Miscellaneous Logical helper functions
aldor
#include "axiom"
#pile
define Domain:Category == with;
+++
+++ A set is often considered to be a collection with "no duplicate elements."
+++ Here we have a slightly different definition which is important to
+++ understand. We define a Set to be an arbitrary collection together with
+++ an equivalence relation "=". Soon this will be made into a mathematical
+++ category where the morphisms are "functions", by which we mean maps
+++ having the special property that a=a' implies f a = f a'. This definition
+++ is more convenient both mathematically and computationally, but you need
+++ to keep in mind that a set may have duplicate elements.
+++
define Set:Category == Join(Domain, Printable) with {
=:(%,%) -> Boolean;
}
+++
+++ A Preorder is a collection with reflexive and transitive <=, but without
+++ necessarily being symmetric (x<=y and y<=x) implying x=y. Since
+++ (x<=y and y<=x) is always an equivalence relation, our definition of
+++ "Set" is always satisfied in any case.
+++
define Preorder:Category == Set with {
<=: (%,%) -> Boolean;
>=: (%,%) -> Boolean;
< : (%,%) -> Boolean;
> : (%,%) -> Boolean;
default {
(x:%) =(y:%):Boolean == (x<=y) and (y<=x);
(x:%)>=(y:%):Boolean == y<=x;
(x:%)< (y:%):Boolean == (x<=y) and ~(x=y);
(x:%)> (y:%):Boolean == (x>=y) and ~(x=y)
}
}
define TotalOrder:Category == Preorder with {
min: (%,%) -> %;
max: (%,%) -> %;
min: Tuple % -> %;
max: Tuple % -> %;
default {
min(x:%,y:%):% == { x<=y => x; y };
max(x:%,y:%):% == { x<=y => y; x };
import from List %;
min(t:Tuple %):% == associativeProduct(%,min,[t]);
max(t:Tuple %):% == associativeProduct(%,max,[t]);
}
}
+++
+++ Countable is the category of collections for which every element in the
+++ collection can be produced. This is done by the generator "elements"
+++ below. Note that there is no guarantee that elements will not produce
+++ "duplicates." In fact, a Countable may not be a Set, so duplicates may
+++ have no meaning. Also, Countable is not guaranteed to terminate.
+++
define Countable:Category == with {
elements: () -> Generator %
}
--
-- I'm using an empty function elements() rather than a constant elements
-- to avoid some compiler problems.
--
+++
+++ CountablyFinite is the same as Countable except that termination is
+++ guaranteed.
+++
define CountablyFinite:Category == Countable with
+++
+++ A "Monoids" is the usual Monoid (we don't use Monoid to avoid clashing
+++ with axllib): a Set with an associative product (associative relative to
+++ the equivalence relation of the Set, of course) and a unit.
+++
define Monoids:Category == Set with {
*: (%,%) -> %;
1: %;
^:(%,Integer) -> %;
monoidProduct: Tuple % -> %; -- associative product
monoidProduct: List % -> %;
default {
(x:%)^(i:Integer):% == {
i=0 => 1;
i<0 => error "Monoid negative powers are not defined.";
associativeProduct(%,*,x for j:Integer in 1..i)
};
monoidProduct(t:Tuple %):% == { import from List %; monoidProduct(t) }
monoidProduct(l:List %):% == {
import from NonNegativeInteger;
#l = 0 => 1;
associativeProduct(%,*,l);
}
}
}
+++
+++ Groups are Groups in the usual mathematical sense. We use "Groups"
+++ rather than "Group" to avoid clashing with axllib.
+++
define Groups:Category == Monoids with {
inv: % -> %
}
+++
+++ Printing is a whole area that I'm going to have a nice categorical
+++ solution for, but still it is convenient to have a low level Printable
+++ signature for debugging purposes.
+++
define Printable:Category == with {
coerce: % -> OutputForm;
coerce: List % -> OutputForm;
coerce: Generator % -> OutputForm;
default {
(t:OutputForm)**(l:List %):OutputForm == {
import from Integer;
empty? l => t;
hconcat(coerce first l, hspace(1)$OutputForm) ** rest l;
};
coerce(l:List %):OutputForm == empty() ** l;
coerce(g:Generator %):OutputForm == {
import from List %;
empty() ** [x for x in g];
}
}
}
+++
+++ This evaluates associative products.
