header {* Addition with fixpoint of successor *} theory Ex3 imports LCF begin axiomatization s :: "'a => 'a" and p :: "'a => 'a => 'a" where p_strict: "p(UU) = UU" and p_s: "p(s(x),y) = s(p(x,y))" declare p_strict [simp] p_s [simp] lemma example: "p(FIX(s),y) = FIX(s)" apply (tactic {* induct_tac @{context} "s" 1 *}) apply simp apply simp done end