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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.
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