(*  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