1 theory IndExamples

     2 imports Main

     3 begin

     4 

     5 section {* Transitive Closure *}

     6 

     7 text {*

     8   Introduction rules:

     9   @{term "trcl R x x"}

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

    11 *}

    12 

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

    14   \<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"

    15 

    16 lemma trcl_induct:

    17   assumes trcl: "trcl R x y"

    18   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"

    19   apply (atomize (full))

    20   apply (cut_tac trcl)

    21   apply (unfold trcl_def)

    22   apply (drule spec [where x=P])

    23   apply assumption

    24   done

    25 

    26 lemma trcl_base: "trcl R x x"

    27   apply (unfold trcl_def)

    28   apply (rule allI impI)+

    29   apply (drule spec)

    30   apply assumption

    31   done

    32 

    33 lemma trcl_step: "R x y \<Longrightarrow> trcl R y z \<Longrightarrow> trcl R x z"

    34   apply (unfold trcl_def)

    35   apply (rule allI impI)+

    36   proof -

    37     case goal1

    38     show ?case

    39       apply (rule goal1(4) [rule_format])

    40       apply (rule goal1(1))

    41       apply (rule goal1(2) [THEN spec [where x=P], THEN mp, THEN mp,

    42 	OF goal1(3-4)])

    43       done

    44   qed

    45 

    46 

    47 

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

    49 

    50 text {*

    51   Introduction rules:

    52   @{term "even 0"}

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

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

    55 *}

    56 

    57 definition "even \<equiv> \<lambda>n.

    58   \<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"

    59 

    60 definition "odd \<equiv> \<lambda>n.

    61   \<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"

    62 

    63 lemma even_induct:

    64   assumes even: "even n"

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

    66   apply (atomize (full))

    67   apply (cut_tac even)

    68   apply (unfold even_def)

    69   apply (drule spec [where x=P])

    70   apply (drule spec [where x=Q])

    71   apply assumption

    72   done

    73 

    74 lemma odd_induct:

    75   assumes odd: "odd n"

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

    77   apply (atomize (full))

    78   apply (cut_tac odd)

    79   apply (unfold odd_def)

    80   apply (drule spec [where x=P])

    81   apply (drule spec [where x=Q])

    82   apply assumption

    83   done

    84 

    85 lemma even0: "even 0"

    86   apply (unfold even_def)

    87   apply (rule allI impI)+

    88   apply assumption

    89   done

    90 

    91 lemma evenS: "odd m \<Longrightarrow> even (Suc m)"

    92   apply (unfold odd_def even_def)

    93   apply (rule allI impI)+

    94   proof -

    95     case goal1

    96     show ?case

    97       apply (rule goal1(3) [rule_format])

    98       apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],

    99 	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])

   100       done

   101   qed

   102 

   103 lemma oddS: "even m \<Longrightarrow> odd (Suc m)"

   104   apply (unfold odd_def even_def)

   105   apply (rule allI impI)+

   106   proof -

   107     case goal1

   108     show ?case

   109       apply (rule goal1(4) [rule_format])

   110       apply (rule goal1(1) [THEN spec [where x=P], THEN spec [where x=Q],

   111 	THEN mp, THEN mp, THEN mp, OF goal1(2-4)])

   112       done

   113   qed

   114 

   115 

   116 

   117 section {* Accessible Part *}

   118 

   119 text {*

   120   Introduction rules:

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

   122 *}

   123 

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

   125 

   126 lemma accpart_induct:

   127   assumes acc: "accpart R x"

   128   shows "(\<And>x. (\<And>y. R y x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P x"

   129   apply (atomize (full))

   130   apply (cut_tac acc)

   131   apply (unfold accpart_def)

   132   apply (drule spec [where x=P])

   133   apply assumption

   134   done

   135 

   136 lemma accpartI: "(\<And>y. R y x \<Longrightarrow> accpart R y) \<Longrightarrow> accpart R x"

   137   apply (unfold accpart_def)

   138   apply (rule allI impI)+

   139   proof -

   140     case goal1

   141     note goal1' = this

   142     show ?case

   143       apply (rule goal1'(2) [rule_format])

   144       proof -

   145         case goal1

   146         show ?case

   147 	  apply (rule goal1'(1) [OF goal1, THEN spec [where x=P],

   148             THEN mp, OF goal1'(2)])

   149 	  done

   150       qed

   151     qed

   152 

   153 end