theory Ind_Examples
imports Main LaTeXsugar
begin

section{* Examples of Inductive Definitions \label{sec:ind-examples} *}

text {*

  In this section, we will give three examples showing how to define inductive
  predicates by hand and prove characteristic properties such as introduction
  rules and an induction rule. From these examples, we will then figure out a
  general method for defining inductive predicates.  It should be noted that
  our aim in this section is not to write proofs that are as beautiful as
  possible, but as close as possible to the ML code producing the proofs that
  we will develop later.  As a first example, we consider the transitive
  closure of a relation @{text R}. It is an inductive predicate characterized
  by the two introduction rules

  \begin{center}
  @{term[mode=Axiom] "trcl R x x"} \hspace{5mm}
  @{term[mode=Rule] "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"}
  \end{center}

  (FIXME first rule should be an ``axiom'')

  Note that the @{text trcl} predicate has two different kinds of parameters: the
  first parameter @{text R} stays \emph{fixed} throughout the definition, whereas
  the second and third parameter changes in the ``recursive call''.

  Since an inductively defined predicate is the least predicate closed under
  a collection of introduction rules, we define the predicate @{text "trcl R x y"} in
  such a way that it holds if and only if @{text "P x y"} holds for every predicate
  @{text P} closed under the rules above. This gives rise to the definition 
*}

definition "trcl R x y \<equiv> 
      \<forall>P. (\<forall>x. P x x) \<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}.

  Since the predicate @{term "trcl R x y"} yields an element of the type of object
  level truth values @{text bool}, the meta-level implications @{text "\<Longrightarrow>"} in the
  above introduction rules have to be converted to object-level implications
  @{text "\<longrightarrow>"}. Moreover, we use object-level universal quantifiers @{text "\<forall>"}
  rather than meta-level universal quantifiers @{text "\<And>"} for quantifying over
  the variable parameters of the introduction rules. Isabelle already offers some
  infrastructure for converting between meta-level and object-level connectives,
  which we will use later on.

  With this definition of the transitive closure, the proof of the (weak) induction
  theorem is almost immediate. It suffices to convert all the meta-level connectives
  in the induction rule to object-level connectives using the @{text atomize} proof
  method, expand the definition of @{text trcl}, eliminate the universal quantifier
  contained in it, and then solve the goal by assumption.
*}

lemma trcl_induct:
  assumes trcl: "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 trcl)
  apply (unfold trcl_def)
  apply (drule spec [where x=P])
  apply assumption
  done
(*<*)
lemma "trcl R x x"
  apply (unfold trcl_def)
  apply (rule allI impI)+
(*>*)

txt {*

  The above induction rule is \emph{weak} in the sense that the induction step may
  only be proved using the assumptions @{term "R x y"} and @{term "P y z"}, but not
  using the additional assumption \mbox{@{term "trcl R y z"}}. A stronger induction rule
  containing this additional assumption can be derived from the weaker one with the
  help of the introduction rules for @{text trcl}.

  We now turn to the proofs of the introduction rules, which are slightly more complicated.
  In order to prove the first introduction rule, we again unfold the definition and
  then apply the introdution rules for @{text "\<forall>"} and @{text "\<longrightarrow>"} as often as possible.
  We then end up in a proof state of the following form:
  
  @{subgoals [display]}

  The two assumptions correspond to the introduction rules, where @{term "trcl R"} has been
  replaced by @{term "P"}. Thus, all we have to do is to eliminate the universal quantifier
  in front of the first assumption, and then solve the goal by assumption:
  *}
(*<*)oops(*>*)

lemma trcl_base: "trcl R x x"
  apply (unfold trcl_def)
  apply (rule allI impI)+
  apply (drule spec)
  apply assumption
  done

(*<*)
lemma "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
  apply (unfold trcl_def)
  apply (rule allI impI)+
(*>*)

txt {*

  Since the second introduction rule has premises, its proof is not as easy as the previous
  one. After unfolding the definitions and applying the introduction rules for @{text "\<forall>"}
  and @{text "\<longrightarrow>"}, we get the proof state
  @{subgoals [display]}
  The third and fourth assumption corresponds to the first and second introduction rule,
  respectively, whereas the first and second assumption corresponds to the premises of
  the introduction rule. Since we want to prove the second introduction rule, we apply
  the fourth assumption to the goal @{term "P x z"}. In order for the assumption to
  be applicable, we have to eliminate the universal quantifiers and turn the object-level
  implications into meta-level ones. This can be accomplished using the @{text rule_format}
  attribute. Applying the assumption produces two new subgoals, which can be solved using
  the first and second assumption. The second assumption again involves a quantifier and
  implications that have to be eliminated before it can be applied. To avoid problems
  with higher order unification, it is advisable to provide an instantiation for the
  universally quantified predicate variable in the assumption.
*}
(*<*)oops(*>*)

