(* Title: HOLCF/IOA/meta_theory/Automata.ML ID: $Id: Automata.ML,v 1.22 2005/09/02 15:24:00 wenzelm Exp $ Author: Olaf Mueller, Tobias Nipkow, Konrad Slind *) (* this modification of the simpset is local to this file *) Delsimps [split_paired_Ex]; val ioa_projections = [asig_of_def, starts_of_def, trans_of_def,wfair_of_def,sfair_of_def]; (* ------------------------------------------------------------------------- *) section "asig_of, starts_of, trans_of"; Goal "((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)"; by (simp_tac (simpset() addsimps ioa_projections) 1); qed "ioa_triple_proj"; Goalw [is_trans_of_def,actions_def, is_asig_def] "[| is_trans_of A; (s1,a,s2):trans_of(A) |] ==> a:act A"; by (REPEAT(etac conjE 1)); by (EVERY1[etac allE, etac impE, atac]); by (Asm_full_simp_tac 1); qed "trans_in_actions"; Goal "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"; by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); qed "starts_of_par"; Goal "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))}"; by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); qed "trans_of_par"; (* ------------------------------------------------------------------------- *) section "actions and par"; Goal "actions(asig_comp a b) = actions(a) Un actions(b)"; by (simp_tac (simpset() addsimps ([actions_def,asig_comp_def]@asig_projections)) 1); by (fast_tac (set_cs addSIs [equalityI]) 1); qed "actions_asig_comp"; Goal "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)"; by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); qed "asig_of_par"; Goal "ext (A1||A2) = \ \ (ext A1) Un (ext A2)"; by (asm_full_simp_tac (simpset() addsimps [externals_def,asig_of_par,asig_comp_def, asig_inputs_def,asig_outputs_def,Un_def,set_diff_def]) 1); by (rtac set_ext 1); by (fast_tac set_cs 1); qed"externals_of_par"; Goal "act (A1||A2) = \ \ (act A1) Un (act A2)"; by (asm_full_simp_tac (simpset() addsimps [actions_def,asig_of_par,asig_comp_def, asig_inputs_def,asig_outputs_def,asig_internals_def,Un_def,set_diff_def]) 1); by (rtac set_ext 1); by (fast_tac set_cs 1); qed"actions_of_par"; Goal "inp (A1||A2) =\ \ ((inp A1) Un (inp A2)) - ((out A1) Un (out A2))"; by (asm_full_simp_tac (simpset() addsimps [actions_def,asig_of_par,asig_comp_def, asig_inputs_def,asig_outputs_def,Un_def,set_diff_def]) 1); qed"inputs_of_par"; Goal "out (A1||A2) =\ \ (out A1) Un (out A2)"; by (asm_full_simp_tac (simpset() addsimps [actions_def,asig_of_par,asig_comp_def, asig_outputs_def,Un_def,set_diff_def]) 1); qed"outputs_of_par"; Goal "int (A1||A2) =\ \ (int A1) Un (int A2)"; by (asm_full_simp_tac (simpset() addsimps [actions_def,asig_of_par,asig_comp_def, asig_inputs_def,asig_outputs_def,asig_internals_def,Un_def,set_diff_def]) 1); qed"internals_of_par"; (* ------------------------------------------------------------------------ *) section "actions and compatibility"; Goal"compatible A B = compatible B A"; by (asm_full_simp_tac (simpset() addsimps [compatible_def,Int_commute]) 1); by Auto_tac; qed"compat_commute"; Goalw [externals_def,actions_def,compatible_def] "[| compatible A1 A2; a:ext A1|] ==> a~:int A2"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"ext1_is_not_int2"; (* just commuting the previous one: better commute compatible *) Goalw [externals_def,actions_def,compatible_def] "[| compatible A2 A1 ; a:ext A1|] ==> a~:int A2"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"ext2_is_not_int1"; bind_thm("ext1_ext2_is_not_act2",ext1_is_not_int2 RS int_and_ext_is_act); bind_thm("ext1_ext2_is_not_act1",ext2_is_not_int1 RS int_and_ext_is_act); Goalw [externals_def,actions_def,compatible_def] "[| compatible A B; x:int A |] ==> x~:ext B"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"intA_is_not_extB"; Goalw [externals_def,actions_def,compatible_def,is_asig_def,asig_of_def] "[| compatible A B; a:int A |] ==> a ~: act B"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"intA_is_not_actB"; (* the only one that needs disjointness of outputs and of internals and _all_ acts *) Goalw [asig_outputs_def,asig_internals_def,actions_def,asig_inputs_def, compatible_def,is_asig_def,asig_of_def] "[| compatible A B; a:out A ;a:act B|] ==> a : inp B"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"outAactB_is_inpB"; (* needed for propagation of input_enabledness from A,B to A||B *) Goalw [asig_outputs_def,asig_internals_def,actions_def,asig_inputs_def, compatible_def,is_asig_def,asig_of_def] "[| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B"; by (Asm_full_simp_tac 1); by (Blast_tac 1); qed"inpAAactB_is_inpBoroutB"; (* ------------------------------------------------------------------------- *) section "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 *) Goalw [input_enabled_def] "[| compatible A B; input_enabled A; input_enabled B|] \ \ ==> input_enabled (A||B)"; by (asm_full_simp_tac (simpset()addsimps[Let_def,inputs_of_par,trans_of_par])1); by (safe_tac set_cs); by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 2); (* a: inp A *) by (case_tac "a:act B" 1); (* a:act B *) by (eres_inst_tac [("x","a")] allE 1); by (Asm_full_simp_tac 1); by (dtac inpAAactB_is_inpBoroutB 1); by (assume_tac 1); by (assume_tac 1); by (eres_inst_tac [("x","a")] allE 1); by (Asm_full_simp_tac 1); by (eres_inst_tac [("x","aa")] allE 1); by (eres_inst_tac [("x","b")] allE 1); by (etac exE 1); by (etac exE 1); by (res_inst_tac [("x","(s2,s2a)")] exI 1); by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); (* a~: act B*) by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); by (eres_inst_tac [("x","a")] allE 1); by (Asm_full_simp_tac 1); by (eres_inst_tac [("x","aa")] allE 1); by (etac exE 1); by (res_inst_tac [("x","(s2,b)")] exI 1); by (Asm_full_simp_tac 1); (* a:inp B *) by (case_tac "a:act A" 1); (* a:act A *) by (eres_inst_tac [("x","a")] allE 1); by (eres_inst_tac [("x","a")] allE 1); by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); by (forw_inst_tac [("A1","A")] (compat_commute RS iffD1) 1); by (dtac inpAAactB_is_inpBoroutB 1); back(); by (assume_tac 1); by (assume_tac 1); by (Asm_full_simp_tac 1); by (rotate_tac ~1 1); by (Asm_full_simp_tac 1); by (eres_inst_tac [("x","aa")] allE 1); by (eres_inst_tac [("x","b")] allE 1); by (etac exE 1); by (etac exE 1); by (res_inst_tac [("x","(s2,s2a)")] exI 1); by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); (* a~: act B*) by (asm_full_simp_tac (simpset() addsimps [inp_is_act]) 1); by (eres_inst_tac [("x","a")] allE 1); by (Asm_full_simp_tac 1); by (eres_inst_tac [("x","a")] allE 1); by (Asm_full_simp_tac 1); by (eres_inst_tac [("x","b")] allE 1); by (etac exE 1); by (res_inst_tac [("x","(aa,s2)")] exI 1); by (Asm_full_simp_tac 1); qed"input_enabled_par"; (* ------------------------------------------------------------------------- *) section "invariants"; val [p1,p2] = goalw (the_context ()) [invariant_def] "[| !!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"; by (rtac allI 1); by (rtac impI 1); by (res_inst_tac [("xa","s")] reachable.induct 1); by (atac 1); by (etac p1 1); by (eres_inst_tac [("s1","sa")] (p2 RS mp) 1); by (REPEAT (atac 1)); qed"invariantI"; val [p1,p2] = goal (the_context ()) "[| !!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"; by (fast_tac (HOL_cs addSIs [invariantI] addSDs [p1,p2]) 1); qed "invariantI1"; Goalw [invariant_def] "[| invariant A P; reachable A s |] ==> P(s)"; by (Blast_tac 1); qed "invariantE"; (* ------------------------------------------------------------------------- *) section "restrict"; val reachable_0 = reachable.reachable_0 and reachable_n = reachable.reachable_n; Goal "starts_of(restrict ioa acts) = starts_of(ioa) & \ \ trans_of(restrict ioa acts) = trans_of(ioa)"; by (simp_tac (simpset() addsimps ([restrict_def]@ioa_projections)) 1); qed "cancel_restrict_a"; Goal "reachable (restrict ioa acts) s = reachable ioa s"; by (rtac iffI 1); by (etac reachable.induct 1); by (asm_full_simp_tac (simpset() addsimps [cancel_restrict_a,reachable_0]) 1); by (etac reachable_n 1); by (asm_full_simp_tac (simpset() addsimps [cancel_restrict_a]) 1); (* <-- *) by (etac reachable.induct 1); by (rtac reachable_0 1); by (asm_full_simp_tac (simpset() addsimps [cancel_restrict_a]) 1); by (etac reachable_n 1); by (asm_full_simp_tac (simpset() addsimps [cancel_restrict_a]) 1); qed "cancel_restrict_b"; Goal "act (restrict A acts) = act A"; by (simp_tac (simpset() addsimps [actions_def,asig_internals_def, asig_outputs_def,asig_inputs_def,externals_def,asig_of_def,restrict_def, restrict_asig_def]) 1); by Auto_tac; qed"acts_restrict"; Goal "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"; by (simp_tac (simpset() addsimps [cancel_restrict_a,cancel_restrict_b,acts_restrict]) 1); qed"cancel_restrict"; (* ------------------------------------------------------------------------- *) section "rename"; Goal "s -a--(rename C f)-> t ==> (? x. Some(x) = f(a) & s -x--C-> t)"; by (asm_full_simp_tac (simpset() addsimps [Let_def,rename_def,trans_of_def]) 1); qed"trans_rename"; Goal "[| reachable (rename C g) s |] ==> reachable C s"; by (etac reachable.induct 1); by (rtac reachable_0 1); by (asm_full_simp_tac (simpset() addsimps [rename_def]@ioa_projections) 1); by (dtac trans_rename 1); by (etac exE 1); by (etac conjE 1); by (etac reachable_n 1); by (assume_tac 1); qed"reachable_rename"; (* ------------------------------------------------------------------------- *) section "trans_of(A||B)"; Goal "[|(s,a,t):trans_of (A||B); a:act A|] \ \ ==> (fst s,a,fst t):trans_of A"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_A_proj"; Goal "[|(s,a,t):trans_of (A||B); a:act B|] \ \ ==> (snd s,a,snd t):trans_of B"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_B_proj"; Goal "[|(s,a,t):trans_of (A||B); a~:act A|]\ \ ==> fst s = fst t"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_A_proj2"; Goal "[|(s,a,t):trans_of (A||B); a~:act B|]\ \ ==> snd s = snd t"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_B_proj2"; Goal "(s,a,t):trans_of (A||B) \ \ ==> a :act A | a :act B"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_AB_proj"; Goal "[|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)"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_AB"; Goal "[|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)"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_A_notB"; Goal "[|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)"; by (asm_full_simp_tac (simpset() addsimps [Let_def,par_def,trans_of_def]) 1); qed"trans_notA_B"; val trans_of_defs1 = [trans_AB,trans_A_notB,trans_notA_B]; val trans_of_defs2 = [trans_A_proj,trans_B_proj,trans_A_proj2, trans_B_proj2,trans_AB_proj]; Goal "((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)))))"; by (simp_tac (simpset() addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq,Let_def]@ ioa_projections)) 1); qed "trans_of_par4"; (* ------------------------------------------------------------------------- *) section "proof obligation generator for IOA requirements"; (* without assumptions on A and B because is_trans_of is also incorporated in ||def *) Goalw [is_trans_of_def] "is_trans_of (A||B)"; by (simp_tac (simpset() addsimps [Let_def,actions_of_par,trans_of_par]) 1); qed"is_trans_of_par"; Goalw [is_trans_of_def] "is_trans_of A ==> is_trans_of (restrict A acts)"; by (asm_simp_tac (simpset() addsimps [cancel_restrict,acts_restrict])1); qed"is_trans_of_restrict"; Goalw [is_trans_of_def,restrict_def,restrict_asig_def] "is_trans_of A ==> is_trans_of (rename A f)"; by (asm_full_simp_tac (simpset() addsimps [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]) 1); by (Blast_tac 1); qed"is_trans_of_rename"; Goal "[| is_asig_of A; is_asig_of B; compatible A B|] \ \ ==> is_asig_of (A||B)"; by (asm_full_simp_tac (simpset() addsimps [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]) 1); by (asm_full_simp_tac (simpset() addsimps [asig_of_def]) 1); by Auto_tac; qed"is_asig_of_par"; Goalw [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] "is_asig_of A ==> is_asig_of (restrict A f)"; by (Asm_full_simp_tac 1); by Auto_tac; qed"is_asig_of_restrict"; Goal "is_asig_of A ==> is_asig_of (rename A f)"; by (asm_full_simp_tac (simpset() addsimps [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]) 1); by Auto_tac; by (ALLGOALS(dres_inst_tac [("s","Some ?x")] sym THEN' Asm_full_simp_tac)); by (ALLGOALS(Blast_tac)); qed"is_asig_of_rename"; Addsimps [is_asig_of_par,is_asig_of_restrict,is_asig_of_rename, is_trans_of_par,is_trans_of_restrict,is_trans_of_rename]; Goalw [compatible_def] "[|compatible A B; compatible A C |]==> compatible A (B||C)"; by (asm_full_simp_tac (simpset() addsimps [internals_of_par, outputs_of_par,actions_of_par]) 1); by Auto_tac; qed"compatible_par"; (* better derive by previous one and compat_commute *) Goalw [compatible_def] "[|compatible A C; compatible B C |]==> compatible (A||B) C"; by (asm_full_simp_tac (simpset() addsimps [internals_of_par, outputs_of_par,actions_of_par]) 1); by Auto_tac; qed"compatible_par2"; Goalw [compatible_def] "[| compatible A B; (ext B - S) Int ext A = {}|] \ \ ==> compatible A (restrict B S)"; by (asm_full_simp_tac (simpset() addsimps [ioa_triple_proj,asig_triple_proj, externals_def,restrict_def,restrict_asig_def,actions_def]) 1); by Auto_tac; qed"compatible_restrict"; Addsimps [split_paired_Ex];