Commit ac82dd65 authored by Guillaume Melquiond's avatar Guillaume Melquiond
Browse files

Prove Znearest_imp.

parent 5fe900f3
......@@ -1747,6 +1747,25 @@ apply Rlt_le.
now apply Znearest_N_strict.
Theorem Znearest_imp :
forall x n,
(Rabs (x - Z2R n) < /2)%R ->
Znearest x = n.
intros x n Hd.
cut (Zabs (Znearest x - n) < 1)%Z.
clear ; zify ; omega.
apply lt_Z2R.
rewrite Z2R_abs, Z2R_minus.
replace (Z2R (Znearest x) - Z2R n)%R with (- (x - Z2R (Znearest x)) + (x - Z2R n))%R by ring.
apply Rle_lt_trans with (1 := Rabs_triang _ _).
replace R1 with (/2 + /2)%R by field.
apply Rplus_le_lt_compat with (2 := Hd).
rewrite Rabs_Ropp.
apply Znearest_N.
Theorem round_N_pt :
forall x,
Rnd_N_pt generic_format x (round Znearest x).
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