Constants, operators and ``type labels'' (retractions with strongly cantorian range) can be defined in Watson. Examples of the definition commands have appeared in example sections above, where this seemed appropriate.
The definition of an atomic constant as a complex term is straightforward:
- defineconstant "four" "2+2"; four: four = 2 + 2 four , 0
The prover checks for defects such as circularity in the proposed definition and, if no error is found, proves a new theorem embodying the definition. When a constant is defined, the name of the new theorem is the same as the name of the defined constant (which cannot have been declared previously).
The issue of stratification arises in the definitions of functions and operations.
- defineconstant "Double@?x" "?x+?x"; Double: Double @ ?x = ?x + ?x Double , 0 - defineconstant "(Comp@?f,?g)@?x" "?f@?g@?x"; Comp: (Comp @ ?f , ?g) @ ?x = ?f @ ?g @ ?x Comp , 0 - defineconstant "Comp2@?f,?g,?x" "?f@?g@?x"; Watson: Format, declaration or stratification failure of proposed definition of Comp2@?f,?g,?x
While reading the two definitions of composition of functions, recall that all operations have the same precedence and group to the right. In the first definition of composition of functions, note that defined functions can have lists of arguments and can be ``curried'' if the types of their arguments warrant this. In the second definition of composition, we see a failure of stratification.
Operators can be defined similarly.
(* defining operators with ``flat'' type *) - defineinfix "OR" "?x|?y" "~(~?x)& ~?y"; OR: ?x | ?y = ~ (~ ?x) & ~ ?y OR , 0 (* defining typed operators *) - showdec "@"; (* show the declaration of the function application operator *) Watson: Reserved operator @ left type: 1 right type: 0 - definetypedinfix "CONV_APP" 0 1 "?x <@ ?y" "?y @ ?x"; (* the integer parameters are the left and right types of the defined operator <@ (the converse of application, which has left type 1 and right type 0, must have left type 0 and right type 1) *) CONV_APP: ?x <@ ?y = ?y @ ?x CONV_APP , 0 (* definition of opaque operators *) - defineopaque "WEIRD" "?x +`? ?y" "(?x@?x)+?y@?y"; WEIRD: ?x +`? ?y = (?x @ ?x) + ?y @ ?y WEIRD , 0
The name of the theorem defining an operator must be supplied as a parameter to the defineinfix or definetypedinfix commands, as we see in the examples. The defineinfix command is used to define ``flat'' infixes (those with left and right type of 0); the definetypedinfix command is used to define infixes with nontrivial left and right type, and takes the types of its arguments as parameters. Completely unstratified operations such as +`? in the example above can also be defined as ``opaque'' operators; the use of opaque operators in definitions and in contexts with bound variables is extremely restricted, but they do have some applications.
The most complex form of definition in Watson is the facility of defining ``type labels'' (retractions onto strongly cantorian ranges). The usual restrictions of non-circularity and stratification in definitions of functions and operators are first modified (the stratification requirements are sharper) then extended with the requirement that the operation being defined be shown to be a retraction. We illustrate the definition of a type by defining the type of booleans (using a new identifier Bool to avoid conflict with the type label bool declared above).
- start "?x||true,false"; - assign "?x" "?x||true,false"; (* sets up theorem that this operation is a retraction *) - ri "(!@CASEINTRO)@?x"; ex(); - top(); downtoleft "?x||?y,?z"; ri "1|-|1"; ex(); - top(); downtoright "?x||?y,?z"; ri "1|-|1"; ex(); - top(); ex(); - p "BOOLRETRACT"; BOOLRETRACT: (?x || true , false) || true , false = ?x || true , false CASEINTRO , 0 - defineconstanttype "BOOLRETRACT" "Bool:?x" "?x||true,false"; Bool: Bool : ?x = ?x || true , false Bool , CASEINTRO , 0
The thing to notice is that the defineconstanttype command takes as an additional parameter a theorem witnessing the fact that the operation being defined is a retraction. The reason that Watson can tell that this retraction has strongly cantorian domain has to do with technical aspects of the way Watson determines whether case expressions are stratified (it implicitly views the hypothesis of a case expression as being restricted to a strongly cantorian domain).
It is possible to define functions and operations which generate type labels (type constructors). There are subtleties with the use of type constructors which make it necessary to declare constructors as ``opaque'' operators.