--- a/src/HOL/ex/Induction_Schema.thy Mon Jul 13 15:23:32 2020 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,48 +0,0 @@
-(* Title: HOL/ex/Induction_Schema.thy
- Author: Alexander Krauss, TU Muenchen
-*)
-
-section \<open>Examples of automatically derived induction rules\<close>
-
-theory Induction_Schema
-imports Main
-begin
-
-subsection \<open>Some simple induction principles on nat\<close>
-
-lemma nat_standard_induct: (* cf. Nat.thy *)
- "\<lbrakk>P 0; \<And>n. P n \<Longrightarrow> P (Suc n)\<rbrakk> \<Longrightarrow> P x"
-by induction_schema (pat_completeness, lexicographic_order)
-
-lemma nat_induct2:
- "\<lbrakk> P 0; P (Suc 0); \<And>k. P k ==> P (Suc k) ==> P (Suc (Suc k)) \<rbrakk>
- \<Longrightarrow> P n"
-by induction_schema (pat_completeness, lexicographic_order)
-
-lemma minus_one_induct:
- "\<lbrakk>\<And>n::nat. (n \<noteq> 0 \<Longrightarrow> P (n - 1)) \<Longrightarrow> P n\<rbrakk> \<Longrightarrow> P x"
-by induction_schema (pat_completeness, lexicographic_order)
-
-theorem diff_induct: (* cf. Nat.thy *)
- "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
- (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
-by induction_schema (pat_completeness, lexicographic_order)
-
-lemma list_induct2': (* cf. List.thy *)
- "\<lbrakk> P [] [];
- \<And>x xs. P (x#xs) [];
- \<And>y ys. P [] (y#ys);
- \<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
- \<Longrightarrow> P xs ys"
-by induction_schema (pat_completeness, lexicographic_order)
-
-theorem even_odd_induct:
- assumes "R 0"
- assumes "Q 0"
- assumes "\<And>n. Q n \<Longrightarrow> R (Suc n)"
- assumes "\<And>n. R n \<Longrightarrow> Q (Suc n)"
- shows "R n" "Q n"
- using assms
-by induction_schema (pat_completeness+, lexicographic_order)
-
-end