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theory CSP_F_semantics(*-------------------------------------------* | CSP-Prover on Isabelle2004 | | December 2004 | | August 2005 (modified) | | | | CSP-Prover on Isabelle2005 | | October 2005 (modified) | | April 2006 (modified) | | March 2007 (modified) | | August 2007 (modified) | | | | CSP-Prover on Isabelle2008 | | June 2008 (modified) | | | | Yoshinao Isobe (AIST JAPAN) | *-------------------------------------------*) theory CSP_F_semantics imports CSP_T_semantics Domain_F_cms begin (***************************************************************** 1. semantic clause 2. 3. 4. *****************************************************************) (********************************************************* semantic clause *********************************************************) consts failures :: "('p,'a) proc => ('p => 'a domF) => 'a setF" primrec "failures(STOP) = (%M. {f. EX X. f = (<>, X) }f)" "failures(SKIP) = (%M. {f. (EX X. f = (<>, X) & X <= Evset) | (EX X. f = (<Tick>, X)) }f)" "failures(DIV) = (%M. {}f)" "failures(a -> P) = (%M. {f. (EX X. f = (<>,X) & Ev a ~: X) | (EX s X. f = (<Ev a> ^^ s, X) & (s,X) :f failures(P) M ) }f)" "failures(? :X -> Pf) = (%M. {f. (EX Y. f = (<>,Y) & Ev`X Int Y = {}) | (EX a s Y. f = (<Ev a> ^^ s, Y) & (s,Y) :f failures(Pf a) M & a : X) }f)" "failures(P [+] Q) = (%M. {f. (EX X. f = (<>,X) & f :f failures(P) M IntF failures(Q) M) | (EX s X. f = (s,X) & f :f failures(P) M UnF failures(Q) M & s ~= <>) | (EX X. f = (<>,X) & <Tick> :t traces(P) (fstF o M) UnT traces(Q) (fstF o M) & X <= Evset) }f)" "failures(P |~| Q) = (%M. failures(P) M UnF failures(Q) M)" "failures(!! :C .. Pf) = (%M. {f. EX c: sumset C. f :f failures(Pf c) M}f)" "failures(IF b THEN P ELSE Q) = (%M. if b then failures(P) M else failures(Q) M)" "failures(P |[X]| Q) = (%M. {f. EX u Y Z. f = (u, Y Un Z) & Y-((Ev`X) Un {Tick})= Z-((Ev`X) Un {Tick}) & (EX s t. u : s |[X]|tr t & (s,Y) :f failures(P) M & (t,Z) :f failures(Q) M) }f)" "failures(P -- X) = (%M. {f. EX s Y. f = (s --tr X, Y) & (s,(Ev`X) Un Y) :f failures(P) M}f)" "failures(P [[r]]) = (%M. {f. EX s t X. f = (t,X) & s [[r]]* t & (s, [[r]]inv X) :f failures(P) M }f)" "failures(P ;; Q) = (%M. {f. (EX t X. f = (t,X) & (t, X Un {Tick}) :f failures(P) M & noTick t) | (EX s t X. f = (s ^^ t,X) & s ^^ <Tick> :t traces(P) (fstF o M) & (t, X) :f failures(Q) M & noTick s) }f)" "failures(P |. n) = (%M. failures(P) M .|. n)" "failures($p) = (%M. sndF (M p))" declare failures.simps [simp del] (*** for dealing with both !nat and !set ***) lemma failures_inv_inj[simp]: "inj g ==> (EX c. (EX n. c = g n & n : N) & f :f failures (Pf (inv g c)) x) = (EX z:N. f :f failures (Pf z) x)" by (auto) lemma Rep_int_choice_failures_nat: "failures(!nat :N .. Pf) = (%M. {f. EX n:N. f :f failures(Pf n) M}f)" apply (simp add: Rep_int_choice_ss_def) apply (simp add: failures.simps) done lemma Rep_int_choice_failures_set: "failures(!set :Xs .. Pf) = (%M. {f. EX X:Xs. f :f failures(Pf X) M}f)" apply (simp add: Rep_int_choice_ss_def) apply (simp add: failures.simps) done lemma Rep_int_choice_failures_com_lm: "(EX z. (EX a. z = {a} & a : X) & f :f failures (Pf (contents z)) M) = (EX a:X. f :f failures (Pf a) M)" apply (auto) apply (rule_tac x="{a}" in exI) apply (auto) done lemma Rep_int_choice_failures_com: "failures(! :X .. Pf) = (%M. {f. EX a:X. f :f failures(Pf a) M}f)" apply (simp add: Rep_int_choice_com_def) apply (simp add: Rep_int_choice_failures_set) apply (simp add: Rep_int_choice_failures_com_lm) done lemma Rep_int_choice_failures_f: "inj g ==> failures(!<g> :X .. Pf) = (%M. {f. EX a:X. f :f failures(Pf a) M}f)" apply (simp add: Rep_int_choice_f_def) apply (simp add: Rep_int_choice_failures_com) done lemmas Rep_int_choice_failures = Rep_int_choice_failures_nat Rep_int_choice_failures_set Rep_int_choice_failures_com Rep_int_choice_failures_f lemmas failures_def = failures.simps Rep_int_choice_failures (********************************************************* semantics *********************************************************) consts semFf :: "('p,'a) proc => ('p => 'a domF) => 'a domF" ("[[_]]Ff") semFfun :: "('p => ('p,'a) proc) => ('p => 'a domF) => ('p => 'a domF)" ("[[_]]Ffun") defs semFf_def : "[[P]]Ff == (%M. (traces(P) (fstF o M) ,, failures(P) M))" semFfun_def: "[[Pf]]Ffun == (%M. %p. [[Pf p]]Ff M)" (* notation (xsymbols) semFf ("[|_|]Ff") notation (xsymbols) semFfun ("[|_|]Ffun") *) (*** relation ***) lemma semFf_semFfun: "(%p. [[Pf p]]Ff M) = [[Pf]]Ffun M" by (simp add: semFfun_def semFf_def) (*------------------------------------------------------------------* M such that [[p]]Ff M = [[PNfun(p)]]Ff M such M is the fixed point of the function [[PNfun(p)]]Ffun *------------------------------------------------------------------*) consts semFfix :: "('p => ('p,'a) proc) => ('p => 'a domF)" ("[[_]]Ffix") defs semFfix_def : "[[Pf]]Ffix == (if (FPmode = CMSmode) then (UFP ([[Pf]]Ffun)) else (LFP ([[Pf]]Ffun)))" (* notation (xsymbols) semFfix ("[|_|]Ffix") *) consts MF :: "('p => 'a domF)" defs MF_def : "MF == [[PNfun]]Ffix" (*** semantics ***) consts semF :: "('p,'a) proc => 'a domF" ("[[_]]F") defs semF_def : "[[P]]F == [[P]]Ff MF" (* notation (xsymbols) semF ("[|_|]F") *) (********************************************************* relations over processes *********************************************************) consts refF :: "('p,'a) proc => ('p => 'a domF) => ('q => 'a domF) => ('q,'a) proc => bool" ("(0_ /<=F[_,_] /_)" [51,0,0,50] 50) eqF :: "('p,'a) proc => ('p => 'a domF) => ('q => 'a domF) => ('q,'a) proc => bool" ("(0_ /=F[_,_] /_)"[51,0,0,50] 50) defs refF_def : "P1 <=F[M1,M2] P2 == [[P2]]Ff M2 <= [[P1]]Ff M1" eqF_def : "P1 =F[M1,M2] P2 == [[P1]]Ff M1 = [[P2]]Ff M2" (*------------------------------------* | X-Symbols | *------------------------------------*) (* notation (xsymbols) refF ("(0_ /\<sqsubseteq>F[_,_] /_)" [51,100,100,50] 50) *) (********************************************************* relations over processes (fixed point) *********************************************************) abbreviation refFfix :: "('p,'a) proc => ('q,'a) proc => bool" ("(0_ /<=F /_)" [51,50] 50) where "P1 <=F P2 == P1 <=F[MF,MF] P2" abbreviation eqFfix :: "('p,'a) proc => ('q,'a) proc => bool" ("(0_ /=F /_)" [51,50] 50) where "P1 =F P2 == P1 =F[MF,MF] P2" (* =F and <=F *) lemma refF_semF: "(P1 <=F P2) = ([[P2]]F <= [[P1]]F)" apply (simp add: refF_def) apply (simp add: semF_def) done lemma eqF_semF: "(P1 =F P2) = ([[P1]]F = [[P2]]F)" apply (simp add: eqF_def) apply (simp add: semF_def) done (*------------------------------------* | X-Symbols | *------------------------------------*) (* notation (xsymbols) refFfix ("(0_ /\<sqsubseteq>F /_)" [51,50] 50) *) (*------------------* | csp law | *------------------*) (*** eq and ref ***) lemma cspF_eq_ref_iff: "(P1 =F[M1,M2] P2) = (P1 <=F[M1,M2] P2 & P2 <=F[M2,M1] P1)" by (auto simp add: refF_def eqF_def) lemma cspF_eq_ref: "P1 =F[M1,M2] P2 ==> P1 <=F[M1,M2] P2" by (simp add: cspF_eq_ref_iff) lemma cspF_ref_eq: "[| P1 <=F[M1,M2] P2 ; P2 <=F[M2,M1] P1 |] ==> P1 =F[M1,M2] P2" by (simp add: cspF_eq_ref_iff) (*** reflexivity ***) lemma cspF_reflex_eq_P[simp]: "P =F[M,M] P" by (simp add: eqF_def) lemma cspF_reflex_eq_STOP[simp]: "STOP =F[M1,M2] STOP" by (simp add: eqF_def semFf_def traces_def failures_def) lemma cspF_reflex_eq_SKIP[simp]: "SKIP =F[M1,M2] SKIP" by (simp add: eqF_def semFf_def traces_def failures_def) lemma cspF_reflex_eq_DIV[simp]: "DIV =F[M1,M2] DIV" by (simp add: eqF_def semFf_def traces_def failures_def) lemmas cspF_reflex_eq = cspF_reflex_eq_P cspF_reflex_eq_STOP cspF_reflex_eq_SKIP cspF_reflex_eq_DIV lemma cspF_reflex_ref_P[simp]: "P <=F[M,M] P" by (simp add: refF_def) lemma cspF_reflex_ref_STOP[simp]: "STOP <=F[M1,M2] STOP" by (simp add: refF_def semFf_def traces_def failures_def) lemma cspF_reflex_ref_SKIP[simp]: "SKIP <=F[M1,M2] SKIP" by (simp add: refF_def semFf_def traces_def failures_def) lemma cspF_reflex_ref_DIV[simp]: "DIV <=F[M1,M2] DIV" by (simp add: refF_def semFf_def traces_def failures_def) lemmas cspF_reflex_ref = cspF_reflex_ref_P cspF_reflex_ref_STOP cspF_reflex_ref_SKIP cspF_reflex_ref_DIV lemmas cspF_reflex = cspF_reflex_eq cspF_reflex_ref (*** symmetry ***) lemma cspF_sym: "P1 =F[M1,M2] P2 ==> P2 =F[M2,M1] P1" by (simp add: eqF_def) lemma cspF_symE: "[| P1 =F[M1,M2] P2 ; P2 =F[M2,M1] P1 ==> Z |] ==> Z" by (simp add: eqF_def) (*** transitivity ***) lemma cspF_trans_left_eq: "[| P1 =F[M1,M2] P2 ; P2 =F[M2,M3] P3 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_trans_left_ref: "[| P1 <=F[M1,M2] P2 ; P2 <=F[M2,M3] P3 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def) lemmas cspF_trans_left = cspF_trans_left_eq cspF_trans_left_ref lemmas cspF_trans = cspF_trans_left lemma cspF_trans_right_eq: "[| P2 =F[M2,M3] P3 ; P1 =F[M1,M2] P2 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_trans_right_ref: "[| P2 <=F[M2,M3] P3 ; P1 <=F[M1,M2] P2 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def) lemmas cspF_trans_rught = cspF_trans_right_eq cspF_trans_right_ref (*** rewrite (eq) ***) lemma cspF_rw_left_eq_MF: "[| P1 =F P2 ; P2 =F P3 |] ==> P1 =F P3" by (simp add: eqF_def) lemma cspF_rw_left_eq: "[| P1 =F[M1,M1] P2 ; P2 =F[M1,M3] P3 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_rw_left_ref_MF: "[| P1 =F P2 ; P2 <=F P3 |] ==> P1 <=F P3" by (simp add: refF_def eqF_def) lemma cspF_rw_left_ref: "[| P1 =F[M1,M1] P2 ; P2 <=F[M1,M3] P3 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def eqF_def) lemmas cspF_rw_left = cspF_rw_left_eq_MF cspF_rw_left_ref_MF cspF_rw_left_eq cspF_rw_left_ref lemma cspF_rw_right_eq: "[| P3 =F[M3,M3] P2 ; P1 =F[M1,M3] P2 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_rw_right_eq_MF: "[| P3 =F P2 ; P1 =F P2 |] ==> P1 =F P3" by (simp add: eqF_def) lemma cspF_rw_right_ref: "[| P3 =F[M3,M3] P2 ; P1 <=F[M1,M3] P2 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def eqF_def) lemma cspF_rw_right_ref_MF: "[| P3 =F P2 ; P1 <=F P2 |] ==> P1 <=F P3" by (simp add: refF_def eqF_def) lemmas cspF_rw_right = cspF_rw_right_eq_MF cspF_rw_right_ref_MF cspF_rw_right_eq cspF_rw_right_ref (*** rewrite (ref) ***) lemma cspF_tr_left_eq: "[| P1 =F[M1,M1] P2 ; P2 =F[M1,M3] P3 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_tr_left_ref: "[| P1 <=F[M1,M1] P2 ; P2 <=F[M1,M3] P3 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def eqF_def) lemmas cspF_tr_left = cspF_tr_left_eq cspF_tr_left_ref lemma cspF_tr_right_eq: "[| P2 =F[M3,M3] P3 ; P1 =F[M1,M3] P2 |] ==> P1 =F[M1,M3] P3" by (simp add: eqF_def) lemma cspF_tr_right_ref: "[| P2 <=F[M3,M3] P3 ; P1 <=F[M1,M3] P2 |] ==> P1 <=F[M1,M3] P3" by (simp add: refF_def eqF_def) lemmas cspF_tr_right = cspF_tr_right_eq cspF_tr_right_ref (*----------------------------------------* | rewriting processes in assumptions | *----------------------------------------*) (*** rewrite (eq) ***) lemma cspF_rw_left_eqE_MF: "[| P1 =F P3 ; P1 =F P2 ; [| P2 =F P3 |] ==> R |] ==> R" apply (subgoal_tac "P2 =F P3") apply (simp) apply (rule cspF_rw_left) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_left_eqE: "[| P1 =F[M1,M3] P3 ; P1 =F[M1,M1] P2 ; [| P2 =F[M1,M3] P3 |] ==> R |] ==> R" apply (subgoal_tac "P2 =F[M1,M3] P3") apply (simp) apply (rule cspF_rw_left) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_left_refE_MF: "[| P1 <=F P3 ; P1 =F P2 ; [| P2 <=F P3 |] ==> R |] ==> R" apply (subgoal_tac "P2 <=F P3") apply (simp) apply (rule cspF_rw_left) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_left_refE: "[| P1 <=F[M1,M3] P3 ; P1 =F[M1,M1] P2 ; [| P2 <=F[M1,M3] P3 |] ==> R |] ==> R" apply (subgoal_tac "P2 <=F[M1,M3] P3") apply (simp) apply (rule cspF_rw_left) apply (rule cspF_sym) apply (simp) apply (simp) done