\DOC PURE_LIST_CONV
\TYPE {PURE_LIST_CONV : {{Aux_thms: thm list, Fold_thms: thm list}} ->
conv}
\SYNOPSIS
Proves theorems about list constants applied to {NIL}, {CONS}, {SNOC},
{APPEND}, {FLAT} and {REVERSE}.
\KEYWORDS
conversion, list.
\DESCRIBE
{PURE_LIST_CONV} 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 theorems must be provided about {CONST1}.
{PURE_LIST_CONV}. These are passed as a record argument.
The {Fold_thms} field of the record should hold 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
}
Given this theorem, no auxiliary theorems and the term
{--`SUM (CONS x l)`--}, a call to {PURE_LIST_CONV} returns the theorem:
{
|- SUM (CONS x l) = x + (SUM l)
}
The {Aux_thms} field of the record argument to {PURE_LIST_CONV} provides
auxiliary theorems concerning the terms {f} and {e} found in the
definition
with respect to {FOLDR} or {FOLDL}. For example, given the theorem:
{
|- MONOID $+ 0
}
and given the term {--`SUM (APPEND l1 l2)`--}, a call to
{PURE_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
= tm2
|
side conditions
| tm2 in result |- tm1
==============|================================|=========================
==
[]
| 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
}
{|- MONOID f e} can be used instead of {|- FCOMM f f},
{|- LEFT_ID f} or {|- RIGHT_ID f}. {|- ASSOC f} can also be used in place
of
{|- FCOMM f f}.
\EXAMPLE
{
- val SUM_FOLDR = theorem "list" "SUM_FOLDR";
val SUM_FOLDR = |- !l. SUM l = FOLDR $+ 0 l
- PURE_LIST_CONV
=
{{Fold_thms = [SUM_FOLDR], Aux_thms = []}} (--`SUM (CONS h t)`--);
|- SUM (CONS h t) = h + SUM t
}
{
- val SUM_FOLDL = theorem "list" "SUM_FOLDL";
val SUM_FOLDL = |- !l. SUM l = FOLDL $+ 0 l
- PURE_LIST_CONV
=
{{Fold_thms = [SUM_FOLDL], Aux_thms = []}} (--`SUM (SNOC h t)`--);
|- SUM (SNOC h t) = SUM t + h
}
{
- val MONOID_ADD_0 = theorem "arithmetic" "MONOID_ADD_0";
val MONOID_ADD_0 = |- MONOID $+ 0
- PURE_LIST_CONV
=
{{Fold_thms = [SUM_FOLDR], Aux_thms = [MONOID_ADD_0]}}
=
(--`SUM (APPEND l1 l2)`--);
|- SUM (APPEND l1 l2) = SUM l1 + SUM l2
}
{
- PURE_LIST_CONV
=
{{Fold_thms = [SUM_FOLDR], Aux_thms = [MONOID_ADD_0]}} (--`SUM (FLAT
l)`--);
|- SUM (FLAT l) = SUM (MAP SUM l)
}
\FAILURE
{PURE_LIST_CONV tm} fails if {tm} is not of the form described above. It
also
fails if no suitable fold definition for {CONST1} is supplied, or if the
required auxiliary theorems as described above are not supplied.
\SEEALSO
LIST_CONV, X_LIST_CONV.
\ENDDOC