Fix some deprecation warnings about ceil_ge (math-comp#1415)

* Fix some deprecation warnings about ceil_ge

reals/real_interval.v

@@ -274,7 +274,7 @@ rewrite in_itv/= => /andP[sx xs]; exists `|ceil (s - x)^-1|%N => //=.
 rewrite in_itv/= sx/= lerBrDl addrC -lerBrDl.
 rewrite -[in X in _ <= X](invrK (s - x)) ler_pV2.
 - rewrite -natr1 natr_absz ger0_norm; last first.
-    by rewrite -ceil_ge0 (lt_le_trans (ltrN10 R))// invr_ge0 subr_ge0 ltW.
+    by rewrite -(ceil0 R) ceil_le// invr_ge0 subr_ge0 ltW.
   by rewrite (@le_trans _ _ (ceil (s - x)^-1)%:~R)// ?lerDl// ceil_ge.
 - by rewrite inE unitfE ltr0n andbT pnatr_eq0.
 - by rewrite inE invr_gt0 subr_gt0 xs andbT unitfE invr_eq0 subr_eq0 gt_eqF.
@@ -298,9 +298,8 @@ Proof.
 apply/seteqP; split=> y; rewrite /= !in_itv/= andbT; last first.
   by move=> [k _ /=]; move: b => [|] /=; rewrite in_itv/= => /andP[//] /ltW.
 move=> xy; exists `|ceil (y - x)|%N => //=; rewrite in_itv/= xy/= -lerBlDl.
-rewrite !natr_absz/= ger0_norm -?ceil_ge0 ?ceil_ge//.
-rewrite (lt_le_trans (ltrN10 R))// subr_ge0.
-by case: b xy => //= /ltW.
+rewrite natr_absz ger0_norm ?ceil_ge//.
+by rewrite -(ceil0 R) ceil_le// subr_ge0 (lteifW xy).
 Qed.
 
 Lemma itv_infty_bnd_bigcup (R : realType) b (x : R) :
@@ -310,7 +309,7 @@ Proof.
 have /(congr1 (fun x => -%R @` x)) := itv_bnd_infty_bigcup (~~ b) (- x).
 rewrite opp_itv_bnd_infty negbK opprK => ->; rewrite image_bigcup.
 apply eq_bigcupr => k _; apply/seteqP; split=> [_ /= -[r rbxk <-]|y/= yxkb].
-   by rewrite oppr_itv/= opprB addrC.
+  by rewrite oppr_itv/= opprB addrC.
 by exists (- y); [rewrite oppr_itv/= negbK opprD opprK|rewrite opprK].
 Qed.
 

theories/lebesgue_integral.v

@@ -1548,19 +1548,15 @@ move=> Dx fxoo; have approx_x n : approx n x = n%:R.
   by rewrite fgen_A0 // ?mulr0 // fxoo leey.
 case/cvg_ex => /= l; have [l0|l0] := leP 0%R l.
 - move=> /cvgrPdist_lt/(_ _ ltr01) -[n _].
-  move=> /(_ (`|ceil l|.+1 + n)%N) /= /(_ (leq_addl _ _)).
-  rewrite approx_x.
-  apply/negP; rewrite -leNgt distrC (le_trans _ (lerB_normD _ _)) //.
-  rewrite normrN lerBrDl addSnnS [leRHS]ger0_norm ?ler0n//.
-  rewrite natrD lerD// ?ler1n// ger0_norm // (le_trans (ceil_ge _)) //.
-  by rewrite -(@gez0_abs (ceil _)) // -ceil_ge0 (lt_le_trans _ l0).
+  move=> /(_ (`|ceil l|.+1 + n)%N) /= /(_ (leq_addl _ _)); apply/negP.
+  rewrite -leNgt approx_x distrC (le_trans _ (lerB_normD _ _))// normrN.
+  rewrite lerBrDl addSnnS natrD [leRHS]ger0_norm// lerD ?ler1n// natr_absz.
+  by rewrite !ger0_norm ?le_ceil// -ceil_ge0; apply: lt_le_trans l0.
 - move=> /cvgrPdist_lt/(_ _ ltr01)[n _].
   move=> /(_ (`|floor l|.+1 + n)%N)/(_ (leq_addl _ _)); apply/negP.
-  rewrite approx_x -leNgt distrC (le_trans _ (lerB_normD _ _))//.
-  rewrite normrN lerBrDl addSnnS [leRHS]ger0_norm ?ler0n//.
-  rewrite natrD lerD ?ler1n// ltr0_norm// (@le_trans _ _ (- floor l)%:~R)//.
-    by rewrite mulrNz lerNl opprK ge_floor.
-  by rewrite -(@lez0_abs (floor _))// -floor_le0// (lt_le_trans l0).
+  rewrite approx_x -leNgt distrC (le_trans _ (lerB_normD _ _))// normrN.
+  rewrite lerBrDl addSnnS natrD [leRHS]ger0_norm// lerD ?ler1n// natr_absz.
+  by rewrite !ltr0_norm -?floor_lt0// mulrNz lerN2 ge_floor.
 Qed.
 
