(* Title: HOL/HOLCF/IOA/ex/TrivEx.thy Author: Olaf Müller *) section \Trivial Abstraction Example\ theory TrivEx imports Abstraction begin datatype action = INC definition C_asig :: "action signature" where "C_asig = ({},{INC},{})" definition C_trans :: "(action, nat)transition set" where "C_trans = {tr. let s = fst(tr); t = snd(snd(tr)) in case fst(snd(tr)) of INC => t = Suc(s)}" definition C_ioa :: "(action, nat)ioa" where "C_ioa = (C_asig, {0}, C_trans,{},{})" definition A_asig :: "action signature" where "A_asig = ({},{INC},{})" definition A_trans :: "(action, bool)transition set" where "A_trans = {tr. let s = fst(tr); t = snd(snd(tr)) in case fst(snd(tr)) of INC => t = True}" definition A_ioa :: "(action, bool)ioa" where "A_ioa = (A_asig, {False}, A_trans,{},{})" definition h_abs :: "nat => bool" where "h_abs n = (n~=0)" axiomatization where MC_result: "validIOA A_ioa (\\\%(b,a,c). b\)" lemma h_abs_is_abstraction: "is_abstraction h_abs C_ioa A_ioa" apply (unfold is_abstraction_def) apply (rule conjI) txt \start states\ apply (simp (no_asm) add: h_abs_def starts_of_def C_ioa_def A_ioa_def) txt \step case\ apply (rule allI)+ apply (rule imp_conj_lemma) apply (simp (no_asm) add: trans_of_def C_ioa_def A_ioa_def C_trans_def A_trans_def) apply (induct_tac "a") apply (simp add: h_abs_def) done lemma TrivEx_abstraction: "validIOA C_ioa (\\\%(n,a,m). n~=0\)" apply (rule AbsRuleT1) apply (rule h_abs_is_abstraction) apply (rule MC_result) apply abstraction apply (simp add: h_abs_def) done end