(* Title: HOLCF/IOA/meta_theory/Automata.thy ID: $Id: Automata.thy,v 1.19 2008/05/07 08:58:51 berghofe Exp $ Author: Olaf Müller, Konrad Slind, Tobias Nipkow *) header {* The I/O automata of Lynch and Tuttle in HOLCF *} theory Automata imports Asig begin defaultsort type types ('a, 's) transition = "'s * 'a * 's" ('a, 's) ioa = "'a signature * 's set * ('a,'s)transition set * ('a set set) * ('a set set)" consts (* IO automata *) asig_of ::"('a,'s)ioa => 'a signature" starts_of ::"('a,'s)ioa => 's set" trans_of ::"('a,'s)ioa => ('a,'s)transition set" wfair_of ::"('a,'s)ioa => ('a set) set" sfair_of ::"('a,'s)ioa => ('a set) set" is_asig_of ::"('a,'s)ioa => bool" is_starts_of ::"('a,'s)ioa => bool" is_trans_of ::"('a,'s)ioa => bool" input_enabled ::"('a,'s)ioa => bool" IOA ::"('a,'s)ioa => bool" (* constraints for fair IOA *) fairIOA ::"('a,'s)ioa => bool" input_resistant::"('a,'s)ioa => bool" (* enabledness of actions and action sets *) enabled ::"('a,'s)ioa => 'a => 's => bool" Enabled ::"('a,'s)ioa => 'a set => 's => bool" (* action set keeps enabled until probably disabled by itself *) en_persistent :: "('a,'s)ioa => 'a set => bool" (* post_conditions for actions and action sets *) was_enabled ::"('a,'s)ioa => 'a => 's => bool" set_was_enabled ::"('a,'s)ioa => 'a set => 's => bool" (* invariants *) invariant :: "[('a,'s)ioa, 's=>bool] => bool" (* binary composition of action signatures and automata *) asig_comp ::"['a signature, 'a signature] => 'a signature" compatible ::"[('a,'s)ioa, ('a,'t)ioa] => bool" par ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "||" 10) (* hiding and restricting *) hide_asig :: "['a signature, 'a set] => 'a signature" "hide" :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa" restrict_asig :: "['a signature, 'a set] => 'a signature" restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa" (* renaming *) rename_set :: "'a set => ('c => 'a option) => 'c set" rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa" notation (xsymbols) par (infixr "\<parallel>" 10) inductive reachable :: "('a, 's) ioa => 's => bool" for C :: "('a, 's) ioa" where reachable_0: "s : starts_of C ==> reachable C s" | reachable_n: "[| reachable C s; (s, a, t) : trans_of C |] ==> reachable C t" abbreviation trans_of_syn ("_ -_--_-> _" [81,81,81,81] 100) where "s -a--A-> t == (s,a,t):trans_of A" notation (xsymbols) trans_of_syn ("_ \<midarrow>_\<midarrow>_--> _" [81,81,81,81] 100) abbreviation "act A == actions (asig_of A)" abbreviation "ext A == externals (asig_of A)" abbreviation int where "int A == internals (asig_of A)" abbreviation "inp A == inputs (asig_of A)" abbreviation "out A == outputs (asig_of A)" abbreviation "local A == locals (asig_of A)" defs (* --------------------------------- IOA ---------------------------------*) asig_of_def: "asig_of == fst" starts_of_def: "starts_of == (fst o snd)" trans_of_def: "trans_of == (fst o snd o snd)" wfair_of_def: "wfair_of == (fst o snd o snd o snd)" sfair_of_def: "sfair_of == (snd o snd o snd o snd)" is_asig_of_def: "is_asig_of A == is_asig (asig_of A)" is_starts_of_def: "is_starts_of A == (~ starts_of A = {})" is_trans_of_def: "is_trans_of A == (!triple. triple:(trans_of A) --> fst(snd(triple)):actions(asig_of A))" input_enabled_def: "input_enabled A == (!a. (a:inputs(asig_of A)) --> (!s1. ? s2. (s1,a,s2):(trans_of A)))" ioa_def: "IOA A == (is_asig_of A & is_starts_of A & is_trans_of A & input_enabled A)" invariant_def: "invariant A P == (!s. reachable A s --> P(s))" (* ------------------------- parallel composition --------------------------*) compatible_def: "compatible A B == (((out A Int out B) = {}) & ((int A Int act B) = {}) & ((int B Int act A) = {}))" asig_comp_def: "asig_comp a1 a2 == (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)), (outputs(a1) Un outputs(a2)), (internals(a1) Un internals(a2))))" par_def: "(A || B) == (asig_comp (asig_of A) (asig_of B), {pr. fst(pr):starts_of(A) & snd(pr):starts_of(B)}, {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) in (a:act A | a:act B) & (if a:act A then (fst(s),a,fst(t)):trans_of(A) else fst(t) = fst(s)) & (if a:act B then (snd(s),a,snd(t)):trans_of(B) else snd(t) = snd(s))}, wfair_of A Un wfair_of B, sfair_of A Un sfair_of B)" (* ------------------------ hiding -------------------------------------------- *) restrict_asig_def: "restrict_asig asig actns == (inputs(asig) Int actns, outputs(asig) Int actns, internals(asig) Un (externals(asig) - actns))" (* Notice that for wfair_of and sfair_of nothing has to be changed, as changes from the outputs to the internals does not touch the locals as a whole, which is of importance for fairness only *) restrict_def: "restrict A actns == (restrict_asig (asig_of A) actns, starts_of A, trans_of A, wfair_of A, sfair_of A)" hide_asig_def: "hide_asig asig actns == (inputs(asig) - actns, outputs(asig) - actns, internals(asig) Un actns)" hide_def: "hide A actns == (hide_asig (asig_of A) actns, starts_of A, trans_of A, wfair_of A, sfair_of A)" (* ------------------------- renaming ------------------------------------------- *) rename_set_def: "rename_set A ren == {b. ? x. Some x = ren b & x : A}" rename_def: "rename ioa ren == ((rename_set (inp ioa) ren, rename_set (out ioa) ren, rename_set (int ioa) ren), starts_of ioa, {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) in ? x. Some(x) = ren(a) & (s,x,t):trans_of ioa}, {rename_set s ren | s. s: wfair_of ioa}, {rename_set s ren | s. s: sfair_of ioa})" (* ------------------------- fairness ----------------------------- *) fairIOA_def: "fairIOA A == (! S : wfair_of A. S<= local A) & (! S : sfair_of A. S<= local A)" input_resistant_def: "input_resistant A == ! W : sfair_of A. ! s a t. reachable A s & reachable A t & a:inp A & Enabled A W s & s -a--A-> t --> Enabled A W t" enabled_def: "enabled A a s == ? t. s-a--A-> t" Enabled_def: "Enabled A W s == ? w:W. enabled A w s" en_persistent_def: "en_persistent A W == ! s a t. Enabled A W s & a ~:W & s -a--A-> t --> Enabled A W t" was_enabled_def: "was_enabled A a t == ? s. s-a--A-> t" set_was_enabled_def: "set_was_enabled A W t == ? w:W. was_enabled A w t" declare split_paired_Ex [simp del] lemmas ioa_projections = asig_of_def starts_of_def trans_of_def wfair_of_def sfair_of_def subsection "asig_of, starts_of, trans_of" lemma ioa_triple_proj: "((asig_of (x,y,z,w,s)) = x) & ((starts_of (x,y,z,w,s)) = y) & ((trans_of (x,y,z,w,s)) = z) & ((wfair_of (x,y,z,w,s)) = w) & ((sfair_of (x,y,z,w,s)) = s)" apply (simp add: ioa_projections) done lemma trans_in_actions: "[| is_trans_of A; (s1,a,s2):trans_of(A) |] ==> a:act A" apply (unfold is_trans_of_def actions_def is_asig_def) apply (erule allE, erule impE, assumption) apply simp done lemma starts_of_par: "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}" apply (simp add: par_def ioa_projections) done lemma trans_of_par: "trans_of(A || B) = {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) in (a:act A | a:act B) & (if a:act A then (fst(s),a,fst(t)):trans_of(A) else fst(t) = fst(s)) & (if a:act B then (snd(s),a,snd(t)):trans_of(B) else snd(t) = snd(s))}" apply (simp add: par_def ioa_projections) done subsection "actions and par" lemma actions_asig_comp: "actions(asig_comp a b) = actions(a) Un actions(b)" apply (simp (no_asm) add: actions_def asig_comp_def asig_projections) apply blast done lemma asig_of_par: "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)" apply (simp add: par_def ioa_projections) done lemma externals_of_par: "ext (A1||A2) = (ext A1) Un (ext A2)" apply (simp add: externals_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def Un_def set_diff_eq) apply blast done lemma actions_of_par: "act (A1||A2) = (act A1) Un (act A2)" apply (simp add: actions_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq) apply blast done lemma inputs_of_par: "inp (A1||A2) = ((inp A1) Un (inp A2)) - ((out A1) Un (out A2))" apply (simp add: actions_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def Un_def set_diff_eq) done lemma outputs_of_par: "out (A1||A2) = (out A1) Un (out A2)" apply (simp add: actions_def asig_of_par asig_comp_def asig_outputs_def Un_def set_diff_eq) done lemma internals_of_par: "int (A1||A2) = (int A1) Un (int A2)" apply (simp add: actions_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq) done subsection "actions and compatibility" lemma compat_commute: "compatible A B = compatible B A" apply (simp add: compatible_def Int_commute) apply auto done lemma ext1_is_not_int2: "[| compatible A1 A2; a:ext A1|] ==> a~:int A2" apply (unfold externals_def actions_def compatible_def) apply simp apply blast done (* just commuting the previous one: better commute compatible *) lemma ext2_is_not_int1: "[| compatible A2 A1 ; a:ext A1|] ==> a~:int A2" apply (unfold externals_def actions_def compatible_def) apply simp apply blast done lemmas ext1_ext2_is_not_act2 = ext1_is_not_int2 [THEN int_and_ext_is_act, standard] lemmas ext1_ext2_is_not_act1 = ext2_is_not_int1 [THEN int_and_ext_is_act, standard] lemma intA_is_not_extB: "[| compatible A B; x:int A |] ==> x~:ext B" apply (unfold externals_def actions_def compatible_def) apply simp apply blast done lemma intA_is_not_actB: "[| compatible A B; a:int A |] ==> a ~: act B" apply (unfold externals_def actions_def compatible_def is_asig_def asig_of_def) apply simp apply blast done (* the only one that needs disjointness of outputs and of internals and _all_ acts *) lemma outAactB_is_inpB: "[| compatible A B; a:out A ;a:act B|] ==> a : inp B" apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def compatible_def is_asig_def asig_of_def) apply simp apply blast done (* needed for propagation of input_enabledness from A,B to A||B *) lemma inpAAactB_is_inpBoroutB: "[| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B" apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def compatible_def is_asig_def asig_of_def) apply simp apply blast done subsection "input_enabledness and par" (* ugly case distinctions. Heart of proof: 1. inpAAactB_is_inpBoroutB ie. internals are really hidden. 