author | wenzelm |
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(* Title: HOLCF/IOA/meta_theory/Automata.thy |
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Author: Olaf Müller, Konrad Slind, Tobias Nipkow |
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*) |
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header {* The I/O automata of Lynch and Tuttle in HOLCF *} |
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theory Automata |
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imports Asig |
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begin |
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defaultsort type |
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types |
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('a, 's) transition = "'s * 'a * 's" |
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('a, 's) ioa = "'a signature * 's set * ('a,'s)transition set * ('a set set) * ('a set set)" |
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consts |
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(* IO automata *) |
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asig_of ::"('a,'s)ioa => 'a signature" |
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starts_of ::"('a,'s)ioa => 's set" |
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trans_of ::"('a,'s)ioa => ('a,'s)transition set" |
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wfair_of ::"('a,'s)ioa => ('a set) set" |
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sfair_of ::"('a,'s)ioa => ('a set) set" |
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is_asig_of ::"('a,'s)ioa => bool" |
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is_starts_of ::"('a,'s)ioa => bool" |
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is_trans_of ::"('a,'s)ioa => bool" |
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input_enabled ::"('a,'s)ioa => bool" |
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IOA ::"('a,'s)ioa => bool" |
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(* constraints for fair IOA *) |
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fairIOA ::"('a,'s)ioa => bool" |
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input_resistant::"('a,'s)ioa => bool" |
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(* enabledness of actions and action sets *) |
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enabled ::"('a,'s)ioa => 'a => 's => bool" |
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Enabled ::"('a,'s)ioa => 'a set => 's => bool" |
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(* action set keeps enabled until probably disabled by itself *) |
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en_persistent :: "('a,'s)ioa => 'a set => bool" |
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(* post_conditions for actions and action sets *) |
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was_enabled ::"('a,'s)ioa => 'a => 's => bool" |
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set_was_enabled ::"('a,'s)ioa => 'a set => 's => bool" |
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(* invariants *) |
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invariant :: "[('a,'s)ioa, 's=>bool] => bool" |
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(* binary composition of action signatures and automata *) |
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asig_comp ::"['a signature, 'a signature] => 'a signature" |
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compatible ::"[('a,'s)ioa, ('a,'t)ioa] => bool" |
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par ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "||" 10) |
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(* hiding and restricting *) |
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hide_asig :: "['a signature, 'a set] => 'a signature" |
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hide :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa" |
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restrict_asig :: "['a signature, 'a set] => 'a signature" |
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restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa" |
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(* renaming *) |
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rename_set :: "'a set => ('c => 'a option) => 'c set" |
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rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa" |
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notation (xsymbols) |
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par (infixr "\<parallel>" 10) |
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inductive |
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reachable :: "('a, 's) ioa => 's => bool" |
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for C :: "('a, 's) ioa" |
