--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/activities/tphols09/IDW/ExampleConstruction.thy	Wed Mar 30 17:27:34 2016 +0100
@@ -0,0 +1,153 @@
+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