theory IndExamples

imports Main

begin

section {* Transitive Closure *}

text {*

  Introduction rules:

  @{term "trcl R x x"}

  @{term "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"}

*}

definition "trcl \<equiv> \<lambda>R x y.

  \<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"

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_base: "trcl R x x"

  apply (unfold trcl_def)

  apply (rule allI impI)+

  apply (drule spec)

  apply assumption

  done

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

section {* Even and Odd Numbers, Mutually Inductive *}

text {*

  Introduction rules:

  @{term "even 0"}

  @{term "odd m \<Longrightarrow> even (Suc m)"}

  @{term "even m \<Longrightarrow> odd (Suc m)"}

*}

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 \<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"

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

lemma odd_induct:

  assumes odd: "odd n"

  shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> Q n"

  apply (atomize (full))

  apply (cut_tac odd)

  apply (unfold odd_def)

  apply (drule spec [where x=P])

  apply (drule spec [where x=Q])

  apply assumption

  done

lemma even0: "even 0"

  apply (unfold even_def)

  apply (rule allI impI)+

  apply assumption

  done

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 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

section {* Accessible Part *}

text {*

  Introduction rules:

  @{term "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"}

*}

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

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)+

  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