# Theory Stfun

```(*  Title:      HOL/TLA/Stfun.thy
Author:     Stephan Merz
*)

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.
*)
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 (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 (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
```