Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7701307392394560546-25px.axm
Heap exhausted during garbage collection: 0 bytes available, 16 requested.
Immobile Object Counts
Gen layout fdefn symbol code Boxed Cons Raw Code SmMix Mixed LgRaw LgCode LgMix Waste% Alloc Trig Dirty GCs Mem-age
2 0 0 0 0 3 10478 1 0 0 1 0 0 0 0.9 340512912 202181376 10483 1 1.3689
3 0 0 0 0 1 6759 1 0 0 0 0 0 0 0.8 219689072 6291456 6675 0 0.0126
4 0 4665 0 4894 113 909 78 0 17 9 0 0 0 1.3 36422448 2000000 79 0 0.0000
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0 2000000 0 0 0.0000
6 749 24064 26788 24347 448 176 63 5 48 16 0 0 74 2.4 26541936 2000000 18 0 0.0000
Tot 749 28729 26788 29241 565 18322 143 5 65 26 0 0 74 1.0 623166368 [99.0% of 629145600 max]
GC control variables:
GC-INHIBIT = true
GC-PENDING = true
STOP-FOR-GC-PENDING = false
fatal error encountered in SBCL pid 1788095 tid 1788095:
Heap exhausted, game over.
Error opening /dev/tty: No such device or address
Checking for foreign routines
FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu"
spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so"
foreign routines found
openServer result -2
FriCAS Computer Algebra System
Version: FriCAS 1.3.12 built with sbcl 2.2.9.debian
Timestamp: Sat 7 Jun 23:54:49 CEST 2025
-----------------------------------------------------------------------------
Issue )copyright to view copyright notices.
Issue )summary for a summary of useful system commands.
Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> Scalar := Expression Integer
$$
Expression
\left(
{Integer}
\right)
\leqno(1)
$$
Type: Type
vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix(
Expression(Integer)) has been added to workspace.
Type: Void
vect [a0,a1,a2,a3]
Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))
$$
\left[
\begin{array}{c}
a0 \
a1 \
a2 \
a3
\end{array}
\right]
\leqno(3)
$$
Type: Matrix(Expression(Integer))
(4) -> ID:=diagonalMatrix([1,1,1,1])
$$
\left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(4)
$$
Type: Matrix(Integer)
(5) -> htrigs2exp == rule
cosh(a) == (exp(a)+exp(-a))/2
sinh(a) == (exp(a)-exp(-a))/2
Type: Void
sinhcosh == rule
?cexp(a)+?cexp(-a) == 2ccosh(a)
?cexp(a)-?cexp(-a) == 2csinh(a)
?cexp(a-b)+?cexp(b-a) == 2ccosh(a-b)
?cexp(a-b)-?cexp(b-a) == 2csinh(a-b)
Type: Void
expandhtrigs == rule
cosh(:x+y) == sinh(x)sinh(y)+cosh(x)cosh(y)
sinh(:x+y) == cosh(x)sinh(y)+sinh(x)cosh(y)
cosh(2x) == 2cosh(x)^2-1
sinh(2x) == 2sinh(x)*cosh(x)
Type: Void
expandhtrigs2 == rule
cosh(2x+2y) == 2cosh(x+y)^2-1
sinh(2x+2y) == 2sinh(x+y)cosh(x+y)
cosh(2x-2y) == 2cosh(x-y)^2-1
sinh(2x-2y) == 2sinh(x-y)cosh(x-y)
Type: Void
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
Function declaration Simplify : Expression(Integer) -> Expression(
Integer) has been added to workspace.
Type: Void
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
Type: Void
is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean
has been added to workspace.
Type: Void
Is?(eq:Equation(Matrix(Scalar))):Boolean ==
(map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber =
zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean
Function declaration Is? : Equation(Matrix(Expression(Integer))) ->
Boolean has been added to workspace.
