Updating files + reorganizing everything

21 files changed, 1444 insertions, 629 deletions

diff --git a/mathcomp/solvable/Make b/mathcomp/solvable/Make
index 41bd3e0..d89d213 100644
--- a/mathcomp/solvable/Make
+++ b/mathcomp/solvable/Make
@@ -1,9 +1,10 @@
 abelian.v
-all.v
+all_solvable.v
 alt.v
 burnside_app.v
 center.v
 commutator.v
+cyclic.v
 extraspecial.v
 extremal.v
 finmodule.v
diff --git a/mathcomp/solvable/abelian.v b/mathcomp/solvable/abelian.v
index 52f92ec..0d87755 100644
--- a/mathcomp/solvable/abelian.v
+++ b/mathcomp/solvable/abelian.v
@@ -1,14 +1,13 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div fintype finfun bigop finset prime binomial.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism action quotient gproduct.
-From mathcomp.algebra
-Require Import cyclic zmodp.
-Require Import gfunctor pgroup gseries nilpotent sylow.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq path div fintype.
+From mathcomp
+Require Import finfun bigop finset prime binomial fingroup morphism perm.
+From mathcomp
+Require Import automorphism action quotient gfunctor gproduct zmodp cyclic.
+From mathcomp
+Require Import pgroup gseries nilpotent sylow.
 
 (******************************************************************************)
 (* Constructions based on abelian groups and their structure, with some       *)
@@ -245,7 +244,7 @@ Lemma exponent1 : exponent [1 gT] = 1%N.
 Proof. by apply/eqP; rewrite -dvdn1 -trivg_exponent eqxx. Qed.
 
 Lemma exponent_dvdn G : exponent G %| #|G|.
-Proof. by apply/dvdn_biglcmP=> x Gx; exact: order_dvdG. Qed.
+Proof. by apply/dvdn_biglcmP=> x Gx; apply: order_dvdG. Qed.
 
 Lemma exponent_gt0 G : 0 < exponent G.
 Proof. exact: dvdn_gt0 (exponent_dvdn G). Qed.
@@ -257,7 +256,7 @@ congr (_ && _); first by rewrite cardG_gt0 exponent_gt0.
 apply: eq_all_r => p; rewrite !mem_primes cardG_gt0 exponent_gt0 /=.
 apply: andb_id2l => p_pr; apply/idP/idP=> pG.
   exact: dvdn_trans pG (exponent_dvdn G).
-by case/Cauchy: pG => // x Gx <-; exact: dvdn_exponent.
+by case/Cauchy: pG => // x Gx <-; apply: dvdn_exponent.
 Qed.
 
 Lemma exponentJ A x : exponent (A :^ x) = exponent A.
@@ -284,7 +283,7 @@ Lemma exponent_cycle x : exponent <[x]> = #[x].
 Proof. by apply/eqP; rewrite eqn_dvd exponent_dvdn dvdn_exponent ?cycle_id. Qed.
 
 Lemma exponent_cyclic X : cyclic X -> exponent X = #|X|.
-Proof. by case/cyclicP=> x ->; exact: exponent_cycle. Qed.
+Proof. by case/cyclicP=> x ->; apply: exponent_cycle. Qed.
 
 Lemma primes_exponent G : primes (exponent G) = primes (#|G|).
 Proof.
@@ -320,7 +319,7 @@ Lemma exponent_Hall pi G H : pi.-Hall(G) H -> exponent H = (exponent G)`_pi.
 Proof.
 move=> hallH; have [sHG piH _] := and3P hallH.
 rewrite -(partn_exponentS sHG) -?(card_Hall hallH) ?part_pnat_id //.
-by apply: pnat_dvd piH; exact: exponent_dvdn.
+by apply: pnat_dvd piH; apply: exponent_dvdn.
 Qed.
 
 Lemma exponent_Zgroup G : Zgroup G -> exponent G = #|G|.
@@ -371,7 +370,7 @@ Proof.
 move=> cGG; apply/group_setP.
 split=> [|x y]; rewrite !inE ?group1 ?expg1n //=.
 case/andP=> Gx /eqP xn /andP[Gy /eqP yn].
-rewrite groupM //= expgMn ?xn ?yn ?mulg1 //; exact: (centsP cGG).
+by rewrite groupM //= expgMn ?xn ?yn ?mulg1 //; apply: (centsP cGG).
 Qed.
 
 Lemma abelian_exponent_gen A : abelian A -> exponent <<A>> = exponent A.
@@ -414,7 +413,7 @@ Qed.
 Lemma cyclic_abelem_prime p X : p.-abelem X -> cyclic X -> X :!=: 1 -> #|X| = p.
 Proof.
 move=> abelX cycX; case/cyclicP: cycX => x -> in abelX *.
-by rewrite cycle_eq1; exact: abelem_order_p abelX (cycle_id x).
+by rewrite cycle_eq1; apply: abelem_order_p abelX (cycle_id x).
 Qed.
 
 Lemma cycle_abelem p x : p.-elt x || prime p -> p.-abelem <[x]> = (#[x] %| p).
@@ -423,7 +422,7 @@ move=> p_xVpr; rewrite /abelem cycle_abelian /=.
 apply/andP/idP=> [[_ xp1] | x_dvd_p].
   by rewrite order_dvdn (exponentP xp1) ?cycle_id.
 split; last exact: dvdn_trans (exponent_dvdn _) x_dvd_p.
-by case/orP: p_xVpr => // /pnat_id; exact: pnat_dvd.
+by case/orP: p_xVpr => // /pnat_id; apply: pnat_dvd.
 Qed.
 
 Lemma exponent2_abelem G : exponent G %| 2 -> 2.-abelem G.
@@ -494,7 +493,7 @@ by rewrite (is_abelem_pgroup (abelem_pgroup abelG)).
 Qed.
 
 Lemma pElemP p A E : reflect (E \subset A /\ p.-abelem E) (E \in 'E_p(A)).
-Proof. by rewrite inE; exact: andP. Qed.
+Proof. by rewrite inE; apply: andP. Qed.
 Implicit Arguments pElemP [p A E].
 
 Lemma pElemS p A B : A \subset B -> 'E_p(A) \subset 'E_p(B).
@@ -602,7 +601,7 @@ apply/and3P; split; last 1 first.
 - apply/imsetP=> [[X /card_p1Elem oX X'0]].
   by rewrite -oX (cardsD1 1) -X'0 group1 cards0 in p_pr.
 - rewrite eqEsubset; apply/andP; split.
-    by apply/bigcupsP=> _ /imsetP[X /pnElemP[sXE _ _] ->]; exact: setSD.
+    by apply/bigcupsP=> _ /imsetP[X /pnElemP[sXE _ _] ->]; apply: setSD.
   apply/subsetP=> x /setD1P[ntx Ex].
   apply/bigcupP; exists <[x]>^#; last by rewrite !inE ntx cycle_id.
   apply/imsetP; exists <[x]>%G; rewrite ?p1ElemE // !inE cycle_subG Ex /=.
@@ -680,7 +679,7 @@ Qed.
 Lemma pmaxElemP p A E :
   reflect (E \in 'E_p(A) /\ forall H, H \in 'E_p(A) -> E \subset H -> H :=: E)
           (E \in 'E*_p(A)).
-Proof. by rewrite [E \in 'E*_p(A)]inE; exact: (iffP maxgroupP). Qed.
+Proof. by rewrite [E \in 'E*_p(A)]inE; apply: (iffP maxgroupP). Qed.
 
 Lemma pmaxElem_exists p A D :
   D \in 'E_p(A) -> {E | E \in 'E*_p(A) & D \subset E}.
@@ -716,7 +715,7 @@ Proof.
 move=> sAB; apply/subsetP=> E; rewrite !inE.
 case/andP=> /maxgroupP[/pElemP[_ abelE] maxE] sEA.
 apply/maxgroupP; rewrite inE sEA; split=> // D EpD.
-by apply: maxE; apply: subsetP EpD; exact: pElemS.
+by apply: maxE; apply: subsetP EpD; apply: pElemS.
 Qed.
 
 Lemma pmaxElemJ p A E x : ((E :^ x)%G \in 'E*_p(A :^ x)) = (E \in 'E*_p(A)).
@@ -736,7 +735,7 @@ Qed.
 Lemma grank_witness G : {B | <<B>> = G & #|B| = 'm(G)}.
 Proof.
 rewrite /gen_rank; case: arg_minP => [|B defG _]; first by rewrite genGid.
-by exists B; first exact/eqP.
+by exists B; first apply/eqP.
 Qed.
 
 Lemma p_rank_witness p G : {E | E \in 'E_p^('r_p(G))(G)}.
@@ -751,7 +750,7 @@ Proof.
 apply: (iffP idP) => [|[E]]; last first.
   by rewrite inE => /andP[Ep_E /eqP <-]; rewrite (bigmax_sup E).
 have [D /pnElemP[sDG abelD <-]] := p_rank_witness p G.
-by case/abelem_pnElem=> // E; exists E; exact: (subsetP (pnElemS _ _ sDG)).
+by case/abelem_pnElem=> // E; exists E; apply: (subsetP (pnElemS _ _ sDG)).
 Qed.
 
 Lemma p_rank_gt0 p H : ('r_p(H) > 0) = (p \in \pi(H)).
@@ -772,7 +771,7 @@ Proof. by move=> EpA_E; rewrite (bigmax_sup E). Qed.
 Lemma p_rank_le_logn p G : 'r_p(G) <= logn p #|G|.
 Proof.
 have [E EpE] := p_rank_witness p G.
-by have [sEG _ <-] := pnElemP EpE; exact: lognSg.
+by have [sEG _ <-] := pnElemP EpE; apply: lognSg.
 Qed.
 
 Lemma p_rank_abelem p G : p.-abelem G -> 'r_p(G) = logn p #|G|.
@@ -931,7 +930,7 @@ Qed.
 
 Lemma morphim_Ldiv n A : f @* 'Ldiv_n(A) \subset 'Ldiv_n(f @* A).
 Proof.
-by apply: subset_trans (morphimI f A _) (setIS _ _); exact: morphim_LdivT.
+by apply: subset_trans (morphimI f A _) (setIS _ _); apply: morphim_LdivT.
 Qed.
 
 Lemma morphim_abelem p G : p.-abelem G -> p.-abelem (f @* G).
@@ -984,7 +983,7 @@ by rewrite sub_morphim_pre ?morphim_sub // morphpre_invm.
 Qed.
 
 Lemma injm_abelem p : p.-abelem (f @* G) = p.-abelem G.
-Proof. by apply/idP/idP; first rewrite -{2}defG; exact: morphim_abelem. Qed.
+Proof. by apply/idP/idP; first rewrite -{2}defG; apply: morphim_abelem. Qed.
 
 Lemma injm_pElem p (E : {group aT}) :
   E \subset D -> ((f @* E)%G \in 'E_p(f @* G)) = (E \in 'E_p(G)).
@@ -1255,7 +1254,7 @@ move=> pG; apply/eqP; rewrite eqEsubset !gen_subG; apply/andP.
 do [split; apply/subsetP=> xpn; case/imsetP=> x] => [|Gx ->]; last first.
   by rewrite Mho_p_elt ?(mem_p_elt pG).
 case/setIdP=> Gx _ ->; have [-> | ntx] := eqVneq x 1; first by rewrite expg1n.
-by rewrite (pdiv_p_elt (mem_p_elt pG Gx) ntx) mem_gen //; exact: mem_imset.
+by rewrite (pdiv_p_elt (mem_p_elt pG Gx) ntx) mem_gen //; apply: mem_imset.
 Qed.
 
 Lemma MhoEabelian p G :
@@ -1264,7 +1263,7 @@ Proof.
 move=> pG cGG; rewrite (MhoE pG); rewrite gen_set_id //; apply/group_setP.
 split=> [|xn yn]; first by apply/imsetP; exists 1; rewrite ?expg1n.
 case/imsetP=> x Gx ->; case/imsetP=> y Gy ->.
-by rewrite -expgMn; [apply: mem_imset; rewrite groupM | exact: (centsP cGG)].
+by rewrite -expgMn; [apply: mem_imset; rewrite groupM | apply: (centsP cGG)].
 Qed.
 
 Lemma trivg_Mho G : 'Mho^n(G) == 1 -> 'Ohm_n(G) == G.
@@ -1274,14 +1273,14 @@ rewrite -{1}(Sylow_gen G) genS //; apply/bigcupsP=> P.
 case/SylowP=> p p_pr /and3P[sPG pP _]; apply/subsetP=> x Px.
 have Gx := subsetP sPG x Px; rewrite inE Gx //=.
 rewrite (sameP eqP set1P) (subsetP Gp1) ?mem_gen //; apply: mem_imset.
-by rewrite inE Gx; exact: pgroup_p (mem_p_elt pP Px).
+by rewrite inE Gx; apply: pgroup_p (mem_p_elt pP Px).
 Qed.
 
 Lemma Mho_p_cycle p x : p.-elt x -> 'Mho^n(<[x]>) = <[x ^+ (p ^ n)]>.
 Proof.
 move=> p_x.
 apply/eqP; rewrite (MhoE p_x) eqEsubset cycle_subG mem_gen; last first.
-  by apply: mem_imset; exact: cycle_id.
+  by apply: mem_imset; apply: cycle_id.
 rewrite gen_subG andbT; apply/subsetP=> _ /imsetP[_ /cycleP[k ->] ->].
 by rewrite -expgM mulnC expgM mem_cycle.
 Qed.
@@ -1332,7 +1331,7 @@ Proof. exact: morphimF. Qed.
 
 Lemma injm_Ohm (f : {morphism D >-> rT}) :
   'injm f -> G \subset D -> f @* 'Ohm_n(G) = 'Ohm_n(f @* G).
-Proof. by move=> injf; exact: injmF. Qed.
+Proof. by move=> injf; apply: injmF. Qed.
 
 Lemma isog_Ohm (H : {group rT}) : G \isog H -> 'Ohm_n(G) \isog 'Ohm_n(H).
 Proof. exact: gFisog. Qed.
@@ -1425,7 +1424,7 @@ Lemma abelem_Ohm1P p G :
   abelian G -> p.-group G -> reflect ('Ohm_1(G) = G) (p.-abelem G).
 Proof.
 move=> cGG pG.
-apply: (iffP idP) => [| <-]; [exact: Ohm1_id | exact: Ohm1_abelem].
+by apply: (iffP idP) => [| <-]; [apply: Ohm1_id | apply: Ohm1_abelem].
 Qed.
 
 Lemma TI_Ohm1 G H : H :&: 'Ohm_1(G) = 1 -> H :&: G = 1.
@@ -1699,7 +1698,7 @@ have{lnOx} lnOy i: lnO i <[x]> = lnO i <[y]>.
   have co_yx': coprime #[y] #[x.`_p^'] by rewrite !order_constt coprime_partC.
   have defX: <[y]> \x <[x.`_p^']> = <[x]>.
     rewrite dprodE ?coprime_TIg //.
-      by rewrite -cycleM ?consttC //; apply: (centsP cXX); exact: mem_cycle.
+      by rewrite -cycleM ?consttC //; apply: (centsP cXX); apply: mem_cycle.
     by apply: (sub_abelian_cent2 cXX); rewrite cycle_subG mem_cycle.
   rewrite -(lnOx i _ _ _ defX) addnC {1}/lnO lognE.
   case: and3P => // [[p_pr _ /idPn[]]]; rewrite -p'natE //.
@@ -1760,7 +1759,7 @@ apply/eqP; rewrite -def_t size_map eqn_leq andbC; apply/andP; split.
 case/lastP def_b: b => // [b' x]; pose p := pdiv #[x].
 have p_pr: prime p.
   have:= abelian_type_gt1 G; rewrite -def_t def_b map_rcons -cats1 all_cat.
-  by rewrite /= andbT => /andP[_]; exact: pdiv_prime.
+  by rewrite /= andbT => /andP[_]; apply: pdiv_prime.
 suffices: all [pred y | logn p #[y] > 0] b.
   rewrite all_count (count_logn_dprod_cycle _ _ defG) -def_b; move/eqP <-.
   by rewrite Ohm0 indexg1 -p_rank_abelian ?p_rank_le_rank.
@@ -1791,9 +1790,9 @@ case p_x: (p_group <[x]>); last first.
   rewrite -order_gt1 /p_group -orderE; set p := pdiv _ => ntx p'x.
   have def_x: <[x.`_p]> \x <[x.`_p^']> = <[x]>.
     have ?: coprime #[x.`_p] #[x.`_p^'] by rewrite !order_constt coprime_partC.
-    have ?: commute x.`_p x.`_p^' by exact: commuteX2.
+    have ?: commute x.`_p x.`_p^' by apply: commuteX2.
     rewrite dprodE ?coprime_TIg -?cycleM ?consttC //.
-    by rewrite cent_cycle cycle_subG; exact/cent1P.
+    by rewrite cent_cycle cycle_subG; apply/cent1P.
   rewrite -(dprod_card (Ohm_dprod n def_x)) -(dprod_card (Mho_dprod n def_x)).
   rewrite mulnCA -mulnA mulnCA mulnA.
   rewrite !{}IHm ?(dprod_card def_x) ?(leq_trans _ lexm) {m lexm}//.
@@ -1801,7 +1800,7 @@ case p_x: (p_group <[x]>); last first.
     rewrite p_part -(expn0 p) ltn_exp2l 1?lognE ?prime_gt1 ?pdiv_prime //.
     by rewrite order_gt0 pdiv_dvd.
   rewrite proper_card // properEneq cycle_subG mem_cycle andbT.
-  by apply: contra (negbT p'x); move/eqP <-; exact: p_elt_constt.
+  by apply: contra (negbT p'x); move/eqP <-; apply: p_elt_constt.
 case/p_groupP: p_x => p p_pr p_x.
 rewrite (Ohm_p_cycle n p_x) (Mho_p_cycle n p_x) -!orderE.
 set k := logn p #[x]; have ox: #[x] = (p ^ k)%N by rewrite -card_pgroup.
@@ -1865,7 +1864,7 @@ Lemma homocyclic_Ohm_Mho n p G :
   p.-group G -> homocyclic G -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G).
 Proof.
 move=> pG /andP[cGG homoG]; set e := exponent G.
-have{pG} p_e: p.-nat e by apply: pnat_dvd pG; exact: exponent_dvdn.
+have{pG} p_e: p.-nat e by apply: pnat_dvd pG; apply: exponent_dvdn.
 have{homoG}: all (pred1 e) (abelian_type G).
   move: homoG; rewrite /abelian_type -(prednK (cardG_gt0 G)) /=.
   by case: (_ && _) (tag _); rewrite //= genGid eqxx.
@@ -1884,7 +1883,7 @@ Lemma Ohm_Mho_homocyclic (n p : nat) G :
 Proof.
 set e := exponent G => cGG pG /andP[n_gt0 n_lte] eq_Ohm_Mho.
 suffices: all (pred1 e) (abelian_type G).
-  by rewrite /homocyclic cGG; exact: all_pred1_constant.
+  by rewrite /homocyclic cGG; apply: all_pred1_constant.
 case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <-.
 elim: b {-3}G defG (subxx G) eq_Ohm_Mho => //= x b IHb H.
 rewrite big_cons => defG; case/dprodP: defG (defG) => [[_ K _ defK]].
@@ -2081,7 +2080,7 @@ case: (ltnP e _) (b_sorted p) => [lt_e_x | le_x_e].
   move: ordb; rewrite (take_nth 1 ltib) -/x rev_rcons all_rcons /= lt_e_x.
   case/andP=> _ /=; move/(order_path_min leq_trans); apply: contraLR.
   rewrite -!has_predC !has_map; case/hasP=> y b_y /= le_y_e; apply/hasP.
-  by exists y; rewrite ?mem_rev //=; apply: contra le_y_e; exact: leq_trans.
+  by exists y; rewrite ?mem_rev //=; apply: contra le_y_e; apply: leq_trans.
 rewrite -(cat_take_drop i b) -map_rev rev_cat !map_cat cat_path.
 case/andP=> ordb _; rewrite count_cat -{1}(size_takel (ltnW ltib)) ltnNge.
 rewrite addnC ((count _ _ =P 0) _) ?count_size //.
diff --git a/mathcomp/solvable/all.v b/mathcomp/solvable/all_solvable.v
index 8c2ea8f..b05a3c4 100644
--- a/mathcomp/solvable/all.v
+++ b/mathcomp/solvable/all_solvable.v
@@ -3,6 +3,7 @@ Require Export alt.
 Require Export burnside_app.
 Require Export center.
 Require Export commutator.
+Require Export cyclic.
 Require Export extraspecial.
 Require Export extremal.
 Require Export finmodule.
diff --git a/mathcomp/solvable/alt.v b/mathcomp/solvable/alt.v
index 3ee2526..18a6588 100644
--- a/mathcomp/solvable/alt.v
+++ b/mathcomp/solvable/alt.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div fintype tuple finfun bigop prime finset.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism quotient action.
-From mathcomp.algebra
-Require Import cyclic.
-Require Import pgroup gseries sylow primitive_action.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div fintype tuple.
+From mathcomp
+Require Import tuple bigop prime finset fingroup morphism perm automorphism.
+From mathcomp
+Require Import quotient action cyclic pgroup gseries sylow primitive_action.
 
