fricas
(1) -> <spad>
--)lisp (defvar |$inclAssertions| nil)
-- in principle d:V[k]->V[k+1], then what is d d = 0?
-- each vector space V[k] has its own zero! But we will
-- only use one zero form for all. ZeroChain()?
fricas
)abbrev category CCHNCAT CoChainCategory
CoChainCategory() : Category == SetCategory with
degree : % -> Polynomial Integer
mainType : % -> String -- trick: get the type
fricas
)abbrev category ZCHNCAT ZeroChainCategory
ZeroChainCategory() : Category == SetCategory with
null : %
fricas
)abbrev domain ZCHN ZeroChain
ZeroChain : Exports == Implementation where
BOP ==> BasicOperator
PINT ==> Polynomial Integer
Exports == Join(CoChainCategory,ZeroChainCategory) with
degree : % -> PINT
d : % -> %
--null : %
Implementation == BOP add
Rep := BOP
null == operator("0"::Symbol)
degree x == 0::PINT
d x == null()
mainType x == "ZeroChain"
fricas
)abbrev domain COCHN CoChain
CoChain(p) : Exports == Implementation where
p:Polynomial Integer
BOP ==> BasicOperator
CTOF ==> CoercibleTo OutputForm
CVTS ==> ConvertibleTo String
PINT ==> Polynomial Integer
Exports == Join(CTOF, CVTS, CoChainCategory) with
coerce : Symbol -> %
degree : % -> PINT
Implementation == BOP add
Rep := BOP
coerce(s:Symbol):% == operator s
degree x == p
mainType x == "CoChain"
fricas
)abbrev domain WEDGE WedgeProduct
WedgeProduct(a,b) : Exports == Implementation where
CCCAT ==> CoChainCategory
OF ==> OutputForm
a:CCCAT
b:CCCAT
CTOF ==> CoercibleTo OutputForm
CVTS ==> ConvertibleTo String
PINT ==> Polynomial Integer
Exports == Join(CTOF, CVTS, CoChainCategory) with
if a has ZeroChainCategory or b has ZeroChainCategory then
_* : (a,b) -> ZeroChain
else
_* : (a,b) -> %
degree : % -> PINT
Implementation == add
if a has ZeroChainCategory or b has ZeroChainCategory then
Rep := ZeroChain
(x:a * y:b):ZeroChain == null()$ZeroChain
else
Rep := Record(left:a,right:b)
x * y == [x,y]@Rep
degree x == degree(x.left)$a + degree(x.right)$b
mainType x == "WedgeProduct"
coerce(x:%):OutputForm == tensor(x.left::OF,x.right::OF)
------ temp using tensor, lack of wedge symbol
fricas
)abbrev domain EXTDF ExteriorDerivative
ExteriorDerivative(a) : Exports == Implementation where
CCCAT ==> CoChainCategory
OF ==> OutputForm
a:CCCAT
CTOF ==> CoercibleTo OutputForm
CVTS ==> ConvertibleTo String
PINT ==> Polynomial Integer
Exports == Join(CTOF, CVTS, CoChainCategory) with
d : % -> ZeroChain
d : a -> %
degree : % -> PINT
Implementation == add
if a has ZeroChainCategory then
Rep := ZeroChain
else
Rep := Record(arg:a)
d(x:%):ZeroChain == null()$ZeroChain
d(x:a):% == [x]@Rep
degree x == 1 + degree(x.arg)$a
mainType x == "ExteriorDerivative"
coerce(x:%):OutputForm ==
dsym:=outputForm("d"::Symbol)$OF
mainType(x.arg)$a = "CoChain" => hconcat(dsym,x.arg::OF)
hconcat(dsym,paren(x.arg::OF))
fricas
)abbrev domain HODGE HodgeStar
HodgeStar(a) : Exports == Implementation where
CCCAT ==> CoChainCategory
OF ==> OutputForm
a:CCCAT
CTOF ==> CoercibleTo OutputForm
CVTS ==> ConvertibleTo String
PINT ==> Polynomial Integer
Exports == Join(CTOF, CVTS, CoChainCategory) with
hodge : a -> %
degree : % -> PINT
Implementation == add
Rep := Record(arg:a)
n:PINT:="spacedim"::Symbol::PINT
degree x == n - degree(x.arg)$a
hodge x == [x]@Rep
mainType x == "HodgeStar"
coerce(x:%):OutputForm ==
dsym:=outputForm("*"::Symbol)$OF
mT:String:=mainType(x.arg)$a
mT ~= "WedgeProduct" => hconcat(dsym,x.arg::OF)
hconcat(dsym,paren(x.arg::OF))</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5250937285633644280-25px001.spad
using old system compiler.