 Lemma ecvg_approx (f0 : forall x, D x -> (0 <= f x)%E) x :
@@ -2350,23 +2346,20 @@ transitivity (\int[mu]_(x in D) limn (g^~ x)).
   - rewrite gt0_mulye//; apply/cvgeyPgey; near=> M.
     have M0 : (0 <= M)%R by [].
     rewrite /g; case: (f x) fx0 => [r r0|_|//]; last first.
-      exists 1%N => // m /= m0.
-      by rewrite mulry gtr0_sg// ?mul1e ?leey// ltr0n.
+      by exists 1%N => // m /= m0; rewrite mulry gtr0_sg// ?ltr0n// mul1e leey.
     near=> n; rewrite lee_fin -ler_pdivrMr//.
     near: n; exists `|ceil (M / r)|%N => // m /=.
     rewrite -(ler_nat R); apply: le_trans.
-    rewrite natr_absz ger0_norm ?ceil_ge// -ceil_ge0// (lt_le_trans (ltrN10 _))//.
-    by rewrite divr_ge0// ?ltW.
+    rewrite natr_absz ger0_norm ?ceil_ge//.
+    by rewrite -(ceil0 R) ceil_le// divr_ge0// ltW.
   - rewrite lt0_mulye//; apply/cvgeNyPleNy; near=> M;
     have M0 : (M <= 0)%R by [].
     rewrite /g; case: (f x) fx0 => [r r0|//|_]; last first.
-      exists 1%N => // m /= m0.
-      by rewrite mulrNy gtr0_sg// ?ltr0n// mul1e ?leNye.
+      by exists 1%N => // m /= m0; rewrite mulrNy gtr0_sg// ?ltr0n// mul1e leNye.
     near=> n; rewrite lee_fin -ler_ndivrMr//.
     near: n; exists `|ceil (M / r)|%N => // m /=.
     rewrite -(ler_nat R); apply: le_trans.
-    rewrite natr_absz ger0_norm ?ceil_ge// -ceil_ge0// (lt_le_trans (ltrN10 _))//.
-    by rewrite -mulrNN mulr_ge0// lerNr oppr0// ltW// invr_lt0.
+    by rewrite pmulrn abszE ceil_ge_int ler_norm.
   - rewrite -fx0 mule0 /g -fx0.
     under eq_fun do rewrite mule0/=. (*TODO: notation broken*)
     exact: cvg_cst.
@@ -2387,9 +2380,8 @@ rewrite -lee_pdivrMr//; last first.
 near: n.
 exists `|ceil (M * (fine (\int[mu]_(x in D) f x))^-1)|%N => //.
 move=> n /=; rewrite -(@ler_nat R) -lee_fin; apply: le_trans.
-rewrite lee_fin natr_absz ger0_norm ?ceil_ge// -ceil_ge0//.
-rewrite (lt_le_trans (ltrN10 _))//.
-by rewrite mulr_ge0// ?invr_ge0//; exact/fine_ge0/integral_ge0.
+rewrite lee_fin natr_absz ger0_norm ?ceil_ge//.
+by rewrite -(ceil0 R) ceil_le// divr_ge0//; exact/fine_ge0/integral_ge0.
 Unshelve. all: by end_near. Qed.
 
