Rearranging variable_names

avs_term_rearranged(AVs, T, AVsR) :-
    term_variables(T, Vs),
    copy_term(Vs+AVs, Vs1+AVs1),
    bind_names(AVs1),
    build_vn_list(Vs, Vs1, AVsR).

bind_names([]).
bind_names([N=V|AVs]) :-
    N = V,
    bind_names(AVs).

build_vn_list([], [], []).
build_vn_list([V|Vs],[N|Ns],NVs) :-
    ( atom(N) ->
      NVs = [N=V|NVs1]
    ; var(N) ->
      NVs = NVs1
    ),
    build_vn_list(Vs, Ns, NVs1).

My previous answer had quadratic runtime complexity. If that's a problem, here is a linear alternative. The reason this is a bit tricky is that unbound variables cannot be used as keys for hashing etc.

As before, we basically rearrange the list AVs such that its elements have the same order as the variables in the list Xs (obtained from term_variables/2). The new idea here is to first compute a (ground) description of the required permutation (APs), and then construct the permutation of AV using this description.

avs_term_rearranged(AVs, T, AVRs) :-
    term_variables(T, Xs),
    copy_term(AVs-Xs, APs-Ps),
    ints(Ps, 0, N),
    functor(Array, a, N),
    distribute(AVs, APs, Array),
    gather(1, N, Array, AVRs).

ints([], N, N).
ints([I|Is], I0, N) :- I is I0+1, ints(Is, I, N).

distribute([], [], _).
distribute([AV|AVs], [_=P|APs], Array) :-
    nonvar(P),
    arg(P, Array, AV),
    distribute(AVs, APs, Array).

gather(I, N, Array, AVRs) :-
    ( I > N ->
        AVRs = []
    ;
        arg(I, Array, AV),
        I1 is I+1,
        ( var(AV) -> AVRs=AVRs1 ; AVRs = [AV|AVRs1] ),
        gather(I1, N, Array, AVRs1)
    ).

Use term_variables/2 on T to obtain a list Xs with variables in the desired order. Then build a list with the elements of AVs, but in that order.

avs_term_rearranged(AVs, T, AVRs) :-
    term_variables(T, Xs),
    pick_in_order(AVs, Xs, AVRs).

pick_in_order([], [], []).
pick_in_order(AVs, [X|Xs], AVRs) :-
    ( pick(AVs, X, AV, AVs1) ->
        AVRs = [AV|AVRs1],
        pick_in_order(AVs1, Xs, AVRs1)
    ;
        pick_in_order(AVs, Xs, AVRs)
    ).

pick([AV|AVs], X, AX, DAVs) :-
    (_=V) = AV,
    ( V==X ->
        AX = AV,
        DAVs = AVs
    ;
        DAVs = [AV|DAVs1],
        pick(AVs, X, AX, DAVs1)
    ).

Notes:

• this is quadratic because pick/4 is linear
• term_variables/2 is not strictly necessary, you could traverse T directly

This version is very short, using member/2 (from the Prolog prologue) for the search. It has very low auxiliary memory consumption. Only the list AVsR is created. No temporary heap-terms are created (on current implementations). It has quadratic overhead in the length of AVs, though.

avs_term_rearranged(AVs, T, AVsR) :-
   term_variables(T, Vs),
   rearrange(Vs, AVs, AVsR).

rearrange([], _, []).
rearrange([V|Vs], AVs, AVsR0) :-
   ( member(AV, AVs),
     AV = (_=Var), V == Var ->
      AVsR0 = [AV|AVsR]
   ;  AVsR0 = AVsR
   ),
   rearrange(Vs, AVs, AVsR).

Another aspect is that the member(AV, AVs) goal is inherently non-deterministic, even if only relatively shallow non-determinism is involved, whereas @jschimpf's (first) version does create a choice point only for the (;)/2 of the if-then-else-implementation. Both versions do not leave any choice points behind.

Here is a version more faithfully simulating @jschimpf's idea. This, however, now creates auxiliary terms that are left on the heap.

rearrange_js([], _, []).
rearrange_js([V|Vs], AVs0, AVsR0) :-
   ( select(AV, AVs0, AVs),
     AV = (_=Var), V == Var ->
      AVsR0 = [AV|AVsR]
   ;  AVsR0 = AVsR,
      AVs0 = AVs
   ),
   rearrange_js(Vs, AVs, AVsR).