CCHNCAT abbreviates category CoChainCategory
------------------------------------------------------------------------
initializing NRLIB CCHNCAT for CoChainCategory
compiling into NRLIB CCHNCAT
;;; *** |CoChainCategory| REDEFINED
Time: 0 SEC.
finalizing NRLIB CCHNCAT
Processing CoChainCategory for Browser database:
--->-->CoChainCategory(constructor): Not documented!!!!
--->-->CoChainCategory((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->CoChainCategory((mainType ((String) %))): Not documented!!!!
--->-->CoChainCategory(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CCHNCAT.NRLIB/CCHNCAT.lsp" (written 03 SEP 2022 03:11:28 PM):
; /var/aw/var/LatexWiki/CCHNCAT.NRLIB/CCHNCAT.fasl written
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
CoChainCategory is now explicitly exposed in frame initial
CoChainCategory will be automatically loaded when needed from
/var/aw/var/LatexWiki/CCHNCAT.NRLIB/CCHNCAT
ZCHNCAT abbreviates category ZeroChainCategory
------------------------------------------------------------------------
initializing NRLIB ZCHNCAT for ZeroChainCategory
compiling into NRLIB ZCHNCAT
;;; *** |ZeroChainCategory| REDEFINED
Time: 0 SEC.
finalizing NRLIB ZCHNCAT
Processing ZeroChainCategory for Browser database:
--->-->ZeroChainCategory(constructor): Not documented!!!!
--->-->ZeroChainCategory((null (%) constant)): Not documented!!!!
--->-->ZeroChainCategory(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ZCHNCAT.NRLIB/ZCHNCAT.lsp" (written 03 SEP 2022 03:11:28 PM):
; /var/aw/var/LatexWiki/ZCHNCAT.NRLIB/ZCHNCAT.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
ZeroChainCategory is now explicitly exposed in frame initial
ZeroChainCategory will be automatically loaded when needed from
/var/aw/var/LatexWiki/ZCHNCAT.NRLIB/ZCHNCAT
ZCHN abbreviates domain ZeroChain
------------------------------------------------------------------------
initializing NRLIB ZCHN for ZeroChain
compiling into NRLIB ZCHN
compiling exported null : () -> $
Time: 0 SEC.
compiling exported degree : $ -> Polynomial Integer
Time: 0.04 SEC.
compiling exported d : $ -> $
Time: 0.01 SEC.
compiling exported mainType : $ -> String
ZCHN;mainType;$S;4 is replaced by ZeroChain
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |ZeroChain| REDEFINED
;;; *** |ZeroChain| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor ZeroChain
Time: 0.05 seconds
--------------non extending category----------------------
.. ZeroChain of cat
(|Join| (|CoChainCategory|) (|ZeroChainCategory|)
(CATEGORY |domain| (SIGNATURE |degree| ((|Polynomial| (|Integer|)) $))
(SIGNATURE |d| ($ $)))) has no
(|OrderedSet|) finalizing NRLIB ZCHN
Processing ZeroChain for Browser database:
--->-->ZeroChain(constructor): Not documented!!!!
--->-->ZeroChain((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->ZeroChain((d (% %))): Not documented!!!!
--->-->ZeroChain(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ZCHN.NRLIB/ZCHN.lsp" (written 03 SEP 2022 03:11:28 PM):
; /var/aw/var/LatexWiki/ZCHN.NRLIB/ZCHN.fasl written
; compilation finished in 0:00:00.014
------------------------------------------------------------------------
ZeroChain is now explicitly exposed in frame initial
ZeroChain will be automatically loaded when needed from
/var/aw/var/LatexWiki/ZCHN.NRLIB/ZCHN
COCHN abbreviates domain CoChain
------------------------------------------------------------------------
initializing NRLIB COCHN for CoChain
compiling into NRLIB COCHN
compiling exported coerce : Symbol -> $
Time: 0 SEC.