 (******************************************************************************)
 (*  Definitions of the symmetric and alternate groups, and some properties.   *)
@@ -189,10 +186,10 @@ have FF (H : {group {perm T}}): H <| 'Alt_T -> H :<>: 1 -> 20 %| #|H|.
 - move=> Hh1 Hh3.
   have [x _]: exists x, x \in T by apply/existsP/eqP; rewrite oT.
   have F2 := Alt_trans T; rewrite oT /= in F2.
-  have F3: [transitive 'Alt_T, on setT | 'P] by exact: ntransitive1 F2.
-  have F4: [primitive 'Alt_T, on setT | 'P] by exact: ntransitive_primitive F2.
+  have F3: [transitive 'Alt_T, on setT | 'P] by apply: ntransitive1 F2.
+  have F4: [primitive 'Alt_T, on setT | 'P] by apply: ntransitive_primitive F2.
   case: (prim_trans_norm F4 Hh1) => F5.
-    case: Hh3; apply/trivgP; exact: subset_trans F5 (aperm_faithful _).
+    by case: Hh3; apply/trivgP; apply: subset_trans F5 (aperm_faithful _).
   have F6: 5 %| #|H| by rewrite -oT -cardsT (atrans_dvd F5).
   have F7: 4 %| #|H|.
     have F7: #|[set~ x]| = 4 by rewrite cardsC1 oT.
@@ -206,10 +203,10 @@ have FF (H : {group {perm T}}): H <| 'Alt_T -> H :<>: 1 -> 20 %| #|H|.
       exact: ntransitive1 F9.
     have F11: [primitive Gx, on [set~ x] | 'P].
       exact: ntransitive_primitive F9.
-    have F12: K \subset Gx by apply: setSI; exact: normal_sub.
+    have F12: K \subset Gx by apply: setSI; apply: normal_sub.
     have F13: K <| Gx by rewrite /(K <| _) F12 normsIG // normal_norm.
     case: (prim_trans_norm F11 F13) => Ksub; last first.
-      apply: dvdn_trans (cardSg F8); rewrite -F7; exact: atrans_dvd Ksub.
+      by apply: dvdn_trans (cardSg F8); rewrite -F7; apply: atrans_dvd Ksub.
     have F14: [faithful Gx, on [set~ x] | 'P].
       apply/subsetP=> g; do 2![case/setIP] => Altg cgx cgx'.
       apply: (subsetP (aperm_faithful 'Alt_T)).
@@ -293,7 +290,7 @@ Lemma rfd_funP (p : {perm T}) (u : T') :
   let p1 := if p x == x then p else 1 in p1 (val u) != x.
 Proof.
 case: (p x =P x) => /= [pxx|_]; last by rewrite perm1 (valP u).
-by rewrite -{2}pxx (inj_eq (@perm_inj _ p)); exact: (valP u).
+by rewrite -{2}pxx (inj_eq (@perm_inj _ p)); apply: (valP u).
 Qed.
 
 Definition rfd_fun p := [fun u => Sub ((_ : {perm T}) _) (rfd_funP p u) : T'].
@@ -366,7 +363,7 @@ have Hcp1: #|[set x | p1 x != x]| <= n.
   by move: Hp; rewrite (cardD1 x1) inE Hx1.
 have ->: p = p1 * tperm x1 (p x1) by rewrite -mulgA tperm2 mulg1.
 rewrite odd_permM odd_tperm eq_sym Hx1 morphM; last 2 first.
-- by rewrite 2!inE; exact/astab1P.
+- by rewrite 2!inE; apply/astab1P.
 - by rewrite 2!inE; apply/astab1P; rewrite -{1}Hpx /= /aperm -permM.
 rewrite odd_permM Hrec //=; congr (_ (+) _).
 pose x2 : T' := Sub x1 nx1x; pose px2 : T' := Sub (p x1) npx1x.
@@ -441,7 +438,7 @@ have F11: [primitive Gx, on [set~ x] | 'P].
 have F12: K \subset Gx by rewrite setSI // normal_sub.
 have F13: K <| Gx by apply/andP; rewrite normsIG.
 have:= prim_trans_norm F11; case/(_ K) => //= => Ksub; last first.
-  have F14: Gx * H = 'Alt_T by exact/(subgroup_transitiveP _ _ F3).
+  have F14: Gx * H = 'Alt_T by apply/(subgroup_transitiveP _ _ F3).
   have: simple Gx.
     by rewrite (isog_simple (rfd_iso x)) Hrec //= card_sig cardC1 Hde.
   case/simpleP=> _ simGx; case/simGx: F13 => /= HH2.
@@ -470,7 +467,7 @@ have Hreg g z: g \in H -> g z = z -> g = 1.
   by rewrite memJ_norm ?(subsetP nH).
 clear K F8 F12 F13 Ksub F14.
 have Hcard: 5 < #|H|.
-  apply: (leq_trans oT); apply dvdn_leq; first by exact: cardG_gt0.
+  apply: (leq_trans oT); apply dvdn_leq; first by apply: cardG_gt0.
   by rewrite -cardsT (atrans_dvd F5).
 case Eh: (pred0b [predD1 H & 1]).
   by move: Hcard; rewrite /pred0b in Eh; rewrite (cardD1 1) group1 (eqP Eh).
@@ -491,7 +488,7 @@ case diff_gx_hx: (g x == h x).
   by rewrite permM -(eqP diff_gx_hx) -permM mulgV perm1.
 case diff_hgx_x: ((h * g) x == x).
   case/negP: diff_h1_g; apply/eqP; apply: (mulgI h); rewrite !gsimp.
-  by apply: (Hreg _ x); [exact: groupM | apply/eqP].
+  by apply: (Hreg _ x); [apply: groupM | apply/eqP].
 case diff_hgx_hx: ((h * g) x == h x).
   case/negP: diff_1_g; apply/eqP.
   by apply: (Hreg _ (h x)) => //; apply/eqP; rewrite -permM.
diff --git a/mathcomp/solvable/burnside_app.v b/mathcomp/solvable/burnside_app.v
index 635fd84..e8fe7dc 100644
--- a/mathcomp/solvable/burnside_app.v
+++ b/mathcomp/solvable/burnside_app.v
@@ -1,12 +1,9 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div choice fintype tuple finfun bigop finset.
-From mathcomp.fingroup
-Require Import fingroup action perm.
-Require Import primitive_action.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
+From mathcomp
+Require Import tuple finfun bigop finset fingroup action perm primitive_action.
 
 (*   Application of the Burside formula to count the number of distinct       *)
 (* colorings of the vertices of a square and a cube.                          *)
@@ -120,7 +117,7 @@ Qed.
 Lemma rot_r1 : forall r, is_rot r -> r = r1 ^+ (r c0).
 Proof.
 move=> r hr;apply: rot_eq_c0 => //;apply/eqP.
-   by symmetry; exact: commuteX.
+   by symmetry; apply: commuteX.
 by case: (r c0); do 4?case => //=; rewrite ?permM !permE  /=.
 Qed.
 
@@ -304,12 +301,12 @@ Definition square_coloring_number4 := #|orbit to rotations @: setT|.
 Definition square_coloring_number8 := #|orbit to isometries @: setT|.
 
 Lemma Fid : 'Fix_to(1) = setT.
-Proof. apply/setP=> x /=; rewrite in_setT; apply/afix1P; exact: act1. Qed.
+Proof. by apply/setP=> x /=; rewrite in_setT; apply/afix1P; apply: act1. Qed.
 
 Lemma card_Fid : #|'Fix_to(1)| = (n ^ 4)%N.
 Proof.
 rewrite -[4]card_ord -[n]card_ord -card_ffun_on Fid cardsE.
-by symmetry; apply: eq_card => f; exact/ffun_onP.
+by symmetry; apply: eq_card => f; apply/ffun_onP.
 Qed.
 
 Definition coin0 (sc : col_squares) : colors := sc c0.
@@ -691,7 +688,7 @@ Qed.
 Definition sop (p : {perm cube}) : seq cube := val (val (val p)).
 
 Lemma sop_inj : injective sop.
-Proof. do 2!apply: (inj_comp val_inj); exact: val_inj. Qed.
+Proof. by do 2!apply: (inj_comp val_inj); apply: val_inj. Qed.
 
 Definition prod_tuple (t1 t2 : seq cube) :=
   map (fun n : 'I_6 => nth F0 t2 n) t1.
@@ -728,7 +725,7 @@ Definition seq_iso_L := [::
 Lemma seqs1 : forall f injf, sop (@perm _ f injf) = map f ecubes.
 Proof.
 move=> f ?; rewrite ecubes_def /sop /= -codom_ffun pvalE.
-apply: eq_codom; exact: permE.
+by apply: eq_codom; apply: permE.
 Qed.
 
 Lemma Lcorrect : seq_iso_L == map sop [:: id3; s05; s14; s23; r05; r14; r23;
@@ -741,7 +738,7 @@ Proof. by move=> p; rewrite /= !inE /= -!(eq_sym p). Qed.
 
 Lemma L_iso : forall p, (p \in dir_iso3) = (sop p \in seq_iso_L).
 Proof.
-move=> p; rewrite (eqP Lcorrect) mem_map ?iso0_1 //; exact: sop_inj.
+by move=> p; rewrite (eqP Lcorrect) mem_map ?iso0_1 //; apply: sop_inj.
 Qed.
 
 Lemma stable : forall x y,
@@ -831,7 +828,7 @@ Canonical iso_group3 := Group group_set_iso3.
 Lemma group_set_diso3 : group_set  dir_iso3.
 Proof.
 apply/group_setP;split;first by   rewrite inE eqxx /=.
-by exact:stable.
+by apply:stable.
 Qed.
 Canonical diso_group3 := Group group_set_diso3.
 
@@ -862,8 +859,8 @@ have -> : s3 = r05 * r14 * r05 by iso_tac.
 have -> : s4 = r05 * r14  * r14 * r14 * r05 by iso_tac.
 have -> : s5 = r14  * r05 * r05 by iso_tac.
 have -> : s6 = r05 * r05 * r14 by iso_tac.
-do ?case/predU1P=> [<-|]; first exact: group1; last (move/eqP => <-);
-   by rewrite ?groupMl ?mem_gen // !inE eqxx ?orbT.
+by do ?case/predU1P=> [<-|]; first exact: group1; last (move/eqP => <-);
+   rewrite ?groupMl ?mem_gen // !inE eqxx ?orbT.
 Qed.
 
 Notation col_cubes := {ffun cube -> colors}.
@@ -887,7 +884,7 @@ Proof. by apply/setP=> x /=; rewrite (sameP afix1P eqP) !inE act1 eqxx. Qed.
 Lemma card_Fid3 : #|'Fix_to_g[1]| = (n ^ 6)%N.
 Proof.
 rewrite -[6]card_ord -[n]card_ord -card_ffun_on Fid3 cardsT.
-by symmetry; apply: eq_card => ff; exact/ffun_onP.
+by symmetry; apply: eq_card => ff; apply/ffun_onP.
 Qed.
 
 Definition col0 (sc : col_cubes) : colors := sc F0.
diff --git a/mathcomp/solvable/center.v b/mathcomp/solvable/center.v
index 6226861..c0c0451 100644
--- a/mathcomp/solvable/center.v
+++ b/mathcomp/solvable/center.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div fintype bigop finset.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism quotient action gproduct.
-From mathcomp.algebra
-Require Import cyclic.
-Require Import gfunctor.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div fintype bigop.
+From mathcomp
+Require Import finset fingroup morphism perm automorphism quotient action.
+From mathcomp
+Require Import gproduct gfunctor cyclic.
 
 (******************************************************************************)
 (* Definition of the center of a group and of external central products:      *)
@@ -63,7 +60,7 @@ Notation "''Z' ( A )" := (center A) : group_scope.
 Notation "''Z' ( H )" := (center_group H) : Group_scope.
 
 Lemma morphim_center : GFunctor.pcontinuous center.
-Proof. move=> gT rT G D f; exact: morphim_subcent. Qed.
+Proof. by move=> gT rT G D f; apply: morphim_subcent. Qed.
 
 Canonical center_igFun := [igFun by fun _ _ => subsetIl _ _ & morphim_center].
 Canonical center_gFun := [gFun by morphim_center].
@@ -103,8 +100,8 @@ Proof. exact: subcentP. Qed.
 Lemma center_sub A : 'Z(A) \subset A.
 Proof. exact: subsetIl. Qed.
 
-Lemma center1 : 'Z(1) = [1 gT].
-Proof. by apply/eqP; rewrite eqEsubset center_sub sub1G. Qed.
+Lemma center1 : 'Z(1) = 1 :> {set gT}. 
+Proof. exact: gF1. Qed.
 
 Lemma centerC A : {in A, centralised 'Z(A)}.
 Proof. by apply/centsP; rewrite centsC subsetIr. Qed.
@@ -137,13 +134,13 @@ by apply: (iffP cent1P) => [|[]].
 Qed.
 
 Lemma subcent1_id x G : x \in G -> x \in 'C_G[x].
-Proof. move=> Gx; rewrite inE Gx; exact/cent1P. Qed.
+Proof. by move=> Gx; rewrite inE Gx; apply/cent1P. Qed.
 
 Lemma subcent1_sub x G : 'C_G[x] \subset G.
 Proof. exact: subsetIl. Qed.
 
 Lemma subcent1C x y G : x \in G -> y \in 'C_G[x] -> x \in 'C_G[y].
-Proof. by move=> Gx /subcent1P[_ cxy]; exact/subcent1P. Qed.
+Proof. by move=> Gx /subcent1P[_ cxy]; apply/subcent1P. Qed.
 
 Lemma subcent1_cycle_sub x G : x \in G -> <[x]> \subset 'C_G[x].
 Proof. by move=> Gx; rewrite cycle_subG ?subcent1_id. Qed.
@@ -171,9 +168,9 @@ Lemma cyclic_factor_abelian H G :
   H \subset 'Z(G) -> cyclic (G / H) -> abelian G.
 Proof.
 move=> sHZ cycGH; apply: cyclic_center_factor_abelian.
-have nG: G \subset 'N(_) := normal_norm (sub_center_normal _).
-have [f <-]:= homgP (homg_quotientS (nG _ sHZ) (nG _ (subxx _)) sHZ).
-exact: morphim_cyclic.
+have /andP[_ nHG]: H <| G := sub_center_normal sHZ.
+have [f <-]:= homgP (homg_quotientS nHG (gFnorm _ G) sHZ).
+exact: morphim_cyclic cycGH.
 Qed.
 
 Section Injm.
@@ -293,7 +290,7 @@ Let kerHK := ker_cprod_by isoZ.
 
 Let injgz : 'injm gz. Proof. by case/isomP: isoZ. Qed.
 Let gzZ : gz @* 'Z(H) = 'Z(K). Proof. by case/isomP: isoZ. Qed.
-Let gzZchar : gz @* 'Z(H) \char 'Z(K). Proof. by rewrite gzZ char_refl. Qed.
+Let gzZchar : gz @* 'Z(H) \char 'Z(K). Proof. by rewrite gzZ. Qed.
 Let sgzZZ : gz @* 'Z(H) \subset 'Z(K) := char_sub gzZchar.
 Let sZH := center_sub H.
 Let sZK := center_sub K.
@@ -557,7 +554,7 @@ by split=> //; apply/isomP; rewrite ker_f' injm_invm im_f' -fC im_invm.
 Qed.
 
 Lemma isog_cprod_by : G \isog C.
-Proof. by have [f [isoG _ _]] := cprod_by_uniq; exact: isom_isog isoG. Qed.
+Proof. by have [f [isoG _ _]] := cprod_by_uniq; apply: isom_isog isoG. Qed.
 
 End Isomorphism.
 
@@ -651,7 +648,7 @@ Lemma Aut_ncprod_full n :
 Proof.
 move=> AutZinG; elim: n => [|n IHn].
   by rewrite center_ncprod0; apply/Aut_sub_fullP=> // g injg gG0; exists g.
-by case: ncprodS => gz isoZ; exact: Aut_cprod_by_full.
+by case: ncprodS => gz isoZ; apply: Aut_cprod_by_full.
 Qed.
 
 End IterCprod.
diff --git a/mathcomp/solvable/commutator.v b/mathcomp/solvable/commutator.v
index 6f6a2b5..7a42a9a 100644
--- a/mathcomp/solvable/commutator.v
+++ b/mathcomp/solvable/commutator.v
@@ -1,12 +1,9 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat.
-From mathcomp.discrete
-Require Import fintype bigop finset prime binomial.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient.
-Require Import gfunctor.
+From mathcomp
+Require Import ssrfun ssrbool eqtype ssrnat fintype bigop finset.
+From mathcomp
+Require Import binomial fingroup morphism automorphism quotient gfunctor.
 
 (******************************************************************************)
 (*   This files contains the proofs of several key properties of commutators, *)
@@ -17,8 +14,8 @@ Require Import gfunctor.
 (*           G^`(0) ==  G                                                     *)
 (*       G^`(n.+1) == [~: G^`(n), G^`(n)]                                     *)
 (* as several classical results involve the (first) derived group G^`(1),     *)
-(* such as the equivalence H <| G /\ G / H abelian <-> G^`(1) \subset H. The  *)
-(* connection between the derived series and solvable groups will only be     *)
+(* such as the equivalence H <| G /\ G / H abelian <-> G^`(1) \subset H.      *)
+(* The connection between the derived series and solvable groups will only be *)
 (* established in nilpotent.v, however.                                       *)
 (******************************************************************************)
 
@@ -49,7 +46,7 @@ Lemma derg1 A : A^`(1) = [~: A, A]. Proof. by []. Qed.
 Lemma dergSn n A : A^`(n.+1) = [~: A^`(n), A^`(n)]. Proof. by []. Qed.
 
 Lemma der_group_set G n : group_set G^`(n).
-Proof. by case: n => [|n]; exact: groupP. Qed.
+Proof. by case: n => [|n]; apply: groupP. Qed.
 
 Canonical derived_at_group G n := Group (der_group_set G n).
 
@@ -130,7 +127,7 @@ Variables (i j : nat) (x y : gT).
 Hypotheses (cxz : commute x [~ x, y]) (cyz : commute y [~ x, y]).
 
 Lemma commXXg : [~ x ^+ i, y ^+ j] = [~ x, y] ^+ (i * j).
-Proof. rewrite expgM commgX commXg //; exact: commuteX. Qed.
+Proof. by rewrite expgM commgX commXg //; apply: commuteX. Qed.
 
 Lemma expMg_Rmul : (y * x) ^+ i = y ^+ i * x ^+ i * [~ x, y] ^+ 'C(i, 2).
 Proof.
@@ -176,10 +173,10 @@ Lemma commg_normal G H : [~: G, H] <| G <*> H.
 Proof. by rewrite /(_ <| _) commg_sub commg_norm. Qed.
 
 Lemma normsRl A G B : A \subset G -> A \subset 'N([~: G, B]).
-Proof. by move=> sAG; exact: subset_trans (commg_norml G B). Qed.
+Proof. by move=> sAG; apply: subset_trans (commg_norml G B). Qed.
 
 Lemma normsRr A G B : A \subset G -> A \subset 'N([~: B, G]).
-Proof. by move=> sAG; exact: subset_trans (commg_normr G B). Qed.
+Proof. by move=> sAG; apply: subset_trans (commg_normr G B). Qed.
 
 Lemma commg_subr G H : ([~: G, H] \subset H) = (G \subset 'N(H)).
 Proof.
@@ -225,7 +222,7 @@ Lemma sub_der1_normal G H : G^`(1) \subset H -> H \subset G -> H <| G.
 Proof. by move=> sG'H sHG; rewrite /(H <| G) sHG sub_der1_norm. Qed.
 
 Lemma sub_der1_abelian G H : G^`(1) \subset H -> abelian (G / H).
-Proof. by move=> sG'H; exact: quotient_cents2r. Qed.
+Proof. by move=> sG'H; apply: quotient_cents2r. Qed.
 
 Lemma der1_min G H : G \subset 'N(H) -> abelian (G / H) -> G^`(1) \subset H.
 Proof. by move=> nHG abGH; rewrite -quotient_cents2. Qed.
@@ -341,7 +338,7 @@ by rewrite !dergSn -IHn morphimR ?(subset_trans (der_sub n G)).
 Qed.
 
