Help:=proc(): print(`F(n,K), Fs(n,K), FnK(n,K),  EvalC(L,x,x0), EvalDC(L,x,x0), EvalMC(L,x,x0), EvalMC1(n,m,x,x0,F)`):
end:


Fs:=proc(n,K) :

if n=0 or n=1 then
 1:
else
Fs(n-1,K)+Fs(n-2,K) mod K:
fi:

end:

F:=proc(n,K) 
option remember:
if n=0 or n=1 then
 1:
else
F(n-1,K)+F(n-2,K) mod K:
fi:

end:

FnK:=proc(n,K) local i,P:
P:=[0,1]:
for i from 1 to n do
 P:=[P[2],P[1]+P[2] mod K]:
od:
P[2]:
end:


#EvalC(L,x,x0): given a list of expressions L representing functions in x, outputs
#the composition L[1](L[2](....L[n](x0))
EvalC:=proc(L,x,x0):

if nops(L)=1 then
 RETURN(subs(x=x0,L[1])):
else
 subs(x=EvalC([op(2..nops(L),L)],x,x0),L[1]):
fi:

end:


#EvalDC(L,x,x0): given a list of expressions L representing functions in x, outputs
#the dreivative of the composition L[1](L[2](....L[n](x0)) at x0
EvalDC:=proc(L,x,x0):

if nops(L)=1 then
 RETURN(subs(x=x0,diff(L[1],x) )):
else
 subs(x=EvalC([op(2..nops(L),L)],x,x0),diff(L[1],x))*EvalDC([op(2..nops(L),L)],x,x0):
fi:

end:


#EvalMC1(n,m,x,x0,F): inputs pos. integer m and n, a symbol x, a numerical (or symbolic) list x0 of length n, and
#a list F , of length m of expressions in x[1], ..., x[n] denoting a transformation from n-dim space to m-dim space
#outputs the evaluation of x0 in F. Try
#EvalMC1(2,3,x,[1,2],[x[1]+x[2],x[1]+2*x[2],x[1]*x[2]]);
EvalMC1:=proc(n,m,x,x0,F) local i:

if not  (type(m,integer) and type(n,integer) and m>0 and n>0 and nops(x0)=n and nops(F)=m) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

subs({seq(x[i]=x0[i],i=1..n)},F):
end:


#EvalMC(L,x,x0): given a list of expressions L representing functions in x[1],x[2], ..., outputs
#the composition L[1](L[2](....L[n](x0)), where x0 is a vector that is compatible
EvalMC:=proc(L,x,x0)local i,A:

if nops(L)=1 then
 RETURN(subs({seq(x[i]=x0[i],i=1..nops(x0))},L[1])):
#else
# A:=EvalMC([op(2..nops(L),L)],
fi:

end: