Olafs new version.
--- a/src/HOL/IOA/meta_theory/Asig.ML Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/Asig.ML Thu Apr 13 17:05:41 1995 +0200
@@ -9,3 +9,11 @@
open Asig;
val asig_projections = [asig_inputs_def, asig_outputs_def, asig_internals_def];
+
+goal Asig.thy "!!a.[| a~:internals(S) ;a~:externals(S)|] ==> a~:actions(S)";
+by (asm_full_simp_tac (list_ss addsimps [externals_def,actions_def]) 1);
+qed"int_and_ext_is_act";
+
+goal Asig.thy "!!a.[|a:externals(S)|] ==> a:actions(S)";
+by (asm_full_simp_tac (list_ss addsimps [externals_def,actions_def]) 1);
+qed"ext_is_act";
\ No newline at end of file
--- a/src/HOL/IOA/meta_theory/IOA.ML Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/IOA.ML Thu Apr 13 17:05:41 1995 +0200
@@ -33,16 +33,16 @@
by (simp_tac (SS setloop (split_tac [expand_if])) 1);
qed "filter_oseq_idemp";
-goalw IOA.thy [mk_behaviour_def,filter_oseq_def]
-"(mk_behaviour A s n = None) = \
+goalw IOA.thy [mk_trace_def,filter_oseq_def]
+"(mk_trace A s n = None) = \
\ (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \
\ & \
-\ (mk_behaviour A s n = Some(a)) = \
+\ (mk_trace A s n = Some(a)) = \
\ (s(n)=Some(a) & a : externals(asig_of(A)))";
by (Option.option.induct_tac "s(n)" 1);
by (ALLGOALS (simp_tac (SS setloop (split_tac [expand_if]))));
by (fast_tac HOL_cs 1);
-qed "mk_behaviour_thm";
+qed "mk_trace_thm";
goalw IOA.thy [reachable_def] "!!A. s:starts_of(A) ==> reachable A s";
by (res_inst_tac [("x","(%i.None,%i.s)")] bexI 1);
@@ -149,3 +149,27 @@
goal IOA.thy "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)";
by(simp_tac (SS addsimps (par_def::ioa_projections)) 1);
qed "asig_of_par";
+
+
+goal IOA.thy "externals(asig_of(A1||A2)) = \
+\ (externals(asig_of(A1)) Un externals(asig_of(A2)))";
+by (asm_full_simp_tac (SS 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";
+
+goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
+ "!! a. [| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
+by (asm_full_simp_tac SS 1);
+by (best_tac (set_cs addEs [equalityCE]) 1);
+qed"ext1_is_not_int2";
+
+goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
+ "!! a. [| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
+by (asm_full_simp_tac SS 1);
+by (best_tac (set_cs addEs [equalityCE]) 1);
+qed"ext2_is_not_int1";
+
+val ext1_ext2_is_not_act2 = ext1_is_not_int2 RS int_and_ext_is_act;
+val ext1_ext2_is_not_act1 = ext2_is_not_int1 RS int_and_ext_is_act;
+
--- a/src/HOL/IOA/meta_theory/IOA.thy Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/IOA.thy Thu Apr 13 17:05:41 1995 +0200
@@ -24,16 +24,17 @@
trans_of ::"('action,'state)ioa => ('action,'state)transition set"
IOA ::"('action,'state)ioa => bool"
- (* Executions, schedules, and behaviours *)
+ (* Executions, schedules, and traces *)
is_execution_fragment,
has_execution ::"[('action,'state)ioa, ('action,'state)execution] => bool"
executions :: "('action,'state)ioa => ('action,'state)execution set"
- mk_behaviour :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
+ mk_trace :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
reachable :: "[('action,'state)ioa, 'state] => bool"
invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
- has_behaviour :: "[('action,'state)ioa, 'action oseq] => bool"
- behaviours :: "('action,'state)ioa => 'action oseq set"
+ has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
+ traces :: "('action,'state)ioa => 'action oseq set"
+ NF :: "'a oseq => 'a oseq"
(* Composition of action signatures and automata *)
compatible_asigs ::"('a => 'action signature) => bool"
@@ -44,7 +45,7 @@
(* binary composition of action signatures and automata *)
compat_asigs ::"['action signature, 'action signature] => bool"
asig_comp ::"['action signature, 'action signature] => 'action signature"
- compat_ioas ::"[('action,'state)ioa, ('action,'state)ioa] => bool"
+ compat_ioas ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
"||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr 10)
(* Filtering and hiding *)
@@ -56,6 +57,8 @@
(* Notions of correctness *)
ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
+ (* Instantiation of abstract IOA by concrete actions *)
+ rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
defs
@@ -90,20 +93,7 @@
"executions(ioa) == {e. snd e 0:starts_of(ioa) & \
\ is_execution_fragment ioa e}"
-
-(* Is a state reachable. Using an inductive definition, this could be defined
- * by the following 2 rules
- *
- * x:starts_of(ioa)
- * ----------------
- * reachable(ioa,x)
- *
- * reachable(ioa,s) & ? (s,a,s'):trans_of(ioa)
- * -------------------------------------------
- * reachable(ioa,s')
- *
- * A direkt definition follows.