 Lemma dergS n G H : G \subset H -> G^`(n) \subset H^`(n).
-Proof. by move=> sGH; elim: n => // n IHn; exact: commgSS. Qed.
+Proof. by move=> sGH; elim: n => // n IHn; apply: commgSS. Qed.
 
 Lemma quotient_der n G H : G \subset 'N(H) -> G^`(n) / H = (G / H)^`(n).
 Proof. exact: morphim_der. Qed.
diff --git a/mathcomp/solvable/cyclic.v b/mathcomp/solvable/cyclic.v
new file mode 100644
index 0000000..be77dd9
--- /dev/null
+++ b/mathcomp/solvable/cyclic.v
@@ -0,0 +1,869 @@
+(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
+Require Import mathcomp.ssreflect.ssreflect.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div fintype bigop.
+From mathcomp
+Require Import prime finset fingroup morphism perm automorphism quotient.
+From mathcomp
+Require Import gproduct ssralg finalg zmodp poly.
+
+(******************************************************************************)
+(*  Properties of cyclic groups.                                              *)
+(* Definitions:                                                               *)
+(* Defined in fingroup.v:                                                     *)
+(*           <[x]> == the cycle (cyclic group) generated by x.                *)
+(*            #[x] == the order of x, i.e., the cardinal of <[x]>.            *)
+(* Defined in prime.v:                                                        *)
+(*       totient n == Euler's totient function                                *)
+(* Definitions in this file:                                                  *)
+(*        cyclic G <=> G is a cyclic group.                                   *)
+(*    metacyclic G <=> G is a metacyclic group (i.e., a cyclic extension of a *)
+(*                     cyclic group).                                         *)
+(*   generator G x <=> x is a generator of the (cyclic) group G.              *)
+(*           Zpm x == the isomorphism mapping the additive group of integers  *)
+(*                    mod #[x] to the cyclic group <[x]>.                     *)
+(*      cyclem x n == the endomorphism y |-> y ^+ n of <[x]>.                 *)
+(*      Zp_unitm x == the isomorphism mapping the multiplicative group of the *)
+(*                    units of the ring of integers mod #[x] to the group of  *)
+(*                    automorphisms of <[x]> (i.e., Aut <[x]>).               *)
+(*                    Zp_unitm x maps u to cyclem x u.                        *)
+(*    eltm dvd_y_x == the smallest morphism (with domain <[x]>) mapping x to  *)
+(*                    y, given a proof dvd_y_x : #[y] %| #[x].                *)
+(*   expg_invn G k == if coprime #|G| k, the inverse of exponent k in G.      *)
+(* Basic results for these notions, plus the classical result that any finite *)
+(* group isomorphic to a subgroup of a field is cyclic, hence that Aut G is   *)
+(* cyclic when G is of prime order.                                           *)
+(******************************************************************************)
+
+Set Implicit Arguments.
+Unset Strict Implicit.
+Unset Printing Implicit Defensive.
+
+Import GroupScope GRing.Theory.
+
+(***********************************************************************)
+(*  Cyclic groups.                                                     *)
+(***********************************************************************)
+
+Section Cyclic.
+
+Variable gT : finGroupType.
+Implicit Types (a x y : gT) (A B : {set gT}) (G K H : {group gT}).
+
+Definition cyclic A := [exists x, A == <[x]>].
+
+Lemma cyclicP A : reflect (exists x, A = <[x]>) (cyclic A).
+Proof. exact: exists_eqP. Qed.
+
+Lemma cycle_cyclic x : cyclic <[x]>.
+Proof. by apply/cyclicP; exists x. Qed.
+
+Lemma cyclic1 : cyclic [1 gT].
+Proof. by rewrite -cycle1 cycle_cyclic. Qed.
+
+(***********************************************************************)
+(* Isomorphism with the additive group                                 *)
+(***********************************************************************)
+
+Section Zpm.
+
+Variable a : gT.
+
+Definition Zpm (i : 'Z_#[a]) := a ^+ i.
+
+Lemma ZpmM : {in Zp #[a] &, {morph Zpm : x y / x * y}}.
+Proof.
+rewrite /Zpm; case: (eqVneq a 1) => [-> | nta] i j _ _.
+  by rewrite !expg1n ?mulg1.
+by rewrite /= {3}Zp_cast ?order_gt1 // expg_mod_order expgD.
+Qed.
+
+Canonical Zpm_morphism := Morphism ZpmM.
+
+Lemma im_Zpm : Zpm @* Zp #[a] = <[a]>.
+Proof.
+apply/eqP; rewrite eq_sym eqEcard cycle_subG /= andbC morphimEdom.
+rewrite (leq_trans (leq_imset_card _ _)) ?card_Zp //= /Zp order_gt1.
+case: eqP => /= [a1 | _]; first by rewrite imset_set1 morph1 a1 set11.
+by apply/imsetP; exists 1%R; rewrite ?expg1 ?inE.
+Qed.
+
+Lemma injm_Zpm : 'injm Zpm.
+Proof.
+apply/injmP/dinjectiveP/card_uniqP.
+rewrite size_map -cardE card_Zp //= {7}/order -im_Zpm morphimEdom /=.
+by apply: eq_card => x; apply/imageP/imsetP=> [] [i Zp_i ->]; exists i.
+Qed.
+
+Lemma eq_expg_mod_order m n : (a ^+ m == a ^+ n) = (m == n %[mod #[a]]).
+Proof.
+have [->|] := eqVneq a 1; first by rewrite order1 !modn1 !expg1n eqxx.
+rewrite -order_gt1 => lt1a; have ZpT: Zp #[a] = setT by rewrite /Zp lt1a.
+have: injective Zpm by move=> i j; apply (injmP injm_Zpm); rewrite /= ZpT inE.
+move/inj_eq=> eqZ; symmetry; rewrite -(Zp_cast lt1a).
+by rewrite -[_ == _](eqZ (inZp m) (inZp n)) /Zpm /= Zp_cast ?expg_mod_order.
+Qed.
+
+Lemma Zp_isom : isom (Zp #[a]) <[a]> Zpm.
+Proof. by apply/isomP; rewrite injm_Zpm im_Zpm. Qed.
+
+Lemma Zp_isog : isog (Zp #[a]) <[a]>.
+Proof. exact: isom_isog Zp_isom. Qed.
+
+End Zpm.
+
+(***********************************************************************)
+(*        Central and direct product of cycles                         *)
+(***********************************************************************)
+
+Lemma cyclic_abelian A : cyclic A -> abelian A.
+Proof. by case/cyclicP=> a ->; apply: cycle_abelian. Qed.
+
+Lemma cycleMsub a b :
+  commute a b -> coprime #[a] #[b] -> <[a]> \subset <[a * b]>.
+Proof.
+move=> cab co_ab; apply/subsetP=> _ /cycleP[k ->].
+apply/cycleP; exists (chinese #[a] #[b] k 0); symmetry.
+rewrite expgMn // -expg_mod_order chinese_modl // expg_mod_order.
+by rewrite /chinese addn0 -mulnA mulnCA expgM expg_order expg1n mulg1.
+Qed.
+
+Lemma cycleM a b :
+  commute a b -> coprime #[a] #[b] -> <[a * b]> = <[a]> * <[b]>.
+Proof.
+move=> cab co_ab; apply/eqP; rewrite eqEsubset -(cent_joinEl (cents_cycle cab)).
+rewrite join_subG {3}cab !cycleMsub // 1?coprime_sym //.
+by rewrite -genM_join cycle_subG mem_gen // mem_imset2 ?cycle_id.
+Qed.
+
+Lemma cyclicM A B :
+    cyclic A -> cyclic B -> B \subset 'C(A) -> coprime #|A| #|B| ->
+  cyclic (A * B).
+Proof.
+move=> /cyclicP[a ->] /cyclicP[b ->]; rewrite cent_cycle cycle_subG => cab coab.
+by rewrite -cycleM ?cycle_cyclic //; apply/esym/cent1P.
+Qed.
+
+Lemma cyclicY K H :
+    cyclic K -> cyclic H -> H \subset 'C(K) -> coprime #|K| #|H| ->
+  cyclic (K <*> H).
+Proof. by move=> cycK cycH cKH coKH; rewrite cent_joinEr // cyclicM. Qed.
+
+(***********************************************************************)
+(*        Order properties                                             *)
+(***********************************************************************)
+
+Lemma order_dvdn a n : #[a] %| n = (a ^+ n == 1).
+Proof. by rewrite (eq_expg_mod_order a n 0) mod0n. Qed.
+
+Lemma order_inf a n : a ^+ n.+1 == 1 -> #[a] <= n.+1.
+Proof. by rewrite -order_dvdn; apply: dvdn_leq. Qed.
+
+Lemma order_dvdG G a : a \in G -> #[a] %| #|G|.
+Proof. by move=> Ga; apply: cardSg; rewrite cycle_subG. Qed.
+
+Lemma expg_cardG G a : a \in G -> a ^+ #|G| = 1.
+Proof. by move=> Ga; apply/eqP; rewrite -order_dvdn order_dvdG. Qed.
+
+Lemma expg_znat G x k : x \in G -> x ^+ (k%:R : 'Z_(#|G|))%R = x ^+ k.
+Proof.
+case: (eqsVneq G 1) => [-> /set1P-> | ntG Gx]; first by rewrite !expg1n.
+apply/eqP; rewrite val_Zp_nat ?cardG_gt1 // eq_expg_mod_order.
+by rewrite modn_dvdm ?order_dvdG.
+Qed.
+
+Lemma expg_zneg G x (k : 'Z_(#|G|)) : x \in G -> x ^+ (- k)%R = x ^- k.
+Proof.
+move=> Gx; apply/eqP; rewrite eq_sym eq_invg_mul -expgD.
+by rewrite -(expg_znat _ Gx) natrD natr_Zp natr_negZp subrr.
+Qed.
+
+Lemma nt_gen_prime G x : prime #|G| -> x \in G^# -> G :=: <[x]>.
+Proof.
+move=> Gpr /setD1P[]; rewrite -cycle_subG -cycle_eq1 => ntX sXG.
+apply/eqP; rewrite eqEsubset sXG andbT.
+by apply: contraR ntX => /(prime_TIg Gpr); rewrite (setIidPr sXG) => ->.
+Qed.
+
+Lemma nt_prime_order p x : prime p -> x ^+ p = 1 -> x != 1 -> #[x] = p.
+Proof.
+move=> p_pr xp ntx; apply/prime_nt_dvdP; rewrite ?order_eq1 //.
+by rewrite order_dvdn xp.
+Qed.
+
+Lemma orderXdvd a n : #[a ^+ n] %| #[a].
+Proof. by apply: order_dvdG; apply: mem_cycle. Qed.
+
+Lemma orderXgcd a n : #[a ^+ n] = #[a] %/ gcdn #[a] n.
+Proof.
+apply/eqP; rewrite eqn_dvd; apply/andP; split.
+  rewrite order_dvdn -expgM -muln_divCA_gcd //.
+  by rewrite expgM expg_order expg1n.
+have [-> | n_gt0] := posnP n; first by rewrite gcdn0 divnn order_gt0 dvd1n.
+rewrite -(dvdn_pmul2r n_gt0) divn_mulAC ?dvdn_gcdl // dvdn_lcm.
+by rewrite order_dvdn mulnC expgM expg_order eqxx dvdn_mulr.
+Qed.
+
+Lemma orderXdiv a n : n %| #[a] -> #[a ^+ n] = #[a] %/ n.
+Proof. by case/dvdnP=> q defq; rewrite orderXgcd {2}defq gcdnC gcdnMl. Qed.
+
+Lemma orderXexp p m n x : #[x] = (p ^ n)%N -> #[x ^+ (p ^ m)] = (p ^ (n - m))%N.
+Proof.
+move=> ox; have [n_le_m | m_lt_n] := leqP n m.
+  rewrite -(subnKC n_le_m) subnDA subnn expnD expgM -ox.
+  by rewrite expg_order expg1n order1.
+rewrite orderXdiv ox ?dvdn_exp2l ?expnB ?(ltnW m_lt_n) //.
+by have:= order_gt0 x; rewrite ox expn_gt0 orbC -(ltn_predK m_lt_n).
+Qed.
+
+Lemma orderXpfactor p k n x :
+  #[x ^+ (p ^ k)] = n -> prime p -> p %| n -> #[x] = (p ^ k * n)%N.
+Proof.
+move=> oxp p_pr dv_p_n.
+suffices pk_x: p ^ k %| #[x] by rewrite -oxp orderXdiv // mulnC divnK.
+rewrite pfactor_dvdn // leqNgt; apply: contraL dv_p_n => lt_x_k.
+rewrite -oxp -p'natE // -(subnKC (ltnW lt_x_k)) expnD expgM.
+rewrite (pnat_dvd (orderXdvd _ _)) // -p_part // orderXdiv ?dvdn_part //.
+by rewrite -{1}[#[x]](partnC p) // mulKn // part_pnat.
+Qed.
+
+Lemma orderXprime p n x :
+  #[x ^+ p] = n -> prime p -> p %| n -> #[x] = (p * n)%N.
+Proof. exact: (@orderXpfactor p 1). Qed.
+
+Lemma orderXpnat m n x : #[x ^+ m] = n -> \pi(n).-nat m -> #[x] = (m * n)%N.
+Proof.
+move=> oxm n_m; have [m_gt0 _] := andP n_m.
+suffices m_x: m %| #[x] by rewrite -oxm orderXdiv // mulnC divnK.
+apply/dvdn_partP=> // p; rewrite mem_primes => /and3P[p_pr _ p_m].
+have n_p: p \in \pi(n) by apply: (pnatP _ _ n_m).
+have p_oxm: p %| #[x ^+ (p ^ logn p m)].
+  apply: dvdn_trans (orderXdvd _ m`_p^'); rewrite -expgM -p_part ?partnC //.
+  by rewrite oxm; rewrite mem_primes in n_p; case/and3P: n_p.
+by rewrite (orderXpfactor (erefl _) p_pr p_oxm) p_part // dvdn_mulr.
+Qed.
+
+Lemma orderM a b :
+  commute a b -> coprime #[a] #[b] -> #[a * b] = (#[a] * #[b])%N.
+Proof. by move=> cab co_ab; rewrite -coprime_cardMg -?cycleM. Qed.
+
+Definition expg_invn A k := (egcdn k #|A|).1.
+
+Lemma expgK G k :
+  coprime #|G| k -> {in G, cancel (expgn^~ k) (expgn^~ (expg_invn G k))}.
+Proof.
+move=> coGk x /order_dvdG Gx; apply/eqP.
+rewrite -expgM (eq_expg_mod_order _ _ 1) -(modn_dvdm 1 Gx).
+by rewrite -(chinese_modl coGk 1 0) /chinese mul1n addn0 modn_dvdm.
+Qed.
+
+Lemma cyclic_dprod K H G :
+  K \x H = G -> cyclic K -> cyclic H -> cyclic G = coprime #|K| #|H| .
+Proof.
+case/dprodP=> _ defKH cKH tiKH cycK cycH; pose m := lcmn #|K| #|H|.
+apply/idP/idP=> [/cyclicP[x defG] | coKH]; last by rewrite -defKH cyclicM.
+rewrite /coprime -dvdn1 -(@dvdn_pmul2l m) ?lcmn_gt0 ?cardG_gt0 //.
+rewrite muln_lcm_gcd muln1 -TI_cardMg // defKH defG order_dvdn.
+have /mulsgP[y z Ky Hz ->]: x \in K * H by rewrite defKH defG cycle_id.
+rewrite -[1]mulg1 expgMn; last exact/commute_sym/(centsP cKH).
+apply/eqP; congr (_ * _); apply/eqP; rewrite -order_dvdn.
+  exact: dvdn_trans (order_dvdG Ky) (dvdn_lcml _ _).
+exact: dvdn_trans (order_dvdG Hz) (dvdn_lcmr _ _).
+Qed.
+
+(***********************************************************************)
+(*        Generator                                                    *)
+(***********************************************************************)
+
+Definition generator (A : {set gT}) a := A == <[a]>.
+
+Lemma generator_cycle a : generator <[a]> a.
+Proof. exact: eqxx. Qed.
+
+Lemma cycle_generator a x : generator <[a]> x -> x \in <[a]>.
+Proof. by move/(<[a]> =P _)->; apply: cycle_id. Qed.
+
+Lemma generator_order a b : generator <[a]> b -> #[a] = #[b].
+Proof. by rewrite /order => /(<[a]> =P _)->. Qed.
+
+End Cyclic.
+
+Arguments Scope cyclic [_ group_scope].
+Arguments Scope generator [_ group_scope group_scope].
+Arguments Scope expg_invn [_ group_scope nat_scope].
+Implicit Arguments cyclicP [gT A].
+Prenex Implicits cyclic Zpm generator expg_invn.
+
+(* Euler's theorem *)
+Theorem Euler_exp_totient a n : coprime a n -> a ^ totient n  = 1 %[mod n].
+Proof.
+case: n => [|[|n']] //; [by rewrite !modn1 | set n := n'.+2] => co_a_n.
+have{co_a_n} Ua: coprime n (inZp a : 'I_n) by rewrite coprime_sym coprime_modl.
+have: FinRing.unit 'Z_n Ua ^+ totient n == 1.
+  by rewrite -card_units_Zp // -order_dvdn order_dvdG ?inE.
+by rewrite -2!val_eqE unit_Zp_expg /= -/n modnXm => /eqP.
+Qed.
+
+Section Eltm.
+
+Variables (aT rT : finGroupType) (x : aT) (y : rT).
+
+Definition eltm of #[y] %| #[x] := fun x_i => y ^+ invm (injm_Zpm x) x_i.
+
+Hypothesis dvd_y_x : #[y] %| #[x].
+
+Lemma eltmE i : eltm dvd_y_x (x ^+ i) = y ^+ i.
+Proof.
+apply/eqP; rewrite eq_expg_mod_order.
+have [x_le1 | x_gt1] := leqP #[x] 1. 
+  suffices: #[y] %| 1 by rewrite dvdn1 => /eqP->; rewrite !modn1.
+  by rewrite (dvdn_trans dvd_y_x) // dvdn1 order_eq1 -cycle_eq1 trivg_card_le1.
+rewrite -(expg_znat i (cycle_id x)) invmE /=; last by rewrite /Zp x_gt1 inE.
+by rewrite val_Zp_nat // modn_dvdm.
+Qed.
+
+Lemma eltm_id : eltm dvd_y_x x = y. Proof. exact: (eltmE 1). Qed.
+
+Lemma eltmM : {in <[x]> &, {morph eltm dvd_y_x : x_i x_j / x_i * x_j}}.
+Proof.
+move=> _ _ /cycleP[i ->] /cycleP[j ->].
+by apply/eqP; rewrite -expgD !eltmE expgD.
+Qed.
+Canonical eltm_morphism := Morphism eltmM.
+
+Lemma im_eltm : eltm dvd_y_x @* <[x]> = <[y]>.
+Proof. by rewrite morphim_cycle ?cycle_id //= eltm_id. Qed.
+
+Lemma ker_eltm : 'ker (eltm dvd_y_x) = <[x ^+ #[y]]>.
+Proof.
+apply/eqP; rewrite eq_sym eqEcard cycle_subG 3!inE mem_cycle /= eltmE.
+rewrite expg_order eqxx (orderE y) -im_eltm card_morphim setIid -orderE.
+by rewrite orderXdiv ?dvdn_indexg //= leq_divRL ?indexg_gt0 ?Lagrange ?subsetIl.
+Qed.
+
+Lemma injm_eltm : 'injm (eltm dvd_y_x) = (#[x] %| #[y]).
+Proof. by rewrite ker_eltm subG1 cycle_eq1 -order_dvdn. Qed.
+
+End Eltm.
+
+Section CycleSubGroup.
+
+Variable gT : finGroupType.
+
+(*  Gorenstein, 1.3.1 (i) *)
+Lemma cycle_sub_group (a : gT) m :
+     m %| #[a] ->
+  [set H : {group gT} | H \subset <[a]> & #|H| == m]
+     = [set <[a ^+ (#[a] %/ m)]>%G].
+Proof.
+move=> m_dv_a; have m_gt0: 0 < m by apply: dvdn_gt0 m_dv_a.
+have oam: #|<[a ^+ (#[a] %/ m)]>| = m.
+  apply/eqP; rewrite [#|_|]orderXgcd -(divnMr m_gt0) muln_gcdl divnK //.
+  by rewrite gcdnC gcdnMr mulKn.
+apply/eqP; rewrite eqEsubset sub1set inE /= cycleX oam eqxx !andbT.
+apply/subsetP=> X; rewrite in_set1 inE -val_eqE /= eqEcard oam.
+case/andP=> sXa /eqP oX; rewrite oX leqnn andbT.
+apply/subsetP=> x Xx; case/cycleP: (subsetP sXa _ Xx) => k def_x.
+have: (x ^+ m == 1)%g by rewrite -oX -order_dvdn cardSg // gen_subG sub1set.
+rewrite {x Xx}def_x -expgM -order_dvdn -[#[a]](Lagrange sXa) -oX mulnC.
+rewrite dvdn_pmul2r // mulnK // => /dvdnP[i ->].
+by rewrite mulnC expgM groupX // cycle_id.
+Qed.
+
+Lemma cycle_subgroup_char a (H : {group gT}) : H \subset <[a]> -> H \char <[a]>.
+Proof.
+move=> sHa; apply: lone_subgroup_char => // J sJa isoJH.
+have dvHa: #|H| %| #[a] by apply: cardSg.
+have{dvHa} /setP Huniq := esym (cycle_sub_group dvHa).
+move: (Huniq H) (Huniq J); rewrite !inE /=.
+by rewrite sHa sJa (card_isog isoJH) eqxx => /eqP<- /eqP<-.
+Qed.
+
+End CycleSubGroup.
+
+(***********************************************************************)
+(*  Reflected boolean property and morphic image, injection, bijection *)
+(***********************************************************************)
+
+Section MorphicImage.
+
+Variables aT rT : finGroupType.
+Variables (D : {group aT}) (f : {morphism D >-> rT}) (x : aT).
+Hypothesis Dx : x \in D.
+
+Lemma morph_order : #[f x] %| #[x].
+Proof. by rewrite order_dvdn -morphX // expg_order morph1. Qed.
+
+Lemma morph_generator A : generator A x -> generator (f @* A) (f x).
+Proof. by move/(A =P _)->; rewrite /generator morphim_cycle. Qed.
+
+End MorphicImage.
+
+Section CyclicProps.
+
+Variables gT : finGroupType.
+Implicit Types (aT rT : finGroupType) (G H K : {group gT}).
+
+Lemma cyclicS G H : H \subset G -> cyclic G -> cyclic H.
+Proof.
+move=> sHG /cyclicP[x defG]; apply/cyclicP.
+exists (x ^+ (#[x] %/ #|H|)); apply/congr_group/set1P.
+by rewrite -cycle_sub_group /order -defG ?cardSg // inE sHG eqxx.
+Qed.
+
+Lemma cyclicJ G x : cyclic (G :^ x) = cyclic G.
+Proof.
+apply/cyclicP/cyclicP=> [[y /(canRL (conjsgK x))] | [y ->]].
+  by rewrite -cycleJ; exists (y ^ x^-1).
+by exists (y ^ x); rewrite cycleJ.
+Qed.
+
+Lemma eq_subG_cyclic G H K :
+  cyclic G -> H \subset G -> K \subset G -> (H :==: K) = (#|H| == #|K|).
+Proof.
+case/cyclicP=> x -> sHx sKx; apply/eqP/eqP=> [-> //| eqHK].
+have def_GHx := cycle_sub_group (cardSg sHx); set GHx := [set _] in def_GHx.
+have []: H \in GHx /\ K \in GHx by rewrite -def_GHx !inE sHx sKx eqHK /=.
+by do 2!move/set1P->.
+Qed.
+
+Lemma cardSg_cyclic G H K :
+  cyclic G -> H \subset G -> K \subset G -> (#|H| %| #|K|) = (H \subset K).
+Proof.
+move=> cycG sHG sKG; apply/idP/idP; last exact: cardSg.
+case/cyclicP: (cyclicS sKG cycG) => x defK; rewrite {K}defK in sKG *.
+case/dvdnP=> k ox; suffices ->: H :=: <[x ^+ k]> by apply: cycleX.
