# How can I substitute something using only map?

For example, I have the equation

x^2 + y^2 == x*y


and I want to apply the rule

y -> s*x


I can do it easily by

x^2 + y^2 == x*y //. y -> s*x


But it seems to me it would be possible to use only Map to change everything. How can I do it that way? In other words, I think Replace is just a kind of syntactic sugar, and want to know how to express Replace by Map.

-
"I think Replace is just a kind of syntactic sugar" - no. No, it's not. – J. M. May 16 '13 at 2:00
@J.M. Agree. But perhaps it could be interesting if the OP tells us why he thinks it is – Dr. belisarius May 16 '13 at 2:04
@J.M Oh then they are totally different? – poorGuy May 16 '13 at 2:07
I think it would be possible if there is a "function" that changes y to s*x. It seems rules are not the function...... – poorGuy May 16 '13 at 2:09
Yes, they act quite differently; Map[f, {1, 2}] will apply the function f on the elements of the list {1, 2} to yield {f[1], f[2]}. That wors differently from Replace[{1, 2}, 1 -> 5, {1}] which replaces the 1 in {1, 2} with 5, to yield {5, 2}. – J. M. May 16 '13 at 2:22

As expressed in the comments, the Replace functions are not merely "syntactic sugar" for Map. The two are quite different. One primary difference is the order in which expressions are visited. See:
How to perform a depth-first preorder traversal of an expression? Another is that Replace will go inside held expressions, while Map does not evaluate:

Hold[1 + 2 + 3] /. {2 -> 7}

Hold[1 + 7 + 3]

Map[Print, Hold[1 + 2 + 3], {2}]

Hold[Print[1] + Print[2] + Print[3]]


With the exception of these important differences, one can do something resembling a ReplaceAll using MapAll, like this:

SetAttributes[f, HoldAll]

f[y] := s*x
f[other_] := other

f //@ (x^2+y^2==x*y)

x^2 + s^2 x^2 == s x^2

-
Nevertheless, the last use of //@ just feels like using a pipe wrench to whack nails... – J. M. May 16 '13 at 3:22
Oh god Thank you! – poorGuy May 16 '13 at 3:29