moved directory src/HOLCF to src/HOL/HOLCF;
added HOLCF theories to src/HOL/IsaMakefile;
(* Title: HOLCF/IOA/meta_theory/Automata.thy
Author: Olaf Müller, Konrad Slind, Tobias Nipkow
*)
header {* The I/O automata of Lynch and Tuttle in HOLCF *}
theory Automata
imports Asig
begin
default_sort 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>_\<longrightarrow> _" [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 (global_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