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where |
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reachable_0: "s : starts_of C ==> reachable C s" |
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| reachable_n: "[| reachable C s; (s, a, t) : trans_of C |] ==> reachable C t" |
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abbreviation |
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trans_of_syn ("_ -_--_-> _" [81,81,81,81] 100) where |
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"s -a--A-> t == (s,a,t):trans_of A" |
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notation (xsymbols) |
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trans_of_syn ("_ \<midarrow>_\<midarrow>_\<longrightarrow> _" [81,81,81,81] 100) |
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abbreviation "act A == actions (asig_of A)" |
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abbreviation "ext A == externals (asig_of A)" |
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abbreviation int where "int A == internals (asig_of A)" |
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abbreviation "inp A == inputs (asig_of A)" |
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abbreviation "out A == outputs (asig_of A)" |
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abbreviation "local A == locals (asig_of A)" |
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defs |
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(* --------------------------------- IOA ---------------------------------*) |
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asig_of_def: "asig_of == fst" |
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starts_of_def: "starts_of == (fst o snd)" |
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trans_of_def: "trans_of == (fst o snd o snd)" |
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wfair_of_def: "wfair_of == (fst o snd o snd o snd)" |
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sfair_of_def: "sfair_of == (snd o snd o snd o snd)" |
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is_asig_of_def: |
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"is_asig_of A == is_asig (asig_of A)" |
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is_starts_of_def: |
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"is_starts_of A == (~ starts_of A = {})" |
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is_trans_of_def: |
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"is_trans_of A == |
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(!triple. triple:(trans_of A) --> fst(snd(triple)):actions(asig_of A))" |
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input_enabled_def: |
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"input_enabled A == |
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(!a. (a:inputs(asig_of A)) --> (!s1. ? s2. (s1,a,s2):(trans_of A)))" |
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ioa_def: |
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"IOA A == (is_asig_of A & |
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is_starts_of A & |
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is_trans_of A & |
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input_enabled A)" |
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invariant_def: "invariant A P == (!s. reachable A s --> P(s))" |
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(* ------------------------- parallel composition --------------------------*) |
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compatible_def: |
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"compatible A B == |
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(((out A Int out B) = {}) & |
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((int A Int act B) = {}) & |
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((int B Int act A) = {}))" |
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asig_comp_def: |
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"asig_comp a1 a2 == |
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(((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)), |
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(outputs(a1) Un outputs(a2)), |
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(internals(a1) Un internals(a2))))" |
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par_def: |
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"(A || B) == |
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(asig_comp (asig_of A) (asig_of B), |
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{pr. fst(pr):starts_of(A) & snd(pr):starts_of(B)}, |
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{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) |
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in (a:act A | a:act B) & |
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(if a:act A then |