compiling exported degree : $ -> Polynomial Integer
Time: 0.01 SEC.
compiling exported mainType : $ -> String
COCHN;mainType;$S;3 is replaced by CoChain
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |CoChain| REDEFINED
;;; *** |CoChain| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor CoChain
Time: 0.01 seconds
--------------non extending category----------------------
.. CoChain(#1) of cat
(|Join| (|CoercibleTo| (|OutputForm|)) (|ConvertibleTo| (|String|))
(|CoChainCategory|)
(CATEGORY |domain| (SIGNATURE |coerce| ($ (|Symbol|)))
(SIGNATURE |degree| ((|Polynomial| (|Integer|)) $)))) has no
(|OrderedSet|) finalizing NRLIB COCHN
Processing CoChain for Browser database:
--->-->CoChain(constructor): Not documented!!!!
--->-->CoChain((coerce (% (Symbol)))): Not documented!!!!
--->-->CoChain((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->CoChain(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COCHN.NRLIB/COCHN.lsp" (written 03 SEP 2022 03:11:28 PM):
; /var/aw/var/LatexWiki/COCHN.NRLIB/COCHN.fasl written
; compilation finished in 0:00:00.011
------------------------------------------------------------------------
CoChain is now explicitly exposed in frame initial
CoChain will be automatically loaded when needed from
/var/aw/var/LatexWiki/COCHN.NRLIB/COCHN
WEDGE abbreviates domain WedgeProduct
------------------------------------------------------------------------
initializing NRLIB WEDGE for WedgeProduct
compiling into NRLIB WEDGE
****** Domain: a already in scope
augmenting a: (ZeroChainCategory)
Local variable Rep type redefined: (Join (CoChainCategory) (ZeroChainCategory) (CATEGORY domain (SIGNATURE degree ((Polynomial (Integer)) $)) (SIGNATURE d ($ $)))) to (Join (OrderedSet) (CATEGORY domain (SIGNATURE name ((Symbol) $)) (SIGNATURE properties ((AssociationList (Symbol) (None)) $)) (SIGNATURE copy ($ $)) (SIGNATURE operator ($ (Symbol))) (SIGNATURE operator ($ (Symbol) (NonNegativeInteger))) (SIGNATURE arity ((Union (NonNegativeInteger) failed) $)) (SIGNATURE nullary? ((Boolean) $)) (SIGNATURE unary? ((Boolean) $)) (SIGNATURE nary? ((Boolean) $)) (SIGNATURE weight ((NonNegativeInteger) $)) (SIGNATURE weight ($ $ (NonNegativeInteger))) (SIGNATURE equality ($ $ (Mapping (Boolean) $ $))) (SIGNATURE comparison ($ $ (Mapping (Boolean) $ $))) (SIGNATURE display ((Union (Mapping (OutputForm) (List (OutputForm))) failed) $)) (SIGNATURE display ($ $ (Mapping (OutputForm) (List (OutputForm))))) (SIGNATURE display ($ $ (Mapping (OutputForm) (OutputForm)))) (SIGNATURE input ($ $ (Mapping (InputForm) (List (InputForm))))) (SIGNATURE input ((Union (Mapping (InputForm) (List (InputForm))) failed) $)) (SIGNATURE is? ((Boolean) $ (Symbol))) (SIGNATURE has? ((Boolean) $ (Symbol))) (SIGNATURE assert ($ $ (Symbol))) (SIGNATURE deleteProperty! ($ $ (Symbol))) (SIGNATURE property ((Union (None) failed) $ (Symbol))) (SIGNATURE setProperty ($ $ (Symbol) (None))) (SIGNATURE setProperties ($ $ (AssociationList (Symbol) (None))))))
compiling exported * : (a,b) -> ZeroChain
Time: 0 SEC.
****** Domain: b already in scope
augmenting b: (ZeroChainCategory)
compiling exported * : (a,b) -> ZeroChain
Time: 0.01 SEC.
Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: left a) (: right b)) a b)) (SIGNATURE ~= ((Boolean) (Record (: left a) (: right b)) (Record (: left a) (: right b)))) (SIGNATURE coerce ((OutputForm) (Record (: left a) (: right b)))) (SIGNATURE elt (a (Record (: left a) (: right b)) left)) (SIGNATURE elt (b (Record (: left a) (: right b)) right)) (SIGNATURE setelt! (a (Record (: left a) (: right b)) left a)) (SIGNATURE setelt! (b (Record (: left a) (: right b)) right b)) (SIGNATURE copy ((Record (: left a) (: right b)) (Record (: left a) (: right b)))))) to (Join (CoChainCategory) (ZeroChainCategory) (CATEGORY domain (SIGNATURE degree ((Polynomial (Integer)) $)) (SIGNATURE d ($ $))))
compiling exported * : (a,b) -> $
WEDGE;*;ab$;3 is replaced by CONS
Time: 0 SEC.
compiling exported degree : $ -> Polynomial Integer
Time: 0 SEC.
compiling exported mainType : $ -> String
WEDGE;mainType;$S;5 is replaced by WedgeProduct
Time: 0 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |WedgeProduct| REDEFINED
;;; *** |WedgeProduct| REDEFINED
Time: 0 SEC.
Warnings:
[1] *: signature of lhs not unique: $ab chosen
[2] degree: left has no value
[3] degree: right has no value
Cumulative Statistics for Constructor WedgeProduct
Time: 0.01 seconds
finalizing NRLIB WEDGE
Processing WedgeProduct for Browser database:
--->-->WedgeProduct(constructor): Not documented!!!!
--->-->WedgeProduct((* ((ZeroChain) a b))): Not documented!!!!
--->-->WedgeProduct((* (% a b))): Not documented!!!!
--->-->WedgeProduct((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->WedgeProduct(): Missing Description
; compiling file "/var/aw/var/LatexWiki/WEDGE.NRLIB/WEDGE.lsp" (written 03 SEP 2022 03:11:28 PM):
; /var/aw/var/LatexWiki/WEDGE.NRLIB/WEDGE.fasl written
; compilation finished in 0:00:00.028
------------------------------------------------------------------------
WedgeProduct is now explicitly exposed in frame initial
WedgeProduct will be automatically loaded when needed from
/var/aw/var/LatexWiki/WEDGE.NRLIB/WEDGE
EXTDF abbreviates domain ExteriorDerivative
------------------------------------------------------------------------
initializing NRLIB EXTDF for ExteriorDerivative
compiling into NRLIB EXTDF
****** Domain: a already in scope
augmenting a: (ZeroChainCategory)
Local variable Rep type redefined: (Join (CoChainCategory) (ZeroChainCategory) (CATEGORY domain (SIGNATURE degree ((Polynomial (Integer)) $)) (SIGNATURE d ($ $)))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: left a) (: right b)) a b)) (SIGNATURE ~= ((Boolean) (Record (: left a) (: right b)) (Record (: left a) (: right b)))) (SIGNATURE coerce ((OutputForm) (Record (: left a) (: right b)))) (SIGNATURE elt (a (Record (: left a) (: right b)) left)) (SIGNATURE elt (b (Record (: left a) (: right b)) right)) (SIGNATURE setelt! (a (Record (: left a) (: right b)) left a)) (SIGNATURE setelt! (b (Record (: left a) (: right b)) right b)) (SIGNATURE copy ((Record (: left a) (: right b)) (Record (: left a) (: right b))))))
Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: arg a)) a)) (SIGNATURE ~= ((Boolean) (Record (: arg a)) (Record (: arg a)))) (SIGNATURE coerce ((OutputForm) (Record (: arg a)))) (SIGNATURE elt (a (Record (: arg a)) arg)) (SIGNATURE setelt! (a (Record (: arg a)) arg a)) (SIGNATURE copy ((Record (: arg a)) (Record (: arg a)))))) to (Join (CoChainCategory) (ZeroChainCategory) (CATEGORY domain (SIGNATURE degree ((Polynomial (Integer)) $)) (SIGNATURE d ($ $))))
compiling exported d : $ -> ZeroChain
Time: 0 SEC.
compiling exported d : a -> $
EXTDF;d;a$;2 is replaced by LIST
Time: 0 SEC.
compiling exported degree : $ -> Polynomial Integer
Time: 0.02 SEC.
compiling exported mainType : $ -> String
EXTDF;mainType;$S;4 is replaced by ExteriorDerivative
Time: 0.01 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |ExteriorDerivative| REDEFINED
;;; *** |ExteriorDerivative| REDEFINED
Time: 0.01 SEC.
