Extended_Finite_State_Machi.../Extended_Finite_State_Machi.../examples/Drinks_Subsumption.thy

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subsection\<open>Example\<close>
text\<open>This theory shows how contexts can be used to prove transition subsumption.\<close>
theory Drinks_Subsumption
imports "Extended_Finite_State_Machine_Inference.Subsumption" "Extended_Finite_State_Machines.Drinks_Machine_2"
begin
lemma stop_at_3: "\<not>obtains 1 c drinks2 3 r t"
proof(induct t arbitrary: r)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
by (simp add: drinks2_def)
qed
lemma no_1_2: "\<not>obtains 1 c drinks2 2 r t"
proof(induct t arbitrary: r)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
apply (simp add: drinks2_def)
apply clarsimp
apply (simp add: drinks2_def[symmetric])
apply (erule disjE)
apply simp
apply (erule disjE)
apply simp
by (simp add: stop_at_3)
qed
lemma no_change_1_1: "obtains 1 c drinks2 1 r t \<Longrightarrow> c = r"
proof(induct t)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
apply (simp add: drinks2_def)
apply clarsimp
apply (simp add: drinks2_def[symmetric])
apply (erule disjE)
apply (simp add: vend_nothing_def apply_updates_def)
by (simp add: no_1_2)
qed
lemma obtains_1: "obtains 1 c drinks2 0 <> t \<Longrightarrow> c $ 2 = Some (Num 0)"
proof(induct t)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
apply (simp add: drinks2_def)
apply (simp add: drinks2_def[symmetric])
apply (simp add: select_def can_take apply_updates_def)
using no_change_1_1 by fastforce
qed
lemma obtains_1_1_2:
"obtains 1 c1 drinks2 1 r t \<Longrightarrow>
obtains 1 c2 drinks 1 r t \<Longrightarrow>
c1 = r \<and> c2 = r"
proof(induct t arbitrary: r)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
apply (simp add: drinks2_def drinks_def)
apply clarsimp
apply (simp add: drinks2_def[symmetric] drinks_def[symmetric])
apply safe
using Cons.prems(1) no_change_1_1 apply blast
apply (simp add: coin_def vend_nothing_def)
using Cons.prems(1) no_change_1_1 apply blast
apply (simp add: vend_fail_def vend_nothing_def apply_updates_def)
using Cons.prems(1) no_change_1_1 apply blast
apply (metis drinks_rejects_future numeral_eq_one_iff obtains.cases obtains_recognises semiring_norm(85))
using no_1_2 apply blast
using no_1_2 apply blast
using Cons.prems(1) no_change_1_1 apply blast
using no_1_2 apply blast
using no_1_2 apply blast
using no_1_2 by blast
qed
lemma obtains_1_c2:
"obtains 1 c1 drinks2 0 <> t \<Longrightarrow> obtains 1 c2 drinks 0 <> t \<Longrightarrow> c2 $ 2 = Some (Num 0)"
proof(induct t)
case Nil
then show ?case
by (simp add: obtains_base)
next
case (Cons a t)
then show ?case
apply (case_tac a)
apply (simp add: obtains_step)
apply clarify
apply (simp add: in_possible_steps[symmetric])
apply (simp add: drinks2_def drinks_def)
apply clarsimp
apply (simp add: drinks2_def[symmetric] drinks_def[symmetric])
apply (simp add: select_def can_take apply_updates_def)
using obtains_1_1_2 by fastforce
qed
lemma directly_subsumes: "directly_subsumes drinks2 drinks 1 1 vend_fail vend_nothing"
apply (rule direct_subsumption[of _ _ _ _ "\<lambda>c2. c2 $ 2 = Some (Num 0)"])
apply (simp add: obtains_1_c2)
apply (rule subsumption)
apply (simp add: vend_fail_def vend_nothing_def)
apply (simp add: vend_fail_def vend_nothing_def can_take value_gt_true)
apply (simp add: vend_fail_def vend_nothing_def)
by (simp add: posterior_separate_def vend_fail_def vend_nothing_def)
lemma directly_subsumes_flip: "directly_subsumes drinks2 drinks 1 1 vend_nothing vend_fail"
apply (rule direct_subsumption[of _ _ _ _ "\<lambda>c2. c2 $ 2 = Some (Num 0)"])
apply (simp add: obtains_1_c2)
apply (rule subsumption)
apply (simp add: vend_fail_def vend_nothing_def)
apply (simp add: vend_fail_def vend_nothing_def can_take value_gt_true)
apply (simp add: vend_fail_def vend_nothing_def can_take value_gt_true)
by (simp add: posterior_separate_def vend_fail_def vend_nothing_def)
end