simplifying expressions - Problem with Simplify, Sqrt, and Set: What's going on?

I am having some trouble getting Mathematica to simplify an expression of the form Sqrt[x]*Sqrt[1/x], where x>0. The problem is that x is assigned to some complicated form by the time Mathematica encounters it, and it fails to recognize that it will simplify. While debugging this problem, I wrote the following code that fails to simplify only when x gets sufficiently complicated. The cases involving x0,x1,x2,x3 will all simplify, but the x4 case will not. What's going on here?

x0 = a;
x1 = a + b;
x2 = a + b*c;
x3 = a + b*c*d;
x4 = a + b*c*d*e;
Simplify[1 == Sqrt[a] Sqrt[1/a], x0 > 0]
Simplify[1 == Sqrt[a + b] Sqrt[1/(a + b)], x1 > 0]

Simplify[1 == Sqrt[a + b*c] Sqrt[1/(a + b*c)], x2 > 0]
Simplify[1 == Sqrt[a + b*c*d] Sqrt[1/(a + b*c*d)], x3 > 0]
Simplify[1 == Sqrt[a + b*c*d*e] Sqrt[1/(a + b*c*d*e)], x4 > 0]

Answer

Probably some internal weirdness with the ComplexityFunction, but:

Simplify[Sqrt[1/(a + b c d e  )] Sqrt[a + b c d e ]==1] // PowerExpand

Simplify[1 == Sqrt[a + b*c*d*e] Sqrt[1/(a + b*c*d*e)], x4 > 0] // PowerExpand

(* 

   True
   True
*)

Comments

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