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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 *)

Lemma hol_light12: forall x y z,
2 <= x /\ x <= 125841 / 50000 /\
2 <= y /\ y <= 125841 / 50000 /\
2 <= z /\ z <= 125841 / 50000
-> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= 0.