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(fst(s),a,fst(t)):trans_of(A) |
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else fst(t) = fst(s)) |
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& |
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(if a:act B then |
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(snd(s),a,snd(t)):trans_of(B) |
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else snd(t) = snd(s))}, |
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wfair_of A Un wfair_of B, |
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sfair_of A Un sfair_of B)" |
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(* ------------------------ hiding -------------------------------------------- *) |
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restrict_asig_def: |
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"restrict_asig asig actns == |
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(inputs(asig) Int actns, |
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outputs(asig) Int actns, |
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internals(asig) Un (externals(asig) - actns))" |
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(* Notice that for wfair_of and sfair_of nothing has to be changed, as |
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changes from the outputs to the internals does not touch the locals as |
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a whole, which is of importance for fairness only *) |
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restrict_def: |
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"restrict A actns == |
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(restrict_asig (asig_of A) actns, |
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starts_of A, |
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trans_of A, |
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wfair_of A, |
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sfair_of A)" |
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hide_asig_def: |
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"hide_asig asig actns == |
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(inputs(asig) - actns, |
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outputs(asig) - actns, |
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internals(asig) Un actns)" |
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hide_def: |
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"hide A actns == |
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(hide_asig (asig_of A) actns, |
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starts_of A, |
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trans_of A, |
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wfair_of A, |
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sfair_of A)" |
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(* ------------------------- renaming ------------------------------------------- *) |
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rename_set_def: |
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"rename_set A ren == {b. ? x. Some x = ren b & x : A}" |
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rename_def: |
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"rename ioa ren == |
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((rename_set (inp ioa) ren, |
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rename_set (out ioa) ren, |
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rename_set (int ioa) ren), |
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starts_of ioa, |
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{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) |
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in |
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? x. Some(x) = ren(a) & (s,x,t):trans_of ioa}, |
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{rename_set s ren | s. s: wfair_of ioa}, |
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{rename_set s ren | s. s: sfair_of ioa})" |
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(* ------------------------- fairness ----------------------------- *) |
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fairIOA_def: |
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"fairIOA A == (! S : wfair_of A. S<= local A) & |
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(! S : sfair_of A. S<= local A)" |
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input_resistant_def: |
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"input_resistant A == ! W : sfair_of A. ! s a t. |
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reachable A s & reachable A t & a:inp A & |
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Enabled A W s & s -a--A-> t |
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--> Enabled A W t" |