+apply/eqP; rewrite (eq_subG_cyclic cycG) ?(subset_trans (cycleX _ _)) //.
+rewrite -orderE orderXdiv orderE ox ?dvdn_mulr ?mulKn //.
+by have:= order_gt0 x; rewrite orderE ox; case k.
+Qed.
+
+Lemma sub_cyclic_char G H : cyclic G -> (H \char G) = (H \subset G).
+Proof.
+case/cyclicP=> x ->; apply/idP/idP => [/andP[] //|].
+exact: cycle_subgroup_char.
+Qed.
+
+Lemma morphim_cyclic rT G H (f : {morphism G >-> rT}) :
+  cyclic H -> cyclic (f @* H).
+Proof.
+move=> cycH; wlog sHG: H cycH / H \subset G.
+  by rewrite -morphimIdom; apply; rewrite (cyclicS _ cycH, subsetIl) ?subsetIr.
+case/cyclicP: cycH sHG => x ->; rewrite gen_subG sub1set => Gx.
+by apply/cyclicP; exists (f x); rewrite morphim_cycle.
+Qed.
+
+Lemma quotient_cycle x H : x \in 'N(H) -> <[x]> / H = <[coset H x]>.
+Proof. exact: morphim_cycle. Qed.
+
+Lemma quotient_cyclic G H : cyclic G -> cyclic (G / H).
+Proof. exact: morphim_cyclic. Qed.
+
+Lemma quotient_generator x G H :
+  x \in 'N(H) -> generator G x -> generator (G / H) (coset H x).
+Proof. by move=> Nx; apply: morph_generator. Qed.
+
+Lemma prime_cyclic G : prime #|G| -> cyclic G.
+Proof.
+case/primeP; rewrite ltnNge -trivg_card_le1.
+case/trivgPn=> x Gx ntx /(_ _ (order_dvdG Gx)).
+rewrite order_eq1 (negbTE ntx) => /eqnP oxG; apply/cyclicP.
+by exists x; apply/eqP; rewrite eq_sym eqEcard -oxG cycle_subG Gx leqnn.
+Qed.
+
+Lemma dvdn_prime_cyclic G p : prime p -> #|G| %| p -> cyclic G.
+Proof.
+move=> p_pr pG; case: (eqsVneq G 1) => [-> | ntG]; first exact: cyclic1.
+by rewrite prime_cyclic // (prime_nt_dvdP p_pr _ pG) -?trivg_card1.
+Qed.
+
+Lemma cyclic_small G : #|G| <= 3 -> cyclic G.
+Proof.
+rewrite 4!(ltnS, leq_eqVlt) -trivg_card_le1 orbA orbC.
+case/predU1P=> [-> | oG]; first exact: cyclic1.
+by apply: prime_cyclic; case/pred2P: oG => ->.
+Qed.
+
+End CyclicProps.
+
+Section IsoCyclic.
+
+Variables gT rT : finGroupType.
+Implicit Types (G H : {group gT}) (M : {group rT}).
+
+Lemma injm_cyclic G H (f : {morphism G >-> rT}) :
+  'injm f -> H \subset G -> cyclic (f @* H) = cyclic H.
+Proof.
+move=> injf sHG; apply/idP/idP; last exact: morphim_cyclic.
+by rewrite -{2}(morphim_invm injf sHG); apply: morphim_cyclic.
+Qed.
+
+Lemma isog_cyclic G M : G \isog M -> cyclic G = cyclic M.
+Proof. by case/isogP=> f injf <-; rewrite injm_cyclic. Qed.
+
+Lemma isog_cyclic_card G M : cyclic G -> isog G M = cyclic M && (#|M| == #|G|).
+Proof.
+move=> cycG; apply/idP/idP=> [isoGM | ].
+  by rewrite (card_isog isoGM) -(isog_cyclic isoGM) cycG /=.
+case/cyclicP: cycG => x ->{G} /andP[/cyclicP[y ->] /eqP oy].
+by apply: isog_trans (isog_symr _) (Zp_isog y); rewrite /order oy Zp_isog.
+Qed.
+
+Lemma injm_generator G H (f : {morphism G >-> rT}) x :
+    'injm f -> x \in G -> H \subset G ->
+  generator (f @* H) (f x) = generator H x.
+Proof.
+move=> injf Gx sHG; apply/idP/idP; last exact: morph_generator.
+rewrite -{2}(morphim_invm injf sHG) -{2}(invmE injf Gx).
+by apply: morph_generator; apply: mem_morphim.
+Qed.
+
+End IsoCyclic.
+
+(* Metacyclic groups. *)
+Section Metacyclic.
+
+Variable gT : finGroupType.
+Implicit Types (A : {set gT}) (G H : {group gT}).
+
+Definition metacyclic A :=
+  [exists H : {group gT}, [&& cyclic H, H <| A & cyclic (A / H)]].
+
+Lemma metacyclicP A : 
+  reflect (exists H : {group gT}, [/\ cyclic H, H <| A & cyclic (A / H)]) 
+          (metacyclic A).
+Proof. exact: 'exists_and3P. Qed.
+
+Lemma metacyclic1 : metacyclic 1.
+Proof. 
+by apply/existsP; exists 1%G; rewrite normal1 trivg_quotient !cyclic1.
+Qed.
+
+Lemma cyclic_metacyclic A : cyclic A -> metacyclic A.
+Proof.
+case/cyclicP=> x ->; apply/existsP; exists (<[x]>)%G.
+by rewrite normal_refl cycle_cyclic trivg_quotient cyclic1.
+Qed.
+
+Lemma metacyclicS G H : H \subset G -> metacyclic G -> metacyclic H.
+Proof.
+move=> sHG /metacyclicP[K [cycK nsKG cycGq]]; apply/metacyclicP.
+exists (H :&: K)%G; rewrite (cyclicS (subsetIr H K)) ?(normalGI sHG) //=.
+rewrite setIC (isog_cyclic (second_isog _)) ?(cyclicS _ cycGq) ?quotientS //.
+by rewrite (subset_trans sHG) ?normal_norm.
+Qed.
+
+End Metacyclic.
+
+Arguments Scope metacyclic [_ group_scope].
+Prenex Implicits metacyclic.
+Implicit Arguments metacyclicP [gT A].
+
+(* Automorphisms of cyclic groups. *)
+Section CyclicAutomorphism.
+
+Variable gT : finGroupType.
+
+Section CycleAutomorphism.
+
+Variable a : gT.
+
+Section CycleMorphism.
+
+Variable n : nat.
+
+Definition cyclem of gT := fun x : gT => x ^+ n.
+
+Lemma cyclemM : {in <[a]> & , {morph cyclem a : x y / x * y}}.
+Proof.
+by move=> x y ax ay; apply: expgMn; apply: (centsP (cycle_abelian a)).
+Qed.
+
+Canonical cyclem_morphism := Morphism cyclemM.
+
+End CycleMorphism.
+
+Section ZpUnitMorphism.
+
+Variable u : {unit 'Z_#[a]}.
+
+Lemma injm_cyclem : 'injm (cyclem (val u) a).
+Proof.
+apply/subsetP=> x /setIdP[ax]; rewrite !inE -order_dvdn.
+case: (eqVneq a 1) => [a1 | nta]; first by rewrite a1 cycle1 inE in ax.
+rewrite -order_eq1 -dvdn1; move/eqnP: (valP u) => /= <-.
+by rewrite dvdn_gcd {2}Zp_cast ?order_gt1 // order_dvdG.
+Qed.
+
+Lemma im_cyclem : cyclem (val u) a @* <[a]> = <[a]>.
+Proof.
+apply/morphim_fixP=> //; first exact: injm_cyclem.
+by rewrite morphim_cycle ?cycle_id ?cycleX.
+Qed.
+
+Definition Zp_unitm := aut injm_cyclem im_cyclem.
+
+End ZpUnitMorphism.
+
+Lemma Zp_unitmM : {in units_Zp #[a] &, {morph Zp_unitm : u v / u * v}}.
+Proof.
+move=> u v _ _; apply: (eq_Aut (Aut_aut _ _)) => [|x a_x].
+  by rewrite groupM ?Aut_aut.
+rewrite permM !autE ?groupX //= /cyclem -expgM.
+rewrite -expg_mod_order modn_dvdm ?expg_mod_order //.
+case: (leqP #[a] 1) => [lea1 | lt1a]; last by rewrite Zp_cast ?order_dvdG.
+by rewrite card_le1_trivg // in a_x; rewrite (set1P a_x) order1 dvd1n.
+Qed.
+
+Canonical Zp_unit_morphism := Morphism Zp_unitmM.
+
+Lemma injm_Zp_unitm : 'injm Zp_unitm.
+Proof.
+case: (eqVneq a 1) => [a1 | nta].
+  by rewrite subIset //= card_le1_trivg ?subxx // card_units_Zp a1 order1.
+apply/subsetP=> /= u /morphpreP[_ /set1P/= um1].
+have{um1}: Zp_unitm u a == Zp_unitm 1 a by rewrite um1 morph1.
+rewrite !autE ?cycle_id // eq_expg_mod_order.
+by rewrite -[n in _ == _ %[mod n]]Zp_cast ?order_gt1 // !modZp inE.
+Qed.
+
+Lemma generator_coprime m : generator <[a]> (a ^+ m) = coprime #[a] m.
+Proof.
+rewrite /generator eq_sym eqEcard cycleX -/#[a] [#|_|]orderXgcd /=.
+apply/idP/idP=> [le_a_am|co_am]; last by rewrite (eqnP co_am) divn1.
+have am_gt0: 0 < gcdn #[a] m by rewrite gcdn_gt0 order_gt0.
+by rewrite /coprime eqn_leq am_gt0 andbT -(@leq_pmul2l #[a]) ?muln1 -?leq_divRL.
+Qed.
+
+Lemma im_Zp_unitm : Zp_unitm @* units_Zp #[a] = Aut <[a]>.
+Proof.
+rewrite morphimEdom; apply/setP=> f; pose n := invm (injm_Zpm a) (f a).
+apply/imsetP/idP=> [[u _ ->] | Af]; first exact: Aut_aut.
+have [a1 | nta] := eqVneq a 1.
+  by rewrite a1 cycle1 Aut1 in Af; exists 1; rewrite // morph1 (set1P Af).
+have a_fa: <[a]> = <[f a]>.
+  by rewrite -(autmE Af) -morphim_cycle ?im_autm ?cycle_id.
+have def_n: a ^+ n = f a.
+  by rewrite -/(Zpm n) invmK // im_Zpm a_fa cycle_id.
+have co_a_n: coprime #[a].-2.+2 n.
+  by rewrite {1}Zp_cast ?order_gt1 // -generator_coprime def_n; apply/eqP.
+exists (FinRing.unit 'Z_#[a] co_a_n); rewrite ?inE //.
+apply: eq_Aut (Af) (Aut_aut _ _) _ => x ax.
+rewrite autE //= /cyclem; case/cycleP: ax => k ->{x}.
+by rewrite -(autmE Af) morphX ?cycle_id //= autmE -def_n -!expgM mulnC.
+Qed.
+
+Lemma Zp_unit_isom : isom (units_Zp #[a]) (Aut <[a]>) Zp_unitm.
+Proof. by apply/isomP; rewrite ?injm_Zp_unitm ?im_Zp_unitm. Qed.
+
+Lemma Zp_unit_isog : isog (units_Zp #[a]) (Aut <[a]>).
+Proof. exact: isom_isog Zp_unit_isom. Qed.
+
+Lemma card_Aut_cycle : #|Aut <[a]>| = totient #[a].
+Proof. by rewrite -(card_isog Zp_unit_isog) card_units_Zp. Qed.
+
+Lemma totient_gen : totient #[a] = #|[set x | generator <[a]> x]|.
+Proof.
+have [lea1 | lt1a] := leqP #[a] 1.
+  rewrite /order card_le1_trivg // cards1 (@eq_card1 _ 1) // => x.
+  by rewrite !inE -cycle_eq1 eq_sym.
+rewrite -(card_injm (injm_invm (injm_Zpm a))) /= ?im_Zpm; last first.
+  by apply/subsetP=> x; rewrite inE; apply: cycle_generator.
+rewrite -card_units_Zp // cardsE card_sub morphim_invmE; apply: eq_card => /= d.
+by rewrite !inE /= qualifE /= /Zp lt1a inE /= generator_coprime {1}Zp_cast.
+Qed.
+
+Lemma Aut_cycle_abelian : abelian (Aut <[a]>).
+Proof. by rewrite -im_Zp_unitm morphim_abelian ?units_Zp_abelian. Qed.
+
+End CycleAutomorphism.
+
+Variable G : {group gT}.
+
+Lemma Aut_cyclic_abelian : cyclic G -> abelian (Aut G).
+Proof. by case/cyclicP=> x ->; apply: Aut_cycle_abelian. Qed.
+
+Lemma card_Aut_cyclic : cyclic G -> #|Aut G| = totient #|G|.
+Proof. by case/cyclicP=> x ->; apply: card_Aut_cycle. Qed.
+
+Lemma sum_ncycle_totient :
+  \sum_(d < #|G|.+1) #|[set <[x]> | x in G & #[x] == d]| * totient d = #|G|.
+Proof.
+pose h (x : gT) : 'I_#|G|.+1 := inord #[x].
+symmetry; rewrite -{1}sum1_card (partition_big h xpredT) //=.
+apply: eq_bigr => d _; set Gd := finset _.
+rewrite -sum_nat_const sum1dep_card -sum1_card (_ : finset _ = Gd); last first.
+  apply/setP=> x; rewrite !inE; apply: andb_id2l => Gx.
+  by rewrite /eq_op /= inordK // ltnS subset_leq_card ?cycle_subG.
+rewrite (partition_big_imset cycle) {}/Gd; apply: eq_bigr => C /=.
+case/imsetP=> x /setIdP[Gx /eqP <-] -> {C d}.
+rewrite sum1dep_card totient_gen; apply: eq_card => y; rewrite !inE /generator.
+move: Gx; rewrite andbC eq_sym -!cycle_subG /order.
+by case: eqP => // -> ->; rewrite eqxx.
+Qed.
+
+End CyclicAutomorphism.
+
+Lemma sum_totient_dvd n : \sum_(d < n.+1 | d %| n) totient d = n.
+Proof.
+case: n => [|[|n']]; try by rewrite big_mkcond !big_ord_recl big_ord0.
+set n := n'.+2; pose x1 : 'Z_n := 1%R.
+have ox1: #[x1] = n by rewrite /order -Zp_cycle card_Zp.
+rewrite -[rhs in _ = rhs]ox1 -[#[_]]sum_ncycle_totient [#|_|]ox1 big_mkcond /=.
+apply: eq_bigr => d _; rewrite -{2}ox1; case: ifP => [|ndv_dG]; last first.
+  rewrite eq_card0 // => C; apply/imsetP=> [[x /setIdP[Gx oxd] _{C}]].
+  by rewrite -(eqP oxd) order_dvdG in ndv_dG.
+move/cycle_sub_group; set Gd := [set _] => def_Gd.
+rewrite (_ : _ @: _ = @gval _ @: Gd); first by rewrite imset_set1 cards1 mul1n.
+apply/setP=> C; apply/idP/imsetP=> [| [gC GdC ->{C}]].
+  case/imsetP=> x /setIdP[_ oxd] ->; exists <[x]>%G => //.
+  by rewrite -def_Gd inE -Zp_cycle subsetT.
+have:= GdC; rewrite -def_Gd => /setIdP[_ /eqP <-].
+by rewrite (set1P GdC) /= mem_imset // inE eqxx (mem_cycle x1).
+Qed.
+
+Section FieldMulCyclic.
+
+(***********************************************************************)
+(* A classic application to finite multiplicative subgroups of fields. *)
+(***********************************************************************)
+
+Import GRing.Theory.
+
+Variables (gT : finGroupType) (G : {group gT}).
+
+Lemma order_inj_cyclic :
+  {in G &, forall x y, #[x] = #[y] -> <[x]> = <[y]>} -> cyclic G.
+Proof.
+move=> ucG; apply: negbNE (contra _ (negbT (ltnn #|G|))) => ncG.
+rewrite -{2}[#|G|]sum_totient_dvd big_mkcond (bigD1 ord_max) ?dvdnn //=.
+rewrite -{1}[#|G|]sum_ncycle_totient (bigD1 ord_max) //= -addSn leq_add //.
+  rewrite eq_card0 ?totient_gt0 ?cardG_gt0 // => C.
+  apply/imsetP=> [[x /setIdP[Gx /eqP oxG]]]; case/cyclicP: ncG.
+  by exists x; apply/eqP; rewrite eq_sym eqEcard cycle_subG Gx -oxG /=.
+elim/big_ind2: _ => // [m1 n1 m2 n2 | d _]; first exact: leq_add.
+set Gd := _ @: _; case: (set_0Vmem Gd) => [-> | [C]]; first by rewrite cards0.
+rewrite {}/Gd => /imsetP[x /setIdP[Gx /eqP <-] _ {C d}].
+rewrite order_dvdG // (@eq_card1 _ <[x]>) ?mul1n // => C.
+apply/idP/eqP=> [|-> {C}]; last by rewrite mem_imset // inE Gx eqxx.
+by case/imsetP=> y /setIdP[Gy /eqP/ucG->].
+Qed.
+
+Lemma div_ring_mul_group_cyclic (R : unitRingType) (f : gT -> R) :
+    f 1 = 1%R -> {in G &, {morph f : u v / u * v >-> (u * v)%R}} ->
+    {in G^#, forall x, f x - 1 \in GRing.unit}%R ->
+  abelian G -> cyclic G.
+Proof.
+move=> f1 fM f1P abelG.
+have fX n: {in G, {morph f : u / u ^+ n >-> (u ^+ n)%R}}.
+  by case: n => // n x Gx; elim: n => //= n IHn; rewrite expgS fM ?groupX ?IHn.
+have fU x: x \in G -> f x \in GRing.unit.
+  by move=> Gx; apply/unitrP; exists (f x^-1); rewrite -!fM ?groupV ?gsimp.
+apply: order_inj_cyclic => x y Gx Gy; set n := #[x] => yn.
+apply/eqP; rewrite eq_sym eqEcard -[#|_|]/n yn leqnn andbT cycle_subG /=.
+suff{y Gy yn} ->: <[x]> = G :&: [set z | #[z] %| n] by rewrite !inE Gy yn /=.
+apply/eqP; rewrite eqEcard subsetI cycle_subG {}Gx /= cardE; set rs := enum _.
+apply/andP; split; first by apply/subsetP=> y xy; rewrite inE order_dvdG.
+pose P : {poly R} := ('X^n - 1)%R; have n_gt0: n > 0 by apply: order_gt0.
+have szP: size P = n.+1 by rewrite size_addl size_polyXn ?size_opp ?size_poly1.
+rewrite -ltnS -szP -(size_map f) max_ring_poly_roots -?size_poly_eq0 ?{}szP //.
+  apply/allP=> fy /mapP[y]; rewrite mem_enum !inE order_dvdn => /andP[Gy].
+  move/eqP=> yn1 ->{fy}; apply/eqP.
+  by rewrite !(hornerE, hornerXn) -fX // yn1 f1 subrr.
+have: uniq rs by apply: enum_uniq.
+have: all (mem G) rs by apply/allP=> y; rewrite mem_enum; case/setIP.
+elim: rs => //= y rs IHrs /andP[Gy Grs] /andP[y_rs]; rewrite andbC.
+move/IHrs=> -> {IHrs}//; apply/allP=> _ /mapP[z rs_z ->].
+have{Grs} Gz := allP Grs z rs_z; rewrite /diff_roots -!fM // (centsP abelG) //.
+rewrite eqxx -[f y]mul1r -(mulgKV y z) fM ?groupM ?groupV //=.
+rewrite -mulNr -mulrDl unitrMl ?fU ?f1P // !inE.
+by rewrite groupM ?groupV // andbT -eq_mulgV1; apply: contra y_rs; move/eqP <-.
+Qed.
+
+Lemma field_mul_group_cyclic (F : fieldType) (f : gT -> F) :
+    {in G &, {morph f : u v / u * v >-> (u * v)%R}} ->
+    {in G, forall x, f x = 1%R <-> x = 1} ->
+  cyclic G.
+Proof.
+move=> fM f1P; have f1 : f 1 = 1%R by apply/f1P.
+apply: (div_ring_mul_group_cyclic f1 fM) => [x|].
+  case/setD1P=> x1 Gx; rewrite unitfE; apply: contra x1.
+  by rewrite subr_eq0 => /eqP/f1P->.
+apply/centsP=> x Gx y Gy; apply/commgP/eqP.
+apply/f1P; rewrite ?fM ?groupM ?groupV //.
+by rewrite mulrCA -!fM ?groupM ?groupV // mulKg mulVg.
+Qed.
+
+End FieldMulCyclic.
+
+Lemma field_unit_group_cyclic (F : finFieldType) (G : {group {unit F}}) :
+  cyclic G.
+Proof.
+apply: field_mul_group_cyclic FinRing.uval _ _ => // u _.
+by split=> /eqP ?; apply/eqP.
+Qed.
+
+Section PrimitiveRoots.
+
+Open Scope ring_scope.
+Import GRing.Theory.
+
+Lemma has_prim_root (F : fieldType) (n : nat) (rs : seq F) :
+    n > 0 -> all n.-unity_root rs -> uniq rs -> size rs >= n ->
+  has n.-primitive_root rs.
+Proof.
+move=> n_gt0 rsn1 Urs; rewrite leq_eqVlt ltnNge max_unity_roots // orbF eq_sym.
+move/eqP=> sz_rs; pose r := val (_ : seq_sub rs).
+have rn1 x: r x ^+ n = 1.
+  by apply/eqP; rewrite -unity_rootE (allP rsn1) ?(valP x).
+have prim_r z: z ^+ n = 1 -> z \in rs.
+  by move/eqP; rewrite -unity_rootE -(mem_unity_roots n_gt0).
+pose r' := SeqSub (prim_r _ _); pose sG_1 := r' _ (expr1n _ _).
+have sG_VP: r _ ^+ n.-1 ^+ n = 1.
+  by move=> x; rewrite -exprM mulnC exprM rn1 expr1n.
+have sG_MP: (r _ * r _) ^+ n = 1 by move=> x y; rewrite exprMn !rn1 mul1r.
+pose sG_V := r' _ (sG_VP _); pose sG_M := r' _ (sG_MP _ _).
+have sG_Ag: associative sG_M by move=> x y z; apply: val_inj; rewrite /= mulrA.
+have sG_1g: left_id sG_1 sG_M by move=> x; apply: val_inj; rewrite /= mul1r.
+have sG_Vg: left_inverse sG_1 sG_V sG_M.
+  by move=> x; apply: val_inj; rewrite /= -exprSr prednK ?rn1.
+pose sgT := BaseFinGroupType _ (FinGroup.Mixin sG_Ag sG_1g sG_Vg).
+pose gT := @FinGroupType sgT sG_Vg.
+have /cyclicP[x gen_x]: @cyclic gT setT.
+  apply: (@field_mul_group_cyclic gT [set: _] F r) => // x _.
+  by split=> [ri1 | ->]; first apply: val_inj.
+apply/hasP; exists (r x); first exact: (valP x).
+have [m prim_x dvdmn] := prim_order_exists n_gt0 (rn1 x).
+rewrite -((m =P n) _) // eqn_dvd {}dvdmn -sz_rs -(card_seq_sub Urs) -cardsT.
+rewrite gen_x (@order_dvdn gT) /(_ == _) /= -{prim_x}(prim_expr_order prim_x).
+by apply/eqP; elim: m => //= m IHm; rewrite exprS expgS /= -IHm.
+Qed.
+
+End PrimitiveRoots.
+
+(***********************************************************************)
+(*  Cycles of prime order                                              *)
+(***********************************************************************)
+
+Section AutPrime.
+
+Variable gT : finGroupType.
+
+Lemma Aut_prime_cycle_cyclic (a : gT) : prime #[a] -> cyclic (Aut <[a]>).
+Proof.
+move=> pr_a; have inj_um := injm_Zp_unitm a; have eq_a := Fp_Zcast pr_a.
+pose fm := cast_ord (esym eq_a) \o val \o invm inj_um.
+apply: (@field_mul_group_cyclic _ _ _ fm) => [f g Af Ag | f Af] /=.
+  by apply: val_inj; rewrite /= morphM ?im_Zp_unitm //= eq_a.
+split=> [/= fm1 |->]; last by apply: val_inj; rewrite /= morph1.
+apply: (injm1 (injm_invm inj_um)); first by rewrite /= im_Zp_unitm.
+by do 2!apply: val_inj; move/(congr1 val): fm1.
+Qed.
+
+Lemma Aut_prime_cyclic (G : {group gT}) : prime #|G| -> cyclic (Aut G).
+Proof.
+move=> pr_G; case/cyclicP: (prime_cyclic pr_G) (pr_G) => x ->.
+exact: Aut_prime_cycle_cyclic.
+Qed.
+
+End AutPrime.
diff --git a/mathcomp/solvable/extraspecial.v b/mathcomp/solvable/extraspecial.v
index 737838b..137518b 100644
--- a/mathcomp/solvable/extraspecial.v
+++ b/mathcomp/solvable/extraspecial.v
@@ -1,17 +1,15 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype bigop finset prime binomial.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism quotient action gproduct.
-From mathcomp.fingroup
-Require Import presentation.
-From mathcomp.algebra
-Require Import ssralg finalg zmodp matrix cyclic.
-Require Import pgroup center gseries commutator gfunctor.
-Require Import nilpotent sylow abelian finmodule maximal extremal.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
+From mathcomp
+Require Import bigop finset prime binomial fingroup morphism perm automorphism.
+From mathcomp
+Require Import presentation quotient action commutator gproduct gfunctor.
+From mathcomp
+Require Import ssralg finalg zmodp cyclic pgroup center gseries.
+From mathcomp
+Require Import nilpotent sylow abelian finmodule matrix maximal extremal.
 
