Prolog Higher-order reduce predicate

Question

We can define a higher-order map predicate as:

map([], [], F).
map([A|As], [B|Bs], F) :-
   call(F, A, B),
   map(As, Bs, F).

Similarly, we can define fold (left) as:

fold([], Acc, Acc, _F).
fold([A|As], B, Acc1, F) :-
   call(F, Acc1, A, Acc2),
   fold(As, B, Acc2, F).

What is the correct definition for reduce (left)? Can we define it as follows?

reduce([A|As], Bs, F) :-   
   fold(As, Bs, A, F).

And reduceback (right) as follows?

reduceback([], Ident, F) :-
   identity(F, Ident).
reduceback([A|As], B, F) :-
   reduceback(As, C, F),
   call(F, C, A, B).

Are these correct?

Answer 1

fold/4 and reduce/3 perform correctly, while without identity/1 reduceback/3 is incomplete. But the control flow seems correct, though

1 ?- fold([1,2,3],S,0,[X,Y,Z]>>(Z is X+Y)).
S = 6.

2 ?- reduce([1,2,3],S,[X,Y,Z]>>(Z is X+Y)).
S = 6.

I've added the declarations

:- meta_predicate fold(+,?,+,3).
:- meta_predicate reduce(+,?,3).

that qualify arguments as closures, and used library(yall) for lambdas...

In Prolog, a common convention is to place output arguments as last, so your definitions are rather unreadable to me....

edit

for symmetry with reduce/3, identity/1 seems useless, the last element could be used instead: so it could be

:- meta_predicate reduceback(+,?,3).

reduceback([Last],Last,_F).
reduceback([A|As],B,F):-
  reduceback(As,C,F),
  call(F,C,A,B).

test:

?- reduceback([1,2,3],S,[X,Y,Z]>>(Z is X+Y)).
S = 6 ;
false.

?- reduceback([1,2,3],S,[X,Y,Z]>>(Z is X-Y)).
S = 0 ;
false.