(* Title: HOLCF/IOA/meta_theory/Automata.ML
ID: $Id$
Author: Olaf Mueller, Tobias Nipkow, Konrad Slind
Copyright 1994, 1996 TU Muenchen
The I/O automata of Lynch and Tuttle.
*)
(* this modification of the simpset is local to this file *)
Delsimps [split_paired_Ex];
open reachable;
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]
"!!A. [| 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]
"!! a. [| compatible A1 A2; a:ext A1|] ==> a~:int A2";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 1);
qed"ext1_is_not_int2";
(* just commuting the previous one: better commute compatible *)
Goalw [externals_def,actions_def,compatible_def]
"!! a. [| compatible A2 A1 ; a:ext A1|] ==> a~:int A2";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 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]
"!! x. [| compatible A B; x:int A |] ==> x~:ext B";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 1);
qed"intA_is_not_extB";
Goalw [externals_def,actions_def,compatible_def,is_asig_def,asig_of_def]
"!! a. [| compatible A B; a:int A |] ==> a ~: act B";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 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]
"!! a. [| compatible A B; a:out A ;a:act B|] ==> a : inp B";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 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]
"!! a. [| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 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]
"!!A. [| 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 thy [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 thy
"[| !!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";
val [p1,p2] = goalw thy [invariant_def]
"[| invariant A P; reachable A s |] ==> P(s)";
br(p2 RS (p1 RS spec RS mp))1;
qed "invariantE";
(* ---------------------------------------------------------------------------------- *)
section "restrict";
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 "!!f. 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 "!!s.[| 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 "!!A.[|(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 "!!A.[|(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 "!!A.[|(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 "!!A.[|(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 "!!A.(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. [|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. [|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. [|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]
"!!A. 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]
"!!A. 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 "!! A. [| 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;
by (REPEAT (best_tac (set_cs addEs [equalityCE]) 1));
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]
"!! A. is_asig_of A ==> is_asig_of (restrict A f)";
by (Asm_full_simp_tac 1);
by Auto_tac;
by (REPEAT (best_tac (set_cs addEs [equalityCE]) 1));
qed"is_asig_of_restrict";
Goal "!! A. 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(best_tac (set_cs addEs [equalityCE])));
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]
"!! A. [|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;
by (REPEAT (best_tac (set_cs addEs [equalityCE]) 1));
qed"compatible_par";
(* better derive by previous one and compat_commute *)
Goalw [compatible_def]
"!! A. [|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;
by (REPEAT (best_tac (set_cs addEs [equalityCE]) 1));
qed"compatible_par2";
Goalw [compatible_def]
"!! A. [| 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 (claset() addEs [equalityCE],simpset()));
qed"compatible_restrict";
Addsimps [split_paired_Ex];