@
 *}
 
 lemma %linenos trcl_base: 
-  shows "trcl R x x"
+shows "trcl R x x"
 apply(unfold trcl_def)
 apply(rule allI impI)+
 apply(drule spec)
@@ -110,9 +91,10 @@
 (*<*)oops(*>*)
 
 text {*
-  The two assumptions correspond to the introduction rules. Thus, all we have
-  to do is to eliminate the universal quantifier in front of the first
-  assumption (Line 5), and then solve the goal by assumption (Line 6).
+  The two assumptions come from the definition of @{term trcl} and correspond
+  to the introduction rules. Thus, all we have to do is to eliminate the
+  universal quantifier in front of the first assumption (Line 5), and then
+  solve the goal by @{text assumption} (Line 6).
 *}
 
 text {*
@@ -123,7 +105,7 @@
 *}
 
 lemma trcl_step: 
-  shows "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
+shows "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
 apply (unfold trcl_def)
 apply (rule allI impI)+
 
@@ -147,9 +129,9 @@
   
 txt {*
   The assumptions @{text "p1"} and @{text "p2"} correspond to the premises of
-  the second introduction rule; the assumptions @{text "r1"} and @{text "r2"}
-  correspond to the introduction rules. We apply @{text "r2"} to the goal
-  @{term "P x z"}. In order for the assumption to be applicable as a rule, we
+  the second introduction rule (unfolded); the assumptions @{text "r1"} and @{text "r2"}
+  come from the definition of @{term trcl}. We apply @{text "r2"} to the goal
+  @{term "P x z"}. In order for this assumption to be applicable as a rule, we
   have to eliminate the universal quantifier and turn the object-level
   implications into meta-level ones. This can be accomplished using the @{text
   rule_format} attribute. So we continue the proof with:
@@ -183,7 +165,7 @@
 *}
 
 lemma trcl_step_blast: 
-  shows "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
+shows "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
 apply(unfold trcl_def)
 apply(blast)
 done
@@ -195,40 +177,28 @@
   declare new intro- or simplification rules that can throw automatic tactics
   off course) and also it is very hard to debug proofs involving automatic
   tactics whenever something goes wrong. Therefore if possible, automatic 
-  tactics should be avoided or sufficiently constrained.
+  tactics should be avoided or be constrained sufficiently.
 
   The method of defining inductive predicates by impredicative quantification
   also generalises to mutually inductive predicates. The next example defines
-  the predicates @{text even} and @{text odd} characterised by the following
-  rules:
- 
+  the predicates @{text even} and @{text odd} given by
+
   \begin{center}\small
   @{prop[mode=Axiom] "even (0::nat)"} \hspace{5mm}
   @{prop[mode=Rule] "odd n \<Longrightarrow> even (Suc n)"} \hspace{5mm}
   @{prop[mode=Rule] "even n \<Longrightarrow> odd (Suc n)"}
   \end{center}
-  
-  The user will state for this inductive definition the specification:
-*}
 
-simple_inductive
-  even and odd
-where
-  even0: "even 0"
-| evenS: "odd n \<Longrightarrow> even (Suc n)"
-| oddS: "even n \<Longrightarrow> odd (Suc n)"
-
-text {*
   Since the predicates @{term even} and @{term odd} are mutually inductive, each 
   corresponding definition must quantify over both predicates (we name them 
   below @{text "P"} and @{text "Q"}).
 *}
 
-definition "even\<iota> \<equiv> 
+definition "even \<equiv> 
   \<lambda>n. \<forall>P Q. P 0 \<longrightarrow> (\<forall>m. Q m \<longrightarrow> P (Suc m)) 
                  \<longrightarrow> (\<forall>m. P m \<longrightarrow> Q (Suc m)) \<longrightarrow> P n"
 
-definition "odd\<iota> \<equiv>
+definition "odd \<equiv>
   \<lambda>n. \<forall>P Q. P 0 \<longrightarrow> (\<forall>m. Q m \<longrightarrow> P (Suc m)) 
                  \<longrightarrow> (\<forall>m. P m \<longrightarrow> Q (Suc m)) \<longrightarrow> Q n"
 