2. inputs_of_par: outputs are no longer inputs of par. This is important here *) lemma input_enabled_par: "[| compatible A B; input_enabled A; input_enabled B|] ==> input_enabled (A||B)" apply (unfold input_enabled_def) apply (simp add: Let_def inputs_of_par trans_of_par) apply (tactic "safe_tac (claset_of @{theory Fun})") apply (simp add: inp_is_act) prefer 2 apply (simp add: inp_is_act) (* a: inp A *) apply (case_tac "a:act B") (* a:act B *) apply (erule_tac x = "a" in allE) apply simp apply (drule inpAAactB_is_inpBoroutB) apply assumption apply assumption apply (erule_tac x = "a" in allE) apply simp apply (erule_tac x = "aa" in allE) apply (erule_tac x = "b" in allE) apply (erule exE) apply (erule exE) apply (rule_tac x = " (s2,s2a) " in exI) apply (simp add: inp_is_act) (* a~: act B*) apply (simp add: inp_is_act) apply (erule_tac x = "a" in allE) apply simp apply (erule_tac x = "aa" in allE) apply (erule exE) apply (rule_tac x = " (s2,b) " in exI) apply simp (* a:inp B *) apply (case_tac "a:act A") (* a:act A *) apply (erule_tac x = "a" in allE) apply (erule_tac x = "a" in allE) apply (simp add: inp_is_act) apply (frule_tac A1 = "A" in compat_commute [THEN iffD1]) apply (drule inpAAactB_is_inpBoroutB) back apply assumption apply assumption apply simp apply (erule_tac x = "aa" in allE) apply (erule_tac x = "b" in allE) apply (erule exE) apply (erule exE) apply (rule_tac x = " (s2,s2a) " in exI) apply (simp add: inp_is_act) (* a~: act B*) apply (simp add: inp_is_act) apply (erule_tac x = "a" in allE) apply (erule_tac x = "a" in allE) apply simp apply (erule_tac x = "b" in allE) apply (erule exE) apply (rule_tac x = " (aa,s2) " in exI) apply simp done subsection "invariants" lemma invariantI: "[| !!s. s:starts_of(A) ==> P(s); !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |] ==> invariant A P" apply (unfold invariant_def) apply (rule allI) apply (rule impI) apply (rule_tac x = "s" in reachable.induct) apply assumption apply blast apply blast done lemma invariantI1: "[| !!s. s : starts_of(A) ==> P(s); !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t) |] ==> invariant A P" apply (blast intro: invariantI) done lemma invariantE: "[| invariant A P; reachable A s |] ==> P(s)" apply (unfold invariant_def) apply blast done subsection "restrict" lemmas reachable_0 = reachable.reachable_0 and reachable_n = reachable.reachable_n lemma cancel_restrict_a: "starts_of(restrict ioa acts) = starts_of(ioa) & trans_of(restrict ioa acts) = trans_of(ioa)" apply (simp add: restrict_def ioa_projections) done lemma cancel_restrict_b: "reachable (restrict ioa acts) s = reachable ioa s" apply (rule iffI) apply (erule reachable.induct) apply (simp add: cancel_restrict_a reachable_0) apply (erule reachable_n) apply (simp add: cancel_restrict_a) (* <-- *) apply (erule reachable.induct) apply (rule reachable_0) apply (simp add: cancel_restrict_a) apply (erule reachable_n) apply (simp add: cancel_restrict_a) done lemma acts_restrict: "act (restrict A acts) = act A" apply (simp (no_asm) add: actions_def asig_internals_def asig_outputs_def asig_inputs_def externals_def asig_of_def restrict_def restrict_asig_def) apply auto done lemma cancel_restrict: "starts_of(restrict ioa acts) = starts_of(ioa) & trans_of(restrict ioa acts) = trans_of(ioa) & reachable (restrict ioa acts) s = reachable ioa s & act (restrict A acts) = act A" apply (simp (no_asm) add: cancel_restrict_a cancel_restrict_b acts_restrict) done subsection "rename" lemma trans_rename: "s -a--(rename C f)-> t ==> (? x. Some(x) = f(a) & s -x--C-> t)" apply (simp add: Let_def rename_def trans_of_def) done lemma reachable_rename: "[| reachable (rename C g) s |] ==> reachable C s" apply (erule reachable.induct) apply (rule reachable_0) apply (simp add: rename_def ioa_projections) apply (drule trans_rename) apply (erule exE) apply (erule conjE) apply (erule reachable_n) apply assumption done subsection "trans_of(A||B)" lemma trans_A_proj: "[|(s,a,t):trans_of (A||B); a:act A|] ==> (fst s,a,fst t):trans_of A" apply (simp add: Let_def par_def trans_of_def) done lemma trans_B_proj: "[|(s,a,t):trans_of (A||B); a:act B|] ==> (snd s,a,snd t):trans_of B" apply (simp add: Let_def par_def trans_of_def) done lemma trans_A_proj2: "[|(s,a,t):trans_of (A||B); a~:act A|] ==> fst s = fst t" apply (simp add: Let_def par_def trans_of_def) done lemma trans_B_proj2: "[|(s,a,t):trans_of (A||B); a~:act B|] ==> snd s = snd t" apply (simp add: Let_def par_def trans_of_def) done lemma trans_AB_proj: "(s,a,t):trans_of (A||B) ==> a :act A | a :act B" apply (simp add: Let_def par_def trans_of_def) done lemma trans_AB: "[|a:act A;a:act B; (fst s,a,fst t):trans_of A;(snd s,a,snd t):trans_of B|] ==> (s,a,t):trans_of (A||B)" apply (simp add: Let_def par_def trans_of_def) done lemma trans_A_notB: "[|a:act A;a~:act B; (fst s,a,fst t):trans_of A;snd s=snd t|] ==> (s,a,t):trans_of (A||B)" apply (simp add: Let_def par_def trans_of_def) done lemma trans_notA_B: "[|a~:act A;a:act B; (snd s,a,snd t):trans_of B;fst s=fst t|] ==> (s,a,t):trans_of (A||B)" apply (simp add: Let_def par_def trans_of_def) done lemmas trans_of_defs1 = trans_AB trans_A_notB trans_notA_B and trans_of_defs2 = trans_A_proj trans_B_proj trans_A_proj2 trans_B_proj2 trans_AB_proj lemma trans_of_par4: "((s,a,t) : trans_of(A || B || C || D)) = ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) | a:actions(asig_of(D))) & (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A) else fst t=fst s) & (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B) else fst(snd(t))=fst(snd(s))) & (if a:actions(asig_of(C)) then (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C) else fst(snd(snd(t)))=fst(snd(snd(s)))) & (if a:actions(asig_of(D)) then (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D) else snd(snd(snd(t)))=snd(snd(snd(s)))))" apply (simp (no_asm) add: par_def actions_asig_comp Pair_fst_snd_eq Let_def ioa_projections) done subsection "proof obligation generator for IOA requirements" (* without assumptions on A and B because is_trans_of is also incorporated in ||def *) lemma is_trans_of_par: "is_trans_of (A||B)" apply (unfold is_trans_of_def) apply (simp add: Let_def actions_of_par trans_of_par) done lemma is_trans_of_restrict: "is_trans_of A ==> is_trans_of (restrict A acts)" apply (unfold is_trans_of_def) apply (simp add: cancel_restrict acts_restrict) done lemma is_trans_of_rename: "is_trans_of A ==> is_trans_of (rename A f)" apply (unfold is_trans_of_def restrict_def restrict_asig_def) apply (simp add: Let_def actions_def trans_of_def asig_internals_def asig_outputs_def asig_inputs_def externals_def asig_of_def rename_def rename_set_def) apply blast done lemma is_asig_of_par: "[| is_asig_of A; is_asig_of B; compatible A B|] ==> is_asig_of (A||B)" apply (simp add: is_asig_of_def asig_of_par asig_comp_def compatible_def asig_internals_def asig_outputs_def asig_inputs_def actions_def is_asig_def) apply (simp add: asig_of_def) apply auto done lemma is_asig_of_restrict: "is_asig_of A ==> is_asig_of (restrict A f)" apply (unfold is_asig_of_def is_asig_def asig_of_def restrict_def restrict_asig_def asig_internals_def asig_outputs_def asig_inputs_def externals_def o_def) apply simp apply auto done lemma is_asig_of_rename: "is_asig_of A ==> is_asig_of (rename A f)" apply (simp add: is_asig_of_def