Datasets:

hoskinson-center
/

proof-pile

	theory ASC_Hoare
	imports ASC_Sufficiency "HOL-Hoare.Hoare_Logic"
	begin

	section \<open> Correctness of the Adaptive State Counting Algorithm in Hoare-Logic \<close>

	text \<open>
	In this section we give an example implementation of the adaptive state counting algorithm in a
	simple WHILE-language and prove that this implementation produces a certain output if and only if
	input FSM @{verbatim M1} is a reduction of input FSM @{verbatim M2}.
	\<close>



	lemma atc_io_reduction_on_sets_from_obs :
	assumes "L\<^sub>i\<^sub>n M1 T \<subseteq> L\<^sub>i\<^sub>n M2 T"
	and "(\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>) \<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	shows "atc_io_reduction_on_sets M1 T \<Omega> M2"
	unfolding atc_io_reduction_on_sets.simps atc_io_reduction_on.simps
	proof
	fix iseq assume "iseq \<in> T"
	have "L\<^sub>i\<^sub>n M1 {iseq} \<subseteq> L\<^sub>i\<^sub>n M2 {iseq}"
	by (metis \<open>iseq \<in> T\<close> assms(1) bot.extremum insert_mono io_reduction_on_subset
	mk_disjoint_insert)
	moreover have "\<forall>io\<in>L\<^sub>i\<^sub>n M1 {iseq}. B M1 io \<Omega> \<subseteq> B M2 io \<Omega>"
	proof
	fix io assume "io \<in> L\<^sub>i\<^sub>n M1 {iseq}"
	then have "io \<in> L\<^sub>i\<^sub>n M2 {iseq}"
	using calculation by blast
	show "B M1 io \<Omega> \<subseteq> B M2 io \<Omega> "
	proof
	fix x assume "x \<in> B M1 io \<Omega>"

	have "io \<in> L\<^sub>i\<^sub>n M1 T"
	using \<open>io \<in> L\<^sub>i\<^sub>n M1 {iseq}\<close> \<open>iseq \<in> T\<close> by auto
	moreover have "(io,x) \<in> {io} \<times> B M1 io \<Omega>"
	using \<open>x \<in> B M1 io \<Omega>\<close> by blast
	ultimately have "(io,x) \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>)"
	by blast

	then have "(io,x) \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	using assms(2) by blast
	then have "(io,x) \<in> {io} \<times> B M2 io \<Omega>"
	by blast
	then show "x \<in> B M2 io \<Omega>"
	by blast
	qed
	qed
	ultimately show "L\<^sub>i\<^sub>n M1 {iseq} \<subseteq> L\<^sub>i\<^sub>n M2 {iseq}
	\<and> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {iseq}. B M1 io \<Omega> \<subseteq> B M2 io \<Omega>)"
	by linarith
	qed


	lemma atc_io_reduction_on_sets_to_obs :
	assumes "atc_io_reduction_on_sets M1 T \<Omega> M2"
	shows "L\<^sub>i\<^sub>n M1 T \<subseteq> L\<^sub>i\<^sub>n M2 T"
	and "(\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>) \<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	proof
	fix x assume "x \<in> L\<^sub>i\<^sub>n M1 T"
	show "x \<in> L\<^sub>i\<^sub>n M2 T"
	using assms unfolding atc_io_reduction_on_sets.simps atc_io_reduction_on.simps
	proof -
	assume a1: "\<forall>iseq\<in>T. L\<^sub>i\<^sub>n M1 {iseq} \<subseteq> L\<^sub>i\<^sub>n M2 {iseq}
	\<and> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {iseq}. B M1 io \<Omega> \<subseteq> B M2 io \<Omega>)"
	have f2: "x \<in> UNION T (language_state_for_input M1 (initial M1))"
	by (metis (no_types) \<open>x \<in> L\<^sub>i\<^sub>n M1 T\<close> language_state_for_inputs_alt_def)
	obtain aas :: "'a list set \<Rightarrow> ('a list \<Rightarrow> ('a \<times> 'b) list set) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> 'a list"
	where
	"\<forall>x0 x1 x2. (\<exists>v3. v3 \<in> x0 \<and> x2 \<in> x1 v3) = (aas x0 x1 x2 \<in> x0 \<and> x2 \<in> x1 (aas x0 x1 x2))"
	by moura
	then have "\<forall>ps f A. (ps \<notin> UNION A f \<or> aas A f ps \<in> A \<and> ps \<in> f (aas A f ps))
	\<and> (ps \<in> UNION A f \<or> (\<forall>as. as \<notin> A \<or> ps \<notin> f as))"
	by blast
	then show ?thesis
	using f2 a1 by (metis (no_types) contra_subsetD language_state_for_input_alt_def
	language_state_for_inputs_alt_def)
	qed
	next
	show "(\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>) \<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	proof
	fix iox assume "iox \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>)"
	then obtain io x where "iox = (io,x)"
	by blast