Type: Void
(13) -> G:=diagonalMatrix [-1,1,1,1]
$$
\left[
\begin{array}{cccc}
-1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(13)
$$
Type: Matrix(Integer)
(14) -> g(x) == transpose(x)*G
Type: Void
g(vect [a0,a1,a2,a3])
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer))
$$
\left[
\begin{array}{cccc}
-a0 & a1 & a2 & a3
\end{array}
\right]
\leqno(15)
$$
Type: Matrix(Expression(Integer))
(16) -> dot(x,y) == (g(x)*y)::Scalar
Type: Void
dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
Compiling function dot with type (Matrix(Expression(Integer)),
Matrix(Expression(Integer))) -> Expression(Integer)
$$
{a3 \ b3}+{a2 \ b2}+{a1 \ b1} -{a0 \ b0}
\leqno(17)
$$
Type: Expression(Integer)
(18) -> tensor(x,y) == x*g(y)
Type: Void
tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
Compiling function tensor with type (Matrix(Expression(Integer)),
Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
$$
\left[
\begin{array}{cccc}
-{a0 \ b0} & {a0 \ b1} & {a0 \ b2} & {a0 \ b3} \
-{a1 \ b0} & {a1 \ b1} & {a1 \ b2} & {a1 \ b3} \
-{a2 \ b0} & {a2 \ b1} & {a2 \ b2} & {a2 \ b3} \
-{a3 \ b0} & {a3 \ b1} & {a3 \ b2} & {a3 \ b3}
\end{array}
\right]
\leqno(19)
$$
Type: Matrix(Expression(Integer))
(20) -> P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];
Type: Matrix(Expression(Integer))
dot(P,P)
$$
-1
\leqno(21)
$$
Type: Expression(Integer)
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];
Type: Matrix(Expression(Integer))
R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];
Type: Matrix(Expression(Integer))
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]
$$
\left[
\begin{array}{c}
{\frac{1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
-{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(24)
$$
Type: Matrix(Expression(Integer))
dot(S,S)
$$
-1
\leqno(25)
$$
Type: Expression(Integer)
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]
$$
\left[
\begin{array}{c}
{\frac{1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t2}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \
-{\frac{t3}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(26)
$$
Type: Matrix(Expression(Integer))
U:=vect [cosh(u),sinh(u),0,0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{u}
\right)}
\
{\sinh
\left(
{u}
\right)}
\
0 \
0
\end{array}
\right]
\leqno(27)
$$
Type: Matrix(Expression(Integer))
simplify dot(U,U)
$$
-1
\leqno(28)
$$
Type: Expression(Integer)
V:=vect [cosh(v),sinh(v),0,0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{v}
\right)}
\
{\sinh
\left(
{v}
\right)}
\
0 \
0
\end{array}
\right]
\leqno(29)
$$
Type: Matrix(Expression(Integer))
Simplify dot(U,V)
Compiling body of rule htrigs2exp to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
Compiling body of rule sinhcosh to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
Compiling function Simplify with type Expression(Integer) ->
Expression(Integer)
$$
-{\cosh
\left(
{{v -u}}
\right)}
\leqno(30)
$$
Type: Expression(Integer)
W:=vect [cosh(w),0,sinh(w),0]
$$
\left[
\begin{array}{c}
{\cosh
\left(
{w}
\right)}
\
0 \
{\sinh
\left(
{w}
\right)}
\
0
\end{array}
\right]
\leqno(31)
$$
Type: Matrix(Expression(Integer))
Simplify dot(U,W)
$$
\frac{{-{\cosh
\left(
{{w+u}}
\right)}
-{\cosh
\left(
{{w -u}}
\right)}}}{2}
\leqno(32)
$$
Type: Expression(Integer)
(33) -> vect [1,0,0,0]
$$
\left[
\begin{array}{c}
1 \
0 \
0 \
0
\end{array}
\right]
\leqno(33)
$$
Type: Matrix(Expression(Integer))
dot(%,%)
$$
-1
\leqno(34)
$$
Type: Expression(Integer)
(35) -> ω(P,Q)==-P/dot(P,Q)-Q
Type: Void
ω(P,Q)
Compiling function ω with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer))) -> Matrix(Expression(Integer))
$$
\left[
\begin{array}{c}
{\frac{{{{\left( {p3 \ q3}+{p2 \ q2}+{p1 \ q1}
\right)}
\ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}}+{{\left(
-{{q3} \sp {2}} -{{q2} \sp {2}} -{{q1} \sp {2}}
\right)}
\ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}}}}}{{{{\sqrt
{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp
{2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \
q1}}}} \
{\frac{{{q1 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q1 \ q3}
-{p2 \ q1 \ q2} -{p1 \ {{q1} \sp {2}}} -p1}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q2 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q2 \ q3}
