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(* Micromega: A reflexive tactic using the Positivstellensatz           *)
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(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
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(* Other (simple) examples *)
Open Scope Q_scope.
Open Scope Z_scope.
Open Scope Q_scope.
Goal forall x : Q, x * x >= 0.
intro; nra.
Qed.
Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.

Lemma motzkin' : forall x y, (x^2+y^2+1)*(x^2*y^4 + x^4*y^2 + 1 - (3 # 1) *x^2*y^2) >= 0.