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Simplify, SetDelayed and Condition ... again

This is True

--- True

Now define a function f[x] that f[x]:=3x/;x>0

1st try


--- f[x]>0
--- f[x]>0
--- f[x]>0

2nd try


--- f[x]>0
--- f[x]>0

3st try

>FullSimplify[f[x]>0,x>0] (* 1 *)

--- f[x]>0
--- f[x]>0

but ...


--- True

FullSimplify is not hold. Why do not work (* 1 *)?


And the really question ... I need to modify Condition
to work better? Or there are another way without
FullSimplify in Condition?

For example, define a norm that




--- Abs[z]

but fail


--- norm[z]


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