(* Title: HOL/ex/Induction_Schema.thy
Author: Alexander Krauss, TU Muenchen
*)
header {* Examples of automatically derived induction rules *}
theory Induction_Schema
imports Main
begin
subsection {* Some simple induction principles on nat *}
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