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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 |