define_new_type_bijections

Drule.define_new_type_bijections :
  {name:string, ABS:string, REP:string, tyax:thm} -> thm

Introduces abstraction and representation functions for a defined type.

The result of making a type definition using new_type_definition is a theorem of the following form:

   |- ?rep:nty->ty. TYPE_DEFINITION P rep

which asserts only the existence of a bijection from the type it defines (in this case, nty) to the corresponding subset of an existing type (here, ty) whose characteristic function is specified by P. To automatically introduce constants that in fact denote this bijection and its inverse, the ML function define_new_type_bijections is provided.

name is the name under which the constant definition (a constant specification, in fact) made by define_new_type_bijections will be stored in the current theory segment. tyax must be a definitional axiom of the form returned by new_type_definition. ABS and REP are the user-specified names for the two constants that are to be defined. These constants are defined so as to denote mutually inverse bijections between the defined type, whose definition is given by tyax, and the representing type of this defined type.

If th is a theorem of the form returned by new_type_definition:

   |- ?rep:newty->ty. TYPE_DEFINITION P rep

then evaluating:

   define_new_type_bijections{name="name",ABS="abs",REP="rep",tyax=th} th

automatically defines two new constants abs:ty->newty and rep:newty->ty such that:

   |- (!a. abs(rep a) = a) /\ (!r. P r = (rep(abs r) = r))

This theorem, which is the defining property for the constants abs and rep, is stored under the name name in the current theory segment. It is also the value returned by define_new_type_bijections. The theorem states that abs is the left inverse of rep and, for values satisfying P, that rep is the left inverse of abs.

Failure

A call define_new_type_bijections{name,ABS,REP,tyax} fails if tyax is not a theorem of the form returned by new_type_definition.

See also

Definition.new_type_definition, Drule.prove_abs_fn_one_one, Drule.prove_abs_fn_onto, Drule.prove_rep_fn_one_one, Drule.prove_rep_fn_onto