(************************************************************************)
(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
(*                                                                      *)
(************************************************************************)
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 *)
(* ------------------------------------------------------------------------- *)
(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
(* ------------------------------------------------------------------------- *)

Lemma hol_light16 : forall x y,
0 <= x /\ 0 <= y /\ (x * y = 1)
-> x + y <= x ^ 2 + y ^ 2.