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(* Title: HOL/HOLCF/IOA/ex/TrivEx2.thy
Author: Olaf Müller
*)
section \<open>Trivial Abstraction Example with fairness\<close>
theory TrivEx2
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
C_live_ioa :: "(action, nat)live_ioa" where
"C_live_ioa = (C_ioa, WF C_ioa {INC})"
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
A_live_ioa :: "(action, bool)live_ioa" where
"A_live_ioa = (A_ioa, WF A_ioa {INC})"
definition
h_abs :: "nat => bool" where
"h_abs n = (n~=0)"
axiomatization where
MC_result: "validLIOA (A_ioa,WF A_ioa {INC}) (\\\%(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 \<open>start states\<close>
apply (simp (no_asm) add: h_abs_def starts_of_def C_ioa_def A_ioa_def)
txt \<open>step case\<close>
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 (no_asm) add: h_abs_def)
done
lemma Enabled_implication:
"!!s. Enabled A_ioa {INC} (h_abs s) ==> Enabled C_ioa {INC} s"
apply (unfold Enabled_def enabled_def h_abs_def A_ioa_def C_ioa_def A_trans_def
C_trans_def trans_of_def)
apply auto
done
lemma h_abs_is_liveabstraction:
"is_live_abstraction h_abs (C_ioa, WF C_ioa {INC}) (A_ioa, WF A_ioa {INC})"
apply (unfold is_live_abstraction_def)
apply auto
txt \<open>is_abstraction\<close>
apply (rule h_abs_is_abstraction)
txt \<open>temp_weakening\<close>
apply abstraction
apply (erule Enabled_implication)
done
lemma TrivEx2_abstraction:
"validLIOA C_live_ioa (\\\%(n,a,m). n~=0\)"
apply (unfold C_live_ioa_def)
apply (rule AbsRuleT2)
apply (rule h_abs_is_liveabstraction)
apply (rule MC_result)
apply abstraction
apply (simp add: h_abs_def)
done
end
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