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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.
(* 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.
(* Motivating example from: Expressiveness + Automation + Soundness: Towards COmbining SMT Solvers and Interactive Proof Assistants *)
Parameter rho : Z.
Parameter rho_ge : rho >= 0.
Parameter correct : Z -> Z -> Prop.
Definition rbound1 (C:Z -> Z -> Z) : Prop :=
forall p s t, correct p t /\ s <= t -> C p t - C p s <= (1-rho)*(t-s).
Definition rbound2 (C:Z -> Z -> Z) : Prop :=
forall p s t, correct p t /\ s <= t ->  (1-rho)*(t-s) <= C p t - C p s.
(* 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.