# Theory TrivEx

theory TrivEx
imports Abstraction
```(*  Title:      HOL/HOLCF/IOA/ex/TrivEx.thy
Author:     Olaf MÃ¼ller
*)

section ‹Trivial Abstraction Example›

theory TrivEx
imports IOA.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")