 (******************************************************************************)
 (* This file contains the fine structure thorems for extraspecial p-groups.   *)
@@ -332,7 +330,7 @@ Lemma Ohm1_extraspecial_odd (gT : finGroupType) (G : {group gT}) :
 Proof.
 move=> pG esG oddG Y; have [spG _] := esG.
 have [defPhiG defG'] := spG; set Z := 'Z(G) in defPhiG defG'.
-have{spG} expG: exponent G %| p ^ 2 by exact: exponent_special.
+have{spG} expG: exponent G %| p ^ 2 by apply: exponent_special.
 have p_pr := extraspecial_prime pG esG.
 have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
 have oZ: #|Z| = p := card_center_extraspecial pG esG.
@@ -341,7 +339,7 @@ have nsYG: Y <| G := Ohm_normal 1 G; have [sYG nYG] := andP nsYG.
 have ntZ: Z != 1 by rewrite -cardG_gt1 oZ.
 have sZY: Z \subset Y.
   by apply: contraR ntZ => ?; rewrite -(setIidPl sZG) TI_Ohm1 ?prime_TIg ?oZ.
-have ntY: Y != 1 by apply: contra ntZ; rewrite -!subG1; exact: subset_trans.
+have ntY: Y != 1 by apply: subG1_contra ntZ.
 have p_odd: odd p by rewrite -oZ (oddSg sZG).
 have expY: exponent Y %| p by rewrite exponent_Ohm1_class2 // nil_class2 defG'.
 rewrite (prime_nt_dvdP p_pr _ expY) -?dvdn1 -?trivg_exponent //.
@@ -372,7 +370,7 @@ have iYG: #|G : Y| = p.
   have [sVG [u Gu notVu]] := properP (maxgroupp maxV).
   without loss [v Vv notYv]: / exists2 v, v \in V & v \notin Y.
     have [->| ] := eqVneq Y V; first by rewrite (p_maximal_index pG).
-    by rewrite eqEsubset sYV => not_sVY; apply; exact/subsetPn.
+    by rewrite eqEsubset sYV => not_sVY; apply; apply/subsetPn.
   pose U := <[u]> <*> <[v]>; have Gv := subsetP sVG v Vv.
   have sUG: U \subset G by rewrite join_subG !cycle_subG Gu.
   have Uu: u \in U by rewrite -cycle_subG joing_subl.
@@ -418,7 +416,7 @@ have iC1U (U : {group gT}) x:
   rewrite -(@dvdn_pmul2l (#|U| * #|'C_G[x]|)) ?muln_gt0 ?cardG_gt0 //.
   have maxCx: maximal 'C_G[x] G.
     rewrite cent1_extraspecial_maximal //; apply: contra not_cUx.
-    by rewrite inE Gx; exact: subsetP (centS sUG) _.
+    by rewrite inE Gx; apply: subsetP (centS sUG) _.
   rewrite {1}mul_cardG setIA (setIidPl sUG) -(p_maximal_index pG maxCx) -!mulnA.
   rewrite !Lagrange ?subsetIl // mulnC dvdn_pmul2l //.
   have [sCxG nCxG] := andP (p_maximal_normal pG maxCx).
@@ -432,7 +430,7 @@ have oCG (U : {group gT}):
     by apply/subsetPn; rewrite eqEsubset sZU subsetI sUG centsC in neZU.
   pose W := 'C_U[x]; have iWU: #|U : W| = p by rewrite iC1U // inE not_cUx.
   have maxW: maximal W U by rewrite p_index_maximal ?subsetIl ?iWU.
-  have ltWU: W \proper U by exact: maxgroupp maxW.
+  have ltWU: W \proper U by apply: maxgroupp maxW.
   have [sWU [u Uu notWu]] := properP ltWU.
   have defU: W * <[u]> = U.
     have nsWU: W <| U := p_maximal_normal (pgroupS sUG pG) maxW.
@@ -548,8 +546,8 @@ have not_cEE: ~~ abelian E.
   by apply: contra not_le_p3_p => cEE; rewrite -oE -oZ -defZE (center_idP _).
 have sES: E \subset S by rewrite -defET mulG_subl.
 have sTS: T \subset S by rewrite -defET mulG_subr.
-have expE: exponent E %| p by exact: dvdn_trans (exponentS sES) expG.
-have expT: exponent T %| p by exact: dvdn_trans (exponentS sTS) expG.
+have expE: exponent E %| p by apply: dvdn_trans (exponentS sES) expG.
+have expT: exponent T %| p by apply: dvdn_trans (exponentS sTS) expG.
 have{expE not_cEE} isoE: E \isog p^{1+2}.
   apply: isog_pX1p2 => //.
   by apply: card_p3group_extraspecial p_pr oE _; rewrite defZE.
@@ -589,7 +587,7 @@ Qed.
 
 Lemma Q8_extraspecial : extraspecial 'Q_8.
 Proof.
-have gt32: 3 > 2 by []; have isoQ: 'Q_8 \isog 'Q_(2 ^ 3) by exact: isog_refl.
+have gt32: 3 > 2 by []; have isoQ: 'Q_8 \isog 'Q_(2 ^ 3) by apply: isog_refl.
 have [[x y] genQ _] := generators_quaternion gt32 isoQ.
 have [_ [defQ' defPhiQ _ _]] := quaternion_structure gt32 genQ isoQ.
 case=> defZ oZ _ _ _ _ _; split; last by rewrite oZ.
@@ -673,9 +671,9 @@ rewrite subEproper; case/predU1P=> [defG1 | ltZG1].
   have [n' n'_gt2 isoG]: exists2 n', n' > 2 & G \isog 'Q_(2 ^ n').
     apply/quaternion_classP; apply/eqP.
     have not_cycG: ~~ cyclic G.
-      by apply: contra (extraspecial_nonabelian esG); exact: cyclic_abelian.
+      by apply: contra (extraspecial_nonabelian esG); apply: cyclic_abelian.
     move: oZ; rewrite defG1; move/prime_Ohm1P; rewrite (negPf not_cycG) /=.
-    by apply=> //; apply: contra not_cycG; move/eqP->; exact: cyclic1.
+    by apply=> //; apply: contra not_cycG; move/eqP->; apply: cyclic1.
   have [n0 n'3]: n = 0%N /\ n' = 3.
     have [[x y] genG _] := generators_quaternion n'_gt2 isoG.
     have n'3: n' = 3.
@@ -689,7 +687,7 @@ rewrite subEproper; case/predU1P=> [defG1 | ltZG1].
   by rewrite center_ncprod0.
 case/andP: ltZG1 => _; rewrite (OhmE _ pG) gen_subG.
 case/subsetPn=> x; case/LdivP=> Gx x2 notZx.
-have ox: #[x] = 2 by exact: nt_prime_order (group1_contra notZx).
+have ox: #[x] = 2 by apply: nt_prime_order (group1_contra notZx).
 have Z'x: x \in G :\: 'Z(G) by rewrite inE notZx.
 have [E [R [[oE oR] [defG ziER]]]] := split1_extraspecial pG esG Z'x.
 case=> defZE defZR [esE Ex] esR.
@@ -744,8 +742,8 @@ Qed.
 Lemma rank_Dn n : 'r_2('D^n) = n.+1.
 Proof.
 elim: n => [|n IHn]; first by rewrite p_rank_abelem ?prime_abelem ?card_pX1p2n.
-have oDDn: #|'D^n.+1| = (2 ^ n.+1.*2.+1)%N by exact: card_pX1p2n.
-have esDDn: extraspecial 'D^n.+1 by exact: pX1p2n_extraspecial.
+have oDDn: #|'D^n.+1| = (2 ^ n.+1.*2.+1)%N by apply: card_pX1p2n.
+have esDDn: extraspecial 'D^n.+1 by apply: pX1p2n_extraspecial.
 do [case: pX1p2S => gz isoZ; set DDn := [set: _]] in oDDn esDDn *.
 have pDDn: 2.-group DDn by rewrite /pgroup oDDn pnat_exp.
 apply/eqP; rewrite eqn_leq; apply/andP; split.
@@ -787,7 +785,7 @@ have esDnQ: extraspecial 'D^n*Q := DnQ_extraspecial n.
 do [case: DnQ_P => gz isoZ; set DnQ := setT] in pDnQ esDnQ *.
 suffices [E]: exists2 E, E \in 'E*_2(DnQ) & logn 2 #|E| = n.+1.
   by rewrite (pmaxElem_extraspecial pDnQ esDnQ); case/pnElemP=> _ _ <-.
-have oZ: #|'Z(DnQ)| = 2 by exact: card_center_extraspecial.
+have oZ: #|'Z(DnQ)| = 2 by apply: card_center_extraspecial.
 pose Dn := cpairg1 isoZ @* 'D^n; pose Q := cpair1g isoZ @* 'Q_8.
 have [injDn injQ] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
 have [E EprE]:= p_rank_witness 2 [group of Dn].
diff --git a/mathcomp/solvable/extremal.v b/mathcomp/solvable/extremal.v
index e233a86..83a9fb8 100644
--- a/mathcomp/solvable/extremal.v
+++ b/mathcomp/solvable/extremal.v
@@ -1,17 +1,15 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype bigop finset prime binomial.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism quotient action gproduct.
-From mathcomp.fingroup
-Require Import presentation.
-From mathcomp.algebra
-Require Import ssralg finalg zmodp cyclic matrix.
-Require Import pgroup center gseries commutator gfunctor.
-Require Import nilpotent sylow abelian finmodule maximal.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div choice fintype.
+From mathcomp
+Require Import bigop finset prime binomial fingroup morphism perm automorphism.
+From mathcomp
+Require Import presentation quotient action commutator gproduct gfunctor.
+From mathcomp
+Require Import ssralg finalg zmodp cyclic pgroup center gseries.
+From mathcomp
+Require Import nilpotent sylow abelian finmodule matrix maximal.
 
