(* Title: HOL/UNITY/Simple/Network.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The Communication Network.
From Misra, "A Logic for Concurrent Programming" (1994), section 5.7.
*)
theory Network imports UNITY begin
(*The state assigns a number to each process variable*)
datatype pvar = Sent | Rcvd | Idle
datatype pname = Aproc | Bproc
types state = "pname * pvar => nat"
locale F_props =
fixes F
assumes rsA: "F \<in> stable {s. s(Bproc,Rcvd) \<le> s(Aproc,Sent)}"
and rsB: "F \<in> stable {s. s(Aproc,Rcvd) \<le> s(Bproc,Sent)}"
and sent_nondec: "F \<in> stable {s. m \<le> s(proc,Sent)}"
and rcvd_nondec: "F \<in> stable {s. n \<le> s(proc,Rcvd)}"
and rcvd_idle: "F \<in> {s. s(proc,Idle) = Suc 0 & s(proc,Rcvd) = m}
co {s. s(proc,Rcvd) = m --> s(proc,Idle) = Suc 0}"
and sent_idle: "F \<in> {s. s(proc,Idle) = Suc 0 & s(proc,Sent) = n}
co {s. s(proc,Sent) = n}"
lemmas (in F_props)
sent_nondec_A = sent_nondec [of _ Aproc]
and sent_nondec_B = sent_nondec [of _ Bproc]
and rcvd_nondec_A = rcvd_nondec [of _ Aproc]
and rcvd_nondec_B = rcvd_nondec [of _ Bproc]
and rcvd_idle_A = rcvd_idle [of Aproc]
and rcvd_idle_B = rcvd_idle [of Bproc]
and sent_idle_A = sent_idle [of Aproc]
and sent_idle_B = sent_idle [of Bproc]
and rs_AB = stable_Int [OF rsA rsB]
and sent_nondec_AB = stable_Int [OF sent_nondec_A sent_nondec_B]
and rcvd_nondec_AB = stable_Int [OF rcvd_nondec_A rcvd_nondec_B]
and rcvd_idle_AB = constrains_Int [OF rcvd_idle_A rcvd_idle_B]
and sent_idle_AB = constrains_Int [OF sent_idle_A sent_idle_B]
and nondec_AB = stable_Int [OF sent_nondec_AB rcvd_nondec_AB]
and idle_AB = constrains_Int [OF rcvd_idle_AB sent_idle_AB]
and nondec_idle = constrains_Int [OF nondec_AB [unfolded stable_def]
idle_AB]
lemma (in F_props)
shows "F \<in> stable {s. s(Aproc,Idle) = Suc 0 & s(Bproc,Idle) = Suc 0 &
s(Aproc,Sent) = s(Bproc,Rcvd) &
s(Bproc,Sent) = s(Aproc,Rcvd) &
s(Aproc,Rcvd) = m & s(Bproc,Rcvd) = n}"
apply (unfold stable_def)
apply (rule constrainsI)
apply (drule constrains_Int [OF rs_AB [unfolded stable_def] nondec_idle,
THEN constrainsD], assumption)
apply simp_all
apply (blast del: le0, clarify)
apply (subgoal_tac "s' (Aproc, Rcvd) = s (Aproc, Rcvd)")
apply (subgoal_tac "s' (Bproc, Rcvd) = s (Bproc, Rcvd)")
apply simp
apply (blast intro: order_antisym le_trans eq_imp_le)+
done
end