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enabled_def: |
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"enabled A a s == ? t. s-a--A-> t" |
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Enabled_def: |
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"Enabled A W s == ? w:W. enabled A w s" |
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en_persistent_def: |
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"en_persistent A W == ! s a t. Enabled A W s & |
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a ~:W & |
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s -a--A-> t |
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--> Enabled A W t" |
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was_enabled_def: |
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"was_enabled A a t == ? s. s-a--A-> t" |
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set_was_enabled_def: |
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"set_was_enabled A W t == ? w:W. was_enabled A w t" |
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declare split_paired_Ex [simp del] |
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lemmas ioa_projections = asig_of_def starts_of_def trans_of_def wfair_of_def sfair_of_def |
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subsection "asig_of, starts_of, trans_of" |
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lemma ioa_triple_proj: |
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"((asig_of (x,y,z,w,s)) = x) & |
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((starts_of (x,y,z,w,s)) = y) & |
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((trans_of (x,y,z,w,s)) = z) & |
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((wfair_of (x,y,z,w,s)) = w) & |
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((sfair_of (x,y,z,w,s)) = s)" |
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apply (simp add: ioa_projections) |
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done |
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lemma trans_in_actions: |
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"[| is_trans_of A; (s1,a,s2):trans_of(A) |] ==> a:act A" |
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apply (unfold is_trans_of_def actions_def is_asig_def) |
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apply (erule allE, erule impE, assumption) |
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apply simp |
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done |
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lemma starts_of_par: |
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"starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}" |
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apply (simp add: par_def ioa_projections) |
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done |
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lemma trans_of_par: |
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"trans_of(A || B) = {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) |
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in (a:act A | a:act B) & |
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(if a:act A then |
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(fst(s),a,fst(t)):trans_of(A) |
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else fst(t) = fst(s)) |
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& |
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(if a:act B then |
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(snd(s),a,snd(t)):trans_of(B) |
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else snd(t) = snd(s))}" |
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apply (simp add: par_def ioa_projections) |
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done |
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subsection "actions and par" |
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lemma actions_asig_comp: |
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"actions(asig_comp a b) = actions(a) Un actions(b)" |
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apply (simp (no_asm) add: actions_def asig_comp_def asig_projections) |
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apply blast |
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done |
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lemma asig_of_par: "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)" |
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apply (simp add: par_def ioa_projections) |
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done |
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lemma externals_of_par: "ext (A1||A2) = |
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(ext A1) Un (ext A2)" |
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apply (simp add: externals_def asig_of_par asig_comp_def |