- *******************************)
+
reachable_def
"reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
@@ -119,21 +109,30 @@
\ | Some(x) => if p x then Some x else None)"
-mk_behaviour_def
- "mk_behaviour(ioa) == filter_oseq(%a.a:externals(asig_of(ioa)))"
+mk_trace_def
+ "mk_trace(ioa) == filter_oseq(%a.a:externals(asig_of(ioa)))"
-(* Does an ioa have an execution with the given behaviour *)
-has_behaviour_def
- "has_behaviour ioa b == \
-\ (? ex:executions(ioa). b = mk_behaviour ioa (fst ex))"
-
+(* Does an ioa have an execution with the given trace *)
+has_trace_def
+ "has_trace ioa b == \
+\ (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
-(* All the behaviours of an ioa *)
-behaviours_def
- "behaviours(ioa) == {b. has_behaviour ioa b}"
+normal_form_def
+ "NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) & \
+\ (!j. j ~: range(f) --> nf(j)= None) & \
+\ (!i. nf(i)=None --> (nf (Suc i)) = None) "
+
+(* All the traces of an ioa *)
+ traces_def
+ "traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
+(*
+ traces_def
+ "traces(ioa) == {tr. has_trace ioa tr}"
+*)
+
compat_asigs_def
"compat_asigs a1 a2 == \
\ (((outputs(a1) Int outputs(a2)) = {}) & \
@@ -179,8 +178,19 @@
ioa_implements_def
- "ioa_implements ioa1 ioa2 == \
-\ (externals(asig_of(ioa1)) = externals(asig_of(ioa2)) & \
-\ behaviours(ioa1) <= behaviours(ioa2))"
+ "ioa_implements ioa1 ioa2 == \
+\ ((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) & \
+\ (outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) & \
+\ traces(ioa1) <= traces(ioa2))"
+
+rename_def
+"rename ioa ren == \
+\ (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))}, \
+\ {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))}, \
+\ {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}), \
+\ 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)})"
end
--- a/src/HOL/IOA/meta_theory/Option.ML Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/Option.ML Thu Apr 13 17:05:41 1995 +0200
@@ -14,3 +14,11 @@
by (Option.option.induct_tac "opt" 1);
by (ALLGOALS(fast_tac HOL_cs));
val optE = store_thm("optE", standard(result() RS disjE));
+
+goal Option.thy "x=None | (? y.x=Some(y))";
+by (Option.option.induct_tac "x" 1);
+by (asm_full_simp_tac list_ss 1);
+by (rtac disjI2 1);
+by (rtac exI 1);
+by (asm_full_simp_tac list_ss 1);
+qed"opt_cases";
\ No newline at end of file
--- a/src/HOL/IOA/meta_theory/Option.thy Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/Option.thy Thu Apr 13 17:05:41 1995 +0200
@@ -7,5 +7,5 @@
*)
Option = Arith +
-datatype 'a option = None | Some('a)
+datatype 'a option = None | Some ('a)
end
--- a/src/HOL/IOA/meta_theory/Solve.ML Thu Apr 13 16:57:18 1995 +0200
+++ b/src/HOL/IOA/meta_theory/Solve.ML Thu Apr 13 17:05:41 1995 +0200
@@ -6,22 +6,26 @@
Weak possibilities mapping (abstraction)
*)
-open Solve;
+open Solve;
-val SS = SS addsimps [mk_behaviour_thm,trans_in_actions];
