# Theory Induction_Schema

```(*  Title:      HOL/Examples/Induction_Schema.thy
Author:     Alexander Krauss, TU Muenchen
*)

section ‹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 *)
"⟦P 0; ⋀n. P n ⟹ P (Suc n)⟧ ⟹ P x"
by induction_schema (pat_completeness, lexicographic_order)

lemma nat_induct2:
"⟦ P 0; P (Suc 0); ⋀k. P k ==> P (Suc k) ==> P (Suc (Suc k)) ⟧
⟹ P n"
by induction_schema (pat_completeness, lexicographic_order)

lemma minus_one_induct:
"⟦⋀n::nat. (n ≠ 0 ⟹ P (n - 1)) ⟹ P n⟧ ⟹ 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 *)
"⟦ P [] [];
⋀x xs. P (x#xs) [];
⋀y ys. P [] (y#ys);
⋀x xs y ys. P xs ys  ⟹ P (x#xs) (y#ys) ⟧
⟹ P xs ys"
by induction_schema (pat_completeness, lexicographic_order)

theorem even_odd_induct:
assumes "R 0"
assumes "Q 0"
assumes "⋀n. Q n ⟹ R (Suc n)"
assumes "⋀n. R n ⟹ Q (Suc n)"
shows "R n" "Q n"
using assms
by induction_schema (pat_completeness+, lexicographic_order)

end
```