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asig_inputs_def asig_outputs_def Un_def set_diff_eq) |
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apply blast |
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done |
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||
306 |
lemma actions_of_par: "act (A1||A2) = |
|
307 |
(act A1) Un (act A2)" |
|
308 |
apply (simp add: actions_def asig_of_par asig_comp_def |
|
26806 | 309 |
asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq) |
19741 | 310 |
apply blast |
311 |
done |
|
312 |
||
313 |
lemma inputs_of_par: "inp (A1||A2) = |
|
314 |
((inp A1) Un (inp A2)) - ((out A1) Un (out A2))" |
|
315 |
apply (simp add: actions_def asig_of_par asig_comp_def |
|
26806 | 316 |
asig_inputs_def asig_outputs_def Un_def set_diff_eq) |
19741 | 317 |
done |
318 |
||
319 |
lemma outputs_of_par: "out (A1||A2) = |
|
320 |
(out A1) Un (out A2)" |
|
321 |
apply (simp add: actions_def asig_of_par asig_comp_def |
|
26806 | 322 |
asig_outputs_def Un_def set_diff_eq) |
19741 | 323 |
done |
324 |
||
325 |
lemma internals_of_par: "int (A1||A2) = |
|
326 |
(int A1) Un (int A2)" |
|
327 |
apply (simp add: actions_def asig_of_par asig_comp_def |
|
26806 | 328 |
asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq) |
19741 | 329 |
done |
330 |
||
331 |
||
332 |
subsection "actions and compatibility" |
|
333 |
||
334 |
lemma compat_commute: "compatible A B = compatible B A" |
|
335 |
apply (simp add: compatible_def Int_commute) |
|
336 |
apply auto |
|
337 |
done |
|
338 |
||
339 |
lemma ext1_is_not_int2: |
|
340 |
"[| compatible A1 A2; a:ext A1|] ==> a~:int A2" |
|
341 |
apply (unfold externals_def actions_def compatible_def) |
|
342 |
apply simp |
|
343 |
apply blast |
|
344 |
done |
|
345 |
||
346 |
(* just commuting the previous one: better commute compatible *) |
|
347 |
lemma ext2_is_not_int1: |
|
348 |
"[| compatible A2 A1 ; a:ext A1|] ==> a~:int A2" |
|
349 |
apply (unfold externals_def actions_def compatible_def) |
|
350 |
apply simp |
|
351 |
apply blast |
|
352 |
done |
|
353 |
||
354 |
lemmas ext1_ext2_is_not_act2 = ext1_is_not_int2 [THEN int_and_ext_is_act, standard] |
|
355 |
lemmas ext1_ext2_is_not_act1 = ext2_is_not_int1 [THEN int_and_ext_is_act, standard] |
|
356 |
||
357 |
lemma intA_is_not_extB: |
|
358 |
"[| compatible A B; x:int A |] ==> x~:ext B" |
|
359 |
apply (unfold externals_def actions_def compatible_def) |
|
360 |
apply simp |
|
361 |
apply blast |
|
362 |
done |
|
363 |
||
364 |
lemma intA_is_not_actB: |
|
365 |
"[| compatible A B; a:int A |] ==> a ~: act B" |
|
366 |
apply (unfold externals_def actions_def compatible_def is_asig_def asig_of_def) |
|
367 |
apply simp |
|
368 |
apply blast |
|
369 |
done |
|
370 |
||
371 |
(* the only one that needs disjointness of outputs and of internals and _all_ acts *) |
|
372 |
lemma outAactB_is_inpB: |
|
373 |
"[| compatible A B; a:out A ;a:act B|] ==> a : inp B" |
|
374 |
apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def |
|
375 |
compatible_def is_asig_def asig_of_def) |
|
376 |
apply simp |
|
377 |
apply blast |
|
378 |
done |
|
379 |
||
380 |
(* needed for propagation of input_enabledness from A,B to A||B *) |
|
381 |
lemma inpAAactB_is_inpBoroutB: |
|
382 |
"[| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B" |
|
383 |
apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def |
|
384 |
compatible_def is_asig_def asig_of_def) |
|
385 |
apply simp |
|
386 |
apply blast |
|
387 |
done |
|
388 |
||
389 |
||
390 |
subsection "input_enabledness and par" |
|
391 |
||
392 |
||
393 |
(* ugly case distinctions. Heart of proof: |
|
394 |
1. inpAAactB_is_inpBoroutB ie. internals are really hidden. |
|
395 |
2. inputs_of_par: outputs are no longer inputs of par. This is important here *) |
|
396 |
lemma input_enabled_par: |
|
397 |
"[| compatible A B; input_enabled A; input_enabled B|] |
|
398 |
==> input_enabled (A||B)" |
|
399 |
apply (unfold input_enabled_def) |
|
400 |
apply (simp add: Let_def inputs_of_par trans_of_par) |
|
32152 | 401 |
apply (tactic "safe_tac (global_claset_of @{theory Fun})") |
19741 | 402 |
apply (simp add: inp_is_act) |
403 |
prefer 2 |
|
404 |
apply (simp add: inp_is_act) |
|
405 |
(* a: inp A *) |
|
406 |
apply (case_tac "a:act B") |
|
407 |
(* a:act B *) |
|
408 |
apply (erule_tac x = "a" in allE) |
|
409 |
apply simp |
|
410 |
apply (drule inpAAactB_is_inpBoroutB) |
|
411 |
apply assumption |
|
412 |
apply assumption |
|
413 |
apply (erule_tac x = "a" in allE) |
|
414 |
apply simp |
|
415 |
apply (erule_tac x = "aa" in allE) |
|
416 |
apply (erule_tac x = "b" in allE) |
|
417 |
apply (erule exE) |
|
418 |
apply (erule exE) |
|
419 |
apply (rule_tac x = " (s2,s2a) " in exI) |
|
420 |
apply (simp add: inp_is_act) |
|
421 |
(* a~: act B*) |
|
422 |
apply (simp add: inp_is_act) |
|
423 |
apply (erule_tac x = "a" in allE) |
|
424 |
apply simp |
|
425 |
apply (erule_tac x = "aa" in allE) |
|
426 |
apply (erule exE) |
|
427 |
apply (rule_tac x = " (s2,b) " in exI) |
|
428 |
apply simp |
|
429 |
