# Cours de mathematiques speciales: algebre by Ramis E., Deschamps C., Odoux J.

By Ramis E., Deschamps C., Odoux J.

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3 Named and Generic Speciﬁcations 41 A generic speciﬁcation deﬁnition SPEC-DEFN with some parameters and some imports is written: spec SN [SP1 ] . . [SPn ] given SP1 , . . , SPm = SP end When the list of imports SP1 , . . , SPm is empty, the deﬁnition is written: spec SN [SP1 ] . . [SPn ] = SP end When the list of parameters SP1 , . . , SPn is empty, the deﬁnition merely names a speciﬁcation and is simply written: spec SN = SP end The terminating ‘end’ keyword is optional. It deﬁnes the name SN to refer to the speciﬁcation that has parameter speciﬁcations SP1 , .

This extends the global environment (which must not already include a deﬁnition for SN ). The well-formedness and semantics of a generic speciﬁcation are essentially as for the imports, extended by the union of the parameter speciﬁcations, extended by the body: { SP1 and . . and SPm } then { SP1 and . . , the above speciﬁcation is implicitly closed. The diﬀerence between declaring parameters and leaving them implicit in an extension is that each parameter has to be provided with a ﬁtting argument speciﬁcation in all references to the speciﬁcation name SN .

Globally or locally declared variables are implicitly universally quantiﬁed in subsequent axioms of the enclosing basic speciﬁcation. Variables are not included in the declared signature. Universal quantiﬁcation over a variable that does not occur free in an axiom is semantically irrelevant, due to the assumption that all carrier sets are non-empty. 1 Global Variable Declarations VAR-ITEMS ::= var-items VAR-DECL+ A list VAR-ITEMS of variable declarations is written: vars VD1 ; . . VDn ; 18 I:2 Basic Speciﬁcations Note that local variable declarations are written in a similar way, but followed directly by a bullet ‘ • ’ instead of the optional semicolon.