 Lemma ge0_integralZr k : (forall x, D x -> 0 <= f x) ->
@@ -2627,19 +2619,17 @@ move=> muD0; pose g : (T -> \bar R)^nat := fun n => cst n%:R%:E.
 have <- : (fun t => limn (g^~ t)) = cst +oo.
   rewrite funeqE => t; apply/cvg_lim => //=.
   apply/cvgeryP/cvgryPge => M; exists `|ceil M|%N => //= m.
-  rewrite /= -(ler_nat R); apply: le_trans.
-  by rewrite (le_trans (ceil_ge _))// natr_absz ler_int ler_norm.
+  by rewrite /= pmulrn ceil_ge_int// -lez_nat abszE; apply/le_trans/ler_norm.
 rewrite monotone_convergence //.
 - under [in LHS]eq_fun do rewrite integral_cstr.
   apply/cvg_lim => //; apply/cvgeyPge => M.
   have [muDoo|muDoo] := ltP (mu D) +oo; last first.
     exists 1%N => // m /= m0; move: muDoo; rewrite leye_eq => /eqP ->.
     by rewrite mulry gtr0_sg ?mul1e ?leey// ltr0n.
   exists `|ceil (M / fine (mu D))|%N => // m /=.
-  rewrite -(ler_nat R) => MDm; rewrite -(@fineK _ (mu D)) ?ge0_fin_numE//.
+  rewrite -lez_nat abszE => MDm; rewrite -(@fineK _ (mu D)) ?ge0_fin_numE//.
   rewrite -lee_pdivrMr; last by rewrite fine_gt0// lt0e muD0 measure_ge0.
-  rewrite lee_fin (le_trans _ MDm)//.
-  by rewrite natr_absz (le_trans (ceil_ge _))// ler_int ler_norm.
+  by rewrite lee_fin pmulrn ceil_ge_int// (le_trans _ MDm)// ler_norm.
 - by move=> n; exact: measurable_cst.
 - by move=> n x Dx; rewrite lee_fin.
 - by move=> t Dt n m nm; rewrite /g lee_fin ler_nat.
@@ -3515,9 +3505,8 @@ apply/negP; rewrite -ltNge.
 rewrite -[X in _ * X](@fineK _ (mu (E `&` D))); last first.
   by rewrite fin_numElt muEDoo (lt_le_trans _ (measure_ge0 _ _)).
 rewrite lte_fin -ltr_pdivrMr.
-  rewrite -natr1 natr_absz ger0_norm.
-    by rewrite (le_lt_trans (ceil_ge _))// ltrDl.
-  by rewrite -ceil_ge0// (lt_le_trans (ltrN10 _))// divr_ge0.
+  rewrite pmulrn floor_lt_int intS ltz1D abszE.
+  by apply: le_trans (ler_norm _); rewrite ceil_floor//= lerDl.
 rewrite -lte_fin fineK.
   rewrite lt0e measure_ge0 andbT.
   suff: E `&` D = E by move=> ->; exact/eqP.
@@ -3780,9 +3769,8 @@ move=> mf; split=> [iDf0|Df0].
           by rewrite lt0e abse_ge0 abse_eq0 ft0 ltey.
         - by rewrite inE unitfE invr_eq0 pnatr_eq0 /= invr_gt0.
       rewrite invrK /m -natr1 natr_absz ger0_norm; last first.
-        by rewrite -ceil_ge0// (lt_le_trans (ltrN10 _)).
-      rewrite (@le_trans _ _ ((fine `|f t|)^-1 + 1)%R) ?lerDl//.
-      by rewrite lerD2r// ceil_ge.
+        by rewrite -(ceil0 R) ceil_le.
+      by rewrite intrD1 ceil_ge_int lerDl.
     by split => //; apply: contraTN nft => /eqP ->; rewrite abse0 -ltNge.
   transitivity (limn (fun n => mu (D `&` [set x | `|f x| >= n.+1%:R^-1%:E]))).
     apply/esym/cvg_lim => //; apply: nondecreasing_cvg_mu.

theories/lebesgue_measure.v

@@ -450,19 +450,15 @@ Proof.
 rewrite [X in measurable X](_ : _ =
   \bigcup_k (D `&` ([set  x | - k%:R%:E <= f x] `&` [set x | f x <= k%:R%:E]))).
   apply: bigcupT_measurable => k; rewrite -(setIid D) setIACA.
-  by apply: measurableI; [exact: emeasurable_fun_c_infty|
-                          exact: emeasurable_fun_infty_c].
+  exact/measurableI/emeasurable_fun_infty_c/emeasurable_fun_c_infty.
 rewrite predeqE => t; split => [/= [Dt ft]|].
-  have [ft0|ft0] := leP 0%R (fine (f t)).
-    exists `|ceil (fine (f t))|%N => //=; split => //; split.
-      by rewrite -{2}(fineK ft)// lee_fin (le_trans _ ft0)// lerNl oppr0.
-    rewrite natr_absz ger0_norm; last first.
-      by rewrite -ceil_ge0 (lt_le_trans _ ft0).
-    by rewrite -(fineK ft) lee_fin ceil_ge.
-  exists `|floor (fine (f t))|%N => //=; split => //; split.
-    rewrite natr_absz ltr0_norm -?floor_lt0// EFinN.
-    by rewrite -{2}(fineK ft) lee_fin mulrNz opprK ge_floor// ?num_real.
-  by rewrite -(fineK ft)// lee_fin (le_trans (ltW ft0)).
+  exists `|ceil `|fine (f t)| |%N => //=; split=> //; split.
+    rewrite -[leRHS](fineK ft) lee_fin lerNl pmulrn abszE ceil_ge_int ger0_norm.
+      by rewrite ceil_le// -normrN ler_norm.
+    by rewrite -(ceil0 R) ceil_le.
+  rewrite -[leLHS](fineK ft) lee_fin pmulrn abszE ceil_ge_int ger0_norm.
+    by rewrite ceil_le// ler_norm.
+  by rewrite -(ceil0 R) ceil_le.
 move=> [n _] [/= Dt [nft fnt]]; split => //; rewrite fin_numElt.
 by rewrite (lt_le_trans _ nft) ?ltNyr//= (le_lt_trans fnt)// ltry.
 Qed.
@@ -718,7 +714,7 @@ rewrite eqEsubset; split=> [_ -> i _/=|]; first by rewrite in_itv /= ltry.
 move=> [r| |/(_ O Logic.I)] // /(_ `|ceil r|%N Logic.I); rewrite /= in_itv /=.
 rewrite andbT lte_fin ltNge.
 have [r0|r0] := ltP 0%R r; last by rewrite (le_trans r0).
-by rewrite natr_absz gtr0_norm// ?ceil_ge// -ceil_gt0.
+by rewrite natr_absz gtr0_norm// ?le_ceil// -ceil_gt0.
 Qed.
 
