# Inequality with assumptions not simplifying

Can someone explain to me why the following does not simplify to True (as all constituents are positive)?

Most special cases including e=0 or a=b,... do simplify correctly.

What I eventually want to do is to simplify expressions of the kind

Abs[a+b/(c+(d-e)^2)],

which should evaluate to a+b/(c+(d-e)^2) under given assumptions.

Any workarounds?

Edit

expr[Abs[a+b/(c+(d-e)^2)]]/.Abs[x_]/;Simplify@Reduce[\$Assumptions&&x>0]:>x

It's not particularly pretty, but it works. Thanks to ssch for the idea!

-

One workaround is to use Reduce instead, and simplify that with FullSimplify:

assum=And @@ Thread[{a, b, c, d, e} > 0];
Reduce[assum && a + b/(c + (d - e)^2) > 0]
(* Out: e > 0 && d > 0 && c > 0 && b > 0 && a > 0 *)
FullSimplify[%, assum]
(* Out: True *)
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Good idea! Should have thought of that... – sebhofer Jul 9 '12 at 18:10
I just realized that this actually doesn't help me. What I would like is Mathematica to simplify Abs[a + b/(c + (d - e)^2)] for given assumptions. – sebhofer Jul 9 '12 at 18:25

Either of the below logical formulations will handle this. Generic simplifiers might or might not react well to logical expressions.

In[21]:= Resolve[
ForAll[{a, b, c, d, e},
Implies[And @@ Thread[{a, b, c, d, e} > 0],
a + b/(c + (d - e)^2) > 0]]]

Out[21]= True

In[20]:= Resolve[
ForAll[{a, b, c, d, e}, And @@ Thread[{a, b, c, d, e} > 0],
a + b/(c + (d - e)^2) > 0]]

Out[20]= True
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That's very interesting to know! I didn't even know Resolve existed. But is there also a way to simplify Abs[...]? Of course I could do a replacement and apply your trick (or ssch's) but that's not very elegant. – sebhofer Jul 9 '12 at 19:24
Might be a straightforward way but none is coming to mind right now. – Daniel Lichtblau Jul 9 '12 at 19:43