-{p2 \ {{q2} \sp {2}}} -{p1 \ q1 \ q2} -p2}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q3 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ {{q3} \sp
{2}}}+{{\left( -{p2 \ q2} -{p1 \ q1}
\right)}
\ q3} -p3}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2
\ q2} -{p1 \ q1}}}}
\end{array}
\right]
\leqno(36)
$$
Type: Matrix(Expression(Integer))
ω(S,T)
$$
\left[
\begin{array}{c}
{\frac{{{{t3} \sp {2}} -{s3 \ t3}+{{t2} \sp {2}} -{s2 \ t2}+{{t1} \sp {2}}
-{s1 \ t1}}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s1 \ {{t3} \sp {2}}}+{s3 \ t1 \ t3} -{s1 \ {{t2} \sp {2}}}+{s2
\ t1 \ t2} -t1+s1}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s2 \ {{t3} \sp {2}}}+{s3 \ t2 \ t3}+{{\left( {s1 \ t1} -1
\right)}
\ t2} -{s2 \ {{t1} \sp {2}}}+s2}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1}
-1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{{{\left( {s2 \ t2}+{s1 \ t1} -1
\right)}
\ t3} -{s3 \ {{t2} \sp {2}}} -{s3 \ {{t1} \sp {2}}}+s3}}{{{\left( {s3 \
t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}}
\end{array}
\right]
\leqno(37)
$$
Type: Matrix(Expression(Integer))
(38) -> map(x+->Simplify x,ω(U,V))
$$
\left[
\begin{array}{c}
{\frac{{-{\cosh
\left(
{{{2 \ v} -u}}
\right)}+{\cosh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
{\frac{{-{\sinh
\left(
{{{2 \ v} -u}}
\right)}+{\sinh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
0 \
0
\end{array}
\right]
\leqno(38)
$$
Type: Matrix(Expression(Integer))
vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]
$$
\left[
\begin{array}{c}
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\cosh
\left(
{v}
\right)}}+{\cosh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\sinh
\left(
{v}
\right)}}+{\sinh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(39)
$$
Type: Matrix(Expression(Integer))
Is?(% = ω(U,V))
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean
$$
true
\leqno(40)
$$
Type: Boolean
map(x+->Simplify x,ω(U,W))
$$
\left[
\begin{array}{c}
{\frac{{-{\cosh
\left(
{{2 \ w}}
\right)}+1}}{{2
\ {\cosh
\left(
{w}
\right)}}}}
\
{\frac{{2 \ {\sinh
\left(
{u}
\right)}}}{{{\cosh
\left(
{{w+u}}
\right)}+{\cosh
\left(
{{w -u}}
\right)}}}}
\
-{\sinh
\left(
{w}
\right)}
\
0
\end{array}
\right]
\leqno(41)
$$
Type: Matrix(Expression(Integer))
(42) -> dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))
$$
\frac{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}}}{{{{p3} \sp {2}}+{{p2}
\sp {2}}+{{p1} \sp {2}}+1}}
\leqno(42)
$$
Type: Expression(Integer)
(43) -> dot(P,ω(Q,P))
$$
0
\leqno(43)
$$
Type: Expression(Integer)
possible dot(ω(Q,P),ω(Q,P))::EXPR Float
Compiling function possible with type Expression(Integer) ->
Expression(Integer)
$$
0.9999995659\_3941083651
\leqno(44)
$$
Type: Expression(Float)
dot(Q,ω(P,Q))
$$
0
\leqno(45)
$$
Type: Expression(Integer)
possible dot(ω(P,Q),ω(P,Q))::EXPR Float
$$
0.9999994628\_9493005142
\leqno(46)
$$
Type: Expression(Float)
(47) -> ω(vect [u0,u1,u2,u3],vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
{\frac{u1}{u0}} \
{\frac{u2}{u0}} \
{\frac{u3}{u0}}
\end{array}
\right]
\leqno(47)
$$
Type: Matrix(Expression(Integer))
ω(R,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-{\frac{r1}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r2}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r3}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(48)
$$
Type: Matrix(Expression(Integer))
ω(S,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-s1 \
-s2 \
-s3
\end{array}
\right]
\leqno(49)
$$
Type: Matrix(Expression(Integer))
map(Simplify, ω(U,vect [1,0,0,0]))
$$
\left[
\begin{array}{c}
0 \
{\frac{{\sinh
\left(
{u}
\right)}}{{\cosh
\left(
{u}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(50)
$$
Type: Matrix(Expression(Integer))
(51) -> ω(vect [1,0,0,0],S)
$$
\left[
\begin{array}{c}
{\frac{{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}}}{{\sqrt {{-{{s3} \sp
{2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(51)
$$
Type: Matrix(Expression(Integer))
ω(vect [1,0,0,0],R)
$$
\left[
\begin{array}{c}
{\frac{{-{{r3} \sp {2}} -{{r2} \sp {2}} -{{r1} \sp {2}}}}{{\sqrt {{{{r3} \sp
{2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
r1 \
r2 \
r3
\end{array}
\right]
\leqno(52)
$$
Type: Matrix(Expression(Integer))
is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))