@@ -238,9 +208,8 @@
 *}
 
 lemma even_induct:
-  assumes "even n"
-  shows "P 0 \<Longrightarrow> 
-             (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
+assumes "even n"
+shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
 apply(atomize (full))
 apply(cut_tac prems)
 apply(unfold even_def)
@@ -252,14 +221,14 @@
 text {*
   The only difference with the proof @{text "trcl_induct"} is that we have to
   instantiate here two universal quantifiers.  We omit the other induction
-  principle that has @{term "Q n"} as conclusion.  The proofs of the
-  introduction rules are also very similar to the ones in the @{text
-  "trcl"}-example. We only show the proof of the second introduction rule.
-
+  principle that has @{prop "even n"} as premise and @{term "Q n"} as conclusion.  
+  The proofs of the introduction rules are also very similar to the ones in 
+  the @{text "trcl"}-example. We only show the proof of the second introduction 
+  rule.
 *}
 
 lemma %linenos evenS: 
-  shows "odd m \<Longrightarrow> even (Suc m)"
+shows "odd m \<Longrightarrow> even (Suc m)"
 apply (unfold odd_def even_def)
 apply (rule allI impI)+
 proof -
@@ -277,6 +246,9 @@
 qed
 
 text {*
+  The interesting lines are 7 to 15. The assumptions fall into two categories:
+  @{text p1} corresponds to the premise of the introduction rule; @{text "r1"}
+  to @{text "r3"} come from the definition of @{text "even"}.
   In Line 13, we apply the assumption @{text "r2"} (since we prove the second
   introduction rule). In Lines 14 and 15 we apply assumption @{text "p1"} (if
   the second introduction rule had more premises we have to do that for all
@@ -284,41 +256,26 @@
   need to be instantiated and then also the implications need to be resolved
   with the other rules.
 
+  Next we define the accessible part of a relation @{text R} given by
+  the single rule:
 
-  As a final example, we define the accessible part of a relation @{text R} characterised 
-  by the introduction rule
-  
   \begin{center}\small
   \mbox{\inferrule{@{term "\<And>y. R y x \<Longrightarrow> accpart R y"}}{@{term "accpart R x"}}}
   \end{center}
 
-  whose premise involves a universal quantifier and an implication. The
-  definition of @{text accpart} is:
+  The definition of @{text "accpart"} is:
 *}
 
 definition "accpart \<equiv> \<lambda>R x. \<forall>P. (\<forall>x. (\<forall>y. R y x \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> P x"
 
 text {*
-  The proof of the induction principle is again straightforward.
-*}
-
-lemma accpart_induct:
-  assumes "accpart R x"
-  shows "(\<And>x. (\<And>y. R y x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x"
-apply(atomize (full))
-apply(cut_tac prems)
-apply(unfold accpart_def)
-apply(drule spec[where x=P])
-apply(assumption)
-done
-
-text {*
-  Proving the introduction rule is a little more complicated, because the quantifier
-  and the implication in the premise. The proof is as follows.
+  The proof of the induction principle is again straightforward and omitted.
+  Proving the introduction rule is a little more complicated, because the 
+  quantifier and the implication in the premise. The proof is as follows.
 *}
 
 lemma %linenos accpartI: 
-  shows "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"
+shows "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"
 apply (unfold accpart_def)
 apply (rule allI impI)+
 proof -
@@ -338,14 +295,32 @@
 qed
 
 text {*
-  In Line 11, applying the assumption @{text "r1"} generates a goal state with
-  the new local assumption @{term "R y x"}, named @{text "r1_prem"} in the 
-  proof above (Line 14). This local assumption is used to solve
-  the goal @{term "P y"} with the help of assumption @{text "p1"}.
+  As you can see, there are now two subproofs. The assumptions fall again into
+  two categories (Lines 7 to 9). In Line 11, applying the assumption @{text
+  "r1"} generates a goal state with the new local assumption @{term "R y x"},
+  named @{text "r1_prem"} in the second subproof (Line 14). This local assumption is
+  used to solve the goal @{term "P y"} with the help of assumption @{text
+  "p1"}.
+
 
-  The point of these examples is to get a feeling what the automatic proofs 
-  should do in order to solve all inductive definitions we throw at them.
-  This is usually the first step in writing a package. We next explain
+  \begin{exercise}
+  Give the definition for the freshness predicate for lambda-terms. The rules
+  for this predicate are:
+  
+  \begin{center}\small
+  @{prop[mode=Rule] "a\<noteq>b \<Longrightarrow> fresh a (Var b)"}\hspace{5mm}
+  @{prop[mode=Rule] "\<lbrakk>fresh a t; fresh a s\<rbrakk> \<Longrightarrow> fresh a (App t s)"}\\[2mm]
+  @{prop[mode=Axiom] "fresh a (Lam a t)"}\hspace{5mm}
+  @{prop[mode=Rule] "\<lbrakk>a\<noteq>b; fresh a t\<rbrakk> \<Longrightarrow> fresh a (Lam b t)"}
+  \end{center}
+
+  From the definition derive the induction principle and the introduction 
+  rules. 
+  \end{exercise}
+
+  The point of all these examples is to get a feeling what the automatic
+  proofs should do in order to solve all inductive definitions we throw at
+  them.  This is usually the first step in writing a package. We next explain
   the parsing and typing part of the package.
 
 *}