Warnings:
[1] degree: arg has no value
[2] coerce: arg has no value
Cumulative Statistics for Constructor ExteriorDerivative
Time: 0.04 seconds
finalizing NRLIB EXTDF
Processing ExteriorDerivative for Browser database:
--->-->ExteriorDerivative(constructor): Not documented!!!!
--->-->ExteriorDerivative((d ((ZeroChain) %))): Not documented!!!!
--->-->ExteriorDerivative((d (% a))): Not documented!!!!
--->-->ExteriorDerivative((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->ExteriorDerivative(): Missing Description
; compiling file "/var/aw/var/LatexWiki/EXTDF.NRLIB/EXTDF.lsp" (written 03 SEP 2022 03:11:29 PM):
; /var/aw/var/LatexWiki/EXTDF.NRLIB/EXTDF.fasl written
; compilation finished in 0:00:00.023
------------------------------------------------------------------------
ExteriorDerivative is now explicitly exposed in frame initial
ExteriorDerivative will be automatically loaded when needed from
/var/aw/var/LatexWiki/EXTDF.NRLIB/EXTDF
HODGE abbreviates domain HodgeStar
------------------------------------------------------------------------
initializing NRLIB HODGE for HodgeStar
compiling into NRLIB HODGE
compiling exported degree : $ -> Polynomial Integer
Time: 0.02 SEC.
compiling exported hodge : a -> $
HODGE;hodge;a$;2 is replaced by LIST
Time: 0 SEC.
compiling exported mainType : $ -> String
HODGE;mainType;$S;3 is replaced by HodgeStar
Time: 0 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |HodgeStar| REDEFINED
;;; *** |HodgeStar| REDEFINED
Time: 0 SEC.
Warnings:
[1] degree: arg has no value
[2] coerce: arg has no value
Cumulative Statistics for Constructor HodgeStar
Time: 0.03 seconds
finalizing NRLIB HODGE
Processing HodgeStar for Browser database:
--->-->HodgeStar(constructor): Not documented!!!!
--->-->HodgeStar((hodge (% a))): Not documented!!!!
--->-->HodgeStar((degree ((Polynomial (Integer)) %))): Not documented!!!!
--->-->HodgeStar(): Missing Description
; compiling file "/var/aw/var/LatexWiki/HODGE.NRLIB/HODGE.lsp" (written 03 SEP 2022 03:11:29 PM):
; /var/aw/var/LatexWiki/HODGE.NRLIB/HODGE.fasl written
; compilation finished in 0:00:00.019
------------------------------------------------------------------------
HodgeStar is now explicitly exposed in frame initial
HodgeStar will be automatically loaded when needed from
/var/aw/var/LatexWiki/HODGE.NRLIB/HODGEfricas
f:=f::CoChain(p)
fricas
g:=g::CoChain(q)
fricas
h:=f*g
Type: WedgeProduct
?(CoChain
?(p),
CoChain
?(q))
fricas
degree h
Type: Polynomial(Integer)
fricas
k:=h*f
Type: WedgeProduct
?(WedgeProduct
?(CoChain
?(p),
CoChain
?(q)),
CoChain
?(p))
fricas
degree k
Type: Polynomial(Integer)
fricas
-- car dom (d f) --> ExteriorDerivative
fricas
-- symbol car dom (d f) = 'ExteriorDerivative::Symbol => ok
degree d f --> p+1
Type: Polynomial(Integer)
fricas
hodge d h --> *d(f # g)
Type: HodgeStar
?(ExteriorDerivative
?(WedgeProduct
?(CoChain
?(p),
CoChain
?(q))))
fricas
r:=degree hodge h
Type: Polynomial(Integer)
fricas
eval(r,[p=7,q=3,'spacedim=15]) --> notice the quote!
Type: Polynomial(Integer)
fricas
-->degree % --> spacedim - q - p - 1
null()$ZeroChain
fricas
z:=d d f
fricas
z*f
fricas
f*z
fricas
CoChain(p) has ZeroChainCategory
Type: Boolean
fricas
ZeroChain has ZeroChainCategory
Type: Boolean