+++
associativeProduct(T:Type,p:(T,T)->T,g:Generator T):T == {
l:List T == [t for t in g];
associativeProduct(T,p,l);
}
associativeProduct(T:Type,p:(T,T)->T,l:List T):T == {
if empty? l then error "Empty product.";
mb(t:T,l:List T):T == { empty? l => t; mb( p(t,first l), rest l) };
mb(first l,rest l)
}
+++
+++ Evaluates the logical "For all ..." construction
+++
forall?(g:Generator Boolean):Boolean == {
q:Boolean := true;
for x:Boolean in g repeat { if ~x then { q := false; break } }
q
}
+++
+++ Evaluates the logical "There exists ..." construction
+++
exists?(g:Generator Boolean):Boolean == {
q:Boolean := false;
for x:Boolean in g repeat { if x then { q := true; break } };
q
}
+++
+++ The category of "Maps". There is no implication that a map is a
+++ function in the sense of x=x' => f x = f x'
+++
define MapCategory(Obj:Category,A:Obj,B:Obj):Category == with {
apply: (%,A) -> B;
hom: (A->B) -> %;
}
+++
+++ One convenient implementation of MapCategory
+++
Map(Obj:Category,A:Obj,B:Obj):MapCategory(Obj,A,B) == add {
Rep ==> A->B;
apply(f:%,a:A):B == (rep f) a;
hom (f:A->B):% == per f
}
+++
+++ This strange function turns any Type into an Aldor Category
+++
define Categorify(T:Type):Category == with {
value: T
}
+++
+++ The null function
+++
null(A:Type,B:Type):(A->B) == (a:A):B +-> error "Attempt to evaluate the null function."
+++
+++ A handy package for composition of morphisms. "o" is meant to suggest morphism composition g "o" f, to be coded "g ** f".
+++
o(Obj:Category,A:Obj,B:Obj,C:Obj): with
**: (B->C,A->B) -> (A->C)
== add
(g:B->C)**(f:A->B):(A->C) == (a:A):C +-> g f a
aldor
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/basics.as using AXIOM-XL compiler and
options
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra
Use the system command )set compiler args to change these
options.
#1 (Warning) Deprecated message prefix: use `ALDOR_' instead of `_AXL'
Compiling Lisp source code from file ./basics.lsp
Issuing )library command for basics
Reading /var/zope2/var/LatexWiki/basics.asy
null is now explicitly exposed in frame initial
null will be automatically loaded when needed from
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Categorify is now explicitly exposed in frame initial
Categorify will be automatically loaded when needed from
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MapCategory is now explicitly exposed in frame initial
MapCategory will be automatically loaded when needed from
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Groups is now explicitly exposed in frame initial
Groups will be automatically loaded when needed from
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Monoids is now explicitly exposed in frame initial
Monoids will be automatically loaded when needed from
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Printable is now explicitly exposed in frame initial
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associativeProduct is now explicitly exposed in frame initial
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CountablyFinite is now explicitly exposed in frame initial
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Countable is now explicitly exposed in frame initial
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Domain is now explicitly exposed in frame initial
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TotalOrder is now explicitly exposed in frame initial
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Preorder is now explicitly exposed in frame initial
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Set is now explicitly exposed in frame initial
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Map is now explicitly exposed in frame initial
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o is now explicitly exposed in frame initial
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/var/zope2/var/LatexWiki/basicsTest Funtions
aldor
#include "axiom"
#pile
#library lbasics "basics.ao"
import from lbasics
T:Domain == add
S:Set == Integer add
P:Preorder == Integer add
TO:TotalOrder == Integer add
--CC:Countable == Integer add
-- import from Integer, Generator(Integer)
-- elements():Generator(Integer) == generator (1..10)
M:Monoids == Integer add
G:Groups == Fraction Integer add
MAPS:MapCategory(SetCategory,Integer,Float) == Map(SetCategory,Integer,Float) add
INTS:Categorify(Integer) == add
value:Integer == 1
Null(n:PositiveInteger):Float==null(PositiveInteger,Float)(n)
sincos(x:Expression Integer):Expression Integer ==
import from o(SetCategory,Expression Integer,Expression Integer,Expression Integer)
( ((a:Expression Integer):Expression Integer +-> sin(a)) **
((b:Expression Integer):Expression Integer +-> cos(b)) ) (x)
aldor
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/7819217643380372071-25px002.as using
AXIOM-XL compiler and options
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra
Use the system command )set compiler args to change these
options.
