Up to index of Isabelle/HOLCF/IOA
theory Automata(* Title: HOLCF/IOA/meta_theory/Automata.thy ID: $Id: Automata.thy,v 1.13 2005/09/02 15:24:00 wenzelm 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" (* reachability and invariants *) reachable :: "('a,'s)ioa => 's set" 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" "||" ::"[('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" syntax "_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool" ("_ -_--_-> _" [81,81,81,81] 100) "reachable" :: "[('a,'s)ioa, 's] => bool" "act" :: "('a,'s)ioa => 'a set" "ext" :: "('a,'s)ioa => 'a set" "int" :: "('a,'s)ioa => 'a set" "inp" :: "('a,'s)ioa => 'a set" "out" :: "('a,'s)ioa => 'a set" "local" :: "('a,'s)ioa => 'a set" syntax (xsymbols) "_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool" ("_ \<midarrow>_\<midarrow>_--> _" [81,81,81,81] 100) "op ||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "\<parallel>" 10) inductive "reachable C" intros reachable_0: "s:(starts_of C) ==> s : reachable C" reachable_n: "[|s:reachable C; (s,a,t):trans_of C|] ==> t:reachable C" translations "s -a--A-> t" == "(s,a,t):trans_of A" "reachable A s" == "s:reachable A" "act A" == "actions (asig_of A)" "ext A" == "externals (asig_of A)" "int A" == "internals (asig_of A)" "inp A" == "inputs (asig_of A)" "out A" == "outputs (asig_of A)" "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" ML {* use_legacy_bindings (the_context ()) *} end
theorem 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
theorem trans_in_actions:
[| is_trans_of A; s1.0 -a--A-> s2.0 |] ==> a ∈ act A
theorem starts_of_par:
starts_of (A || B) = {p. fst p ∈ starts_of A ∧ snd p ∈ starts_of B}
theorem 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)}
theorem actions_asig_comp:
actions (asig_comp a b) = actions a ∪ actions b
theorem asig_of_par:
asig_of (A || B) = asig_comp (asig_of A) (asig_of B)
theorem externals_of_par:
ext (A1.0 || A2.0) = ext A1.0 ∪ ext A2.0
theorem actions_of_par:
act (A1.0 || A2.0) = act A1.0 ∪ act A2.0
theorem inputs_of_par:
inp (A1.0 || A2.0) = inp A1.0 ∪ inp A2.0 - (out A1.0 ∪ out A2.0)
theorem outputs_of_par:
out (A1.0 || A2.0) = out A1.0 ∪ out A2.0
theorem internals_of_par:
int (A1.0 || A2.0) = int A1.0 ∪ int A2.0
theorem compat_commute:
compatible A B = compatible B A
theorem ext1_is_not_int2:
[| compatible A1.0 A2.0; a ∈ ext A1.0 |] ==> a ∉ int A2.0
theorem ext2_is_not_int1:
[| compatible A2.0 A1.0; a ∈ ext A1.0 |] ==> a ∉ int A2.0
theorem ext1_ext2_is_not_act2:
[| compatible A1.0 A2.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
theorem ext1_ext2_is_not_act1:
[| compatible A2.0 A1.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
theorem intA_is_not_extB:
[| compatible A B; x ∈ int A |] ==> x ∉ ext B
theorem intA_is_not_actB:
[| compatible A B; a ∈ int A |] ==> a ∉ act B
theorem outAactB_is_inpB:
[| compatible A B; a ∈ out A; a ∈ act B |] ==> a ∈ inp B
theorem inpAAactB_is_inpBoroutB:
[| compatible A B; a ∈ inp A; a ∈ act B |] ==> a ∈ inp B ∨ a ∈ out B
theorem input_enabled_par:
[| compatible A B; input_enabled A; input_enabled B |] ==> input_enabled (A || B)
theorem 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
theorem 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
theorem invariantE:
[| invariant A P; reachable A s |] ==> P s
theorem cancel_restrict_a:
starts_of (restrict ioa acts) = starts_of ioa ∧ trans_of (restrict ioa acts) = trans_of ioa
theorem cancel_restrict_b:
reachable (restrict ioa acts) s = reachable ioa s
theorem acts_restrict:
act (restrict A acts) = act A
theorem 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
theorem trans_rename:
s -a--rename C f-> t ==> ∃x. Some x = f a ∧ s -x--C-> t
theorem reachable_rename:
reachable (rename C g) s ==> reachable C s
theorem trans_A_proj:
[| s -a--(A || B)-> t; a ∈ act A |] ==> fst s -a--A-> fst t
theorem trans_B_proj:
[| s -a--(A || B)-> t; a ∈ act B |] ==> snd s -a--B-> snd t
theorem trans_A_proj2:
[| s -a--(A || B)-> t; a ∉ act A |] ==> fst s = fst t
theorem trans_B_proj2:
[| s -a--(A || B)-> t; a ∉ act B |] ==> snd s = snd t
theorem trans_AB_proj:
s -a--(A || B)-> t ==> a ∈ act A ∨ a ∈ act B
theorem 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
theorem trans_A_notB:
[| a ∈ act A; a ∉ act B; fst s -a--A-> fst t; snd s = snd t |] ==> s -a--(A || B)-> t
theorem trans_notA_B:
[| a ∉ act A; a ∈ act B; snd s -a--B-> snd t; fst s = fst t |] ==> s -a--(A || B)-> t
theorem 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))))
theorem is_trans_of_par:
is_trans_of (A || B)
theorem is_trans_of_restrict:
is_trans_of A ==> is_trans_of (restrict A acts)
theorem is_trans_of_rename:
is_trans_of A ==> is_trans_of (rename A f)
theorem is_asig_of_par:
[| is_asig_of A; is_asig_of B; compatible A B |] ==> is_asig_of (A || B)
theorem is_asig_of_restrict:
is_asig_of A ==> is_asig_of (restrict A f)
theorem is_asig_of_rename:
is_asig_of A ==> is_asig_of (rename A f)
theorem compatible_par:
[| compatible A B; compatible A C |] ==> compatible A (B || C)
theorem compatible_par2:
[| compatible A C; compatible B C |] ==> compatible (A || B) C
theorem compatible_restrict:
[| compatible A B; (ext B - S) ∩ ext A = {} |] ==> compatible A (restrict B S)