theory PolicySystemSAC

lemma abd_reads : "x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) = {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}" apply (rule subset_antisym) defer apply (rule abd_reads_all_bw) apply (simp) apply (rule subsetI) apply (erule subjectReads.induct) 
     (* warning: slow *)
      by (simp, blast?)+

(* Proof state trace *)
proof (prove)
goal (1 subgoal):
 1. x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) = {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} proof (prove)
goal (2 subgoals):
 1. x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) \<subseteq> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 2. x \<in> {NicA, NicB, NicD} \<Longrightarrow> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} \<subseteq> subjectReads SACAuthGraph (partition_label x) proof (prove)
goal (2 subgoals):
 1. x \<in> {NicA, NicB, NicD} \<Longrightarrow> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} \<subseteq> subjectReads SACAuthGraph (partition_label x)
 2. x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) \<subseteq> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} proof (prove)
goal (2 subgoals):
 1. x \<in> {NicA, NicB, NicD} \<Longrightarrow> x \<in> {NicA, NicB, NicD}
 2. x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) \<subseteq> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} proof (prove)
goal (1 subgoal):
 1. x \<in> {NicA, NicB, NicD} \<Longrightarrow> subjectReads SACAuthGraph (partition_label x) \<subseteq> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} proof (prove)
goal (1 subgoal):
 1. \<And>xa. \<lbrakk>x \<in> {NicA, NicB, NicD}; xa \<in> subjectReads SACAuthGraph (partition_label x)\<rbrakk> \<Longrightarrow> xa \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3} proof (prove)
goal (13 subgoals):
 1. \<And>xa l'. \<lbrakk>x \<in> {NicA, NicB, NicD}; (partition_label x, Read, l') \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> l' \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 2. \<And>xa. x \<in> {NicA, NicB, NicD} \<Longrightarrow> partition_label x \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 3. \<And>xa auth ep. \<lbrakk>x \<in> {NicA, NicB, NicD}; (partition_label x, auth, ep) \<in> SACAuthGraph; auth \<in> {SyncSend, Receive}\<rbrakk> \<Longrightarrow> ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 4. \<And>xa a auth' ep t auth. \<lbrakk>x \<in> {NicA, NicB, NicD}; (a, auth', ep) \<in> SACAuthGraph; auth' \<in> {Notify, SyncSend, Reset}; (t, auth, ep) \<in> SACAuthGraph; auth \<in> {SyncSend, Receive}; t \<in> subjectReads SACAuthGraph (partition_label x); t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}\<rbrakk> \<Longrightarrow> ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 5. \<And>xa b t. \<lbrakk>x \<in> {NicA, NicB, NicD}; b \<in> subjectReads SACAuthGraph (partition_label x); b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Write, b) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 6. \<And>xa t p. \<lbrakk>x \<in> {NicA, NicB, NicD}; t \<in> subjectReads SACAuthGraph (partition_label x); t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Read, p) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> p \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 7. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, SyncSend, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 8. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, Call, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 9. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, Receive, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 10. \<And>xa t' t. \<lbrakk>x \<in> {NicA, NicB, NicD}; t' \<in> subjectReads SACAuthGraph (partition_label x); t' \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Reply, t') \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
A total of 13 subgoals... proof (prove)
goal (13 subgoals):
 1. \<And>xa l'. \<lbrakk>x \<in> {NicA, NicB, NicD}; (partition_label x, Read, l') \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> l' \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 2. \<And>xa. x \<in> {NicA, NicB, NicD} \<Longrightarrow> partition_label x \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 3. \<And>xa auth ep. \<lbrakk>x \<in> {NicA, NicB, NicD}; (partition_label x, auth, ep) \<in> SACAuthGraph; auth \<in> {SyncSend, Receive}\<rbrakk> \<Longrightarrow> ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 4. \<And>xa a auth' ep t auth. \<lbrakk>x \<in> {NicA, NicB, NicD}; (a, auth', ep) \<in> SACAuthGraph; auth' \<in> {Notify, SyncSend, Reset}; (t, auth, ep) \<in> SACAuthGraph; auth \<in> {SyncSend, Receive}; t \<in> subjectReads SACAuthGraph (partition_label x); t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}\<rbrakk> \<Longrightarrow> ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 5. \<And>xa b t. \<lbrakk>x \<in> {NicA, NicB, NicD}; b \<in> subjectReads SACAuthGraph (partition_label x); b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Write, b) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 6. \<And>xa t p. \<lbrakk>x \<in> {NicA, NicB, NicD}; t \<in> subjectReads SACAuthGraph (partition_label x); t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Read, p) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> p \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 7. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, SyncSend, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 8. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, Call, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 9. \<And>xa ep b. \<lbrakk>x \<in> {NicA, NicB, NicD}; ep \<in> subjectReads SACAuthGraph (partition_label x); ep \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (b, Receive, ep) \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> b \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
 10. \<And>xa t' t. \<lbrakk>x \<in> {NicA, NicB, NicD}; t' \<in> subjectReads SACAuthGraph (partition_label x); t' \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}; (t, Reply, t') \<in> SACAuthGraph\<rbrakk> \<Longrightarrow> t \<in> {partition_label NicB, partition_label RM, partition_label R, partition_label NicA, partition_label NicD, partition_label EP, partition_label SC, partition_label NicC, partition_label NTFN1, partition_label NTFN2, partition_label NTFN3}
A total of 13 subgoals...