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+Require Export Gappa_tactic.
+Require Export Reals.
+
+Open Scope Z_scope.
+Open Scope R_scope.
+
+Lemma test_base10 : 
+  forall x y:R,
+  0 <= x <= 4 -> 
+  0 <= x * (24 * powerRZ 10 (-1)) <= 10.
+Proof.
+  gappa.
+Qed.
+
+(*
+@rnd = float< ieee_32, zr >;
+a = rnd(a_); b = rnd(b_);
+{ a in [3.2,3.3] /\ b in [1.4,1.9] ->
+  rnd(a - b) - (a - b) in [0,0] }
+*)
+
+Definition rnd := gappa_rounding (rounding_float roundZR 43 (120)).
+
+Lemma test_float3 :
+  forall a_ b_ a b : R,
+  a = rnd a_ ->
+  b = rnd b_ ->
+  52 / 16 <= a <= 53 / 16 ->
+  22 / 16 <= b <= 30 / 16 ->
+  0 <= rnd (a - b) - (a - b) <= 0.
+Proof.
+  unfold rnd.
+  gappa.
+Qed.
+
+Lemma test_float2 :
+  forall x y:R,
+  0 <= x <= 1 ->
+  0 <= y <= 1 ->
+  0 <= gappa_rounding (rounding_float roundNE 53 (1074)) (x+y) <= 2. 
+Proof.
+  gappa.
+Qed.
+
+Lemma test_float1 :
+  forall x y:R,
+  0 <= gappa_rounding (rounding_fixed roundDN (0)) x -
+           gappa_rounding (rounding_fixed roundDN (0)) y <= 0 ->
+  Rabs (x - y) <= 1.
+Proof.
+  gappa.
+Qed.
+
+Lemma test1 : 
+  forall x y:R,
+  0 <= x <= 1 -> 
+  0 <= -y <= 1 ->  
+  0 <= x  * (-y) <= 1.
+Proof.
+  gappa.
+Qed.
+
+Lemma test2 : 
+  forall x y:R,
+  3/4 <= x <= 3 -> 
+  0 <= sqrt x <= 1775 * (powerRZ 2 (-10)).
+Proof.
+  gappa.
+Qed.
+
+Lemma test3 : 
+  forall x y z:R,
+  0 <= x - y <= 3 -> 
+  -2 <= y - z <= 4 ->
+  -2 <= x - z <= 7.
+Proof.
+  gappa.
+Qed.
+
+Lemma test4 : 
+  forall x1 x2 y1 y2 : R,
+  1 <= Rabs y1 <= 1000 ->
+  1 <= Rabs y2 <= 1000 ->
+  - powerRZ 2 (-53) <= (x1 - y1) / y1 <= powerRZ 2 (-53) ->
+  - powerRZ 2 (-53) <= (x2 - y2) / y2 <= powerRZ 2 (-53) ->
+  - powerRZ 2 (-51) <= (x1 * x2 - y1 * y2) / (y1 * y2) <= powerRZ 2 (-51).
+Proof.
+  gappa.
+Qed.
+
+