Commit a85b73a8 authored by Guillaume Melquiond's avatar Guillaume Melquiond
Browse files

Rnd_NA_monotone

parent 221e165b
......@@ -10,6 +10,24 @@ apply Rge_minus.
now apply Rle_ge.
Qed.
Lemma Rabs_eq_Rabs :
forall x y : R,
Rabs x = Rabs y -> x = y \/ x = Ropp y.
Proof.
intros x y H.
unfold Rabs in H.
destruct (Rcase_abs x) as [_|_].
assert (H' := f_equal Ropp H).
rewrite Ropp_involutive in H'.
rewrite H'.
destruct (Rcase_abs y) as [_|_].
left.
apply Ropp_involutive.
now right.
rewrite H.
now destruct (Rcase_abs y) as [_|_] ; [right|left].
Qed.
Section Z2R.
Fixpoint P2R (p : positive) :=
......
......@@ -448,6 +448,68 @@ intros x Hx.
now apply Rnd_N_pt_involutive with F.
Qed.
Theorem Rnd_NA_pt_monotone :
forall F : R -> Prop,
F 0 ->
forall x y f g : R,
Rnd_NA_pt F x f -> Rnd_NA_pt F y g ->
x <= y -> f <= g.
Proof.
intros F HF x y f g (Hf,Hx) (Hg,Hy) [Hxy|Hxy].
now apply Rnd_N_pt_monotone with F x y.
apply Req_le.
rewrite <- Hxy in Hg, Hy.
clear y Hxy.
assert (K: f = g \/ f = -g).
apply Rabs_eq_Rabs.
apply Rle_antisym.
now apply Hy.
now apply Hx.
destruct K as [K|K].
exact K.
rewrite K.
rewrite K in Hf.
clear f Hx Hy K.
unfold Rnd_N_pt in Hf, Hg.
assert (L: g + x = g - x \/ g + x = x - g).
rewrite <- (Ropp_minus_distr g x).
apply Rabs_eq_Rabs.
rewrite <- Rabs_Ropp.
rewrite Ropp_plus_distr.
fold (-g - x).
apply Rle_antisym.
now apply Hf.
now apply Hg.
destruct L as [L|L].
assert (g = 0).
apply Rnd_N_pt_involutive with F.
replace 0 with x.
exact Hg.
apply Rmult_eq_reg_l with 2.
rewrite Rmult_0_r.
rewrite <- (Rminus_diag_eq _ _ L).
ring.
now apply (Z2R_neq 2 0).
exact HF.
rewrite H.
apply Ropp_0.
apply Rplus_eq_reg_l with x.
fold (x - g).
rewrite <- L.
apply Rplus_comm.
Qed.
Theorem Rnd_NA_monotone :
forall F : R -> Prop,
F 0 ->
forall rnd : R -> R,
Rnd_NA F rnd ->
MonotoneP rnd.
Proof.
intros F rnd Hr x y Hxy.
now apply Rnd_NA_pt_monotone with F.
Qed.
Theorem Rnd_NA_pt_involutive :
forall F : R -> Prop,
forall x f : R,
......
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