 (******************************************************************************)
 (*    This file contains the definition and properties of extremal p-groups;  *)
@@ -253,10 +251,10 @@ have mV: {in A, {morph m : a / a^-1 >-> (a^-1)%R}}.
   by rewrite -mM ?groupV ?mulVg.
 have inv_m (u : 'Z_q) : coprime q u -> {a | a \in A & m a = u}.
   rewrite -?unitZpE // natr_Zp -im_m => m_u.
-  by exists (iinv m_u); [exact: mem_iinv | rewrite f_iinv].
+  by exists (iinv m_u); [apply: mem_iinv | rewrite f_iinv].
 have [cycF ffulF]: cyclic F /\ [faithful F, on 'Ohm_1(G) | [Aut G]].
   have Um0 a: ((m a)%:R : 'F_p) \in GRing.unit.
-    have: m a \in GRing.unit by exact: valP.
+    have: m a \in GRing.unit by apply: valP.
     by rewrite -{1}[m a]natr_Zp unitFpE ?unitZpE // {1}/q oG coprime_pexpl.
   pose fm0 a := FinRing.unit 'F_p (Um0 a).
   have natZqp u: (u%:R : 'Z_q)%:R = u %:R :> 'F_p.
@@ -271,7 +269,7 @@ have [cycF ffulF]: cyclic F /\ [faithful F, on 'Ohm_1(G) | [Aut G]].
       by rewrite {1}[q]oG coprime_pexpl // -unitFpE // natZqp natr_Zp.
     by exists a => //; apply: val_inj; rewrite /= m_a natZqp natr_Zp.
   have [x1 defG1]: exists x1, 'Ohm_1(G) = <[x1]>.
-    by apply/cyclicP; exact: cyclicS (Ohm_sub _ _) cycG.
+    by apply/cyclicP; apply: cyclicS (Ohm_sub _ _) cycG.
   have ox1: #[x1] = p by rewrite orderE -defG1 (Ohm1_cyclic_pgroup_prime _ pG).
   have Gx1: x1 \in G by rewrite -cycle_subG -defG1 Ohm_sub.
   have ker_m0: 'ker m0 = 'C('Ohm_1(G) | [Aut G]).
@@ -293,7 +291,7 @@ have [cycF ffulF]: cyclic F /\ [faithful F, on 'Ohm_1(G) | [Aut G]].
   have inj_m0: 'ker_F m0 \subset [1] by rewrite setIC ker_m0_P tiPF.
   split; last by rewrite /faithful -ker_m0.
   have isogF: F \isog [set: {unit 'F_p}].
-    have sFA: F \subset A by exact: pcore_sub.
+    have sFA: F \subset A by apply: pcore_sub.
     apply/isogP; exists (restrm_morphism sFA m0); first by rewrite ker_restrm.
     apply/eqP; rewrite eqEcard subsetT card_injm ?ker_restrm //= oF.
     by rewrite card_units_Fp.
@@ -441,7 +439,7 @@ Lemma Grp_modular_group :
   'Mod_(p ^ n) \isog Grp (x : y : (x ^+ q, y ^+ p, x ^ y = x ^+ r.+1)).
 Proof.
 rewrite /modular_gtype def_p def_q def_r; apply: Extremal.Grp => //.
-set B := <[_]>; have Bb: Zp1 \in B by exact: cycle_id.
+set B := <[_]>; have Bb: Zp1 \in B by apply: cycle_id.
 have oB: #|B| = q by rewrite -orderE order_Zp1 Zp_cast.
 have cycB: cyclic B by rewrite cycle_cyclic.
 have pB: p.-group B by rewrite /pgroup oB pnat_exp ?pnat_id.
@@ -472,7 +470,7 @@ have [Gx Gy]: x \in G /\ y \in G.
 have notXy: y \notin <[x]>.
   apply: contraL ltqm; rewrite -cycle_subG -oG -defG; move/mulGidPl->.
   by rewrite -leqNgt dvdn_leq ?(ltnW q_gt1) // order_dvdn xq.
-have oy: #[y] = p by exact: nt_prime_order (group1_contra notXy).
+have oy: #[y] = p by apply: nt_prime_order (group1_contra notXy).
 exists (x, y) => //=; split; rewrite ?inE ?notXy //.
 apply/eqP; rewrite -(eqn_pmul2r p_gt0) -expnSr -{1}oy (ltn_predK n_gt2) -/m.
 by rewrite -TI_cardMg ?defG ?oG // setIC prime_TIg ?cycle_subG // -orderE oy.
@@ -528,7 +526,7 @@ have defZ: 'Z(G) = <[x ^+ p]>.
   rewrite (dvdn_prime_cyclic p_pr) // card_quotient //.
   rewrite -(dvdn_pmul2l (cardG_gt0 'Z(G))) Lagrange // oG -def_m dvdn_pmul2r //.
   case/p_natP: (pgroupS sZG pG) gtZr => k ->.
-  by rewrite ltn_exp2l // def_n1; exact: dvdn_exp2l.
+  by rewrite ltn_exp2l // def_n1; apply: dvdn_exp2l.
 have Zxr: x ^+ r \in 'Z(G) by rewrite /r def_n expnS expgM defZ mem_cycle.
 have rxy: [~ x, y] = x ^+ r by rewrite /commg xy expgS mulKg.
 have defG': G^`(1) = <[x ^+ r]>.
@@ -542,7 +540,7 @@ have nil2_G: nil_class G = 2.
 have XYp: {in X & <[y]>, forall z t,
    (z * t) ^+ p \in z ^+ p *: <[x ^+ r ^+ 'C(p, 2)]>}.
 - move=> z t Xz Yt; have Gz := subsetP sXG z Xz; have Gt := subsetP sYG t Yt.
-  have Rtz: [~ t, z] \in G^`(1) by exact: mem_commg.
+  have Rtz: [~ t, z] \in G^`(1) by apply: mem_commg.
   have cGtz: [~ t, z] \in 'C(G) by case/setIP: (subsetP sG'Z _ Rtz).
   rewrite expMg_Rmul /commute ?(centP cGtz) //.
   have ->: t ^+ p = 1 by apply/eqP; rewrite -order_dvdn -oy order_dvdG.
@@ -646,7 +644,7 @@ Lemma card_dihedral : #|'D_m| = m.
 Proof. by rewrite /('D_m)%type def_q card_ext_dihedral ?mul1n. Qed.
 
 Lemma Grp_dihedral : 'D_m \isog Grp (x : y : (x ^+ q, y ^+ 2, x ^ y = x^-1)).
-Proof. by rewrite /('D_m)%type def_q; exact: Grp_ext_dihedral. Qed.
+Proof. by rewrite /('D_m)%type def_q; apply: Grp_ext_dihedral. Qed.
 
 Lemma Grp'_dihedral : 'D_m \isog Grp (x : y : (x ^+ 2, y ^+ 2, (x * y) ^+ q)).
 Proof.
@@ -873,7 +871,7 @@ have{defG} defG: <[x]> * <[y]> = G.
 have notXy: y \notin <[x]>.
   apply: contraL ltqm => Xy; rewrite -leqNgt -oG -defG mulGSid ?cycle_subG //.
   by rewrite dvdn_leq // order_dvdn xq.
-have oy: #[y] = 2 by exact: nt_prime_order (group1_contra notXy).
+have oy: #[y] = 2 by apply: nt_prime_order (group1_contra notXy).
 have ox: #[x] = q.
   apply: double_inj; rewrite -muln2 -oy -mul2n def2q -oG -defG TI_cardMg //.
   by rewrite setIC prime_TIg ?cycle_subG // -orderE oy.
@@ -895,7 +893,7 @@ have{defG} defG: <[x]> * <[y]> = G.
 have notXy: y \notin <[x]>.
   apply: contraL ltqm => Xy; rewrite -leqNgt -oG -defG mulGSid ?cycle_subG //.
   by rewrite dvdn_leq // order_dvdn xq.
-have oy: #[y] = 2 by exact: nt_prime_order (group1_contra notXy).
+have oy: #[y] = 2 by apply: nt_prime_order (group1_contra notXy).
 have ox: #[x] = q.
   apply: double_inj; rewrite -muln2 -oy -mul2n def2q -oG -defG TI_cardMg //.
   by rewrite setIC prime_TIg ?cycle_subG // -orderE oy.
@@ -1042,7 +1040,7 @@ have maxMt: {in G :\: X, forall t, maximal <<t ^: G>> G}.
 have X'xy: x * y \in G :\: X by rewrite !inE !groupMl ?cycle_id ?notXy.
 have ti_yG_xyG: [disjoint yG & xyG].
   apply/pred0P=> t; rewrite /= /yG /xyG !def_tG //; apply/andP=> [[yGt]].
-  rewrite rcoset_sym (rcoset_transl yGt) mem_rcoset mulgK; move/order_dvdG.
+  rewrite rcoset_sym (rcoset_eqP yGt) mem_rcoset mulgK; move/order_dvdG.
   by rewrite -orderE ox2 ox gtnNdvd.
 have s_tG_X': {in G :\: X, forall t, t ^: G \subset G :\: X}.
   by move=> t X't /=; rewrite class_sub_norm // normsD ?normG.
@@ -1066,7 +1064,7 @@ split.
   apply/eqP; rewrite eqEsubset Ohm_sub -{1}defXY mulG_subG !cycle_subG.
   by rewrite -(groupMr _ (sX'G1 y X'y)) !sX'G1.
 - split=> //= H; apply/idP/idP=> [maxH |]; last first.
-    by case/or3P; move/eqP->; rewrite ?maxMt.
+    by case/or3P=> /eqP->; rewrite ?maxMt.
   have [sHG nHG]:= andP (p_maximal_normal pG maxH).
   have oH: #|H| = q. 
     apply: double_inj; rewrite -muln2 -(p_maximal_index pG maxH) Lagrange //.
@@ -1075,7 +1073,7 @@ split.
   case sHX: (H \subset X) => //=; case/subsetPn: sHX => t Ht notXt.
   have: t \in yG :|: xyG by rewrite defX' inE notXt (subsetP sHG).
   rewrite !andbT !gen_subG /yG /xyG.
-  by case/setUP; move/class_transr <-; rewrite !class_sub_norm ?Ht ?orbT.
+  by case/setUP; move/class_eqP <-; rewrite !class_sub_norm ?Ht ?orbT.
 rewrite eqn_leq n_gt1; case: leqP n2_abelG => //= n_gt2 _.
 have ->: 'Z(G) = <[x ^+ r]>.
   apply/eqP; rewrite eqEcard andbC -orderE oxr -{1}(setIidPr (center_sub G)).
@@ -1240,14 +1238,14 @@ have oMt: {in G :\: X, forall t, #|<<t ^: G>>| = q}.
   rewrite quotient_cycle ?(subsetP (nXiG 2)) //= -defPhi.
   rewrite -orderE (abelem_order_p (Phi_quotient_abelem pG)) ?mem_quotient //.
   apply: contraNneq notXt; move/coset_idr; move/implyP=> /=.
-  by rewrite defPhi ?(subsetP (nXiG 2)) //; apply: subsetP; exact: cycleX.
+  by rewrite defPhi ?(subsetP (nXiG 2)) //; apply: subsetP; apply: cycleX.
 have maxMt: {in G :\: X, forall t, maximal <<t ^: G>> G}.
   move=> t X't; rewrite /= p_index_maximal -?divgS ?sMtG ?oMt //.
   by rewrite oG -def2q mulnK.
 have X'xy: x * y \in G :\: X by rewrite !inE !groupMl ?cycle_id ?notXy.
 have ti_yG_xyG: [disjoint yG & xyG].
   apply/pred0P=> t; rewrite /= /yG /xyG !def_tG //; apply/andP=> [[yGt]].
-  rewrite rcoset_sym (rcoset_transl yGt) mem_rcoset mulgK; move/order_dvdG.
+  rewrite rcoset_sym (rcoset_eqP yGt) mem_rcoset mulgK; move/order_dvdG.
   by rewrite -orderE ox2 ox gtnNdvd.
 have s_tG_X': {in G :\: X, forall t, t ^: G \subset G :\: X}.
   by move=> t X't /=; rewrite class_sub_norm // normsD ?normG.
@@ -1286,7 +1284,7 @@ rewrite pprodE //; split=> // [|||n_gt3].
   case sHX: (H \subset X) => //=; case/subsetPn: sHX => z Hz notXz.
   have: z \in yG :|: xyG by rewrite defX' inE notXz (subsetP sHG).
   rewrite !andbT !gen_subG /yG /xyG.
-  by case/setUP=> /class_transr <-; rewrite !class_sub_norm ?Hz ?orbT.
+  by case/setUP=> /class_eqP <-; rewrite !class_sub_norm ?Hz ?orbT.
 have isoMt: {in G :\: X, forall z, <<z ^: G>> \isog 'Q_q}.
   have n1_gt2: n.-1 > 2 by rewrite -(subnKC n_gt3).
   move=> z X'z /=; rewrite isogEcard card_quaternion ?oMt // leqnn andbT.
@@ -1385,7 +1383,7 @@ have defG1: 'Mho^1(G) = <[x ^+ 2]>.
     by case/cycleP: Xz => i ->; rewrite -expgM mulnC expgM mem_cycle.
   have{Xz Gz} [xi Xxi ->{z}]: exists2 xi, xi \in X & z = xi * y.
     have Uvy: y \in <[u]> :* v by rewrite defUv -(defU x).
-    apply/rcosetP; rewrite /X defU // (rcoset_transl Uvy) defUv.
+    apply/rcosetP; rewrite /X defU // (rcoset_eqP Uvy) defUv.
     by rewrite inE -(defU x) ?Xz.
   rewrite expn1 expgS {2}(conjgC xi) -{2}[y]/(y ^+ 2.-1) -{1}oy -invg_expg.
   rewrite mulgA mulgK invXX' // -expgS prednK // /r -(subnKC n_gt3) expnS.
@@ -1427,14 +1425,14 @@ have oMt: {in G :\: X, forall t, #|<<t ^: G>>| = q}.
   rewrite quotient_cycle ?(subsetP (nXiG 2)) //= -defPhi -orderE.
   rewrite (abelem_order_p (Phi_quotient_abelem pG)) ?mem_quotient //.
   apply: contraNneq notXt; move/coset_idr; move/implyP=> /=.
-  by rewrite /= defPhi (subsetP (nXiG 2)) //; apply: subsetP; exact: cycleX.
+  by rewrite /= defPhi (subsetP (nXiG 2)) //; apply: subsetP; apply: cycleX.
 have maxMt: {in G :\: X, forall t, maximal <<t ^: G>> G}.
   move=> t X't /=; rewrite p_index_maximal -?divgS ?sMtG ?oMt //.
   by rewrite oG -def2q mulnK.
 have X'xy: x * y \in G :\: X by rewrite !inE !groupMl ?cycle_id ?notXy.
 have ti_yG_xyG: [disjoint yG & xyG].
   apply/pred0P=> t; rewrite /= /yG /xyG !def_tG //; apply/andP=> [[yGt]].
-  rewrite rcoset_sym (rcoset_transl yGt) mem_rcoset mulgK; move/order_dvdG.
+  rewrite rcoset_sym (rcoset_eqP yGt) mem_rcoset mulgK; move/order_dvdG.
   by rewrite -orderE ox2 ox gtnNdvd.
 have s_tG_X': {in G :\: X, forall t, t ^: G \subset G :\: X}.
   by move=> t X't /=; rewrite class_sub_norm // normsD ?normG.
@@ -1481,7 +1479,7 @@ split.
   case sHX: (H \subset X) => //=; case/subsetPn: sHX => t Ht notXt.
   have: t \in yG :|: xyG by rewrite defX' inE notXt (subsetP sHG).
   rewrite !andbT !gen_subG /yG /xyG.
-  by case/setUP=> /class_transr <-; rewrite !class_sub_norm ?Ht ?orbT.
+  by case/setUP=> /class_eqP <-; rewrite !class_sub_norm ?Ht ?orbT.
 have n1_gt2: n.-1 > 2 by [rewrite -(subnKC n_gt3)]; have n1_gt1 := ltnW n1_gt2.
 rewrite !isogEcard card_2dihedral ?card_quaternion ?oMt // leqnn !andbT.
 have invX2X': {in G :\: X, forall t, x ^+ 2 ^ t == x ^- 2}.
@@ -1686,7 +1684,7 @@ have:= semidihedral_structure n_gt3 genG isoG oy.
 rewrite -/X -/q -/r -/yG -/xyG -/My -/Mxy.
 case=> [[defG oxy invXX'] nilG [defZ oZ [-> ->] defMho] [[defX' tiX'] maxG]].
 case=> isoMy isoMxy defX; do 2!split=> //.
-  by apply/dihedral_classP; exists n.-1; first exact: ltnW.
+  by apply/dihedral_classP; exists n.-1; first apply: ltnW.
 by apply/quaternion_classP; exists n.-1.
 Qed.
 
@@ -1724,7 +1722,7 @@ pose Z := <[x ^+ r]>.
 have defZ: Z = 'Ohm_1(X) by rewrite defX (Ohm_p_cycle _ p_x) ox subn1 pfactorK.
 have oZ: #|Z| = p by rewrite -orderE orderXdiv ox -def_pr ?dvdn_mull ?mulnK.
 have cGZ: Z \subset 'C(G).
-  have nsZG: Z <| G by rewrite defZ (char_normal_trans (Ohm_char 1 _)).
+  have nsZG: Z <| G by rewrite defZ gFnormal_trans.
   move/implyP: (meet_center_nil (pgroup_nil pG) nsZG).
   rewrite -cardG_gt1 oZ p_gt1 setIA (setIidPl (normal_sub nsZG)).
   by apply: contraR; move/prime_TIg=> -> //; rewrite oZ.
@@ -1867,9 +1865,9 @@ Proof.
 move=> pG /SCN_P[nsUG scUG] not_cGG cycU; have [sUG nUG] := andP nsUG.
 have [cUU pU] := (cyclic_abelian cycU, pgroupS sUG pG).
 have ltUG: ~~ (G \subset U).
-  by apply: contra not_cGG => sGU; exact: abelianS cUU.
+  by apply: contra not_cGG => sGU; apply: abelianS cUU.
 have ntU: U :!=: 1.
-  by apply: contra ltUG; move/eqP=> U1; rewrite -(setIidPl (cents1 G)) -U1 scUG.
+  by apply: contraNneq ltUG => U1; rewrite -scUG subsetIidl U1 cents1.
 have [p_pr _ [n oU]] := pgroup_pdiv pU ntU.
 have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
 have [u defU] := cyclicP cycU; have Uu: u \in U by rewrite defU cycle_id.
@@ -1879,7 +1877,7 @@ have modM1 (M : {group gT}):
     [/\ U \subset M, #|M : U| = p & extremal_class M = ModularGroup] ->
   M :=: 'C_M('Mho^1(U)) /\ 'Ohm_1(M)%G \in 'E_p^2(M).
 - case=> sUM iUM /modular_group_classP[q q_pr {n oU}[n n_gt23 isoM]].
-  have n_gt2: n > 2 by exact: leq_trans (leq_addl _ _) n_gt23.
+  have n_gt2: n > 2 by apply: leq_trans (leq_addl _ _) n_gt23.
   have def_n: n = (n - 3).+3 by rewrite -{1}(subnKC n_gt2).
   have oM: #|M| = (q ^ n)%N by rewrite (card_isog isoM) card_modular_group.
   have pM: q.-group M by rewrite /pgroup oM pnat_exp pnat_id.
@@ -1905,7 +1903,7 @@ have [_ _ [[|k] oGs]] := pgroup_pdiv pGs ntGs.
     by right; exists G; case: (modM1 G) => // <- ->; rewrite Ohm_char.
   by left; case: eqP ext2G => // <-.
 pose M := 'C_G('Mho^1(U)); right; exists [group of M].
-have sMG: M \subset G by exact: subsetIl.
+have sMG: M \subset G by apply: subsetIl.
 have [pM nUM] := (pgroupS sMG pG, subset_trans sMG nUG).
 have sUM: U \subset M by rewrite subsetI sUG sub_abelian_cent ?Mho_sub.
 pose A := Aut U; have cAA: abelian A by rewrite Aut_cyclic_abelian.
@@ -1986,9 +1984,9 @@ have not_cMM: ~~ abelian M.
   by rewrite (setIidPl _) ?indexgg // -scUG subsetI sMG sub_abelian_cent.
 have modM: extremal_class M = ModularGroup.
   have sU1Z: 'Mho^1(U) \subset 'Z(M).
-    by rewrite subsetI (subset_trans (Mho_sub 1 U)) // centsC subsetIr.
-  case: (predU1P (maximal_cycle_extremal _ _ _ _ iUM)) => //=; rewrite -/M.
-  case/andP; move/eqP=> p2 ext2M; rewrite p2 add1n in n_gt01.
+    by rewrite subsetI gFsub_trans // centsC subsetIr.
+  have /maximal_cycle_extremal/predU1P[] //= := iUM; rewrite -/M.
+  case/andP=> /eqP-p2 ext2M; rewrite p2 add1n in n_gt01.
   suffices{sU1Z}: #|'Z(M)| = 2.
     move/eqP; rewrite eqn_leq leqNgt (leq_trans _ (subset_leq_card sU1Z)) //.
     by rewrite defU1 -orderE (orderXexp 1 ou) subn1 p2 (ltn_exp2l 1).
@@ -2005,9 +2003,9 @@ have modM: extremal_class M = ModularGroup.
   by case: (m == 2) => [|[]//]; move/abelem_abelian->.
 split=> //.
   have [//|_] := modM1 [group of M]; rewrite !inE -andbA /=.
-  by case/andP; move/subset_trans->.
+  by case/andP=> /subset_trans->.
 have{cycGs} [cycGs | [p2 [c fGs_c u_c]]] := cycGs.
-  suffices ->: 'Ohm_1(M) = 'Ohm_1(G) by exact: Ohm_char.
+  suffices ->: 'Ohm_1(M) = 'Ohm_1(G) by apply: Ohm_char.
   suffices sG1M: 'Ohm_1(G) \subset M.
     by apply/eqP; rewrite eqEsubset -{2}(Ohm_id 1 G) !OhmS.
   rewrite -(quotientSGK _ sUM) ?(subset_trans (Ohm_sub _ G)) //= defMs.
@@ -2017,13 +2015,12 @@ have{cycGs} [cycGs | [p2 [c fGs_c u_c]]] := cycGs.
   rewrite -(part_pnat_id (pgroupS (Ohm_sub _ _) pGs)) p_part (leq_exp2l _ 1) //.
   by rewrite -p_rank_abelian -?rank_pgroup -?abelian_rank1_cyclic.
 suffices charU1: 'Mho^1(U) \char G^`(1).
-  rewrite (char_trans (Ohm_char _ _)) // subcent_char ?char_refl //.
-  exact: char_trans (der_char 1 G).
+  by rewrite gFchar_trans // subcent_char ?(char_trans charU1) ?gFchar.
 suffices sUiG': 'Mho^1(U) \subset G^`(1).
-  have cycG': cyclic G^`(1) by rewrite (cyclicS _ cycU) // der1_min.
-  by case/cyclicP: cycG' sUiG' => zs ->; exact: cycle_subgroup_char.
+  have /cyclicP[zs cycG']: cyclic G^`(1) by rewrite (cyclicS _ cycU) ?der1_min.
+  by rewrite cycG' in sUiG' *; apply: cycle_subgroup_char.
 rewrite defU1 cycle_subG p2 -groupV invMg -{2}u_c.
-by case/morphimP: fGs_c => _ _ /morphimP[z _ Gz ->] ->; rewrite fG ?mem_commg.
+by have [_ _ /morphimP[z _ Gz ->] ->] := morphimP fGs_c; rewrite fG ?mem_commg.
 Qed.
 
