Verifikation reaktiver Systeme - Universität Kaiserslautern

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- g‘A /\ B‘;
> val it =
Proof manager status: 1 proof.
1. Incomplete:
Initial goal:
A /\ B
: proofs
- e(CONJ_TAC);
OK..
2 subgoals:
> val it =
B
A
: goalstack
Library The HOL system comes with a rich collection of theories containing
predefined types, functions and tactics and already proven theorems. Figure
1 shows the various theories along with their dependencies. All of them are
derived from the initial theory containing the five axioms and the eight primitive
inference rules stated above.
To use a library, it must be loaded (load ẍyzTheory¨). For an easier access,
its internal declarations can be made public by opening (open xyzTheory). New
theories are created by new theory and contain all theorems that are stored with
the store thm (oder some other storing) function.
Defining new types Whereas defining new functions and carrying out proofs is
quite intuitive, the definition of new types needs more explanation. New types
are always derived from existing ones, i.e. to build a new type, existing type
constants and operators are used to specify a representation type whose range is
restricted possibly restricted to form the set of elements the new type contains.
Figure 2 illustrates the construction: From the elements of a representation
set R, a subset is chosen by a predicate P . The abstractions of those elements
form the set of the new type A. Moreover, for each a ∈ A thereisarepresenting
element r ∈ R.
new_type_definition (tyop, |- ?x. t x)
The ML function new type definition implements the primitive HOL rule
of definition for introducing new type constants into the logic, tyop is the name
of the new type, x is an element of the representation type and t is a predicate
with domain of this type and range of booleans. The new_type_defintion function
must be given a proof which shows that the type contains an element.
Let’s consider the example of defining a type containing three elements (taken
from [3]). This type can be derived from the type bool # bool (the product of
two booleans). In order to get a type containing three elements (and not the