listLib.X_LIST

X_LIST_CONV is a version of LIST_CONV which can be passed auxiliary theorems about user-defined constants as an argument. It takes a term of the form:

   CONST1 ... (CONST2 ...) ...

where CONST1 and CONST2 are operators on lists and CONST2 returns a list result. It can be one of NIL, CONS, SNOC, APPEND, FLAT or REVERSE. The form of the resulting theorem depends on CONST1 and CONST2. Some auxiliary information must be provided about CONST1. X_LIST_CONV maintains a database of such auxiliary information. It initially holds information about the constants in the system. However, additional information can be supplied by the user as new constants are defined. The main information that is needed is a theorem defining the constant in terms of FOLDR or FOLDL. The definition should have the form:

   |- CONST1 ...l... = fold f e l

where fold is either FOLDR or FOLDL, f is a function, e a base element and l a list variable. For example, a suitable theorem for SUM is

   |- SUM l = FOLDR $+ 0 l

Knowing this theorem and given the term --`SUM (CONS x l)`--, X_LIST_CONV returns the theorem:

   |- SUM (CONS x l) = x + (SUM l)

Other auxiliary theorems that are needed concern the terms f and e found in the definition with respect to FOLDR or FOLDL. For example, knowing the theorem:

   |- MONOID $+ 0

and given the term --`SUM (APPEND l1 l2)`--, X_LIST_CONV returns the theorem

   |- SUM (APPEND l1 l2) = (SUM l1) + (SUM l2)

The following table shows the form of the theorem returned and the auxiliary theorems needed if CONST1 is defined in terms of FOLDR.

    CONST2       |  side conditions               | tm2 in result |- tm1 = tm2
   ==============|================================|===========================
   []            | NONE                           | e                         
   [x]           | NONE                           | f x e                     
   CONS x l      | NONE                           | f x (CONST1 l)            
   SNOC x l      | e is a list variable           | CONST1 (f x e) l          
   APPEND l1 l2  | e is a list variable           | CONST1 (CONST1 l1) l2     
   APPEND l1 l2  | |- FCOMM g f, |- LEFT_ID g e   | g (CONST1 l1) (CONST2 l2) 
   FLAT l1       | |- FCOMM g f, |- LEFT_ID g e,  |                           
                 | |- CONST3 l = FOLDR g e l      | CONST3 (MAP CONST1 l)     
   REVERSE l     | |- COMM f, |- ASSOC f          | CONST1 l                  
   REVERSE l     | f == (\x l. h (g x) l)         |                           
                 | |- COMM h, |- ASSOC h          | CONST1 l

The following table shows the form of the theorem returned and the auxiliary theorems needed if CONST1 is defined in terms of FOLDL.

    CONST2       |  side conditions               | tm2 in result |- tm1 = tm2
   ==============|================================|===========================
   []            | NONE                           | e                         
   [x]           | NONE                           | f x e                     
   SNOC x l      | NONE                           | f x (CONST1 l)            
   CONS x l      | e is a list variable           | CONST1 (f x e) l          
   APPEND l1 l2  | e is a list variable           | CONST1 (CONST1 l1) l2     
   APPEND l1 l2  | |- FCOMM f g, |- RIGHT_ID g e  | g (CONST1 l1) (CONST2 l2) 
   FLAT l1       | |- FCOMM f g, |- RIGHT_ID g e, |                           
                 | |- CONST3 l = FOLDR g e l      | CONST3 (MAP CONST1 l)     
   REVERSE l     | |- COMM f, |- ASSOC f          | CONST1 l                  
   REVERSE l     | f == (\l x. h l (g x))         |                           
                 | |- COMM h, |- ASSOC h          | CONST1 l

Auxiliary theorems are held in a user-updatable database. In particular, definitions of constants in terms of FOLDR and FOLDL, and monoid, commutativity, associativity, left identity, right identity and binary function commutativity theorems are stored. The database can be updated by the user to allow LIST_CONV to prove theorems about new constants. This is done by calling set_list_thm_database. The database can be inspected by calling list_thm_database. The database initially holds FOLDR/L theorems for the following system constants: APPEND, FLAT, LENGTH, NULL, REVERSE, MAP, FILTER, ALL_EL, SUM, SOME_EL, IS_EL, AND_EL, OR_EL, PREFIX, SUFFIX, SNOC and FLAT combined with REVERSE. It also holds auxiliary theorems about their step functions and base elements. Rather than updating the database, additional theorems can be passed to X_LIST_CONV as an argument. It takes a record with one field, Fold_thms, for fold definitions, and one, Aux_thms, for theorems about step functions and base elements.