theory IndExamples+
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imports Main+
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begin+
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+
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section {* Transitive Closure *}+
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+
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text {*+
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  Introduction rules:+
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  @{term "trcl R x x"}+
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  @{term "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"}+
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*}+
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+
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definition "trcl \<equiv> \<lambda>R x y.+
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  \<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"+
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+
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lemma trcl_induct:+
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  assumes trcl: "trcl R x y"+
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  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"+
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  apply (atomize (full))+
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  apply (cut_tac trcl)+
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  apply (unfold trcl_def)+
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  apply (drule spec [where x=P])+
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  apply assumption+
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  done+
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+
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lemma trcl_base: "trcl R x x"+
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  apply (unfold trcl_def)+
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  apply (rule allI impI)++
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  apply (drule spec)+
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  apply assumption+
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  done+
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+
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lemma trcl_step: "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"+
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  apply (unfold trcl_def)+
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  apply (rule allI impI)++
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  proof -+
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    case goal1+
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    show ?case+
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      apply (rule goal1(4) [rule_format])+
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      apply (rule goal1(1))+
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      apply (rule goal1(2) [THEN spec [where x=P], THEN mp, THEN mp,+
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	OF goal1(3-4)])+
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      done+
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  qed+
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+
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+
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+
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section {* Even and Odd Numbers, Mutually Inductive *}+
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+
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text {*+
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  Introduction rules:+
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  @{term "even 0"}+
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  @{term "odd m \<Longrightarrow> even (Suc m)"}+
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  @{term "even m \<Longrightarrow> odd (Suc m)"}+
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*}+
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+
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definition "even \<equiv> \<lambda>n.+
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  \<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"+
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+
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definition "odd \<equiv> \<lambda>n.+
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  \<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"+
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+
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lemma even_induct:+
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  assumes even: "even n"+
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  shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"+
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  apply (atomize (full))+
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  apply (cut_tac even)+
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  apply (unfold even_def)+
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  apply (drule spec [where x=P])+
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  apply (drule spec [where x=Q])+
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  apply assumption+
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  done+
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+
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lemma odd_induct:+
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  assumes odd: "odd n"+
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  shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> Q n"+
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  apply (atomize (full))+
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  apply (cut_tac odd)+
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  apply (unfold odd_def)+
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  apply (drule spec [where x=P])+
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  apply (drule spec [where x=Q])+
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  apply assumption+
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  done+
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+
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lemma even0: "even 0"+
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  apply (unfold even_def)+
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  apply (rule allI impI)++
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  apply assumption+
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  done+
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+
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lemma evenS: "odd m \<Longrightarrow> even (Suc m)"+
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  apply (unfold odd_def even_def)+
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  apply (rule allI impI)++
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  proof -+
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    case goal1+
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    show ?case+
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      apply (rule goal1(3) [rule_format])+
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      apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],+
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	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])+
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      done+
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  qed+
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+
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lemma oddS: "even m \<Longrightarrow> odd (Suc m)"+
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  apply (unfold odd_def even_def)+
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  apply (rule allI impI)++
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  proof -+
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    case goal1+
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    show ?case+
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      apply (rule goal1(4) [rule_format])+
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      apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],+
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	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])+
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      done+
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  qed+
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+
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section {* Accessible Part *}+
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+
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text {*+
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  Introduction rules:+
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  @{term "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"}+
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*}+
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+
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definition "accpart \<equiv> \<lambda>R x. \<forall>P. (\<forall>x. (\<forall>y. R y x \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> P x"+
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+
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lemma accpart_induct:+
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  assumes acc: "accpart R x"+
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  shows "(\<And>x. (\<And>y. R y x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x"+
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  apply (atomize (full))+
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  apply (cut_tac acc)+
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  apply (unfold accpart_def)+
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  apply (drule spec [where x=P])+
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  apply assumption+
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  done+
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+
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lemma accpartI: "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"+
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  apply (unfold accpart_def)+
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  apply (rule allI impI)++
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  proof -+
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    case goal1+
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    note goal1' = this+
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    show ?case+
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      apply (rule goal1'(2) [rule_format])+
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      proof -+
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        case goal1+
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        show ?case+
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	  apply (rule goal1'(1) [OF goal1, THEN spec [where x=P],+
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            THEN mp, OF goal1'(2)])+
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	  done+
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      qed+
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    qed+
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+
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end+
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