rename_def rename_set_def asig_internals_def asig_outputs_def asig_inputs_def actions_def is_asig_def asig_of_def) apply auto apply (drule_tac [!] s = "Some ?x" in sym) apply auto done lemmas [simp] = is_asig_of_par is_asig_of_restrict is_asig_of_rename is_trans_of_par is_trans_of_restrict is_trans_of_rename lemma compatible_par: "[|compatible A B; compatible A C |]==> compatible A (B||C)" apply (unfold compatible_def) apply (simp add: internals_of_par outputs_of_par actions_of_par) apply auto done (* better derive by previous one and compat_commute *) lemma compatible_par2: "[|compatible A C; compatible B C |]==> compatible (A||B) C" apply (unfold compatible_def) apply (simp add: internals_of_par outputs_of_par actions_of_par) apply auto done lemma compatible_restrict: "[| compatible A B; (ext B - S) Int ext A = {}|] ==> compatible A (restrict B S)" apply (unfold compatible_def) apply (simp add: ioa_triple_proj asig_triple_proj externals_def restrict_def restrict_asig_def actions_def) apply auto done declare split_paired_Ex [simp] end
lemma ioa_projections:
asig_of == fst
starts_of == fst o snd
trans_of == fst o snd o snd
wfair_of == fst o snd o snd o snd
sfair_of == snd o snd o snd o snd
lemma ioa_triple_proj:
asig_of (x, y, z, w, s) = x ∧
starts_of (x, y, z, w, s) = y ∧
trans_of (x, y, z, w, s) = z ∧
wfair_of (x, y, z, w, s) = w ∧ sfair_of (x, y, z, w, s) = s
lemma trans_in_actions:
[| is_trans_of A; s1.0 -a--A-> s2.0 |] ==> a ∈ act A
lemma starts_of_par:
starts_of (A || B) = {p. fst p ∈ starts_of A ∧ snd p ∈ starts_of B}
lemma trans_of_par:
trans_of (A || B) =
{tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr)
in (a ∈ act A ∨ a ∈ act B) ∧
(if a ∈ act A then fst s -a--A-> fst t else fst t = fst s) ∧
(if a ∈ act B then snd s -a--B-> snd t else snd t = snd s)}
lemma actions_asig_comp:
actions (asig_comp a b) = actions a ∪ actions b
lemma asig_of_par:
asig_of (A || B) = asig_comp (asig_of A) (asig_of B)
lemma externals_of_par:
ext (A1.0 || A2.0) = ext A1.0 ∪ ext A2.0
lemma actions_of_par:
act (A1.0 || A2.0) = act A1.0 ∪ act A2.0
lemma inputs_of_par:
inp (A1.0 || A2.0) = inp A1.0 ∪ inp A2.0 - (out A1.0 ∪ out A2.0)
lemma outputs_of_par:
out (A1.0 || A2.0) = out A1.0 ∪ out A2.0
lemma internals_of_par:
Automata.int (A1.0 || A2.0) = Automata.int A1.0 ∪ Automata.int A2.0
lemma compat_commute:
compatible A B = compatible B A
lemma ext1_is_not_int2:
[| compatible A1.0 A2.0; a ∈ ext A1.0 |] ==> a ∉ Automata.int A2.0
lemma ext2_is_not_int1:
[| compatible A2.0 A1.0; a ∈ ext A1.0 |] ==> a ∉ Automata.int A2.0
lemma ext1_ext2_is_not_act2:
[| compatible A1.0 A2.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
lemma ext1_ext2_is_not_act1:
[| compatible A2.0 A1.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
lemma intA_is_not_extB:
[| compatible A B; x ∈ Automata.int A |] ==> x ∉ ext B
lemma intA_is_not_actB:
[| compatible A B; a ∈ Automata.int A |] ==> a ∉ act B
lemma outAactB_is_inpB:
[| compatible A B; a ∈ out A; a ∈ act B |] ==> a ∈ inp B
lemma inpAAactB_is_inpBoroutB:
[| compatible A B; a ∈ inp A; a ∈ act B |] ==> a ∈ inp B ∨ a ∈ out B
lemma input_enabled_par:
[| compatible A B; input_enabled A; input_enabled B |]
==> input_enabled (A || B)
lemma invariantI:
[| !!s. s ∈ starts_of A ==> P s;
!!s t a. [| reachable A s; P s |] ==> s -a--A-> t --> P t |]
==> invariant A P
lemma invariantI1:
[| !!s. s ∈ starts_of A ==> P s;
!!s t a. reachable A s ==> P s --> s -a--A-> t --> P t |]
==> invariant A P