||
430 |
(* a:inp B *) |
|
431 |
apply (case_tac "a:act A") |
|
432 |
(* a:act A *) |
|
433 |
apply (erule_tac x = "a" in allE) |
|
434 |
apply (erule_tac x = "a" in allE) |
|
435 |
apply (simp add: inp_is_act) |
|
436 |
apply (frule_tac A1 = "A" in compat_commute [THEN iffD1]) |
|
437 |
apply (drule inpAAactB_is_inpBoroutB) |
|
438 |
back |
|
439 |
apply assumption |
|
440 |
apply assumption |
|
441 |
apply simp |
|
442 |
apply (erule_tac x = "aa" in allE) |
|
443 |
apply (erule_tac x = "b" in allE) |
|
444 |
apply (erule exE) |
|
445 |
apply (erule exE) |
|
446 |
apply (rule_tac x = " (s2,s2a) " in exI) |
|
447 |
apply (simp add: inp_is_act) |
|
448 |
(* a~: act B*) |
|
449 |
apply (simp add: inp_is_act) |
|
450 |
apply (erule_tac x = "a" in allE) |
|
451 |
apply (erule_tac x = "a" in allE) |
|
452 |
apply simp |
|
453 |
apply (erule_tac x = "b" in allE) |
|
454 |
apply (erule exE) |
|
455 |
apply (rule_tac x = " (aa,s2) " in exI) |
|
456 |
apply simp |
|
457 |
done |
|
458 |
||
459 |
||
460 |
subsection "invariants" |
|
461 |
||
462 |
lemma invariantI: |
|
463 |
"[| !!s. s:starts_of(A) ==> P(s); |
|
464 |
!!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |] |
|
465 |
==> invariant A P" |
|
466 |
apply (unfold invariant_def) |
|
467 |
apply (rule allI) |
|
468 |
apply (rule impI) |
|
23778 | 469 |
apply (rule_tac x = "s" in reachable.induct) |
19741 | 470 |
apply assumption |
471 |
apply blast |
|
472 |
apply blast |
|
473 |
done |
|
474 |
||
475 |
lemma invariantI1: |
|
476 |
"[| !!s. s : starts_of(A) ==> P(s); |
|
477 |
!!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t) |
|
478 |
|] ==> invariant A P" |
|
479 |
apply (blast intro: invariantI) |
|
480 |
done |
|
481 |
||
482 |
lemma invariantE: "[| invariant A P; reachable A s |] ==> P(s)" |
|
483 |
apply (unfold invariant_def) |
|
484 |
apply blast |
|
485 |
done |
|
486 |
||
487 |
||
488 |
subsection "restrict" |
|
489 |
||
490 |
||
491 |
lemmas reachable_0 = reachable.reachable_0 |
|
492 |
and reachable_n = reachable.reachable_n |
|
493 |
||
494 |
lemma cancel_restrict_a: "starts_of(restrict ioa acts) = starts_of(ioa) & |
|
495 |
trans_of(restrict ioa acts) = trans_of(ioa)" |
|
496 |
apply (simp add: restrict_def ioa_projections) |
|
497 |
done |
|
498 |
||
499 |
lemma cancel_restrict_b: "reachable (restrict ioa acts) s = reachable ioa s" |
|
500 |
apply (rule iffI) |
|
501 |
apply (erule reachable.induct) |
|
502 |
apply (simp add: cancel_restrict_a reachable_0) |
|
503 |
apply (erule reachable_n) |
|
504 |
apply (simp add: cancel_restrict_a) |
|
505 |
(* <-- *) |
|
506 |
apply (erule reachable.induct) |
|
507 |
apply (rule reachable_0) |
|
508 |
apply (simp add: cancel_restrict_a) |
|
509 |
apply (erule reachable_n) |
|
510 |
apply (simp add: cancel_restrict_a) |
|
511 |
done |
|
512 |
||
513 |
lemma acts_restrict: "act (restrict A acts) = act A" |
|
514 |
apply (simp (no_asm) add: actions_def asig_internals_def |
|
515 |
asig_outputs_def asig_inputs_def externals_def asig_of_def restrict_def restrict_asig_def) |
|
516 |
apply auto |
|
517 |
done |
|
518 |
||
519 |
lemma cancel_restrict: "starts_of(restrict ioa acts) = starts_of(ioa) & |
|
520 |
trans_of(restrict ioa acts) = trans_of(ioa) & |
|
521 |
reachable (restrict ioa acts) s = reachable ioa s & |
|
522 |
act (restrict A acts) = act A" |
|
523 |
apply (simp (no_asm) add: cancel_restrict_a cancel_restrict_b acts_restrict) |
|
524 |
done |
|
525 |
||
526 |
||
527 |
subsection "rename" |
|
528 |
||
529 |
lemma trans_rename: "s -a--(rename C f)-> t ==> (? x. Some(x) = f(a) & s -x--C-> t)" |
|
530 |
apply (simp add: Let_def rename_def trans_of_def) |
|
531 |
done |
|
532 |
||
533 |
||
534 |
lemma reachable_rename: "[| reachable (rename C g) s |] ==> reachable C s" |
|
535 |
apply (erule reachable.induct) |
|
536 |
apply (rule reachable_0) |
|
537 |
apply (simp add: rename_def ioa_projections) |
|
538 |
apply (drule trans_rename) |
|
539 |
apply (erule exE) |
|
540 |
apply (erule conjE) |
|
541 |
apply (erule reachable_n) |
|
542 |
apply assumption |
|
543 |
done |
|
544 |
||
545 |
||
546 |
subsection "trans_of(A||B)" |
|
547 |
||
548 |
||
549 |
lemma trans_A_proj: "[|(s,a,t):trans_of (A||B); a:act A|] |
|
550 |
==> (fst s,a,fst t):trans_of A" |
|
551 |
apply (simp add: Let_def par_def trans_of_def) |
|
552 |
done |
|
553 |
||
554 |
lemma trans_B_proj: "[|(s,a,t):trans_of (A||B); a:act B|] |
|
555 |
==> (snd s,a,snd t):trans_of B" |
|
556 |
apply (simp add: Let_def par_def trans_of_def) |
|
557 |
done |
|
558 |
||
559 |
lemma trans_A_proj2: "[|(s,a,t):trans_of (A||B); a~:act A|] |
|
560 |
==> fst s = fst t" |
|
561 |
apply (simp add: Let_def par_def trans_of_def) |
|
562 |
done |
|
563 |
||
564 |
lemma trans_B_proj2: "[|(s,a,t):trans_of (A||B); a~:act B|] |
|
565 |
==> snd s = snd t" |
|
566 |
apply (simp add: Let_def par_def trans_of_def) |
|
567 |
done |
|
568 |
||
569 |
lemma trans_AB_proj: "(s,a,t):trans_of (A||B) |
|
570 |
==> a :act A | a :act B" |
|
571 |