 (* This is Aschbacher, exercise (8.4) *)
@@ -2038,7 +2035,7 @@ have [X maxX]: {X | [max X | X <| G & abelian X]}.
 have cycX: cyclic X by rewrite dn_G_1; case/andP: (maxgroupp maxX).
 have scX: X \in 'SCN(G) := max_SCN pG maxX.
 have [[p2 _ cG] | [M [_ _ _]]] := cyclic_SCN pG scX not_cGG cycX; last first.
-  rewrite 2!inE -andbA; case/and3P=> sEG abelE dimE_2 charE.
+  rewrite 2!inE -andbA => /and3P[sEG abelE dimE_2] charE.
   have:= dn_G_1 _ (char_normal charE) (abelem_abelian abelE).
   by rewrite (abelem_cyclic abelE) (eqP dimE_2).
 have [n oG] := p_natP pG; right; rewrite p2 cG /= in oG *.
@@ -2113,9 +2110,9 @@ move=> pG sympG; have [H [charH]] := Thompson_critical pG.
 have sHG := char_sub charH; have pH := pgroupS sHG pG.
 set U := 'Z(H) => sPhiH_U sHG_U defU; set Z := 'Ohm_1(U).
 have sZU: Z \subset U by rewrite Ohm_sub.
-have charU: U \char G := char_trans (center_char H) charH.
+have charU: U \char G := gFchar_trans _ charH.
 have cUU: abelian U := center_abelian H.
-have cycU: cyclic U by exact: sympG.
+have cycU: cyclic U by apply: sympG.
 have pU: p.-group U := pgroupS (char_sub charU) pG.
 have cHU: U \subset 'C(H) by rewrite subsetIr.
 have cHsHs: abelian (H / Z).
@@ -2130,11 +2127,10 @@ have [K /=] := inv_quotientS nsZH (Ohm_sub 1 (H / Z)); fold Z => defKs sZK sKH.
 have nsZK: Z <| K := normalS sZK sKH nsZH; have [_ nZK] := andP nsZK.
 have abelKs: p.-abelem (K / Z) by rewrite -defKs Ohm1_abelem ?quotient_pgroup.
 have charK: K \char G.
-  have charZ: Z \char H := char_trans (Ohm_char _ _) (center_char H).
+  have charZ: Z \char H := gFchar_trans _ (center_char H).
   rewrite (char_trans _ charH) // (char_from_quotient nsZK) //.
   by rewrite -defKs Ohm_char.
-have cycZK: cyclic 'Z(K).
-  by rewrite sympG ?center_abelian ?(char_trans (center_char _)).
+have cycZK: cyclic 'Z(K) by rewrite sympG ?center_abelian ?gFchar_trans.
 have [cKK | not_cKK] := orP (orbN (abelian K)).
   have defH: U = H.
     apply: center_idP; apply: cyclic_factor_abelian (Ohm_sub 1 _) _.
@@ -2159,14 +2155,14 @@ have [cKK | not_cKK] := orP (orbN (abelian K)).
     by apply: (p3group_extraspecial pG); rewrite // p2 oG pfactorK.
   exists G; [by right | exists ('Z(G))%G; rewrite cprod_center_id setIA setIid].
   by split=> //; left; rewrite prime_cyclic; case: esG.
-have ntK: K :!=: 1 by apply: contra not_cKK; move/eqP->; exact: abelian1.
+have ntK: K :!=: 1 by apply: contra not_cKK => /eqP->; apply: abelian1.
 have [p_pr _ _] := pgroup_pdiv (pgroupS sKH pH) ntK.
 have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
 have oZ: #|Z| = p.
   apply: Ohm1_cyclic_pgroup_prime => //=; apply: contra ntK; move/eqP.
   by move/(trivg_center_pgroup pH)=> GH; rewrite -subG1 -GH.
 have sZ_ZK: Z \subset 'Z(K).
-  by rewrite subsetI sZK (subset_trans (Ohm_sub _ _ )) // subIset ?centS ?orbT.
+  by rewrite subsetI sZK gFsub_trans // subIset ?centS ?orbT.
 have sZsKs: 'Z(K) / Z \subset K / Z by rewrite quotientS ?center_sub.
 have [Es /= splitKs] := abelem_split_dprod abelKs sZsKs.
 have [_ /= defEsZs cEsZs tiEsZs] := dprodP splitKs.
@@ -2187,7 +2183,7 @@ have sEH := subset_trans sEK sKH; have pE := pgroupS sEH pH.
 have esE: extraspecial E.
   split; last by rewrite defZE oZ.
   have sPhiZ: 'Phi(E) \subset Z.
-    rewrite -quotient_sub1 ?(subset_trans (Phi_sub _)) ?(quotient_Phi pE) //.
+    rewrite -quotient_sub1 ?gFsub_trans ?(quotient_Phi pE) //.
     rewrite subG1 (trivg_Phi (quotient_pgroup _ pE)) /= -defEs.
     by rewrite (abelemS sEsKs) //= -defKs Ohm1_abelem ?quotient_pgroup.
   have sE'Phi: E^`(1) \subset 'Phi(E) by rewrite (Phi_joing pE) joing_subl.
@@ -2207,8 +2203,8 @@ have{defE'} sEG_E': [~: E, G] \subset E^`(1).
   have Uge: [~ g, e] \in U by rewrite (subsetP sHG_U) ?mem_commg.
   rewrite inE Uge inE -commgX; last by red; rewrite -(centsP cHU).
   have sZ_ZG: Z \subset 'Z(G).
-    have charZ: Z \char G := char_trans (Ohm_char _ _) charU.
-    move/implyP: (meet_center_nil (pgroup_nil pG) (char_normal charZ)).
+    have charZ: Z \char G := gFchar_trans _ charU.
+    have/implyP:= meet_center_nil (pgroup_nil pG) (char_normal charZ).
     rewrite -cardG_gt1 oZ prime_gt1 //=; apply: contraR => not_sZ_ZG.
     by rewrite prime_TIg ?oZ.
   have: e ^+ p \in 'Z(G).
@@ -2217,7 +2213,7 @@ have{defE'} sEG_E': [~: E, G] \subset E^`(1).
   by case/setIP=> _ /centP cGep; apply/commgP; red; rewrite cGep.
 have sEG: E \subset G := subset_trans sEK (char_sub charK).
 set R := 'C_G(E).
-have{sEG_E'} defG: E \* R = G by exact: (critical_extraspecial pG).
+have{sEG_E'} defG: E \* R = G by apply: (critical_extraspecial pG).
 have [_ defER cRE] := cprodP defG.
 have defH: E \* 'C_H(E) = H by rewrite -(setIidPr sHG) setIAC (cprod_modl defG).
 have{defH} [_ defH cRH_E] := cprodP defH.
@@ -2232,7 +2228,7 @@ have cRH_RH: abelian 'C_H(E).
   have:= Ohm_dprod 1 defHs; rewrite /= defKs (Ohm1_id (abelemS sEsKs abelKs)).
   rewrite dprodC; case/dprodP=> _ defEsRHs1 cRHs1Es tiRHs1Es.
   have sRHsHs: 'C_H(E) / Z \subset H / Z by rewrite quotientS ?subsetIl.
-  have cRHsRHs: abelian ('C_H(E) / Z) by exact: abelianS cHsHs.
+  have cRHsRHs: abelian ('C_H(E) / Z) by apply: abelianS cHsHs.
   have pHs: p.-group (H / Z) by rewrite quotient_pgroup.
   rewrite abelian_rank1_cyclic // (rank_pgroup (pgroupS sRHsHs pHs)).
   rewrite p_rank_abelian // -(leq_add2r (logn p #|Es|)) -lognM ?cardG_gt0 //.
@@ -2259,14 +2255,12 @@ have [[p2 _ ext2R] | [M []]] := cyclic_SCN pR scUR not_cRR cycU.
   exists E; [by right | exists [group of R]; split=> //; right].
   by rewrite dvdn_leq ?(@pfactor_dvdn 2 4) ?cardG_gt0 // -{2}p2.
 rewrite /= -/R => defM iUM modM _ _; pose N := 'C_G('Mho^1(U)).
-have charZN2: 'Z('Ohm_2(N)) \char G.
-  rewrite (char_trans (center_char _)) // (char_trans (Ohm_char _ _)) //.
-  by rewrite subcent_char ?char_refl // (char_trans (Mho_char _ _)).
+have charZN2: 'Z('Ohm_2(N)) \char G by rewrite !(gFchar_trans, subcent_char).
 have:= sympG _ charZN2 (center_abelian _).
 rewrite abelian_rank1_cyclic ?center_abelian // leqNgt; case/negP.
 have defN: E \* M = N.
-  rewrite defM (cprod_modl defG) // centsC (subset_trans (Mho_sub 1 _)) //.
-  by rewrite /= -/U -defRH subsetIr.
+  rewrite defM (cprod_modl defG) // centsC gFsub_trans //= -/U.
+  by rewrite -defRH subsetIr.
 case/modular_group_classP: modM => q q_pr [n n_gt23 isoM].
 have{n_gt23} n_gt2 := leq_trans (leq_addl _ _) n_gt23.
 have n_gt1 := ltnW n_gt2; have n_gt0 := ltnW n_gt1.
@@ -2276,14 +2270,14 @@ have{q_pr} defq: q = p; last rewrite {q}defq in genM modM isoM.
   by have [-> _ _ _] := genM; rewrite Euclid_dvdX // dvdn_prime2 //; case: eqP.
 have [oM Mx ox X'y] := genM; have [My _] := setDP X'y; have [oy _] := modM.
 have [sUM sMR]: U \subset M /\ M \subset R.
-  by rewrite defM subsetI sUR subsetIl centsC (subset_trans (Mho_sub _ _)).
+  by rewrite defM subsetI sUR subsetIl centsC gFsub_trans.
 have oU1: #|'Mho^1(U)| = (p ^ n.-2)%N.
   have oU: #|U| = (p ^ n.-1)%N.
     by rewrite -(divg_indexS sUM) iUM oM -subn1 expnB.
   case/cyclicP: cycU pU oU => u -> p_u ou.
   by rewrite (Mho_p_cycle 1 p_u) -orderE (orderXexp 1 ou) subn1.
 have sZU1: Z \subset 'Mho^1(U).
-  rewrite -(cardSg_cyclic cycU) ?Ohm_sub ?Mho_sub // oZ oU1.
+  rewrite -(cardSg_cyclic cycU) ?gFsub // oZ oU1.
   by rewrite -(subnKC n_gt2) expnS dvdn_mulr.
 case/modular_group_structure: genM => // _ [defZM _ oZM] _ _.
 have:= n_gt2; rewrite leq_eqVlt eq_sym !xpair_eqE andbC.
@@ -2319,7 +2313,7 @@ have defN2: (E <*> X2) \x <[y]> = 'Ohm_2(N).
   rewrite /= (OhmE 2 pN) sub_gen /=; last 1 first.
     by rewrite subsetI -defN mulG_subl sub_LdivT expE.
   rewrite -cent_joinEl // -genM_join genS // -defN.
-  apply/subsetP=> ez; case/setIP; case/imset2P=> e z Ee Mz ->{ez}.
+  apply/subsetP=> _ /setIP[/imset2P[e z Ee Mz ->]].
   rewrite inE expgMn; last by red; rewrite -(centsP cME).
   rewrite (exponentP expE) // mul1g => zp2; rewrite mem_mulg //=.
   by rewrite (OhmE 2 (pgroupS sMR pR)) mem_gen // !inE Mz.
diff --git a/mathcomp/solvable/finmodule.v b/mathcomp/solvable/finmodule.v
index dd84def..2b2aa5f 100644
--- a/mathcomp/solvable/finmodule.v
+++ b/mathcomp/solvable/finmodule.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype bigop prime finset.
-From mathcomp.fingroup
-Require Import fingroup morphism perm action gproduct.
-From mathcomp.algebra
-Require Import ssralg finalg cyclic.
-Require Import commutator.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq path div choice.
+From mathcomp
+Require Import fintype bigop ssralg finset fingroup morphism perm.
+From mathcomp
+Require Import finalg action gproduct commutator cyclic.
 
 (******************************************************************************)
 (*  This file regroups constructions and results that are based on the most   *)
@@ -17,7 +14,7 @@ Require Import commutator.
 (* splitting and transitivity theorem, from which we will later derive the    *)
 (* Schur-Zassenhaus theorem and the elementary abelian special case of        *)
 (* Maschke's theorem, the coprime abelian centraliser/commutator trivial      *)
-(* intersection theorem, from which we will derive that p-groups under coprime*)
+(* intersection theorem, which is used to show that p-groups under coprime    *)
 (* action factor into special groups, and the construction of the transfer    *)
 (* homomorphism and its expansion relative to a cycle, from which we derive   *)
 (* the Higman Focal Subgroup and the Burnside Normal Complement theorems.     *)
@@ -26,7 +23,7 @@ Require Import commutator.
 (* needed much outside this file, which contains all the results that exploit *)
 (* this construction.                                                         *)
 (*   FiniteModule defines the Z[N(A)]-module associated with a finite abelian *)
-(* abelian group A, given a proof abelA : abelian A) :                        *)
+(* abelian group A, given a proof (abelA : abelian A) :                       *)
 (*  fmod_of abelA == the type of elements of the module (similar to but       *)
 (*                   distinct from [subg A]).                                 *)
 (*   fmod abelA x == the injection of x into fmod_of abelA if x \in A, else 0 *)
@@ -95,16 +92,16 @@ Definition fmod_opp u := sub2f u^-1.
 Definition fmod_add u v := sub2f (u * v).
 
 Fact fmod_add0r : left_id (sub2f 1) fmod_add.
-Proof. move=> u; apply: val_inj; exact: mul1g. Qed.
+Proof. by move=> u; apply: val_inj; apply: mul1g. Qed.
 
 Fact fmod_addrA : associative fmod_add.
-Proof. move=> u v w; apply: val_inj; exact: mulgA. Qed.
+Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed.
 
 Fact fmod_addNr : left_inverse (sub2f 1) fmod_opp fmod_add.
-Proof. move=> u; apply: val_inj; exact: mulVg. Qed.
+Proof. by move=> u; apply: val_inj; apply: mulVg. Qed.
 
 Fact fmod_addrC : commutative fmod_add.
-Proof. case=> x Ax [y Ay]; apply: val_inj; exact: (centsP abelA). Qed.
+Proof. by case=> x Ax [y Ay]; apply: val_inj; apply: (centsP abelA). Qed.
 
 Definition fmod_zmodMixin := 
   ZmodMixin fmod_addrA fmod_addrC fmod_add0r fmod_addNr.
@@ -149,7 +146,7 @@ Qed.
 
 Lemma injm_fmod : 'injm fmod.
 Proof. 
-apply/injmP=> x y Ax Ay []; move/val_inj; exact: (injmP (injm_subg A)).
+by apply/injmP=> x y Ax Ay []; move/val_inj; apply: (injmP (injm_subg A)).
 Qed.
 
 Notation "u ^@ x" := (actr u x) : ring_scope.
@@ -275,9 +272,9 @@ have PpP x: pP x \in P by rewrite -mem_rcoset rcoset_repr rcoset_refl.
 have rPmul x y: x \in P -> rP (x * y) = rP y.
   by move=> Px; rewrite /rP rcosetM rcoset_id.
 pose pQ x := remgr H Q x; pose rH x := pQ (pP x) * rP x.
-have pQhq: {in H & Q, forall h q, pQ (h * q) = q} by exact: remgrMid.
+have pQhq: {in H & Q, forall h q, pQ (h * q) = q} by apply: remgrMid.
 have pQmul: {in P &, {morph pQ : x y / x * y}}.
-  apply: remgrM; [exact/complP | exact: normalS (nsHG)].
+  by apply: remgrM; [apply/complP | apply: normalS (nsHG)].
 have HrH x: rH x \in H :* x.
   by rewrite rcoset_sym mem_rcoset invMg mulgA mem_divgr // eqHQ PpP.
 have GrH x: x \in G -> rH x \in G.
@@ -292,7 +289,7 @@ pose nu y := (\sum_(Px in rcosets P G) mu (repr Px) y)%R.
 have rHmul: {in G &, forall x y, rH (x * y) = rH x * rH y * val (mu x y)}.
   move=> x y Gx Gy; rewrite /= fmodK ?mulKVg // -mem_lcoset lcoset_sym.
   rewrite -norm_rlcoset; last by rewrite nHG ?GrH ?groupM.
-  by rewrite (rcoset_transl (HrH _)) -rcoset_mul ?nHG ?GrH // mem_mulg.
+  by rewrite (rcoset_eqP (HrH _)) -rcoset_mul ?nHG ?GrH // mem_mulg.
 have actrH a x: x \in G -> (a ^@ rH x = a ^@ x)%R.
   move=> Gx; apply: val_inj; rewrite /= !fmvalJ ?nHG ?GrH //.
   case/rcosetP: (HrH x) => b /(fmodK abelH) <- ->; rewrite conjgM.
@@ -338,7 +335,7 @@ exists (Morphism fM @* G)%G; apply/complP; split.
   apply/trivgP/subsetP=> x /setIP[Hx /morphimP[y _ Gy eq_x]].
   apply/set1P; move: Hx; rewrite {x}eq_x /= groupMr ?subgP //.
   rewrite -{1}(mulgKV y (rH y)) groupMl -?mem_rcoset // => Hy.
-  by rewrite -(mulg1 y) /f nu_Hmul // rH_Hmul //; exact: (morph1 (Morphism fM)).
+  by rewrite -(mulg1 y) /f nu_Hmul // rH_Hmul //; apply: (morph1 (Morphism fM)).
 apply/setP=> x; apply/mulsgP/idP=> [[h y Hh fy ->{x}] | Gx].
   rewrite groupMl; last exact: (subsetP sHG).
   case/morphimP: fy => z _ Gz ->{h Hh y}.
@@ -519,7 +516,7 @@ have HGgHzg: Hzg \in HG :* <[g]>.
   by rewrite mem_mulg ?set11 // -rcosetE mem_imset.
 have Hzg_x: x \in Hzg by rewrite (repr_mem_pblock trX).
 exists x; first by rewrite (repr_mem_transversal trX).
-case/mulsgP: Hzg_x => y u /rcoset_transl <- /(orbit_act 'Rs) <- -> /=.
+case/mulsgP: Hzg_x => y u /rcoset_eqP <- /(orbit_act 'Rs) <- -> /=.
 by rewrite rcosetE -rcosetM.
 Qed.
 
@@ -543,7 +540,7 @@ pose pcyc x := pcycle (actperm 'Rs g) (H :* x).
 pose traj x := traject (actperm 'Rs g) (H :* x) #|pcyc x|.
 have Hgr_eq x: H_g_rcosets x = pcyc x.
   by rewrite /H_g_rcosets -orbitRs -pcycle_actperm ?inE.
-have pcyc_eq x: pcyc x =i traj x by exact: pcycle_traject.
+have pcyc_eq x: pcyc x =i traj x by apply: pcycle_traject.
 have uniq_traj x: uniq (traj x) by apply: uniq_traject_pcycle.
 have n_eq x: n_ x = #|pcyc x| by rewrite -Hgr_eq.
 have size_traj x: size (traj x) = n_ x by rewrite n_eq size_traject.
@@ -572,7 +569,7 @@ have trY: is_transversal Y HG G.
     by rewrite -[x](mulgK (g ^+ i)) mem_mulg ?rcoset_refl // groupV mem_cycle.
   apply/set1P; rewrite /y eq_xx'; congr (_ * _ ^+ _) => //; apply/eqP.
   rewrite -(@nth_uniq _ (H :* x) (traj x)) ?size_traj // ?eq_xx' //.
-  by rewrite !nth_traj ?(rcoset_transl Hy_x'gj) // -eq_xx'.
+  by rewrite !nth_traj ?(rcoset_eqP Hy_x'gj) // -eq_xx'.
 have rYE x i : x \in X -> i < n_ x -> rY (H :* x :* g ^+ i) = x * g ^+ i.
   move=> Xx lt_i_x; rewrite -rcosetM; apply: (canLR_in (pblockK trY 1)).
     by apply/bigcupP; exists x => //; apply/imsetP; exists (Ordinal lt_i_x).
diff --git a/mathcomp/solvable/frobenius.v b/mathcomp/solvable/frobenius.v
index 64fe7e6..cc589a0 100644
--- a/mathcomp/solvable/frobenius.v
+++ b/mathcomp/solvable/frobenius.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat.
-From mathcomp.discrete
-Require Import div fintype bigop prime finset.
-From mathcomp.fingroup
-Require Import fingroup morphism perm action quotient gproduct.
-From mathcomp.algebra
-Require Import cyclic.
-Require Import center pgroup nilpotent sylow hall abelian.
+From mathcomp
+Require Import ssrfun ssrbool eqtype ssrnat div fintype bigop prime.
+From mathcomp
+Require Import finset fingroup morphism perm action quotient gproduct.
+From mathcomp
+Require Import cyclic center pgroup nilpotent sylow hall abelian.
 