lemma invariantE:
[| invariant A P; reachable A s |] ==> P s
lemma reachable_0:
s ∈ starts_of C ==> reachable C s
and reachable_n:
[| reachable C s; s -a--C-> t |] ==> reachable C t
lemma cancel_restrict_a:
starts_of (restrict ioa acts) = starts_of ioa ∧
trans_of (restrict ioa acts) = trans_of ioa
lemma cancel_restrict_b:
reachable (restrict ioa acts) s = reachable ioa s
lemma acts_restrict:
act (restrict A acts) = act A
lemma cancel_restrict:
starts_of (restrict ioa acts) = starts_of ioa ∧
trans_of (restrict ioa acts) = trans_of ioa ∧
reachable (restrict ioa acts) s = reachable ioa s ∧
act (restrict A acts) = act A
lemma trans_rename:
s -a--rename C f-> t ==> ∃x. Some x = f a ∧ s -x--C-> t
lemma reachable_rename:
reachable (rename C g) s ==> reachable C s
lemma trans_A_proj:
[| s -a--(A || B)-> t; a ∈ act A |] ==> fst s -a--A-> fst t
lemma trans_B_proj:
[| s -a--(A || B)-> t; a ∈ act B |] ==> snd s -a--B-> snd t
lemma trans_A_proj2:
[| s -a--(A || B)-> t; a ∉ act A |] ==> fst s = fst t
lemma trans_B_proj2:
[| s -a--(A || B)-> t; a ∉ act B |] ==> snd s = snd t
lemma trans_AB_proj:
s -a--(A || B)-> t ==> a ∈ act A ∨ a ∈ act B
lemma trans_AB:
[| a ∈ act A; a ∈ act B; fst s -a--A-> fst t; snd s -a--B-> snd t |]
==> s -a--(A || B)-> t
lemma trans_A_notB:
[| a ∈ act A; a ∉ act B; fst s -a--A-> fst t; snd s = snd t |]
==> s -a--(A || B)-> t
lemma trans_notA_B:
[| a ∉ act A; a ∈ act B; snd s -a--B-> snd t; fst s = fst t |]
==> s -a--(A || B)-> t
lemma trans_of_defs1:
[| a ∈ act A; a ∈ act B; fst s -a--A-> fst t; snd s -a--B-> snd t |]
==> s -a--(A || B)-> t
[| a ∈ act A; a ∉ act B; fst s -a--A-> fst t; snd s = snd t |]
==> s -a--(A || B)-> t
[| a ∉ act A; a ∈ act B; snd s -a--B-> snd t; fst s = fst t |]
==> s -a--(A || B)-> t
and trans_of_defs2:
[| s -a--(A || B)-> t; a ∈ act A |] ==> fst s -a--A-> fst t
[| s -a--(A || B)-> t; a ∈ act B |] ==> snd s -a--B-> snd t
[| s -a--(A || B)-> t; a ∉ act A |] ==> fst s = fst t
[| s -a--(A || B)-> t; a ∉ act B |] ==> snd s = snd t
s -a--(A || B)-> t ==> a ∈ act A ∨ a ∈ act B
lemma trans_of_par4:
s -a--(A || B || C || D)-> t =
((a ∈ act A ∨ a ∈ act B ∨ a ∈ act C ∨ a ∈ act D) ∧
(if a ∈ act A then fst s -a--A-> fst t else fst t = fst s) ∧
(if a ∈ act B then fst (snd s) -a--B-> fst (snd t)
else fst (snd t) = fst (snd s)) ∧
(if a ∈ act C then fst (snd (snd s)) -a--C-> fst (snd (snd t))
else fst (snd (snd t)) = fst (snd (snd s))) ∧
(if a ∈ act D then snd (snd (snd s)) -a--D-> snd (snd (snd t))
else snd (snd (snd t)) = snd (snd (snd s))))
lemma is_trans_of_par:
is_trans_of (A || B)
lemma is_trans_of_restrict:
is_trans_of A ==> is_trans_of (restrict A acts)
lemma is_trans_of_rename:
is_trans_of A ==> is_trans_of (rename A f)
lemma is_asig_of_par:
[| is_asig_of A; is_asig_of B; compatible A B |] ==> is_asig_of (A || B)
lemma is_asig_of_restrict:
is_asig_of A ==> is_asig_of (restrict A f)
lemma is_asig_of_rename:
is_asig_of A ==> is_asig_of (rename A f)
lemma
[| is_asig_of A; is_asig_of B; compatible A B |] ==> is_asig_of (A || B)
is_asig_of A ==> is_asig_of (restrict A f)
is_asig_of A ==> is_asig_of (rename A f)
is_trans_of (A || B)
is_trans_of A ==> is_trans_of (restrict A acts)
is_trans_of A ==> is_trans_of (rename A f)
lemma compatible_par:
[| compatible A B; compatible A C |] ==> compatible A (B || C)
lemma compatible_par2:
[| compatible A C; compatible B C |] ==> compatible (A || B) C
lemma compatible_restrict:
[| compatible A B; (ext B - S) ∩ ext A = {} |] ==> compatible A (restrict B S)