apply (simp add: Let_def par_def trans_of_def) |
|
572 |
done |
|
573 |
||
574 |
lemma trans_AB: "[|a:act A;a:act B; |
|
575 |
(fst s,a,fst t):trans_of A;(snd s,a,snd t):trans_of B|] |
|
576 |
==> (s,a,t):trans_of (A||B)" |
|
577 |
apply (simp add: Let_def par_def trans_of_def) |
|
578 |
done |
|
579 |
||
580 |
lemma trans_A_notB: "[|a:act A;a~:act B; |
|
581 |
(fst s,a,fst t):trans_of A;snd s=snd t|] |
|
582 |
==> (s,a,t):trans_of (A||B)" |
|
583 |
apply (simp add: Let_def par_def trans_of_def) |
|
584 |
done |
|
585 |
||
586 |
lemma trans_notA_B: "[|a~:act A;a:act B; |
|
587 |
(snd s,a,snd t):trans_of B;fst s=fst t|] |
|
588 |
==> (s,a,t):trans_of (A||B)" |
|
589 |
apply (simp add: Let_def par_def trans_of_def) |
|
590 |
done |
|
591 |
||
592 |
lemmas trans_of_defs1 = trans_AB trans_A_notB trans_notA_B |
|
593 |
and trans_of_defs2 = trans_A_proj trans_B_proj trans_A_proj2 trans_B_proj2 trans_AB_proj |
|
594 |
||
595 |
||
596 |
lemma trans_of_par4: |
|
597 |
"((s,a,t) : trans_of(A || B || C || D)) = |
|
598 |
((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) | |
|
599 |
a:actions(asig_of(D))) & |
|
600 |
(if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A) |
|
601 |
else fst t=fst s) & |
|
602 |
(if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B) |
|
603 |
else fst(snd(t))=fst(snd(s))) & |
|
604 |
(if a:actions(asig_of(C)) then |
|
605 |
(fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C) |
|
606 |
else fst(snd(snd(t)))=fst(snd(snd(s)))) & |
|
607 |
(if a:actions(asig_of(D)) then |
|
608 |
(snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D) |
|
609 |
else snd(snd(snd(t)))=snd(snd(snd(s)))))" |
|
610 |
apply (simp (no_asm) add: par_def actions_asig_comp Pair_fst_snd_eq Let_def ioa_projections) |
|
611 |
done |
|
612 |
||
613 |
||
614 |
subsection "proof obligation generator for IOA requirements" |
|
615 |
||
616 |
(* without assumptions on A and B because is_trans_of is also incorporated in ||def *) |
|
617 |
lemma is_trans_of_par: "is_trans_of (A||B)" |
|
618 |
apply (unfold is_trans_of_def) |
|
619 |
apply (simp add: Let_def actions_of_par trans_of_par) |
|
620 |
done |
|
621 |
||
622 |
lemma is_trans_of_restrict: |
|
623 |
"is_trans_of A ==> is_trans_of (restrict A acts)" |
|
624 |
apply (unfold is_trans_of_def) |
|
625 |
apply (simp add: cancel_restrict acts_restrict) |
|
626 |
done |
|
627 |
||
628 |
lemma is_trans_of_rename: |
|
629 |
"is_trans_of A ==> is_trans_of (rename A f)" |
|
630 |
apply (unfold is_trans_of_def restrict_def restrict_asig_def) |
|
631 |
apply (simp add: Let_def actions_def trans_of_def asig_internals_def |
|
632 |
asig_outputs_def asig_inputs_def externals_def asig_of_def rename_def rename_set_def) |
|
633 |
apply blast |
|
634 |
done |
|
635 |
||
636 |
lemma is_asig_of_par: "[| is_asig_of A; is_asig_of B; compatible A B|] |
|
637 |
==> is_asig_of (A||B)" |
|
638 |
apply (simp add: is_asig_of_def asig_of_par asig_comp_def compatible_def |
|
639 |
asig_internals_def asig_outputs_def asig_inputs_def actions_def is_asig_def) |
|
640 |
apply (simp add: asig_of_def) |
|
641 |
apply auto |
|
642 |
done |
|
643 |
||
644 |
lemma is_asig_of_restrict: |
|
645 |
"is_asig_of A ==> is_asig_of (restrict A f)" |
|
646 |
apply (unfold is_asig_of_def is_asig_def asig_of_def restrict_def restrict_asig_def |
|
647 |
asig_internals_def asig_outputs_def asig_inputs_def externals_def o_def) |
|
648 |
apply simp |
|
649 |
apply auto |
|
650 |
done |
|
651 |
||
652 |
lemma is_asig_of_rename: "is_asig_of A ==> is_asig_of (rename A f)" |
|
653 |
apply (simp add: is_asig_of_def rename_def rename_set_def asig_internals_def |
|
654 |
asig_outputs_def asig_inputs_def actions_def is_asig_def asig_of_def) |
|
655 |
apply auto |
|
656 |
apply (drule_tac [!] s = "Some ?x" in sym) |
|
657 |
apply auto |
|
658 |
done |
|
659 |
||
660 |
lemmas [simp] = is_asig_of_par is_asig_of_restrict |
|
661 |
is_asig_of_rename is_trans_of_par is_trans_of_restrict is_trans_of_rename |
|
662 |
||
663 |
||
664 |
lemma compatible_par: |
|
665 |
"[|compatible A B; compatible A C |]==> compatible A (B||C)" |
|
666 |
apply (unfold compatible_def) |
|
667 |
apply (simp add: internals_of_par outputs_of_par actions_of_par) |
|
668 |
apply auto |
|
669 |
done |
|
670 |
||
671 |
(* better derive by previous one and compat_commute *) |
|
672 |
lemma compatible_par2: |
|
673 |
"[|compatible A C; compatible B C |]==> compatible (A||B) C" |
|
674 |
apply (unfold compatible_def) |
|
675 |
apply (simp add: internals_of_par outputs_of_par actions_of_par) |
|
676 |
apply auto |
|
677 |
done |
|
678 |
||
679 |
lemma compatible_restrict: |
|
680 |
"[| compatible A B; (ext B - S) Int ext A = {}|] |
|
681 |
==> compatible A (restrict B S)" |
|
682 |
apply (unfold compatible_def) |
|
683 |
apply (simp add: ioa_triple_proj asig_triple_proj externals_def |
|
684 |
restrict_def restrict_asig_def actions_def) |
|
685 |
apply auto |
|
686 |
done |
|
687 |
||
688 |
||
689 |
declare split_paired_Ex [simp] |
|
3071 | 690 |
|
691 |
end |