lemma trcl_step: "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"
  apply (unfold trcl_def)
  apply (rule allI impI)+
  proof -
    case goal1
    show ?case
      apply (rule goal1(4) [rule_format])
      apply (rule goal1(1))
      apply (rule goal1(2) [THEN spec [where x=P], THEN mp, THEN mp,
	OF goal1(3-4)])
      done
  qed

text {*

  This method of defining inductive predicates easily generalizes to mutually inductive
  predicates, like the predicates @{text even} and @{text odd} characterized by the
  following introduction rules:
 
  \begin{center}
  @{term[mode=Axiom] "even (0::nat)"} \hspace{5mm}
  @{term[mode=Rule] "odd m \<Longrightarrow> even (Suc m)"} \hspace{5mm}
  @{term[mode=Rule] "even m \<Longrightarrow> odd (Suc m)"}
  \end{center}
  
  Since the predicates are mutually inductive, each of the definitions contain two
  quantifiers over the predicates @{text P} and @{text Q}.
*}

definition "even n \<equiv> 
  \<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 n \<equiv>
  \<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"

text {*
  For proving the induction rule, we use exactly the same technique as in the transitive
  closure example:
*}

lemma even_induct:
  assumes even: "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 even)
  apply (unfold even_def)
  apply (drule spec [where x=P])
  apply (drule spec [where x=Q])
  apply assumption
  done

text {*
  A similar induction rule having @{term "Q n"} as a conclusion can be proved for
  the @{text odd} predicate.
  The proofs of the introduction rules are also very similar to the ones in the
  previous example. We only show the proof of the second introduction rule,
  since it is almost the same as the one for the third introduction rule,
  and the proof of the first rule is trivial.
*}

lemma evenS: "odd m \<Longrightarrow> even (Suc m)"
  apply (unfold odd_def even_def)
  apply (rule allI impI)+
  proof -
    case goal1
    show ?case
      apply (rule goal1(3) [rule_format])
      apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],
	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])
      done
  qed
(*<*)
lemma even0: "even 0"
  apply (unfold even_def)
  apply (rule allI impI)+
  apply assumption
  done

lemma oddS: "even m \<Longrightarrow> odd (Suc m)"
  apply (unfold odd_def even_def)
  apply (rule allI impI)+
  proof -
    case goal1
    show ?case
      apply (rule goal1(4) [rule_format])
      apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],
	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])
      done
  qed
(*>*)

text {*
  As a final example, we will consider the definition of the accessible
  part of a relation @{text R} characterized by the introduction rule
  
  \begin{center}
  @{term[mode=Rule] "(\<forall>y. R y x \<longrightarrow> accpart R y) \<Longrightarrow> accpart R x"}
  \end{center}

  whose premise involves a universal quantifier and an implication. The
  definition of @{text accpart} is as follows:
*}

definition "accpart R x \<equiv> \<forall>P. (\<forall>x. (\<forall>y. R y x \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> P x"

text {*
  The proof of the induction theorem is again straightforward:
*}

lemma accpart_induct:
  assumes acc: "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 acc)
  apply (unfold accpart_def)
  apply (drule spec [where x=P])
  apply assumption
  done
(*<*)
lemma accpartI: "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"
  apply (unfold accpart_def)
  apply (rule allI impI)+
(*>*)

txt {*

  Proving the introduction rule is a little more complicated, due to the quantifier
  and the implication in the premise. We first convert the meta-level universal quantifier
  and implication to their object-level counterparts. Unfolding the definition of
  @{text accpart} and applying the introduction rules for @{text "\<forall>"} and @{text "\<longrightarrow>"}
  yields the following proof state:
  @{subgoals [display]}
  Applying the second assumption produces a proof state with the new local assumption
  @{term "R y x"}, which will then be used to solve the goal @{term "P y"} using the
  first assumption.
*}
(*<*)oops(*>*)

lemma accpartI: "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"
  apply (unfold accpart_def)
  apply (rule allI impI)+
  proof -
    case goal1
    note goal1' = this
    show ?case
      apply (rule goal1'(2) [rule_format])
      proof -
        case goal1
        show ?case
	  apply (rule goal1'(1) [OF goal1, THEN spec [where x=P],
            THEN mp, OF goal1'(2)])
	  done
      qed
    qed

end