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(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import Lra.
Require Import Reals.
Open Scope R_scope.
Lemma cst_test : 5^5 = 5 * 5 * 5 *5 *5.
Proof.
lra.
Qed.
Lemma yplus_minus : forall x y,
0 = x + y -> 0 = x -y -> 0 = x /\ 0 = y.
Proof.
intros.
lra.
Qed.
(* Other (simple) examples *)
Lemma binomial : forall x y, ((x+y)^2 = x^2 + 2 *x*y + y^2).
Proof.
intros.
lra.
Qed.
Lemma hol_light19 : forall m n, 2 * m + n = (n + m) + m.
Proof.
intros ; lra.
Qed.
Lemma vcgen_25 : forall
(n : R)
(m : R)
(jt : R)
(j : R)
(it : R)
(i : R)
(H0 : 1 * it + (-2%R ) * i + (-1%R ) = 0)
(H : 1 * jt + (-2 ) * j + (-1 ) = 0)
(H1 : 1 * n + (-10 ) = 0)
(H2 : 0 <= (-4028 ) * i + (6222 ) * j + (705 ) * m + (-16674 ))
(H3 : 0 <= (-418 ) * i + (651 ) * j + (94 ) * m + (-1866 ))
(H4 : 0 <= (-209 ) * i + (302 ) * j + (47 ) * m + (-839 ))
(H5 : 0 <= (-1 ) * i + 1 * j + (-1 ))
(H6 : 0 <= (-1 ) * j + 1 * m + (0 ))
(H7 : 0 <= (1 ) * j + (5 ) * m + (-27 ))
(H8 : 0 <= (2 ) * j + (-1 ) * m + (2 ))
(H9 : 0 <= (7 ) * j + (10 ) * m + (-74 ))
(H10 : 0 <= (18 ) * j + (-139 ) * m + (1188 ))
(H11 : 0 <= 1 * i + (0 ))
(H13 : 0 <= (121 ) * i + (810 ) * j + (-7465 ) * m + (64350 )),
(( 1 ) = (-2 ) * i + it).
Proof.
intros.
lra.
Qed.
Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
Proof.
intros.
psatz R 3.
Qed.
Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
Proof.
intros.
nra.
Qed.
Lemma motzkin' : forall x y, (x^2+y^2+1)*(x^2*y^4 + x^4*y^2 + 1 - (3 ) *x^2*y^2) >= 0.
Proof.
intros ; psatz R 2.
Qed.
Lemma l1 : forall x y z : R, Rabs (x - z) <= Rabs (x - y) + Rabs (y - z).
intros; split_Rabs; lra.
Qed.
(* Bug 5073 *)
Lemma opp_eq_0_iff a : -a = 0 <-> a = 0.
Proof.
lra.
Qed.
(* From L. Théry *)
Goal forall (x y : R), x = 0 -> x * y = 0.
Proof.
intros.
nra.
Qed.
Goal forall (x y : R), 2*x = 0 -> x * y = 0.
Proof.
intros.
nra.
Qed.
Goal forall (x y: R), - x*x >= 0 -> x * y = 0.
Proof.
intros.
nra.
Qed.
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