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authorGravatar Théo Zimmermann <theo.zimmermann@univ-paris-diderot.fr>2018-03-04 12:04:03 +0100
committerGravatar Théo Zimmermann <theo.zimmermann@univ-paris-diderot.fr>2018-03-04 12:04:03 +0100
commit39ec5b14518c35450aa0d9be2ab1012f5bd2f8e3 (patch)
tree6c9e4de49bfede9c18878beadf2b1700e20c037a /theories
parent6131f89f6b91c45e641dd877df8719fa77987453 (diff)
Remove all uses of Focus in standard library.
Diffstat (limited to 'theories')
-rw-r--r--theories/Reals/Ranalysis5.v62
1 files changed, 25 insertions, 37 deletions
diff --git a/theories/Reals/Ranalysis5.v b/theories/Reals/Ranalysis5.v
index 61c0debb7..de1e2f303 100644
--- a/theories/Reals/Ranalysis5.v
+++ b/theories/Reals/Ranalysis5.v
@@ -27,46 +27,34 @@ Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub,
(forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
(forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y).
Proof.
-intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub.
- assert (x_encad : f lb <= x <= f ub).
- split ; [assumption | apply Rle_trans with (r2:=y) ; [apply Rlt_le|] ; assumption].
- assert (y_encad : f lb <= y <= f ub).
- split ; [apply Rle_trans with (r2:=x) ; [|apply Rlt_le] ; assumption | assumption].
- assert (Temp1 : lb <= lb) by intuition ; assert (Temp2 : ub <= ub) by intuition.
- assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)).
- assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)).
- clear Temp1 Temp2.
- case (Rlt_dec (g x) (g y)).
- intuition.
+ intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub.
+ assert (x_encad : f lb <= x <= f ub) by lra.
+ assert (y_encad : f lb <= y <= f ub) by lra.
+ assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)).
+ assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)).
+ case (Rlt_dec (g x) (g y)); [ easy |].
intros Hfalse.
- assert (Temp := Rnot_lt_le _ _ Hfalse).
- assert (Hcontradiction : y <= x).
- replace y with (id y) by intuition ; replace x with (id x) by intuition ;
- rewrite <- f_eq_g. rewrite <- f_eq_g.
- assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y).
+ assert (Temp := Rnot_lt_le _ _ Hfalse).
+ enough (y <= x) by lra.
+ replace y with (id y) by easy.
+ replace x with (id x) by easy.
+ rewrite <- f_eq_g by easy.
+ rewrite <- f_eq_g by easy.
+ assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y). {
intros m n lb_le_m m_le_n n_lt_ub.
case (m_le_n).
- intros ; apply Rlt_le ; apply f_incr ; [| | apply Rlt_le] ; assumption.
- intros Hyp ; rewrite Hyp ; apply Req_le ; reflexivity.
- apply f_incr2.
- intuition. intuition.
- Focus 3. intuition.
- Focus 2. intuition.
- Focus 2. intuition. Focus 2. intuition.
- assert (Temp2 : g x <> ub).
- intro Hf.
- assert (Htemp : (comp f g) x = f ub).
- unfold comp ; rewrite Hf ; reflexivity.
- rewrite f_eq_g in Htemp ; unfold id in Htemp.
- assert (Htemp2 : x < f ub).
- apply Rlt_le_trans with (r2:=y) ; intuition.
- clear -Htemp Htemp2. fourier.
- intuition. intuition.
- clear -Temp2 gx_encad.
- case (proj2 gx_encad).
- intuition.
- intro Hfalse ; apply False_ind ; apply Temp2 ; assumption.
- apply False_ind. clear - Hcontradiction x_lt_y. fourier.
+ - intros; apply Rlt_le, f_incr, Rlt_le; assumption.
+ - intros Hyp; rewrite Hyp; apply Req_le; reflexivity.
+ }
+ apply f_incr2; intuition.
+ enough (g x <> ub) by lra.
+ intro Hf.
+ assert (Htemp : (comp f g) x = f ub). {
+ unfold comp; rewrite Hf; reflexivity.
+ }
+ rewrite f_eq_g in Htemp by easy.
+ unfold id in Htemp.
+ fourier.
Qed.
Lemma derivable_pt_id_interv : forall (lb ub x:R),