--- 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.
*}