(* Title: HOL/TLA/Stfun.thy Author: Stephan Merz Copyright: 1998 University of Munich *) section ‹States and state functions for TLA as an "intensional" logic› theory Stfun imports Intensional begin typedecl state instance state :: world .. type_synonym 'a stfun = "state ⇒ 'a" type_synonym stpred = "bool stfun" consts (* 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" basevars, which is defined here. *) stvars :: "'a stfun ⇒ bool" syntax "_PRED" :: "lift ⇒ 'a" ("PRED _") "_stvars" :: "lift ⇒ bool" ("basevars _") translations "PRED P" => "(P::state ⇒ _)" "_stvars" == "CONST stvars" (* Base variables may be assigned arbitrary (type-correct) values. Note that 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, "basevars (x,x)" would definitely be inconsistent. *) overloading stvars ≡ stvars begin definition stvars :: "(state ⇒ 'a) ⇒ bool" where basevars_def: "stvars vs == range vs = UNIV" end lemma basevars: "⋀vs. basevars vs ⟹ ∃u. vs u = c" apply (unfold basevars_def) apply (rule_tac b = c and f = vs in rangeE) apply auto done lemma base_pair1: "⋀x y. basevars (x,y) ⟹ basevars x" apply (simp (no_asm) add: basevars_def) apply (rule equalityI) apply (rule subset_UNIV) apply (rule subsetI) apply (drule_tac c = "(xa, _) " in basevars) apply auto done lemma base_pair2: "⋀x y. basevars (x,y) ⟹ basevars y" apply (simp (no_asm) add: basevars_def) apply (rule equalityI) apply (rule subset_UNIV) apply (rule subsetI) apply (drule_tac c = "(_, xa) " in basevars) apply auto done lemma base_pair: "⋀x y. basevars (x,y) ⟹ basevars x & basevars y" apply (rule conjI) apply (erule base_pair1) apply (erule base_pair2) done (* Since the unit type has just one value, any state function can be regarded as "base". The following axiom can sometimes be useful because it gives a trivial solution for "basevars" premises. *) lemma unit_base: "basevars (v::unit stfun)" apply (unfold basevars_def) apply auto done lemma baseE: "⟦ basevars v; ⋀x. v x = c ⟹ Q ⟧ ⟹ Q" apply (erule basevars [THEN exE]) apply blast done (* ------------------------------------------------------------------------------- The following shows that there should not be duplicates in a "stvars" tuple: *) lemma "⋀v. basevars (v::bool stfun, v) ⟹ False" apply (erule baseE) apply (subgoal_tac "(LIFT (v,v)) x = (True, False)") prefer 2 apply assumption apply simp done end