	have "io \<in> L\<^sub>i\<^sub>n M1 T"
	using \<open>iox = (io, x)\<close> \<open>iox \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>)\<close> by blast
	have "(io,x) \<in> {io} \<times> B M1 io \<Omega>"
	using \<open>iox = (io, x)\<close> \<open>iox \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>)\<close> by blast
	then have "x \<in> B M1 io \<Omega>"
	by blast

	then have "x \<in> B M2 io \<Omega>"
	using assms unfolding atc_io_reduction_on_sets.simps atc_io_reduction_on.simps
	by (metis (no_types, lifting) UN_E \<open>io \<in> L\<^sub>i\<^sub>n M1 T\<close> language_state_for_input_alt_def
	language_state_for_inputs_alt_def subsetCE)
	then have "(io,x) \<in> {io} \<times>B M2 io \<Omega>"
	by blast
	then have "(io,x) \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	using \<open>io \<in> L\<^sub>i\<^sub>n M1 T\<close> by auto
	then show "iox \<in> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
	using \<open>iox = (io, x)\<close> by auto
	qed
	qed

	lemma atc_io_reduction_on_sets_alt_def :
	shows "atc_io_reduction_on_sets M1 T \<Omega> M2 =
	(L\<^sub>i\<^sub>n M1 T \<subseteq> L\<^sub>i\<^sub>n M2 T
	\<and> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>)
	\<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>))"
	using atc_io_reduction_on_sets_to_obs[of M1 T \<Omega> M2]
	and atc_io_reduction_on_sets_from_obs[of M1 T M2 \<Omega>]
	by blast





