MAMMAL:

arms(dog, 0).
arms(human, 2).
legs(human, 2).
mammal(horse).
mammal(X) :- legs(X,4), arms(X,0).
mammal(X) :- legs(X,2), arms(X,2).

ANCESTOR:

parent(joan, john).
parent(joan, jean).
parent(jean, jan).
ancestor(X,Y) :- parent(X,Z),ancestor(Z,Y).
ancestor(X,X).

FACTORIAL (no cut):

fac(0,1).
fac(N,X) :- M is N-1, fac(M,Y), X is N*Y.

FACTORIAL (with cut);

fac(0,1).
fac(N,X) :- M is N-1, fac(M,Y), !, X is N*Y.

NATURAL (no cut):

natural(0).
natural(X) :- X < 0, fail.
natural(X) :- Y is X - 1, natural(Y).

NATURAL (with cut):

natural(0).
natural(X) :- X < 0, !, fail.
natural(X) :- Y is X - 1, natural(Y).

GCD:

gcd(X,0,X).
gcd(X,Y,Z) :- X>Y, gcd(Y,X,Z), !.
gcd(X,Y,Z) :- X =< Y, Y1 is Y mod X, gcd(X, Y1, Z), !.

MEMBER:

ourmember(X, [X|_]).
ourmember(X, [_|Y]) :- ourmember(X, Y).

APPEND:

ourappend([], L, L).
ourappend([X | L1], L2, [X | L3]) :- ourappend(L1, L2, L3).

FLATTEN:

flatten([],[]).
flatten(X,[X]) :- atom(X).
flatten([F|L1],L2) :- flatten(F,FF), flatten(L1,L1F), ourappend(FF, L1F, L2).