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--- a +++ b/thys/TLA/State.thy @@ -0,0 +1,208 @@ +(* Title: A Definitional Encoding of TLA in Isabelle/HOL + Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> + Stephan Merz <Stephan.Merz at loria.fr> + Year: 2011 + Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> +*) + +header {* Representing state in TLA* *} + +theory State +imports Liveness +begin + +text{* + We adopt the hidden state appraoch, as used in the existing + Isabelle/HOL TLA embedding \cite{Merz98}. This approach is also used + in \cite{Ehmety01}. + Here, a state space is defined by its projections, and everything else is + unknown. Thus, a variable is a projection of the state space, and has the same + type as a state function. Moreover, strong typing is achieved, since the projection + function may have any result type. To achieve this, the state space is represented + by an undefined type, which is an instance of the @{text world} class to enable + use with the @{text Intensional} theory. +*} + +typedecl state + +instance state :: world .. + +type_synonym 'a statefun = "(state,'a) stfun" +type_synonym statepred = "bool statefun" +type_synonym 'a tempfun = "(state,'a) formfun" +type_synonym temporal = "state formula" + +text {* + Formalizing type state would require formulas to be tagged with + their underlying state space and would result in a system that is + much harder to use. (Unlike Hoare logic or Unity, TLA has quantification + over state variables, and therefore one usually works with different + state spaces within a single specification.) Instead, state is just + an anonymous type whose only purpose is to provide Skolem constants. + Moreover, we do not define a type of state variables separate from that + of arbitrary state functions, again in order to simplify the definition + of flexible quantification later on. Nevertheless, we need to distinguish + state variables, mainly to define the enabledness of actions. The user + identifies (tuples of) ``base'' state variables in a specification via the + ``meta predicate'' @{text basevars}, which is defined here. +*} + +definition stvars :: "'a statefun \<Rightarrow> bool" +where basevars_def: "stvars \<equiv> surj" + +syntax + "PRED" :: "lift \<Rightarrow> 'a" ("PRED _") + "_stvars" :: "lift \<Rightarrow> bool" ("basevars _") + +translations + "PRED P" \<rightharpoonup> "(P::state => _)" + "_stvars" \<rightleftharpoons> "CONST stvars" + +text {* + Base variables may be assigned arbitrary (type-correct) values. + In the following lemma, note that @{text vs} may be a tuple of variables. + The correct identification of base variables is up to the user who must + take care not to introduce an inconsistency. For example, @{term "basevars (x,x)"} + would definitely be inconsistent. +*} + +lemma basevars: "basevars vs \<Longrightarrow> \<exists>u. vs u = c" +proof (unfold basevars_def surj_def) + assume "\<forall>y. \<exists>x. y = vs x" + then obtain x where "c = vs x" by blast + thus "\<exists>u. vs u = c" by blast +qed + +lemma baseE: + assumes H1: "basevars v" and H2:"\<And>x. v x = c \<Longrightarrow> Q" + shows "Q" + using H1[THEN basevars] H2 by auto + +text {* A variant written for sequences rather than single states. *} +lemma first_baseE: + assumes H1: "basevars v" and H2: "\<And>x. v (first x) = c \<Longrightarrow> Q" + shows "Q" + using H1[THEN basevars] H2 by (force simp: first_def) + +lemma base_pair1: + assumes h: "basevars (x,y)" + shows "basevars x" +proof (auto simp: basevars_def) + fix c + from h[THEN basevars] obtain s where "(LIFT (x,y)) s = (c, arbitrary)" by auto + thus "c \<in> range x" by auto +qed + +lemma base_pair2: + assumes h: "basevars (x,y)" + shows "basevars y" +proof (auto simp: basevars_def) + fix d + from h[THEN basevars] obtain s where "(LIFT (x,y)) s = (arbitrary, d)" by auto + thus "d \<in> range y" by auto +qed + +lemma base_pair: "basevars (x,y) \<Longrightarrow> basevars x \<and> basevars y" + by (auto elim: base_pair1 base_pair2) + +text {* + Since the @{typ unit} type has just one value, any state function of unit type + satisfies the predicate @{text basevars}. The following theorem can sometimes + be useful because it gives a trivial solution for @{text basevars} premises. +*} + +lemma unit_base: "basevars (v::state \<Rightarrow> unit)" + by (auto simp: basevars_def) + +text {* + A pair of the form @{text "(x,x)"} will generally not satisfy the predicate + @{text basevars} -- except for pathological cases such as @{text "x::unit"}. +*} +lemma + fixes x :: "state \<Rightarrow> bool" + assumes h1: "basevars (x,x)" + shows "False" +proof - + from h1 have "\<exists>u. (LIFT (x,x)) u = (False,True)" by (rule basevars) + thus False by auto +qed + +lemma + fixes x :: "state \<Rightarrow> nat" + assumes h1: "basevars (x,x)" + shows "False" +proof - + from h1 have "\<exists>u. (LIFT (x,x)) u = (0,1)" by (rule basevars) + thus False by auto +qed + +text {* + The following theorem reduces the reasoning about the existence of a + state sequence satisfiyng an enabledness predicate to finding a suitable + value @{text c} at the successor state for the base variables of the + specification. This rule is intended for reasoning about standard TLA + specifications, where @{text Enabled} is applied to actions, not arbitrary + pre-formulas. +*} +lemma base_enabled: + assumes h1: "basevars vs" + and h2: "\<And>u. vs (first u) = c \<Longrightarrow> ((first s) ## u) \<Turnstile> F" + shows "s \<Turnstile> Enabled F" +using h1 proof (rule first_baseE) + fix t + assume "vs (first t) = c" + hence "((first s) ## t) \<Turnstile> F" by (rule h2) + thus "s \<Turnstile> Enabled F" unfolding enabled_def by blast +qed + + +subsection "Temporal Quantifiers" + +text{* + In \cite{Lamport94}, Lamport gives a stuttering invariant definition + of quantification over (flexible) variables. It relies on similarity + of two sequences (as supported in our @{theory Sequence} theory), and + equivalence of two sequences up to a variable (the bound variable). + However, sequence equaivalence up to a variable, requires state + equaivalence up to a variable. Our state representation above does not + support this, hence we cannot encode Lamport's definition in our theory. + Thus, we need to axiomatise quantification over (flexible) variables. + Note that with a state representation supporting this, our theory should + allow such an encoding. +*} + +consts + EEx :: "('a statefun \<Rightarrow> temporal) \<Rightarrow> temporal" (binder "Eex " 10) + AAll :: "('a statefun \<Rightarrow> temporal) \<Rightarrow> temporal" (binder "Aall " 10) + +syntax + "_EEx" :: "[idts, lift] \<Rightarrow> lift" ("(3EEX _./ _)" [0,10] 10) + "_AAll" :: "[idts, lift] \<Rightarrow> lift" ("(3AALL _./ _)" [0,10] 10) + +translations + "_EEx v A" == "Eex v. A" + "_AAll v A" == "Aall v. A" + +syntax (xsymbols) + "_EEx" :: "[idts, lift] => lift" ("(3\<exists>\<exists> _./ _)" [0,10] 10) + "_AAll" :: "[idts, lift] => lift" ("(3\<forall>\<forall> _./ _)" [0,10] 10) + +axiomatization where + eexI: "\<turnstile> F x \<longrightarrow> (\<exists>\<exists> x. F x)" +and eexE: "\<lbrakk>s \<Turnstile> (\<exists>\<exists> x. F x) ; basevars vs; (!! x. \<lbrakk> basevars (x,vs); s \<Turnstile> F x \<rbrakk> \<Longrightarrow> s \<Turnstile> G)\<rbrakk> + \<Longrightarrow> (s \<Turnstile> G)" +and all_def: "\<turnstile> (\<forall>\<forall> x. F x) = (\<not>(\<exists>\<exists> x. \<not>(F x)))" +and eexSTUT: "STUTINV F x \<Longrightarrow> STUTINV (\<exists>\<exists> x. F x)" +and history: "\<turnstile> (I \<and> \<box>[A]_v) = (\<exists>\<exists> h. ($h = ha) \<and> I \<and> \<box>[A \<and> h$=hb]_(h,v))" + +lemmas eexI_unl = eexI[unlift_rule] --{* @{text "w \<Turnstile> F x \<Longrightarrow> w \<Turnstile> (\<exists>\<exists> x. F x)"} *} + +text {* + @{text tla_defs} can be used to unfold TLA definitions into lowest predicate level. + This is particularly useful for reasoning about enabledness of formulas. +*} +lemmas tla_defs = unch_def before_def after_def first_def second_def suffix_def + tail_def nexts_def app_def angle_actrans_def actrans_def + + +end