	lemma asc_algorithm_correctness:
	"VARS tsN cN rmN obs obsI obs\<^sub>\<Omega> obsI\<^sub>\<Omega> iter isReduction
	{
	OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>
	}
	tsN := {};
	cN := V;
	rmN := {};
	obs := L\<^sub>i\<^sub>n M2 cN;
	obsI := L\<^sub>i\<^sub>n M1 cN;
	obs\<^sub>\<Omega> := (\<Union>io\<in>L\<^sub>i\<^sub>n M2 cN. {io} \<times> B M2 io \<Omega>);
	obsI\<^sub>\<Omega> := (\<Union>io\<in>L\<^sub>i\<^sub>n M1 cN. {io} \<times> B M1 io \<Omega>);
	iter := 1;
	WHILE (cN \<noteq> {} \<and> obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>)
	INV {
	0 < iter
	\<and> tsN = TS M2 M1 \<Omega> V m (iter-1)
	\<and> cN = C M2 M1 \<Omega> V m iter
	\<and> rmN = RM M2 M1 \<Omega> V m (iter-1)
	\<and> obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN)
	\<and> obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN)
	\<and> obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)
	\<and> obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)
	\<and> OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>
	}
	DO
	iter := iter + 1;
	rmN := {xs' \<in> cN .
	(\<not> (L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'}))
	\<or> (\<forall> io \<in> L\<^sub>i\<^sub>n M1 {xs'} .
	(\<exists> V'' \<in> N io M1 V .
	(\<exists> S1 .
	(\<exists> vs xs .
	io = (vs@xs)
	\<and> mcp (vs@xs) V'' vs
	\<and> S1 \<subseteq> nodes M2
	\<and> (\<forall> s1 \<in> S1 . \<forall> s2 \<in> S1 .
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall> io1 \<in> RP M2 s1 vs xs V'' .
	\<forall> io2 \<in> RP M2 s2 vs xs V'' .
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega> ))
	\<and> m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'' ))))};
	tsN := tsN \<union> cN;
	cN := append_set (cN - rmN) (inputs M2) - tsN;
	obs := obs \<union> L\<^sub>i\<^sub>n M2 cN;
	obsI := obsI \<union> L\<^sub>i\<^sub>n M1 cN;
	obs\<^sub>\<Omega> := obs\<^sub>\<Omega> \<union> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 cN. {io} \<times> B M2 io \<Omega>);
	obsI\<^sub>\<Omega> := obsI\<^sub>\<Omega> \<union> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 cN. {io} \<times> B M1 io \<Omega>)
	OD;
	isReduction := ((obsI \<subseteq> obs) \<and> (obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>))
	{
	isReduction = M1 \<preceq> M2 \<comment>\<open>variable isReduction is used only as a return value,
	it is true if and only if M1 is a reduction of M2\<close>
	}"
	proof (vcg)
	assume precond : "OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>"
	have "{} = TS M2 M1 \<Omega> V m (1-1)"
	"V = C M2 M1 \<Omega> V m 1"
	"{} = RM M2 M1 \<Omega> V m (1-1)"
	"L\<^sub>i\<^sub>n M2 V = L\<^sub>i\<^sub>n M2 ({} \<union> V)"
	"L\<^sub>i\<^sub>n M1 V = L\<^sub>i\<^sub>n M1 ({} \<union> V)"
	"(\<Union>io\<in>L\<^sub>i\<^sub>n M2 V. {io} \<times> B M2 io \<Omega>)
	= (\<Union>io\<in>L\<^sub>i\<^sub>n M2 ({} \<union> V). {io} \<times> B M2 io \<Omega>)"
	"(\<Union>io\<in>L\<^sub>i\<^sub>n M1 V. {io} \<times> B M1 io \<Omega>)
	= (\<Union>io\<in>L\<^sub>i\<^sub>n M1 ({} \<union> V). {io} \<times> B M1 io \<Omega>)"
	using precond by auto
	moreover have "OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega> "
	using precond by assumption
	ultimately show "0 < (1::nat) \<and>
	{} = TS M2 M1 \<Omega> V m (1 - 1) \<and>
	V = C M2 M1 \<Omega> V m 1 \<and>
	{} = RM M2 M1 \<Omega> V m (1 - 1) \<and>
	L\<^sub>i\<^sub>n M2 V = L\<^sub>i\<^sub>n M2 ({} \<union> V) \<and>
	L\<^sub>i\<^sub>n M1 V = L\<^sub>i\<^sub>n M1 ({} \<union> V) \<and>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M2 V. {io} \<times> B M2 io \<Omega>)
	= (\<Union>io\<in>L\<^sub>i\<^sub>n M2 ({} \<union> V). {io} \<times> B M2 io \<Omega>) \<and>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M1 V. {io} \<times> B M1 io \<Omega>)
	= (\<Union>io\<in>L\<^sub>i\<^sub>n M1 ({} \<union> V). {io} \<times> B M1 io \<Omega>) \<and>
	OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>"
	by linarith+
	next
	fix tsN cN rmN obs obsI obs\<^sub>\<Omega> obsI\<^sub>\<Omega> iter isReduction
	assume precond : "(0 < iter \<and>
	tsN = TS M2 M1 \<Omega> V m (iter - 1) \<and>
	cN = C M2 M1 \<Omega> V m iter \<and>
	rmN = RM M2 M1 \<Omega> V m (iter - 1) \<and>
	obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN) \<and>
	obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<and>
	obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>) \<and>
	obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>) \<and>
	OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>)
	\<and> cN \<noteq> {} \<and> obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>"
	then have "0 < iter"
	"OFSM M1"
	"OFSM M2"
	"asc_fault_domain M2 M1 m"
	"test_tools M2 M1 FAIL PM V \<Omega>"
	"cN \<noteq> {}"
	"obsI \<subseteq> obs"
	"tsN = TS M2 M1 \<Omega> V m (iter-1)"
	"cN = C M2 M1 \<Omega> V m iter"
	"rmN = RM M2 M1 \<Omega> V m (iter-1)"
	"obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN)"
	"obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN)"
	"obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)"
	"obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)"
	by linarith+


