> # The computations of Czedli and Romanowska's paper:
# " Some modes with new algebraic closures " (Submitted: Febuary 4, 2011)
#

> # "Maple V Release 5 Version 5.00 Nov 27 1997" , other Maple versions may require a different syntax!

> #

> # Note the syntax (and change if you wish):
# a semicolon after a command displays the result ,
# a colon after a command does not display anything .

> # Lines beginning with # are just comments.

> #

> #

> ###################################################################################################

> #
# Proving the identity (13) [in TeX source: \label{f:maple-a} ] :

> # (x1 y2 p) (x1 x3 p y1 p )h (x2 y3 p) t = (x1 x2 p y1 p) (x1 y3 p) h (x2 x3 p y2 p) t
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts to x y p (Polish notation) to a polynomial.

> mh:=proc(x,y): expand((x+y)/2) end: # "mh" is "mp" with p=1/2

> mt:=proc(x,y): expand(2*x/3+y/3) end: # "mt" is "mp" with p=1/3

> aleft := mt( mh( mp(x1,y2), mp(mp(x1,x3),y1) ) , mp(x2,y3) );

> aright:= mt( mh( mp(mp(x1,x2),y1) , mp(x1,y3) ) , mp(mp(x2,x3),y2) );

> if_zero_then_the_identity_is_proved:=aleft-aright;

> # We have obtained 0. Q.e.d.

> #

> #

> #

> #

> ###################################################################################################

> #

> #

> # Proving the identity (31) [in TeX source: \label{f:maple-b} ] :

> # If r=(q - p)/((1 - p)q) and u= p/(p + q), then (b) (abs yq )u = (abr bs yp ) (b) u
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts x y p (Polish notation) to a polynomial.

> mq:=proc(x,y): expand((1-q)*x+q*y) end: # The same as "mp" with q instead of p.

> ms:=proc(x,y): expand((1-s)*x+s*y) end: # The same as "mp" with s instead of p.

> mr:=proc(x,y): subs(r=(q - p)/((1 - p)*q), expand((1-r)*x+r*y)) end: # The same as "mp" with the given r instead of p.

> mu:=proc(x,y): subs(u=p/(p + q), expand((1-u)*x+u*y)) end: # The same as "mp" with the given u instead of p.

> bleft :=simplify( mu( b , mq(ms(a,b) , y ) ) );

> bright:=simplify( mu( mp( ms(mr(a,b),b), y) , b) );

> if_zero_then_the_identity_is_proved:=bleft-bright;

> # We have obtained 0. Q.e.d.

> #

> #

> #

> #

> ###################################################################################################

> #

> #

> #

> # Proving the identity (32) [in TeX source: \label{f:maple-c} ] :

> # If r=(q - p)/((1 - p)q) and u= p/(p + q), then (abr cp) (b) u = (b) (acq) u
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts x y p (Polish notation) to a polynomial.

> mq:=proc(x,y): expand((1-q)*x+q*y) end: # The same as "mp" with q instead of p.

> mr:=proc(x,y): subs(r=(q - p)/((1 - p)*q), expand((1-r)*x+r*y)) end: # The same as "mp" with the given r instead of p.

> mu:=proc(x,y): subs(u=p/(p + q), expand((1-u)*x+u*y)) end: # The same as "mp" with the given u instead of p.

> cleft := simplify( mu( mp(mr(a,b),c) , b) );

> cright:= simplify( mu(b, mq(a,c)) );

> if_zero_then_the_identity_is_proved:=cleft-cright;

> # We have obtained 0. Q.e.d.

> #

> #

> #

> #

> #

> ###################################################################################################

> #

> #

> #

> # Proving the identity (35) [in TeX source: \label{f:maple-d} ] :

> # If q=1-p, r=q/p=(1-p)/p, h=1/2 and w = 1/(2p+1), then (abr) (b)h (deq) w = (d) (aer) h (b) w.
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts x y p (Polish notation) to a polynomial.

> mq:=proc(x,y): subs(q=1-p, expand((1-q)*x+q*y)) end: # The same as "mp" with the given q instead of p.

> mr:=proc(x,y): subs(r=(1-p)/p, expand((1-r)*x+r*y)) end: # The same as "mp" with the given r instead of p.

> mh:=proc(x,y): subs(h=1/2, expand((1-h)*x+h*y)) end: # The same as "mp" with the given h instead of p.

> mw:=proc(x,y): subs(w=1/(2*p+1), expand((1-w)*x+w*y)) end: # The same as "mp" with the given w instead of p.

> dleft := mw( mh(mr(a,b),b) , mq(d,e) );

> dright:= mw( mh(d,mr(a,e)) ,b);

> if_zero_then_the_identity_is_proved:=simplify(dleft-dright);

> # We have obtained 0. Q.e.d.

> #

> #

> #

> #

> #

> ###################################################################################################

> #

> #

> #

> # Proving the identity (40) [in TeX source: \label{f:maple-e} ] :

> # If h = 1/2, q = 1-p and r = q/p=(1-p)/p, then (abr eq) (b)(xy p)q h = (b) (aaxr r yq)(e)q h
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts x y p (Polish notation) to a polynomial.

> mh:=proc(x,y): subs(h=1/2, expand((1-h)*x+h*y)) end: # The same as "mp" with the given h instead of p.

> mq:=proc(x,y): subs(q=1-p, expand((1-q)*x+q*y)) end: # The same as "mp" with the given q instead of p.

> mr:=proc(x,y): subs(r=(1-p)/p, expand((1-r)*x+r*y)) end: # The same as "mp" with the given r instead of p.

> eleft := mh( mq(mr(a,b),e) , mq(b, mp(x,y)) );

> eright:= mh( b , mq(mq(mr(a,mr(a,x)) ,y) , e) );

> if_zero_then_the_identity_is_proved:=eleft-eright;

> # We have obtained 0. Q.e.d.

> #

> #

> #

> #

> #

> ###################################################################################################

> #

> #

> #

> # Proving the identity (42) [in TeX source: \label{f:maple-f} ] :

> # If q = 1-p and r = q/p=(1-p)/p, then aacp r = acq
#

> restart;

> # We define some procedures:

> mp:=proc(x,y): expand((1-p)*x+p*y) end: # "mp" converts x y p (Polish notation) to a polynomial.

> mq:=proc(x,y): subs(q=1-p, expand((1-q)*x+q*y)) end: # The same as "mp" with the given q instead of p.

> mr:=proc(x,y): subs(r=(1-p)/p, expand((1-r)*x+r*y)) end: # The same as "mp" with the given r instead of p.

> fleft :=mr(a,mp(a,c));

> fright:=mq(a,c);

> if_zero_then_the_identity_is_proved:=fleft-fright;

> # We have obtained 0. Q.e.d.
# End of Maple worksheet.