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theory CSP_F_failures(*-------------------------------------------* | CSP-Prover on Isabelle2004 | | December 2004 | | June 2005 (modified) | | August 2005 (modified) | | | | CSP-Prover on Isabelle2005 | | October 2005 (modified) | | April 2006 (modified) | | March 2007 (modified) | | August 2007 (modified) | | | | Yoshinao Isobe (AIST JAPAN) | *-------------------------------------------*) theory CSP_F_failures imports CSP_F_semantics begin (* The following simplification rules are deleted in this theory file *) (* because they unexpectly rewrite UnionT and InterT. *) (* disj_not1: (~ P | Q) = (P --> Q) *) declare disj_not1 [simp del] (********************************************************* setF *********************************************************) (*--------------------------------* | STOP | *--------------------------------*) (*** STOP_setF ***) lemma STOP_setF: "{f. EX X. f = (<>, X)} : setF" by (simp add: setF_def HC_F2_def) (*** STOP ***) lemma in_failures_STOP : "(f :f failures(STOP) M) = (EX X. f = (<>, X))" apply (simp add: failures_def) by (simp add: CollectF_open_memF STOP_setF) (*--------------------------------* | SKIP | *--------------------------------*) (*** SKIP_setF ***) lemma SKIP_setF: "{f. (EX X. f = (<>, X) & X <= Evset) | (EX X. f = (<Tick>, X)) } : setF" by (auto simp add: setF_def HC_F2_def) (*** SKIP ***) lemma in_failures_SKIP : "(f :f failures(SKIP) M) = ((EX X. f = (<>, X) & X <= Evset) | (EX X. f = (<Tick>, X)))" apply (simp add: failures_def) by (simp add: CollectF_open_memF SKIP_setF) (*--------------------------------* | DIV | *--------------------------------*) (*** DIV ***) lemma in_failures_DIV : "(f ~:f failures(DIV) M)" by (simp add: failures_def) (*--------------------------------* | Act_prefix | *--------------------------------*) (*** Act_prefix_setF ***) lemma Act_prefix_setF: "{f. (EX X. f = (<>,X) & Ev a ~: X) | (EX s X. f = (<Ev a> ^^ s, X) & (s,X) :f F) } : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE disjE, force) apply (elim conjE exE, simp) apply (rule memF_F2, simp_all) done (*** Act_prefix ***) lemma in_failures_Act_prefix: "(f :f failures(a -> P) M) = ((EX X. f = (<>,X) & Ev a ~: X) | (EX s X. f = (<Ev a> ^^ s, X) & (s,X) :f failures(P) M))" apply (simp add: failures_def) by (simp add: CollectF_open_memF Act_prefix_setF) (*--------------------------------* | Ext_pre_choice | *--------------------------------*) (*** Ext_pre_choice_setF ***) lemma Ext_pre_choice_setF: "{f. (EX Y. f = (<>,Y) & Ev`X Int Y = {}) | (EX a s Y. f = (<Ev a> ^^ s, Y) & (s,Y) :f Ff a & a : X) } : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE disjE, force) apply (elim conjE exE, simp) apply (rule memF_F2, simp_all) done (*** Ext_pre_choice ***) lemma in_failures_Ext_pre_choice: "(f :f failures(? :X -> Pf) M) = ((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))" apply (simp add: failures_def) by (simp add: CollectF_open_memF Ext_pre_choice_setF) (*--------------------------------* | Ext_choice | *--------------------------------*) (*** Ext_choice_setF ***) lemma Ext_choice_setF: "{f. (EX X. f = (<>, X)) & f :f F & f :f E | (EX s. (EX X. f = (s, X)) & (f :f F | f :f E) & s ~= <>) | (EX X. f = (<>, X) & (<Tick> :t T | <Tick> :t S) & X <= Evset)} : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE disjE) apply (simp_all) (* part1 *) apply (rule disjI1) apply (rule conjI) apply (rule memF_F2, simp_all) apply (rule memF_F2, simp_all) (* part2 *) apply (rule disjI1) apply (rule memF_F2, simp_all) apply (rule disjI2) apply (rule memF_F2, simp_all) (* part3 *) apply (auto) done (*** Ext_choice ***) lemma in_failures_Ext_choice: "(f :f failures(P [+] Q) M) = ((EX X. f = (<>, X)) & f :f failures P M & f :f failures Q M | (EX s. (EX X. f = (s, X)) & (f :f failures P M | f :f failures Q M) & s ~= <>) | (EX X. f = (<>, X) & (<Tick> :t traces P (fstF o M) | <Tick> :t traces Q (fstF o M)) & X <= Evset))" apply (simp add: failures_def) by (simp only: CollectF_open_memF Ext_choice_setF) (*--------------------------------* | Int_choice | *--------------------------------*) (*** Int_choice ***) lemma in_failures_Int_choice: "(f :f failures(P |~| Q) M) = (f :f failures(P) M | f :f failures(Q) M)" by (simp add: failures_def) (*--------------------------------* | Rep_int_choice | *--------------------------------*) (*** Rep_int_choice_setF ***) lemma Rep_int_choice_setF: "{f. EX a:X. f :f Ff a} : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE bexE) apply (rule_tac x="a" in bexI) apply (rule memF_F2, simp_all) done (*** Union_proc ***) lemma in_failures_Union_proc: "f :f {f. EX a:X. f :f Ff a}f = (EX a:X. f :f Ff a)" by (simp add: CollectF_open_memF Rep_int_choice_setF) lemma in_failures_UNIV_Union_proc: "f :f {f. EX a. f :f Ff a}f = (EX a. f :f Ff a)" apply (insert in_failures_Union_proc[of f UNIV Ff]) by (simp) (*** Rep_int_choice ***) lemma in_failures_Rep_int_choice_sum: "(f :f failures(!! :C .. Pf) M) = (EX c: sumset C. f :f failures(Pf c) M)" apply (simp add: failures_def) by (simp add: in_failures_Union_proc) lemma in_failures_Rep_int_choice_nat: "(f :f failures(!nat :N .. Pf) M) = (EX n:N. f :f failures(Pf n) M)" by (simp add: Rep_int_choice_ss_def in_failures_Rep_int_choice_sum) lemma in_failures_Rep_int_choice_set: "(f :f failures(!set :Xs .. Pf) M) = (EX X:Xs. f :f failures(Pf X) M)" by (simp add: Rep_int_choice_ss_def in_failures_Rep_int_choice_sum) lemma in_failures_Rep_int_choice_com: "(f :f failures(! :X .. Pf) M) = (EX a:X. f :f failures(Pf a) M)" apply (simp add: failures_def) by (simp add: in_failures_Union_proc) lemma in_failures_Rep_int_choice_f: "inj g ==> (f :f failures(!<g> :X .. Pf) M) = (EX a:X. f :f failures(Pf a) M)" apply (simp add: failures_def) by (simp add: in_failures_Union_proc) lemmas in_failures_Rep_int_choice = in_failures_Rep_int_choice_sum in_failures_Rep_int_choice_nat in_failures_Rep_int_choice_set in_failures_Rep_int_choice_com in_failures_Rep_int_choice_f (*--------------------------------* | IF | *--------------------------------*) (*** IF ***) lemma in_failures_IF: "(f :f failures(IF b THEN P ELSE Q) M) = (if b then f :f failures(P) M else f :f failures(Q) M)" by (simp add: failures_def) (*--------------------------------* | Parallel | *--------------------------------*) (*** Parallel_setF ***) lemma Parallel_setF : "{(u, Y Un Z) |u Y Z. Y - insert Tick (Ev ` X) = Z - insert Tick (Ev ` X) & (EX s t. u : s |[X]|tr t & (s, Y) :f F & (t, Z) :f E)} : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE exE, simp) apply (rename_tac u Y Z Y1 Y2 s t) apply (rule_tac x="Z Int (Y1 - (Y2 - insert Tick (Ev ` X))) Un Z Int (Y2 - insert Tick (Ev ` X))" in exI) apply (rule_tac x="Z Int (Y2 - (Y2 - insert Tick (Ev ` X))) Un Z Int (Y2 - insert Tick (Ev ` X))" in exI) (* (s,Y), Z <= Y, Y = Y1 Un Y2, Z = Z1 Un Z2 *) apply (rule conjI, force) apply (rule conjI, force) apply (rule_tac x="s" in exI) apply (rule_tac x="t" in exI) apply (simp) apply (rule conjI) apply (rule memF_F2, simp, force)+ done lemma in_failures_Parallel: "(f :f failures(P |[X]| Q) M) = (EX u Y Z. f = (u, Y Un Z) & Y - insert Tick (Ev ` X) = Z - insert Tick (Ev ` X) & (EX s t. u : s |[X]|tr t & (s,Y) :f failures(P) M & (t,Z) :f failures(Q) M))" apply (simp add: failures_def) by (simp only: CollectF_open_memF Parallel_setF) (*--------------------------------* | Hiding | *--------------------------------*) (*** Hiding_setF ***) lemma Hiding_setF : "{f. EX s Y. f = (s --tr X, Y) & (s,(Ev`X) Un Y) :f F} : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE exE, simp) apply (rename_tac t Y Z s) apply (rule_tac x="s" in exI) apply (simp) by (rule memF_F2, simp, force) lemma in_failures_Hiding: "(f :f failures(P -- X) M) = (EX s Y. f = (s --tr X, Y) & (s,(Ev`X) Un Y) :f failures(P) M)" apply (simp add: failures_def) by (simp only: CollectF_open_memF Hiding_setF) (*--------------------------------* | Renaming | *--------------------------------*) (*** Renaming_setF ***) lemma Renaming_setF : "{f. EX s t X. f = (t,X) & (s [[r]]* t) & (s, [[r]]inv X) :f F } : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE exE) apply (rule_tac x="sa" in exI) apply (simp) apply (rule memF_F2, simp) apply (rule ren_inv_sub, simp) done lemma in_failures_Renaming: "(f :f failures(P[[r]]) M) = (EX s t X. f = (t,X) & s [[r]]* t & (s, [[r]]inv X) :f failures(P) M)" apply (simp add: failures_def) by (simp only: CollectF_open_memF Renaming_setF) (*--------------------------------* | Seq_compo | *--------------------------------*) (*** Seq_compo_setF ***) lemma Seq_compo_setF : "{f. (EX t X. f = (t, X) & (t, insert Tick X) :f F & noTick t) | (EX s t X. f = (s ^^ t, X) & s ^^ <Tick> :t T & (t, X) :f E & noTick s)} : setF" apply (simp add: setF_def HC_F2_def) apply (intro allI impI) apply (elim conjE exE disjE) (* case 1 *) apply (rule disjI1, simp) apply (rule memF_F2, simp, force) (* case 2 *) apply (rule disjI2) apply (rule_tac x="sa" in exI) apply (rule_tac x="t" in exI) apply (simp) apply (rule memF_F2, simp, simp) done lemma in_failures_Seq_compo: "(f :f failures(P ;; Q) M) = ((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))" apply (simp add: failures_def) by (simp only: CollectF_open_memF Seq_compo_setF) (*--------------------------------* | Depth_rest | *--------------------------------*) (*** Depth_rest ***) lemma in_failures_Depth_rest: "(f :f failures(P |. n) M) = (EX t X. f = (t,X) & (t, X) :f failures(P) M & (lengtht t < n | lengtht t = n & (EX s. t = s ^^ <Tick> & noTick s)))" apply (simp add: failures_def) apply (subgoal_tac "EX t X. f = (t, X)") apply (elim exE) apply (simp add: in_rest_setF) apply (auto) done (*--------------------------------* | Proc_name | *--------------------------------*) (*** Proc_name ***) lemma in_failures_Proc_name: "(f :f failures($p) M) = (f :f sndF(M p))" apply (simp add: failures_def) done (*--------------------------------* | alias | *--------------------------------*) lemmas failures_setF = STOP_setF SKIP_setF Act_prefix_setF Ext_pre_choice_setF Ext_choice_setF Rep_int_choice_setF Parallel_setF Hiding_setF Renaming_setF Seq_compo_setF lemmas in_failures = in_failures_STOP in_failures_SKIP in_failures_DIV in_failures_Act_prefix in_failures_Ext_pre_choice in_failures_Ext_choice in_failures_Int_choice in_failures_Rep_int_choice in_failures_IF in_failures_Parallel in_failures_Hiding in_failures_Renaming in_failures_Seq_compo in_failures_Union_proc in_failures_UNIV_Union_proc in_failures_Depth_rest in_failures_Proc_name (****************** to add them again ******************) declare disj_not1 [simp] end