	obtain k where "iter = Suc k"
	using gr0_implies_Suc[OF \<open>0 < iter\<close>] by blast
	then have "cN = C M2 M1 \<Omega> V m (Suc k)"
	"tsN = TS M2 M1 \<Omega> V m k"
	using \<open>cN = C M2 M1 \<Omega> V m iter\<close> \<open>tsN = TS M2 M1 \<Omega> V m (iter-1)\<close> by auto
	have "TS M2 M1 \<Omega> V m iter = TS M2 M1 \<Omega> V m (Suc k)"
	"C M2 M1 \<Omega> V m iter = C M2 M1 \<Omega> V m (Suc k)"
	"RM M2 M1 \<Omega> V m iter = RM M2 M1 \<Omega> V m (Suc k)"
	using \<open>iter = Suc k\<close> by presburger+


	have rmN_calc[simp] : "{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')} =
	RM M2 M1 \<Omega> V m iter"
	proof -


	have "{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')} =
	{xs' \<in> C M2 M1 \<Omega> V m (Suc k).
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs ((TS M2 M1 \<Omega> V m k) \<union> V) S1 \<Omega> V'')}"
	using \<open>cN = C M2 M1 \<Omega> V m (Suc k)\<close> \<open>tsN = TS M2 M1 \<Omega> V m k\<close> by blast

	moreover have "{xs' \<in> C M2 M1 \<Omega> V m (Suc k).
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs ((TS M2 M1 \<Omega> V m k) \<union> V) S1 \<Omega> V'')} =
	RM M2 M1 \<Omega> V m (Suc k)"
	using RM.simps(2)[of M2 M1 \<Omega> V m k] by blast
	ultimately have "{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')} =
	RM M2 M1 \<Omega> V m (Suc k)"
	by presburger
	then show ?thesis
	using \<open>iter = Suc k\<close> by presburger
	qed
	moreover have "RM M2 M1 \<Omega> V m iter = RM M2 M1 \<Omega> V m (iter + 1 - 1)" by simp
	ultimately have rmN_calc' : "{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')} =
	RM M2 M1 \<Omega> V m (iter + 1 - 1)" by presburger

	have "tsN \<union> cN = TS M2 M1 \<Omega> V m (Suc k)"
	proof (cases k)
	case 0
	then show ?thesis
	using \<open>tsN = TS M2 M1 \<Omega> V m k\<close> \<open>cN = C M2 M1 \<Omega> V m (Suc k)\<close> by auto
	next
	case (Suc nat)
	then have "TS M2 M1 \<Omega> V m (Suc k) = TS M2 M1 \<Omega> V m k \<union> C M2 M1 \<Omega> V m (Suc k)"
	using TS.simps(3) by blast
	moreover have "tsN \<union> cN = TS M2 M1 \<Omega> V m k \<union> C M2 M1 \<Omega> V m (Suc k)"
	using \<open>tsN = TS M2 M1 \<Omega> V m k\<close> \<open>cN = C M2 M1 \<Omega> V m (Suc k)\<close> by auto
	ultimately show ?thesis
	by auto
	qed
	then have tsN_calc : "tsN \<union> cN = TS M2 M1 \<Omega> V m iter"
	using \<open>iter = Suc k\<close> by presburger


	have cN_calc : "append_set
	(cN -
	{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN) =
	C M2 M1 \<Omega> V m (iter + 1)"
	proof -
	have "append_set
	(cN -
	{xs' \<in> cN.
	\<not> io_reduction_on M1 {xs'} M2 \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN) =
	append_set
	((C M2 M1 \<Omega> V m iter) -
	(RM M2 M1 \<Omega> V m iter))
	(inputs M2) -
	(TS M2 M1 \<Omega> V m iter) "
	using \<open>cN = C M2 M1 \<Omega> V m iter\<close> \<open>tsN \<union> cN = TS M2 M1 \<Omega> V m iter\<close> rmN_calc by presburger
	moreover have "append_set
	((C M2 M1 \<Omega> V m iter) -
	(RM M2 M1 \<Omega> V m iter))
	(inputs M2) -
	(TS M2 M1 \<Omega> V m iter) = C M2 M1 \<Omega> V m (iter + 1)"
	proof -
	have "C M2 M1 \<Omega> V m (iter + 1) = C M2 M1 \<Omega> V m ((Suc k) + 1)"
	using \<open>iter = Suc k\<close> by presburger+
	moreover have "(Suc k) + 1 = Suc (Suc k)"
	by simp
	ultimately have "C M2 M1 \<Omega> V m (iter + 1) = C M2 M1 \<Omega> V m (Suc (Suc k))"
	by presburger