+val SS = SS addsimps [mk_trace_thm,trans_in_actions];
-goalw Solve.thy [is_weak_pmap_def,behaviours_def]
+goalw Solve.thy [is_weak_pmap_def,traces_def]
"!!f. [| IOA(C); IOA(A); externals(asig_of(C)) = externals(asig_of(A)); \
-\ is_weak_pmap f C A |] ==> behaviours(C) <= behaviours(A)";
+\ is_weak_pmap f C A |] ==> traces(C) <= traces(A)";
- by (simp_tac(SS addsimps [has_behaviour_def])1);
+ by (simp_tac(SS addsimps [has_trace_def])1);
by (safe_tac set_cs);
- (* give execution of abstract automata *)
- by (res_inst_tac[("x","(mk_behaviour A (fst ex),%i.f(snd ex i))")] bexI 1);
+ (* choose same trace, therefore same NF *)
+ by (res_inst_tac[("x","mk_trace C (fst ex)")] exI 1);
+ by (asm_full_simp_tac set_ss 1);
- (* Behaviours coincide *)
- by (asm_simp_tac (SS addsimps [mk_behaviour_def,filter_oseq_idemp])1);
+ (* give execution of abstract automata *)
+ by (res_inst_tac[("x","(mk_trace A (fst ex),%i.f(snd ex i))")] bexI 1);
+
+ (* Traces coincide *)
+ by (asm_simp_tac (SS addsimps [mk_trace_def,filter_oseq_idemp])1);
(* Use lemma *)
by (forward_tac [states_of_exec_reachable] 1);
@@ -43,3 +47,224 @@
by (eres_inst_tac [("x","a")] allE 1);
by (asm_full_simp_tac SS 1);
qed "trace_inclusion";
+
+(* Lemmata *)
+
+val prems = goal HOL.thy "(P ==> Q-->R) ==> P&Q --> R";
+ by(fast_tac (HOL_cs addDs prems) 1);
+val imp_conj_lemma = result();
+
+
+(* fist_order_tautology of externals_of_par *)
+goal IOA.thy "a:externals(asig_of(A1||A2)) = \
+\ (a:externals(asig_of(A1)) & a:externals(asig_of(A2)) | \
+\ a:externals(asig_of(A1)) & a~:externals(asig_of(A2)) | \
+\ a~:externals(asig_of(A1)) & a:externals(asig_of(A2)))";
+by (asm_full_simp_tac (SS addsimps [externals_def,asig_of_par,asig_comp_def,asig_inputs_def,asig_outputs_def]) 1);
+ by (fast_tac set_cs 1);
+val externals_of_par_extra = result();
+
+goal Solve.thy "!!s.[| reachable (C1||C2) s |] ==> reachable C1 (fst s)";
+by (asm_full_simp_tac (SS addsimps [reachable_def]) 1);
+by (etac bexE 1);
+by (res_inst_tac [("x",
+ "(filter_oseq (%a.a:actions(asig_of(C1))) \
+\ (fst ex), \
+\ %i.fst (snd ex i))")] bexI 1);
+(* fst(s) is in projected execution *)
+ by (simp_tac SS 1);
+ by (fast_tac HOL_cs 1);
+(* projected execution is indeed an execution *)
+by (asm_full_simp_tac (SS addsimps [executions_def,is_execution_fragment_def,
+ par_def, starts_of_def,trans_of_def]) 1);
+by (simp_tac (SS addsimps [filter_oseq_def]) 1);
+ by (REPEAT(rtac allI 1));
+ by (cut_inst_tac [("x","fst ex n")] opt_cases 1);
+ by (etac disjE 1);
+(* case 1: action sequence of || had already a None *)
+by (asm_full_simp_tac SS 1);
+ by (REPEAT(etac exE 1));
+ by (case_tac "y:actions(asig_of(C1))" 1);
+(* case 2: action sequence of || had Some(a) and
+ filtered sequence also *)
+by (asm_full_simp_tac SS 1);
+ by (rtac impI 1);
+ by (REPEAT(hyp_subst_tac 1));
+ by (asm_full_simp_tac SS 1);
+(* case 3: action sequence pf || had Some(a) but
+ filtered sequence has None *)
