Theory Handshake

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theory Handshake
imports UNITY_Main

(*  Title:      HOL/UNITY/Handshake.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge

Handshake Protocol

From Misra, "Asynchronous Compositions of Programs", Section 5.3.2
*)


theory Handshake imports "../UNITY_Main" begin

record state =
BB :: bool
NF :: nat
NG :: nat


definition
(*F's program*)
cmdF :: "(state*state) set"
where "cmdF = {(s,s'). s' = s (|NF:= Suc(NF s), BB:=False|) & BB s}"


definition
F :: "state program"
where "F = mk_total_program ({s. NF s = 0 & BB s}, {cmdF}, UNIV)"


definition
(*G's program*)
cmdG :: "(state*state) set"
where "cmdG = {(s,s'). s' = s (|NG:= Suc(NG s), BB:=True|) & ~ BB s}"


definition
G :: "state program"
where "G = mk_total_program ({s. NG s = 0 & BB s}, {cmdG}, UNIV)"


definition
(*the joint invariant*)
invFG :: "state set"
where "invFG = {s. NG s <= NF s & NF s <= Suc (NG s) & (BB s = (NF s = NG s))}"



declare F_def [THEN def_prg_Init, simp]
G_def [THEN def_prg_Init, simp]

cmdF_def [THEN def_act_simp, simp]
cmdG_def [THEN def_act_simp, simp]

invFG_def [THEN def_set_simp, simp]



lemma invFG: "(F Join G) : Always invFG"
apply (rule AlwaysI)
apply force
apply (rule constrains_imp_Constrains [THEN StableI])
apply auto
apply (unfold F_def, safety)
apply (unfold G_def, safety)
done

lemma lemma2_1: "(F Join G) : ({s. NF s = k} - {s. BB s}) LeadsTo
({s. NF s = k} Int {s. BB s})"

apply (rule stable_Join_ensures1[THEN leadsTo_Basis, THEN leadsTo_imp_LeadsTo])
apply (unfold F_def, safety)
apply (unfold G_def, ensures_tac "cmdG")
done

lemma lemma2_2: "(F Join G) : ({s. NF s = k} Int {s. BB s}) LeadsTo
{s. k < NF s}"

apply (rule stable_Join_ensures2[THEN leadsTo_Basis, THEN leadsTo_imp_LeadsTo])
apply (unfold F_def, ensures_tac "cmdF")
apply (unfold G_def, safety)
done

lemma progress: "(F Join G) : UNIV LeadsTo {s. m < NF s}"
apply (rule LeadsTo_weaken_R)
apply (rule_tac f = "NF" and l = "Suc m" and B = "{}"
in GreaterThan_bounded_induct)

(*The inductive step is (F Join G) : {x. NF x = ma} LeadsTo {x. ma < NF x}*)
apply (auto intro!: lemma2_1 lemma2_2
intro: LeadsTo_Trans LeadsTo_Diff simp add: vimage_def)

done

end