	have "C M2 M1 \<Omega> V m (Suc (Suc k))
	= append_set (C M2 M1 \<Omega> V m (Suc k) - RM M2 M1 \<Omega> V m (Suc k)) (inputs M2)
	- TS M2 M1 \<Omega> V m (Suc k)"
	using C.simps(3)[of M2 M1 \<Omega> V m k] by linarith
	show ?thesis
	using Suc_eq_plus1
	\<open>C M2 M1 \<Omega> V m (Suc (Suc k))
	= append_set (C M2 M1 \<Omega> V m (Suc k) - RM M2 M1 \<Omega> V m (Suc k)) (inputs M2)
	- TS M2 M1 \<Omega> V m (Suc k)\<close>
	\<open>iter = Suc k\<close>
	by presburger
	qed

	ultimately show ?thesis
	by presburger
	qed


	have obs_calc : "obs \<union>
	L\<^sub>i\<^sub>n M2
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)) =
	L\<^sub>i\<^sub>n M2
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)))"
	proof -
	have "\<And>A. L\<^sub>i\<^sub>n M2 (tsN \<union> cN \<union> A) = obs \<union> L\<^sub>i\<^sub>n M2 A"
	by (metis (no_types) language_state_for_inputs_union precond)
	then show ?thesis
	by blast
	qed


	have obsI_calc : "obsI \<union>
	L\<^sub>i\<^sub>n M1
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)) =
	L\<^sub>i\<^sub>n M1
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)))"
	proof -
	have "\<And>A. L\<^sub>i\<^sub>n M1 (tsN \<union> cN \<union> A) = obsI \<union> L\<^sub>i\<^sub>n M1 A"
	by (metis (no_types) language_state_for_inputs_union precond)
	then show ?thesis
	by blast
	qed

	have obs\<^sub>\<Omega>_calc : "obs\<^sub>\<Omega> \<union>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M2
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)).
	{io} \<times> B M2 io \<Omega>) =
	(\<Union>io\<in>L\<^sub>i\<^sub>n M2
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))).
	{io} \<times> B M2 io \<Omega>)"
	using \<open>obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN)\<close>
	\<open>obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)\<close>
	obs_calc
	by blast

	have obsI\<^sub>\<Omega>_calc : "obsI\<^sub>\<Omega> \<union>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M1
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)).
	{io} \<times> B M1 io \<Omega>) =
	(\<Union>io\<in>L\<^sub>i\<^sub>n M1
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))).
	{io} \<times> B M1 io \<Omega>)"
	using \<open>obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN)\<close>
	\<open>obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)\<close>
	obsI_calc
	by blast



	have "0 < iter + 1"
	using \<open>0 < iter\<close> by simp
	have "tsN \<union> cN = TS M2 M1 \<Omega> V m (iter + 1 - 1)"
	using tsN_calc by simp