lemmas cspF_rw_leftE = cspF_rw_left_eqE_MF cspF_rw_left_refE_MF cspF_rw_left_eqE cspF_rw_left_refE (* right *) lemma cspF_rw_right_eqE_MF: "[| P1 =F P3 ; P3 =F P2 ; [| P1 =F P2 |] ==> R |] ==> R" apply (subgoal_tac "P1 =F P2") apply (simp) apply (rule cspF_rw_right) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_right_eqE: "[| P1 =F[M1,M3] P3 ; P3 =F[M3,M3] P2 ; [| P1 =F[M1,M3] P2 |] ==> R |] ==> R" apply (subgoal_tac "P1 =F[M1,M3] P2") apply (simp) apply (rule cspF_rw_right) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_right_refE_MF: "[| P1 <=F P3 ; P3 =F P2 ; [| P1 <=F P2 |] ==> R |] ==> R" apply (subgoal_tac "P1 <=F P2") apply (simp) apply (rule cspF_rw_right) apply (rule cspF_sym) apply (simp) apply (simp) done lemma cspF_rw_right_refE: "[| P1 <=F[M1,M3] P3 ; P3 =F[M3,M3] P2 ; [| P1 <=F[M1,M3] P2 |] ==> R |] ==> R" apply (subgoal_tac "P1 <=F[M1,M3] P2") apply (simp) apply (rule cspF_rw_right) apply (rule cspF_sym) apply (simp) apply (simp) done lemmas cspF_rw_rightE = cspF_rw_right_eqE_MF cspF_rw_right_refE_MF cspF_rw_right_eqE cspF_rw_right_refE (*-----------------------------------------* | noPN | *-----------------------------------------*) lemma failures_noPN_Constant_lm: "noPN P --> (EX F. failures P = (%M. F))" apply (induct_tac P) apply (simp add: failures_def, force) apply (simp add: failures_def, force) apply (simp add: failures_def, force) apply (simp add: failures_def, force) (* Ext_pre_choice *) apply (intro impI) apply (simp) apply (subgoal_tac "ALL x. EX F. failures (fun2 x) = (%M. F)") apply (simp add: choice_ALL_EX) apply (erule exE) apply (simp add: failures_def) apply (force) apply (simp) (* Ext_choice *) apply (intro impI) apply (simp) apply (elim exE) apply (simp add: failures_def) apply (subgoal_tac "ALL P. noPN P --> (EX T. traces P = (%M. T))") apply (frule_tac x="proc1" in spec) apply (drule_tac x="proc2" in spec) apply (simp) apply (elim exE) apply (simp) apply (force) apply (simp add: traces_noPN_Constant) apply (simp add: failures_def, force) (* Rep_int_choice_nat *) apply (intro impI) apply (simp) apply (subgoal_tac "ALL x. EX F. failures (fun x) = (%M. F)") apply (simp add: choice_ALL_EX) apply (erule exE) apply (simp add: failures_def) apply (force) apply (simp) (* IF *) apply (simp add: failures_def) apply (case_tac "bool") apply (simp) apply (simp) apply (simp add: failures_def, force) apply (simp add: failures_def, force) apply (simp add: failures_def, force) (* Seq_comp *) apply (intro impI) apply (simp) apply (elim exE) apply (simp add: failures_def) apply (subgoal_tac "ALL P. noPN P --> (EX T. traces P = (%M. T))") apply (drule_tac x="proc1" in spec) apply (simp) apply (elim exE) apply (simp) apply (force) apply (simp add: traces_noPN_Constant) apply (simp add: failures_def, force) apply (simp add: failures_def) done lemma failures_noPN_Constant: "noPN P ==> (EX F. failures P = (%M. F))" apply (simp add: failures_noPN_Constant_lm) done end