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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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Open Scope Z_scope.
(* false in Q : x=1/2 and n=1 *)
Goal forall a1 da na b1 db nb,
a1 * da = na ->
b1 * db = nb  ->
a1 * b1 *  da * db = na * nb.
(* From Laurent Théry *)
Goal forall (x y : Z), x = 0 -> x * y = 0.
Goal forall (x y : Z), x = 0 -> x * y = 0.
Goal forall (x y : Z), 2*x = 0 -> x * y = 0.
Goal forall (x y: Z), - x*x >= 0 -> x * y = 0.
(* Rule of signs *)
(* Other (simple) examples *)
(* from hol_light/Examples/sos.ml *)
(* ------------------------------------------------------------------------- *)
(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
(* ------------------------------------------------------------------------- *)

Lemma hol_light24 : forall x1 y1 x2 y2, x1 >= 0 -> x2 >= 0 -> y1 >= 0 -> y2 >= 0 ->
((x1 + y1) ^2 + x1 + 1 = (x2 + y2) ^ 2 + x2 + 1)
-> (x1 + y1 = x2 + y2).