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TERMINATION. R e c a l l the n o n - t e r m i n a t i o n p r o o f in l. 3 t h a t took the form of s h o w i n g that if t e r m i n a t i o n d i d o c c u r then a false assertion w o u l d b e realised. g. -~t~/ze). 28 unavailable in Hoare's notationt expresses "~ terminates". Thus - 7 ~ -~ -I [ w h i l e ~ do a ] f a l s e formalises the p r i n c i p l e that a W ~ £ - c o m m a n d h a l t s i f its test condition is false. Indeed in this situation execution of the ~ ) ~ - c o m m a n d requires no action, so a statement w i l l be true after e x e c u t i o n just in case i t is true before.

P . 1 3 3 of Enderton (1972)). However we get over this difficulty by turning the greater com- plexity of our language to full advantage: the assertion r used in places where ~g has been traditionally employed. [r := g]~ will be explicitly Thus our system constitutes something of a departure even in its treatment of the standard quantifiers. EQUIVALENCE OF PROGRAMS. As an instance of r C a ++ [r := a]~ we have reflecting the fact that the command (r := r) changes nothing, and so is "equivalent" to S ~ p .

Regard A as a o n e - p l a c e d r e l a t i o n on A, and e x t e n d to a function dA : A + L I ~+ having if a E A . dA represents the subset of "defined" e l e m e n t s of A + (i ~e. the members o f A) . Since in general dA(v s(s)) = v sCD~) we may read DE as "e is defined". Indeed ~sDe iff v 8(e) ~ 09. O c c a s s i o n a l l y we informally say "£ is u n d e f i n e d in 8" to mean t h a t 98 (e) = 0~. A l t h o u g h ~a, as a B o o l e a n expression, is p a r t of the s y n t a x of i n t e r n a l logic, its i n t e r p r e t a t i o n as "s is defined" is an e x t e r n a l matter.

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