+ by (asm_full_simp_tac SS 1);
+qed"comp1_reachable";
+
+
+(* Exact copy of proof of comp1_reachable for the second
+ component of a parallel composition. *)
+goal Solve.thy "!!s.[| reachable (C1||C2) s|] ==> reachable C2 (snd s)";
+by (asm_full_simp_tac (SS addsimps [reachable_def]) 1);
+by (etac bexE 1);
+by (res_inst_tac [("x",
+ "(filter_oseq (%a.a:actions(asig_of(C2)))\
+\ (fst ex), \
+\ %i.snd (snd ex i))")] bexI 1);
+(* fst(s) is in projected execution *)
+ by (simp_tac list_ss 1);
+ by (fast_tac HOL_cs 1);
+(* projected execution is indeed an execution *)
+ by (asm_full_simp_tac (SS addsimps [executions_def,is_execution_fragment_def,
+ par_def, starts_of_def,trans_of_def]) 1);
+ by (simp_tac (SS addsimps [filter_oseq_def]) 1);
+ by (REPEAT(rtac allI 1));
+ by (cut_inst_tac [("x","fst ex n")] opt_cases 1);
+ by (etac disjE 1);
+ by (asm_full_simp_tac SS 1);
+ by (REPEAT(etac exE 1));
+ by (case_tac "y:actions(asig_of(C2))" 1);
+ by (asm_full_simp_tac SS 1);
+ by (rtac impI 1);
+ by (REPEAT(hyp_subst_tac 1));
+ by (asm_full_simp_tac SS 1);
+ by (asm_full_simp_tac SS 1);
+qed"comp2_reachable";
+
+
+(* Composition of possibility-mappings *)
+
+goalw Solve.thy [is_weak_pmap_def] "!!f g.[| is_weak_pmap f C1 A1 & \
+\ externals(asig_of(A1))=externals(asig_of(C1)) &\
+\ is_weak_pmap g C2 A2 & \
+\ externals(asig_of(A2))=externals(asig_of(C2)) & \
+\ compat_ioas C1 C2 & compat_ioas A1 A2 |] \
+\ ==> is_weak_pmap (%p.(f(fst(p)),g(snd(p)))) (C1||C2) (A1||A2)";
+ by (rtac conjI 1);
+(* start_states *)
+ by (asm_full_simp_tac (SS addsimps [par_def, starts_of_def]) 1);
+ by (fast_tac set_cs 1);
+(* transitions *)
+by (REPEAT (rtac allI 1));
+by (rtac imp_conj_lemma 1);
+by (REPEAT(etac conjE 1));
+by (simp_tac (SS addsimps [externals_of_par_extra]) 1);
+by (simp_tac (SS addsimps [par_def]) 1);
+by (asm_full_simp_tac (SS addsimps [trans_of_def]) 1);
+by (rtac (expand_if RS ssubst) 1);
+by (rtac conjI 1);
+by (rtac impI 1);
+by (etac disjE 1);
+(* case 1 a:e(A1) | a:e(A2) *)
+by (asm_full_simp_tac (SS addsimps [comp1_reachable,comp2_reachable,
+ ext_is_act]) 1);
+by (etac disjE 1);
+(* case 2 a:e(A1) | a~:e(A2) *)
+by (asm_full_simp_tac (SS addsimps [comp1_reachable,comp2_reachable,
+ ext_is_act,ext1_ext2_is_not_act2]) 1);
+(* case 3 a:~e(A1) | a:e(A2) *)
+by (asm_full_simp_tac (SS addsimps [comp1_reachable,comp2_reachable,
+ ext_is_act,ext1_ext2_is_not_act1]) 1);
+(* case 4 a:~e(A1) | a~:e(A2) *)
+by (rtac impI 1);
+by (subgoal_tac "a~:externals(asig_of(A1)) & a~:externals(asig_of(A2))" 1);
+(* delete auxiliary subgoal *)
+by (asm_full_simp_tac SS 2);
+by (fast_tac HOL_cs 2);
+by (simp_tac (SS addsimps [conj_disj_distribR] addcongs [conj_cong]
+ setloop (split_tac [expand_if])) 1);
+by(REPEAT((resolve_tac [conjI,impI] 1 ORELSE etac conjE 1) THEN
+ asm_full_simp_tac(SS addsimps[comp1_reachable,
+ comp2_reachable])1));
+qed"fxg_is_weak_pmap_of_product_IOA";
+
+