	from \<open>0 < iter + 1\<close>
	\<open>tsN \<union> cN = TS M2 M1 \<Omega> V m (iter + 1 - 1)\<close>
	cN_calc
	rmN_calc'
	obs_calc
	obsI_calc
	obs\<^sub>\<Omega>_calc
	obsI\<^sub>\<Omega>_calc
	\<open>OFSM M1\<close>
	\<open>OFSM M2\<close>
	\<open>asc_fault_domain M2 M1 m\<close>
	\<open>test_tools M2 M1 FAIL PM V \<Omega>\<close>
	show "0 < iter + 1 \<and>
	tsN \<union> cN = TS M2 M1 \<Omega> V m (iter + 1 - 1) \<and>
	append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN) =
	C M2 M1 \<Omega> V m (iter + 1) \<and>
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''.
	B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')} =
	RM M2 M1 \<Omega> V m (iter + 1 - 1) \<and>
	obs \<union>
	L\<^sub>i\<^sub>n M2
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)) =
	L\<^sub>i\<^sub>n M2
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))) \<and>
	obsI \<union>
	L\<^sub>i\<^sub>n M1
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)) =
	L\<^sub>i\<^sub>n M1
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))) \<and>
	obs\<^sub>\<Omega> \<union>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M2
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)).
	{io} \<times> B M2 io \<Omega>) =
	(\<Union>io\<in>L\<^sub>i\<^sub>n M2
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))).
	{io} \<times> B M2 io \<Omega>) \<and>
	obsI\<^sub>\<Omega> \<union>
	(\<Union>io\<in>L\<^sub>i\<^sub>n M1
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN)).
	{io} \<times> B M1 io \<Omega>) =
	(\<Union>io\<in>L\<^sub>i\<^sub>n M1
	(tsN \<union> cN \<union>
	(append_set
	(cN -
	{xs' \<in> cN.
	\<not> L\<^sub>i\<^sub>n M1 {xs'} \<subseteq> L\<^sub>i\<^sub>n M2 {xs'} \<or>
	(\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}.
	\<exists>V''\<in>N io M1 V.
	\<exists>S1 vs xs.
	io = vs @ xs \<and>
	mcp (vs @ xs) V'' vs \<and>
	S1 \<subseteq> nodes M2 \<and>
	(\<forall>s1\<in>S1.
	\<forall>s2\<in>S1.
	s1 \<noteq> s2 \<longrightarrow>
	(\<forall>io1\<in>RP M2 s1 vs xs V''.
	\<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and>
	m < LB M2 M1 vs xs (tsN \<union> V) S1 \<Omega> V'')})
	(inputs M2) -
	(tsN \<union> cN))).
	{io} \<times> B M1 io \<Omega>) \<and>
	OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>"
	by linarith
	next
	fix tsN cN rmN obs obsI obs\<^sub>\<Omega> obsI\<^sub>\<Omega> iter isReduction
	assume precond : "(0 < iter \<and>
	tsN = TS M2 M1 \<Omega> V m (iter - 1) \<and>
	cN = C M2 M1 \<Omega> V m iter \<and>
	rmN = RM M2 M1 \<Omega> V m (iter - 1) \<and>
	obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN) \<and>
	obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<and>
	obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>) \<and>
	obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>) \<and>
	OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>) \<and>
	\<not> (cN \<noteq> {} \<and> obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>)"
	then have "0 < iter"
	"OFSM M1"
	"OFSM M2"
	"asc_fault_domain M2 M1 m"
	"test_tools M2 M1 FAIL PM V \<Omega>"
	"cN = {} \<or> \<not> obsI \<subseteq> obs \<or> \<not> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>"
	"tsN = TS M2 M1 \<Omega> V m (iter-1)"
	"cN = C M2 M1 \<Omega> V m iter"
	"rmN = RM M2 M1 \<Omega> V m (iter-1)"
	"obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN)"
	"obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN)"
	"obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)"
	"obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)"
	by linarith+



	show "(obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>) = M1 \<preceq> M2"
	proof (cases "cN = {}")
	case True
	then have "C M2 M1 \<Omega> V m iter = {}"
	using \<open>cN = C M2 M1 \<Omega> V m iter\<close> by auto

	have "is_det_state_cover M2 V"
	using \<open>test_tools M2 M1 FAIL PM V \<Omega>\<close> by auto
	then have "[] \<in> V"
	using det_state_cover_initial[of M2 V] by simp
	then have "V \<noteq> {}"
	by blast
	have "Suc 0 < iter"
	proof (rule ccontr)
	assume "\<not> Suc 0 < iter"
	then have "iter = Suc 0"
	using \<open>0 < iter\<close> by auto
	then have "C M2 M1 \<Omega> V m (Suc 0) = {}"
	using \<open>C M2 M1 \<Omega> V m iter = {}\<close> by auto
	moreover have "C M2 M1 \<Omega> V m (Suc 0) = V"
	by auto
	ultimately show"False"
	using \<open>V \<noteq> {}\<close> by blast
	qed

	obtain k where "iter = Suc k"
	using gr0_implies_Suc[OF \<open>0 < iter\<close>] by blast
	then have "Suc 0 < Suc k"
	using \<open>Suc 0 < iter\<close> by auto
	then have "0 < k"
	by simp
	then obtain k' where "k = Suc k'"
	using gr0_implies_Suc by blast
	have "iter = Suc (Suc k')"
	using \<open>iter = Suc k\<close> \<open>k = Suc k'\<close> by simp

