By Fred Kröger
Creation to the temporal common sense of - specifically paral- lel - programs.Divided into 3 major elements: - Presenta- tion of the natural temporal common sense: language, semantics, and evidence idea; - illustration of courses and their right- ties in the language of temporal good judgment; - software of the logical gear to the verification of software right- ties together with a brand new embedding of Hoare's good judgment into the temporal framework.
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Proof a) We show the following assertion: if CEU(1F) then f-# -+ oc. The claim a) then follows immediately by (prop) and (T 19'), formally derived at the end of the previous section. Let CEU(1F). We distinguish the six cases according to the defmition of u: =OCE1F =f-# -+oC by (prop). C=-,AEU2(1F) =-,OAE1F = f-# -+ -, oA by (prop) with (ax 1). =f-# -+ 0-' A C=OAEU3(1F) =OAE1F = f-# -+ oA by (prop) with (ax 3). = f-# -+ ooA C=-,OAEU4 (1F) =-,OAE1F and AE1F =f-# -+-,oA /\ A by (prop) =f-# -+-,ooA with (ax3') =f-# -+ 0-, oA with (ax 1).
0, ~, ... Ali, and so on. If So, ... , A'; is finite then we take the path from §" to A'; in the same way and iterate the whole construction by choosing an arbitrary path in T(A';). Let §"O, §"l' ... i. Then AE~ and by induction (as above) we get -'OAE~ for every j:2:. i. Therefore So, %t. i, and 'OAEA';. 6a) there are only finitely many formulas of this kind in T(§"), hence IDA will be used in the above construction of some A';+l' k~l. So we get -,AEA';+l and this is a contradiction. Therefore, §"O, §"l' ...
However, we can show that 'v'xA ~ Ax(t) is valid in the new context of Sf TP if the substitution of t for x does not create new occurrences of local variables in the scope of a temporal operator in A. for every K' with => K;(A) =t ~' (y) = ~ (y) lSI for K' with ~'(x)=S(~·~·)(t) and lSI => K;(AAt» =K;(A) =>K;(AAt»=t. for y=\=x forthis K' for every y =\= x ~'(y)=~(y) lSI 0 A term t with the property that its substitution for x in A does not create new occurrences of local variables in the scope of a temporal operator is called substitutable for x in A.