+goal Solve.thy "!!s.[| reachable (rename C g) s |] ==> reachable C s";
+by (asm_full_simp_tac (SS addsimps [reachable_def]) 1);
+by (etac bexE 1);
+by (res_inst_tac [("x",
+ "((%i. case (fst ex i) \
+\ of None => None\
+\ | Some(x) => g x) ,snd ex)")] bexI 1);
+by (simp_tac SS 1);
+(* execution is indeed an execution of C *)
+by (asm_full_simp_tac (SS addsimps [executions_def,is_execution_fragment_def,
+ par_def, starts_of_def,trans_of_def,rename_def]) 1);
+by (REPEAT(rtac allI 1));
+by (cut_inst_tac [("x","fst ex n")] opt_cases 1);
+ by (etac disjE 1);
+(* case 1: action sequence of rename C had already a None *)
+by (asm_full_simp_tac SS 1);
+(* case 2 *)
+by (REPEAT(etac exE 1));
+by (etac conjE 1);
+by (eres_inst_tac [("x","n")] allE 1);
+by (eres_inst_tac [("x","y")] allE 1);
+by (asm_full_simp_tac SS 1);
+by (etac exE 1);
+by (etac conjE 1);
+by (dtac sym 1);
+by (dtac sym 1);
+by (dtac sym 1);
+by (asm_full_simp_tac SS 1);
+by (safe_tac HOL_cs);
+qed"reachable_rename_ioa";
+
+
+(* HOL Lemmata - must already exist ! *)
+goal HOL.thy "(~(A|B)) =(~A&~B)";
+ by (fast_tac HOL_cs 1);
+val disj_demorgan = result();
+goal HOL.thy "(~(A&B)) =(~A|~B)";
+ by (fast_tac HOL_cs 1);
+val conj_demorgan = result();
+goal HOL.thy "(~(? x.P x)) =(! x. (~ (P x)))";
+ by (fast_tac HOL_cs 1);
+val neg_ex = result();
+
+
+goal Solve.thy "!!f.[| is_weak_pmap f C A |]\
+\ ==> (is_weak_pmap f (rename C g) (rename A g))";
+by (asm_full_simp_tac (SS addsimps [is_weak_pmap_def]) 1);
+by (rtac conjI 1);
+by (asm_full_simp_tac (SS addsimps [rename_def,starts_of_def]) 1);
+by (fast_tac set_cs 1);
+by (REPEAT (rtac allI 1));
+by (rtac imp_conj_lemma 1);
+by (simp_tac (SS addsimps [rename_def]) 1);
+by (asm_full_simp_tac (SS addsimps [externals_def,asig_inputs_def,asig_outputs_def,asig_of_def,trans_of_def]) 1);
+by (safe_tac set_cs);
+by (rtac (expand_if RS ssubst) 1);
+ by (rtac conjI 1);
+ by (rtac impI 1);
+ by (etac disjE 1);
+ by (etac exE 1);
+by (etac conjE 1);
+(* x is input *)
+ by (dtac sym 1);
+ by (dtac sym 1);
+by (asm_full_simp_tac SS 1);
+by (REPEAT (hyp_subst_tac 1));
+by (cut_inst_tac [("C","C"),("g","g"),("s","s")] reachable_rename_ioa 1);
+by (assume_tac 1);
+by (asm_full_simp_tac SS 1);
+(* x is output *)
+ by (etac exE 1);
+by (etac conjE 1);
+ by (dtac sym 1);
+ by (dtac sym 1);
+by (asm_full_simp_tac SS 1);
+by (REPEAT (hyp_subst_tac 1));
+by (cut_inst_tac [("C","C"),("g","g"),("s","s")] reachable_rename_ioa 1);
+by (assume_tac 1);
+by (asm_full_simp_tac SS 1);
+(* x is internal *)
+by (simp_tac (SS addsimps [disj_demorgan,neg_ex,conj_demorgan] addcongs [conj_cong]) 1);
+by (rtac impI 1);
+by (etac conjE 1);
+by (cut_inst_tac [("C","C"),("g","g"),("s","s")] reachable_rename_ioa 1);
+by (assume_tac 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","t")] allE 1);
+by (eres_inst_tac [("x","x")] allE 1);
+by (eres_inst_tac [("x","x")] allE 1);
+by (asm_full_simp_tac SS 1);
+qed"rename_through_pmap";
+
+
+