	have "TS M2 M1 \<Omega> V m (Suc (Suc k')) = TS M2 M1 \<Omega> V m (Suc k') \<union> C M2 M1 \<Omega> V m (Suc (Suc k'))"
	using TS.simps(3)[of M2 M1 \<Omega> V m k'] by blast
	then have "TS M2 M1 \<Omega> V m iter = TS M2 M1 \<Omega> V m (Suc k')"
	using True \<open>cN = C M2 M1 \<Omega> V m iter\<close> \<open>iter = Suc (Suc k')\<close> by blast
	moreover have "Suc k' = iter - 1"
	using \<open>iter = Suc (Suc k')\<close> by presburger
	ultimately have "TS M2 M1 \<Omega> V m iter = TS M2 M1 \<Omega> V m (iter - 1)"
	by auto
	then have "tsN = TS M2 M1 \<Omega> V m iter"
	using \<open>tsN = TS M2 M1 \<Omega> V m (iter-1)\<close> by simp

	then have "TS M2 M1 \<Omega> V m iter = TS M2 M1 \<Omega> V m (iter - 1)"
	using \<open>tsN = TS M2 M1 \<Omega> V m (iter - 1)\<close> by auto
	then have "final_iteration M2 M1 \<Omega> V m (iter-1)"
	using \<open>0 < iter\<close> by auto

	have "M1 \<preceq> M2 = atc_io_reduction_on_sets M1 tsN \<Omega> M2"
	using asc_main_theorem[OF \<open>OFSM M1\<close> \<open>OFSM M2\<close>
	\<open>asc_fault_domain M2 M1 m\<close>
	\<open>test_tools M2 M1 FAIL PM V \<Omega>\<close>
	\<open>final_iteration M2 M1 \<Omega> V m (iter-1)\<close>]
	using \<open>tsN = TS M2 M1 \<Omega> V m (iter - 1)\<close>
	by blast
	moreover have "tsN \<union> cN = tsN"
	using \<open>cN = {}\<close> by blast
	ultimately have "M1 \<preceq> M2 = atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	by presburger

	have "obsI \<subseteq> obs \<equiv> L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<subseteq> L\<^sub>i\<^sub>n M2 (tsN \<union> cN)"
	by (simp add: \<open>obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN)\<close> \<open>obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN)\<close>)

	have "obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega> \<equiv> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)
	\<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)"
	by (simp add: \<open>obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)\<close>
	\<open>obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)\<close>)


	have "(obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>) = atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	proof
	assume "obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>"
	show "atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	using atc_io_reduction_on_sets_from_obs[of M1 "tsN \<union> cN" M2 \<Omega>]
	using \<open>obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>\<close> \<open>obsI \<subseteq> obs \<equiv> L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<subseteq> L\<^sub>i\<^sub>n M2 (tsN \<union> cN)\<close>
	\<open>obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega> \<equiv> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)
	\<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)\<close>
	by linarith
	next
	assume "atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	show "obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>"
	using atc_io_reduction_on_sets_to_obs[of M1 \<open>tsN \<union> cN\<close> \<Omega> M2]
	\<open>atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2\<close>
	\<open>obsI \<subseteq> obs \<equiv> L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<subseteq> L\<^sub>i\<^sub>n M2 (tsN \<union> cN)\<close>
	\<open>obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega> \<equiv> (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>)
	\<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>)\<close>
	by blast
	qed
	then show ?thesis
	using \<open>M1 \<preceq> M2 = atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2\<close> by linarith
	next
	case False


	then have "\<not> obsI \<subseteq> obs \<or> \<not> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>"
	using \<open>cN = {} \<or> \<not> obsI \<subseteq> obs \<or> \<not> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>\<close> by auto

	have "\<not> atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	using atc_io_reduction_on_sets_to_obs[of M1 "tsN \<union> cN" \<Omega> M2]
	\<open>\<not> obsI \<subseteq> obs \<or> \<not> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>\<close> precond
	by fastforce

	have "\<not> M1 \<preceq> M2"
	proof
	assume "M1 \<preceq> M2"
	have "atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2"
	using asc_soundness[OF \<open>OFSM M1\<close> \<open>OFSM M2\<close>] \<open>M1 \<preceq> M2\<close> by blast
	then show "False"
	using \<open>\<not> atc_io_reduction_on_sets M1 (tsN \<union> cN) \<Omega> M2\<close> by blast
	qed

	then show ?thesis
	using \<open>\<not> obsI \<subseteq> obs \<or> \<not> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>\<close> by blast

	qed
	qed


	end