Compiling function is? with type Equation(Expression(Integer)) ->
Boolean
$$
true
\leqno(53)
$$
Type: Boolean
(54) -> L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q)
Type: Void
Is?(L(P,P) = ID)
Compiling function L with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer))) -> Matrix(Expression(Integer))
$$
true
\leqno(55)
$$
Type: Boolean
Is?(L(P,Q)*L(Q,P) = ID)
$$
true
\leqno(56)
$$
Type: Boolean
Is?(L(P,Q)*Q=P)
$$
true
\leqno(57)
$$
Type: Boolean
Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))
$$
true
\leqno(58)
$$
Type: Boolean
(59) -> B(P,Q,X) == ID -
( tensor(
2X, dot(P-Q,P-Q)X - 2dot(X,P)(P-Q)
) +
tensor( _
P-Q, 2dot(X,X)(P-Q)+4dot(X,Q)X
)
) / ( _
dot(X,X)dot(P-Q,P-Q)+4dot(X,P)*dot(X,Q) _
)
Type: Void
Is?(B(P,P,R) = ID)
Compiling function B with type (Matrix(Expression(Integer)), Matrix(
Expression(Integer)), Matrix(Expression(Integer))) -> Matrix(
Expression(Integer))
$$
true
\leqno(60)
$$
Type: Boolean
Is?(B(P,Q,R)*B(Q,P,R) = ID)
$$
true
\leqno(61)
$$
Type: Boolean
Is?(B(P,Q,R)*P=Q)
$$
true
\leqno(62)
$$
Type: Boolean
--Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))
Is?(L(P,Q)=B(Q,P,qQ+pP))
$$
true
\leqno(63)
$$
Type: Boolean
(64) -> map(x+->simplify expandhtrigs2 Simplify x, L(U,V))
Compiling body of rule expandhtrigs2 to compute value of type
Ruleset(Integer,Integer,Expression(Integer))
$$
\left[
\begin{array}{cccc}
{\cosh
\left(
{{v -u}}
\right)}
& -{\sinh
\left(
{{v -u}}
\right)}
& 0 & 0 \
-{\sinh
\left(
{{v -u}}
\right)}
& {\cosh
\left(
{{v -u}}
\right)}
& 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(64)
$$
Type: Matrix(Expression(Integer))
map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W))
Compiling body of rule expandhtrigs to compute value of type Ruleset
(Integer,Integer,Expression(Integer))
$$
\left[
\begin{array}{cccc}
{\frac{{{{\left( {2 \ {{{\cosh
\left(
{u}
\right)}}
\sp {2}}} -1
\right)}
\ {{{\cosh
\left(
{w}
\right)}}
\sp {2}}}+{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}
-{{{\cosh
\left(
{u}
\right)}}
\sp {2}}+1}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( {\cosh
\left(
{w}
\right)}+{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{u}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( {{\left( -{2 \ {{{\cosh
\left(
{u}
\right)}}
\sp {2}}}+1
\right)}
\ {\cosh
\left(
{w}
\right)}}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
{\frac{{{\left( {2 \ {\cosh
\left(
{u}
\right)}
\ {{{\cosh
\left(
{w}
\right)}}
\sp {2}}}+{\cosh
\left(
{w}
\right)}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{u}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+{{{\cosh
\left(
{u}
\right)}}
\sp {2}}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\left( -{2 \ {\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}
-1
\right)}
\ {\sinh
\left(
{u}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
{\frac{{{\left( -{\cosh
\left(
{w}
\right)}
-{\cosh
\left(
{u}
\right)}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{\sinh
\left(
{u}
\right)}
\ {\sinh
\left(
{w}
\right)}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& {\frac{{{{{\cosh
\left(
{w}
\right)}}
\sp {2}}+{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}}}{{{{\cosh
\left(
{u}
\right)}
\ {\cosh
\left(
{w}
\right)}}+1}}}
& 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(65)
$$
Type: Matrix(Expression(Integer))
(66) -> Is?(L(R,P)*L(P,Q) = L(R,Q))
$$
false
\leqno(66)
$$
Type: Boolean
RQ:=aR+bQ;
Type: Matrix(Expression(Integer))
rq:=solve(dot(RQ,RQ)=-1,b); #rq
$$
2
\leqno(68)
$$
Type: PositiveInteger
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
dot(RQ1,RQ1)
$$
-1
\leqno(70)
$$
Type: Expression(Integer)
Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))
$$
true
\leqno(71)
$$
Type: Boolean
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
Is?(RQ1=RQ2)
$$
false
\leqno(73)
$$
Type: Boolean
dot(RQ2,RQ2)
$$
-1
\leqno(74)
$$
Type: Expression(Integer)
Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))
$$
true
\leqno(75)
$$
Type: Boolean
(76) -> LRPQ := L(R,P)*L(P,Q);
Type: Matrix(Expression(Integer))
Is?(LRPQQ = L(R,Q)Q)
$$
true
\leqno(77)
$$
Type: Boolean
is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))
Welcome to LDB, a low-level debugger for the Lisp runtime environment.
(GC in progress, oldspace=2, newspace=3)
ldb>