#1 (Warning) Deprecated message prefix: use `ALDOR_' instead of `_AXL'
"/var/zope2/var/LatexWiki/7819217643380372071-25px002.as", line 22:
((b:Expression Integer):Expression Integer +-> cos(b)) ) (x)
.............................................................^
[L22 C62] #2 (Warning) Suspicious juxtaposition. Check for missing `;'.
Check indentation if you are using `#pile'.
Compiling Lisp source code from file
./7819217643380372071-25px002.lsp
Issuing )library command for 7819217643380372071-25px002
Reading /var/zope2/var/LatexWiki/7819217643380372071-25px002.asy
TO is now explicitly exposed in frame initial
TO will be automatically loaded when needed from
/var/zope2/var/LatexWiki/7819217643380372071-25px002
M is now explicitly exposed in frame initial
M will be automatically loaded when needed from
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P is now explicitly exposed in frame initial
P will be automatically loaded when needed from
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S is now explicitly exposed in frame initial
S will be automatically loaded when needed from
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G is now explicitly exposed in frame initial
G will be automatically loaded when needed from
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INTS is now explicitly exposed in frame initial
INTS will be automatically loaded when needed from
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MAPS is now explicitly exposed in frame initial
MAPS will be automatically loaded when needed from
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T is now explicitly exposed in frame initial
T will be automatically loaded when needed from
/var/zope2/var/LatexWiki/7819217643380372071-25px002axiom
)show T
T is a domain constructor
Abbreviation for T is T
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for T
------------------------------- Operations --------------------------------
Warning: T is being redefined. T is a domain constructor.
Abbreviation for T is T
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for T
------------------------------- Operations --------------------------------
axiom
)show S
S is a domain constructor
Abbreviation for S is S
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for S
------------------------------- Operations --------------------------------
?=? : (%,%) -> Boolean coerce : % -> OutputForm
coerce : List % -> OutputForm
?**? : (OutputForm,List %) -> OutputForm
coerce : Generator % -> OutputForm
axiom
)show P
P is a domain constructor
Abbreviation for P is P
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for P
------------------------------- Operations --------------------------------
?<? : (%,%) -> Boolean ?<=? : (%,%) -> Boolean
?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean
?>=? : (%,%) -> Boolean coerce : % -> OutputForm
coerce : List % -> OutputForm
?**? : (OutputForm,List %) -> OutputForm
coerce : Generator % -> OutputForm
axiom
)show TO
TO is a domain constructor
Abbreviation for TO is TO
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for TO
------------------------------- Operations --------------------------------
?<? : (%,%) -> Boolean ?<=? : (%,%) -> Boolean
?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean
?>=? : (%,%) -> Boolean coerce : % -> OutputForm
coerce : List % -> OutputForm max : (%,%) -> %
max : Tuple % -> % min : (%,%) -> %
min : Tuple % -> %
?**? : (OutputForm,List %) -> OutputForm
coerce : Generator % -> OutputForm
--)show CC
axiom
)show MAPS
MAPS is a domain constructor
Abbreviation for MAPS is MAPS
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for MAPS
------------------------------- Operations --------------------------------
apply : (%,Integer) -> Float hom : (Integer -> Float) -> %
axiom
)show INTS
INTS is a domain constructor
Abbreviation for INTS is INTS
This constructor is exposed in this frame.
Issue )edit 7819217643380372071-25px002.as to see algebra source code for INTS
------------------------------- Operations --------------------------------
value : () -> Integer
axiom
sincos(x::Expression Integer)
>> System error:
|newGoGet| [or a callee] requires less than one argument.
axiom
Null(1)
>> System error:
Attempt to evaluate the null function.
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