 End erealwithrays.
@@ -2857,9 +2853,9 @@ have finite_set_F i : finite_set (F i).
     - by move=> /= x [n Fni Bnx]; exists n => //; exists i.
   have {CFi Fir2} := le_trans MC (le_trans CFi Fir2).
   apply/negP; rewrite -ltNge lebesgue_measure_ball// lte_fin.
-  rewrite -(@natr1 _ `| _ |%N) natr_absz ger0_norm; last first.
-    by rewrite -ceil_ge0// (lt_le_trans (ltrN10 _)).
-  by rewrite -ltr_pdivrMr// -ltrBlDr (lt_le_trans _ (ceil_ge _))// ltrBlDr ltrDl.
+  rewrite -[M%:R]natr1 natr_absz ger0_norm; last first.
+    by rewrite -(ceil0 R) ceil_le.
+  by rewrite -ltr_pdivrMr// intrD1 floor_lt_int ltzD1 ceil_floor// lerDl.
 have mur2_fin_num_ : mu (ball (0:R) (r%:num + 2))%R < +oo.
   by rewrite lebesgue_measure_ball// ltry.
 have FE : \sum_(n <oo) \esum_(i in F n) mu (closure (B i)) =

theories/normedtype.v

@@ -642,7 +642,7 @@ Proof.
 split=> [/cvgryPge|/cvgnyPge] Foo.
   by apply/cvgnyPge => A; near do rewrite -(@ler_nat R); apply: Foo.
 apply/cvgryPgey; near=> A; near=> n.
-rewrite (le_trans (@ceil_ge R A))// (ler_int _ _ (f n)) [ceil _]intEsign.
+rewrite pmulrn ceil_le_int// [ceil _]intEsign.
 by rewrite le_gtF ?expr0 ?mul1r ?lez_nat -?ceil_ge0//; near: n; apply: Foo.
 Unshelve. all: by end_near. Qed.
 

theories/realfun.v

@@ -246,13 +246,11 @@ have y_p : y_ n @[n --> \oo] --> p.
   apply/cvgrPdist_lt => e e0; near=> t.
   rewrite ltr0_norm// ?subr_lt0// opprB.
   rewrite /y_ /sval/=; case: cid2 => //= x /andP[_ + _].
-  rewrite ltrBlDr => /lt_le_trans; apply.
-  rewrite addrC lerD2r -(invrK e) lef_pV2// ?posrE ?invr_gt0//.
+  rewrite -ltrBlDl => /lt_le_trans; apply.
+  rewrite -(invrK e) lef_pV2// ?posrE ?invr_gt0//.
   near: t.
-  exists `|ceil e^-1|%N => // k /= ek.
-  rewrite (le_trans (ceil_ge _))// (@le_trans _ _ `|ceil e^-1|%:~R)//.
-    by rewrite ger0_norm -?ceil_ge0// (lt_le_trans (ltrN10 _))// invr_ge0// ltW.
-  by move: ek;rewrite -(leq_add2r 1) !addn1 -(ltr_nat R) => /ltW.
+  exists `|ceil e^-1|%N => // k /=; rewrite pmulrn ceil_ge_int// -lez_nat abszE.
+  by move=> /(le_trans (ler_norm _)) /le_trans; apply; rewrite lez_nat leqnSn.
 have /fine_cvgP[[m _ mfy_] /= _] := h _ (conj py_ y_p).
 near \oo => n.
 have mn : (m <= n)%N by near: n; exists m.