 (******************************************************************************)
 (*  Definition of Frobenius groups, some basic results, and the Frobenius     *)
@@ -19,8 +16,7 @@ Require Import center pgroup nilpotent sylow hall abelian.
 (*       condition.                                                           *)
 (*    semiprime K H <->                                                       *)
 (*       the internal action of H on K is "prime", i.e., an element of K that *)
-(*       centralises a nontrivial element of H must actually centralise all   *)
-(*       of H.                                                                *)
+(*       centralises a nontrivial element of H must centralise all of H.      *)
 (*    normedTI A G L <=>                                                      *)
 (*       A is nonempty, strictly disjoint from its conjugates in G, and has   *)
 (*       normaliser L in G.                                                   *)
@@ -36,8 +32,8 @@ Require Import center pgroup nilpotent sylow hall abelian.
 (*       G is (isomorphic to) a Frobenius group with complement H; same as    *)
 (*       above, but without the semi-direct product. The proof that this form *)
 (*       is equivalent to the above (i.e., the existence of Frobenius         *)
-(*       kernels) requires chareacter theory and will only be proved in the   *)
-(*       vcharacter module.                                                   *)
+(*       kernels) requires character theory and will only be proved in the    *)
+(*       vcharacter.v file.                                                   *)
 (*    [Frobenius G] <=> G is a Frobenius group.                               *)
 (*    Frobenius_action G H S to <->                                           *)
 (*       The action to of G on S defines an isomorphism of G with a           *)
@@ -306,7 +302,7 @@ Lemma Frobenius_actionP G H :
 Proof.
 apply: (iffP andP) => [[neqHG] | [sT S to [ffulG transG regG ntH [u Su defH]]]].
   case/normedTI_P=> nzH /subsetIP[sHG _] tiHG.
-  suffices: Frobenius_action G H (rcosets H G) 'Rs by exact: HasFrobeniusAction.
+  suffices: Frobenius_action G H (rcosets H G) 'Rs by apply: HasFrobeniusAction.
   pose Hfix x := 'Fix_(rcosets H G | 'Rs)[x].
   have regG: {in G^#, forall x, #|Hfix x| <= 1}.
     move=> x /setD1P[ntx Gx].
@@ -314,12 +310,12 @@ apply: (iffP andP) => [[neqHG] | [sT S to [ffulG transG regG ntH [u Su defH]]]].
     rewrite -(cards1 Hy) => /setIP[/imsetP[y Gy ->{Hy}] cHyx].
     apply/subset_leq_card/subsetP=> _ /setIP[/imsetP[z Gz ->] cHzx].
     rewrite -!sub_astab1 !astab1_act !sub1set astab1Rs in cHyx cHzx *.
-    rewrite !rcosetE; apply/set1P/rcoset_transl; rewrite mem_rcoset.
+    rewrite !rcosetE; apply/set1P/rcoset_eqP; rewrite mem_rcoset.
     apply: tiHG; [by rewrite !in_group | apply/pred0Pn; exists (x ^ y^-1)].
     by rewrite conjD1g !inE conjg_eq1 ntx -mem_conjg cHyx conjsgM memJ_conjg.
   have ntH: H :!=: 1 by rewrite -subG1 -setD_eq0.
   split=> //; first 1 last; first exact: transRs_rcosets.
-    by exists (H : {set gT}); rewrite ?orbit_refl // astab1Rs (setIidPr sHG).
+    by exists (val H); rewrite ?orbit_refl // astab1Rs (setIidPr sHG).
   apply/subsetP=> y /setIP[Gy cHy]; apply: contraR neqHG => nt_y.
   rewrite (index1g sHG) //; apply/eqP; rewrite eqn_leq indexg_gt0 andbT.
   apply: leq_trans (regG y _); last by rewrite setDE 2!inE Gy nt_y /=.
@@ -351,7 +347,7 @@ Lemma FrobeniusWcompl : [Frobenius G with complement H].
 Proof. by case/andP: frobG. Qed.
 
 Lemma FrobeniusW : [Frobenius G].
-Proof. by apply/existsP; exists H; exact: FrobeniusWcompl. Qed.
+Proof. by apply/existsP; exists H; apply: FrobeniusWcompl. Qed.
 
 Lemma Frobenius_context :
   [/\ K ><| H = G, K :!=: 1, H :!=: 1, K \proper G & H \proper G].
@@ -408,7 +404,7 @@ by rewrite inE (subsetP (Frobenius_cent1_ker K1y)) // inE cent1C (subsetP sHG).
 Qed.
 
 Lemma Frobenius_reg_compl : semiregular H K.
-Proof. by apply: semiregular_sym; exact: Frobenius_reg_ker. Qed.
+Proof. by apply: semiregular_sym; apply: Frobenius_reg_ker. Qed.
 
 Lemma Frobenius_dvd_ker1 : #|H| %| #|K|.-1.
 Proof.
@@ -490,11 +486,11 @@ Qed.
 
 Lemma Frobenius_ker_dvd_ker1 G K :
   [Frobenius G with kernel K] -> #|G : K| %| #|K|.-1.
-Proof. case/existsP=> H; exact: Frobenius_index_dvd_ker1. Qed.
+Proof. by case/existsP=> H; apply: Frobenius_index_dvd_ker1. Qed.
 
 Lemma Frobenius_ker_coprime G K :
   [Frobenius G with kernel K] -> coprime #|K| #|G : K|.
-Proof. case/existsP=> H; exact: Frobenius_index_coprime. Qed.
+Proof. by case/existsP=> H; apply: Frobenius_index_coprime. Qed.
 
 Lemma Frobenius_semiregularP G K H :
     K ><| H = G -> K :!=: 1 -> H :!=: 1 ->
@@ -547,7 +543,7 @@ move=> ntH1 sH1H frobG; have [defG ntK _ _ _] := Frobenius_context frobG.
 apply/Frobenius_semiregularP=> //.
   have [_ _ /(subset_trans sH1H) nH1K tiHK] := sdprodP defG.
   by rewrite sdprodEY //; apply/trivgP; rewrite -tiHK setIS.
-by apply: sub_in1 (Frobenius_reg_ker frobG); exact/subsetP/setSD.
+by apply: sub_in1 (Frobenius_reg_ker frobG); apply/subsetP/setSD.
 Qed.
 
 Lemma Frobenius_kerP G K :
@@ -557,7 +553,7 @@ Lemma Frobenius_kerP G K :
 Proof.
 apply: (iffP existsP) => [[H frobG] | [ntK ltKG nsKG regK]].
   have [/sdprod_context[nsKG _ _ _ _] ntK _ ltKG _] := Frobenius_context frobG.
-  by split=> //; exact: Frobenius_cent1_ker frobG.
+  by split=> //; apply: Frobenius_cent1_ker frobG.
 have /andP[sKG nKG] := nsKG.
 have hallK: Hall G K.
   rewrite /Hall sKG //= coprime_sym coprime_pi' //.
@@ -568,7 +564,7 @@ have hallK: Hall G K.
   have /trivgPn[z]: P :&: K :&: 'Z(P) != 1.
     by rewrite meet_center_nil ?(pgroup_nil pP) ?(normalGI sPG nsKG).
   rewrite !inE -andbA -sub_cent1=> /and4P[_ Kz _ cPz] ntz.
-  by apply: subset_trans (regK z _); [exact/subsetIP | exact/setD1P].
+  by apply: subset_trans (regK z _); [apply/subsetIP | apply/setD1P].
 have /splitsP[H /complP[tiKH defG]] := SchurZassenhaus_split hallK nsKG.
 have [_ sHG] := mulG_sub defG; have nKH := subset_trans sHG nKG. 
 exists H; apply/Frobenius_semiregularP; rewrite ?sdprodE //.
@@ -604,7 +600,7 @@ Proof.
 move=> defG FrobG.
 have partG: partition (gval K |: (H^# :^: K)) G.
   apply: Frobenius_partition; apply/andP; rewrite defG; split=> //.
-  by apply/Frobenius_actionP; exact: HasFrobeniusAction FrobG.
+  by apply/Frobenius_actionP; apply: HasFrobeniusAction FrobG.
 have{FrobG} [ffulG transG regG ntH [u Su defH]]:= FrobG.
 apply/setP=> x; rewrite !inE; have [-> | ntx] := altP eqP; first exact: group1. 
 rewrite /= -(cover_partition partG) /cover.
@@ -677,7 +673,8 @@ Qed.
 Lemma injm_Frobenius_ker K sGD injf : 
   [Frobenius G with kernel K] -> [Frobenius f @* G with kernel f @* K].
 Proof.
-case/existsP=> H frobG; apply/existsP; exists (f @* H)%G; exact: injm_Frobenius.
+case/existsP=> H frobG; apply/existsP.
+by exists (f @* H)%G; apply: injm_Frobenius.
 Qed.
 
 Lemma injm_Frobenius_group sGD injf : [Frobenius G] -> [Frobenius f @* G].
@@ -698,7 +695,7 @@ rewrite ltnS oG mulnK // => leqm.
 have:= q_gt0; rewrite leq_eqVlt => /predU1P[q1 | lt1q].
   rewrite -(mul1n n) q1 -oG (setIidPl _) //.
   by apply/subsetP=> x Gx; rewrite inE -order_dvdn order_dvdG.
-pose p := pdiv q; have pr_p: prime p by exact: pdiv_prime.
+pose p := pdiv q; have pr_p: prime p by apply: pdiv_prime.
 have lt1p: 1 < p := prime_gt1 pr_p; have p_gt0 := ltnW lt1p.
 have{leqm} lt_qp_mq: q %/ p < mq by apply: leq_trans leqm; rewrite ltn_Pdiv.
 have: n %| #|'Ldiv_(p * n)(G)|.
diff --git a/mathcomp/solvable/gfunctor.v b/mathcomp/solvable/gfunctor.v
index 25295ff..517010f 100644
--- a/mathcomp/solvable/gfunctor.v
+++ b/mathcomp/solvable/gfunctor.v
@@ -1,10 +1,8 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat.
-From mathcomp.discrete
-Require Import fintype bigop finset.
-From mathcomp.fingroup
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat fintype bigop finset.
+From mathcomp
 Require Import fingroup morphism automorphism quotient gproduct.
 
 (******************************************************************************)
@@ -121,7 +119,7 @@ Definition iso_continuous :=
    'injm phi -> phi @* F G \subset F (phi @* G).
 
 Lemma continuous_is_iso_continuous : continuous -> iso_continuous.
-Proof. by move=> Fcont gT hT G phi inj_phi; exact: Fcont. Qed.
+Proof. by move=> Fcont gT hT G phi inj_phi; apply: Fcont. Qed.
 
 (* Functoriality on Grp with partial morphisms. *)
 Definition pcontinuous :=
@@ -129,7 +127,7 @@ Definition pcontinuous :=
     phi @* F G \subset F (phi @* G).
 
 Lemma pcontinuous_is_continuous : pcontinuous -> continuous.
-Proof. by move=> Fcont gT hT G; exact: Fcont. Qed.
+Proof. by move=> Fcont gT hT G; apply: Fcont. Qed.
 
 (* Heredity with respect to inclusion *)
 Definition hereditary :=
@@ -269,6 +267,10 @@ Variable F : GFunctor.iso_map.
 Lemma gFsub gT (G : {group gT}) : F gT G \subset G.
 Proof. by case: F gT G. Qed.
 
+Lemma gFsub_trans gT (G : {group gT}) (A : pred_class) :
+  G \subset A -> F gT G \subset A.
+Proof. exact/subset_trans/gFsub. Qed.
+
 Lemma gF1 gT : F gT 1 = 1. Proof. exact/trivgP/gFsub. Qed.
 
 Lemma gFiso_cont : GFunctor.iso_continuous F.
@@ -282,32 +284,44 @@ by rewrite -morphimEsub ?gFsub ?gFiso_cont ?injm_autm.
 Qed.
 
 Lemma gFnorm gT (G : {group gT}) : G \subset 'N(F gT G).
-Proof. by rewrite char_norm ?gFchar. Qed.
+Proof. exact/char_norm/gFchar. Qed.
+
+Lemma gFnorms gT (G : {group gT}) : 'N(G) \subset 'N(F gT G).
+Proof. exact/char_norms/gFchar. Qed.
 
 Lemma gFnormal gT (G : {group gT}) : F gT G <| G.
-Proof. by rewrite char_normal ?gFchar. Qed.
+Proof. exact/char_normal/gFchar. Qed.
+
+Lemma gFchar_trans gT (G H : {group gT}) : H \char G -> F gT H \char G.
+Proof. exact/char_trans/gFchar. Qed.
+
+Lemma gFnormal_trans gT (G H : {group gT}) : H <| G -> F gT H <| G.
+Proof. exact/char_normal_trans/gFchar. Qed.
+
+Lemma gFnorm_trans gT (A : pred_class) (G : {group gT}) :
+  A \subset 'N(G) -> A \subset 'N(F gT G).
+Proof. by move/subset_trans/(_ (gFnorms G)). Qed.
 
 Lemma injmF_sub gT rT (G D : {group gT}) (f : {morphism D >-> rT}) :
   'injm f -> G \subset D -> f @* (F gT G) \subset F rT (f @* G).
 Proof.
-move=> injf sGD; apply/eqP; rewrite -(setIidPr (gFsub G)).
-by rewrite-{3}(setIid G) -!(morphim_restrm sGD) gFiso_cont // injm_restrm.
+move=> injf sGD; have:= gFiso_cont (injm_restrm sGD injf).
+by rewrite im_restrm morphim_restrm (setIidPr _) ?gFsub.
 Qed.
 
 Lemma injmF gT rT (G D : {group gT}) (f : {morphism D >-> rT}) :
   'injm f -> G \subset D -> f @* (F gT G) = F rT (f @* G).
 Proof.
-move=> injf sGD; apply/eqP; rewrite eqEsubset injmF_sub //=.
-rewrite -{2}(morphim_invm injf sGD) -[f @* F _ _](morphpre_invm injf).
-have Fsubs := subset_trans (gFsub _).
-by rewrite -sub_morphim_pre (injmF_sub, Fsubs) ?morphimS ?injm_invm.
+move=> injf sGD; have [sfGD injf'] := (morphimS f sGD, injm_invm injf).
+apply/esym/eqP; rewrite eqEsubset -(injmSK injf') ?gFsub_trans //.
+by rewrite !(subset_trans (injmF_sub _ _)) ?morphim_invm // gFsub_trans.
 Qed.
 
 Lemma gFisom gT rT (G D : {group gT}) R (f : {morphism D >-> rT}) :
   G \subset D -> isom G (gval R) f -> isom (F gT G) (F rT R) f.
 Proof.
-case/(restrmP f)=> g [gf _ _ _]; rewrite -{f}gf.
-by case/isomP=> injg <-; rewrite sub_isom ?gFsub ?injmF.
+case/(restrmP f)=> g [gf _ _ _]; rewrite -{f}gf => /isomP[injg <-].
+by rewrite sub_isom ?gFsub ?injmF.
 Qed.
 
 Lemma gFisog gT rT (G : {group gT}) (R : {group rT}) :
@@ -440,7 +454,7 @@ Section Composition.
 Variables (F1 : GFunctor.mono_map) (F2 : GFunctor.map).
 
 Lemma gFcomp_closed : GFunctor.closed (F1 \o F2).
-Proof. by move=> gT G; rewrite (subset_trans (gFsub _ _)) ?gFsub. Qed.
+Proof. by move=> gT G; rewrite !gFsub_trans. Qed.
 
 Lemma gFcomp_cont : GFunctor.continuous (F1 \o F2).
 Proof.
diff --git a/mathcomp/solvable/gseries.v b/mathcomp/solvable/gseries.v
index 718f074..3878413 100644
--- a/mathcomp/solvable/gseries.v
+++ b/mathcomp/solvable/gseries.v
@@ -1,13 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype bigop finset.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient action.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq path fintype bigop.
+From mathcomp
+Require Import finset fingroup morphism automorphism quotient action.
+From mathcomp
 Require Import commutator center.
-
 (******************************************************************************)
 (*           H <|<| G   <=> H is subnormal in G, i.e., H <| ... <| G.         *)
 (* invariant_factor A H G <=> A normalises both H and G, and H <| G.          *)
@@ -173,7 +171,7 @@ Lemma invariant_subnormal A G H :
 Proof.
 move=> nGA nHA /andP[]; move: #|G| => m.
 elim: m => [|m IHm] in G nGA * => sHG.
-  by rewrite eq_sym; exists [::]; last exact/eqP.
+  by rewrite eq_sym; exists [::]; last apply/eqP.
 rewrite iterSr; set K := <<_>>.
 have nKA: A \subset 'N(K) by rewrite norms_gen ?norms_class_support.
 have sHK: H \subset K by rewrite sub_gen ?sub_class_support.
@@ -297,7 +295,7 @@ apply/maxgroupP/maxgroupP; rewrite morphpre_proper //= => [] [ltMG maxM].
 split=> // H ltHG sMH.
 have dH: H \subset D := subset_trans (proper_sub ltHG) (subsetIl D _).
 have defH: f @*^-1 (f @* H) = H.
-  by apply: morphimGK dH; apply: subset_trans sMH; exact: ker_sub_pre.
+  by apply: morphimGK dH; apply: subset_trans sMH; apply: ker_sub_pre.
 rewrite -defH morphpre_proper ?morphimS // in ltHG.
 by rewrite -defH [f @* H]maxM // -(morphpreK dM) morphimS.
 Qed.
@@ -391,12 +389,12 @@ Qed.
 
 Lemma maxnormal_proper A B C : maxnormal A B C -> A \proper B.
 Proof.
-by case/maxsetP=> /and3P[gA pAB _] _; exact: (sub_proper_trans (subset_gen A)).
+by case/maxsetP=> /and3P[gA pAB _] _; apply: (sub_proper_trans (subset_gen A)).
 Qed.
 
 Lemma maxnormal_sub A B C : maxnormal A B C -> A \subset B.
 Proof.
-by move=> maxA; rewrite proper_sub //; exact: (maxnormal_proper maxA).
+by move=> maxA; rewrite proper_sub //; apply: (maxnormal_proper maxA).
 Qed.
 
 Lemma ex_maxnormal_ntrivg G : G :!=: 1-> {N : {group gT} | maxnormal N G G}.
@@ -416,7 +414,7 @@ wlog suffices: H K {maxH} maxK nsHG nsKG cHK ltHK_G / H \subset K.
   by move=> IH; rewrite eqEsubset !IH // -cHK.
 have{maxK} /maxgroupP[_ maxK] := maxK.
 apply/joing_idPr/maxK; rewrite ?joing_subr //= comm_joingE //.
-by rewrite properEneq ltHK_G; exact: normalM.
+by rewrite properEneq ltHK_G; apply: normalM.
 Qed.
 
 Lemma maxnormal_minnormal G L M :
@@ -425,7 +423,7 @@ Lemma maxnormal_minnormal G L M :
 Proof.
 move=> nMG nGL /maxgroupP[/andP[/andP[sMG ltMG] nML] maxM]; apply/mingroupP.
 rewrite -subG1 quotient_sub1 ?ltMG ?quotient_norms //.
-split=> // Hb /andP[ntHb nHbL]; have nsMG: M <| G by exact/andP.
+split=> // Hb /andP[ntHb nHbL]; have nsMG: M <| G by apply/andP.
 case/inv_quotientS=> // H defHb sMH sHG; rewrite defHb; congr (_ / M).
 apply/eqP; rewrite eqEproper sHG /=; apply: contra ntHb => ltHG.
 have nsMH: M <| H := normalS sMH sHG nsMG.
@@ -458,14 +456,14 @@ apply: (iffP mingroupP); rewrite normG andbT => [[ntG simG]].
   split=> // N /andP[sNG nNG].
   by case: (eqsVneq N 1) => [|ntN]; [left | right; apply: simG; rewrite ?ntN].
 split=> // N /andP[ntN nNG] sNG.
-by case: (simG N) ntN => // [|->]; [exact/andP | case/eqP].
+by case: (simG N) ntN => // [|->]; [apply/andP | case/eqP].
 Qed.
 
 Lemma quotient_simple gT (G H : {group gT}) :
   H <| G -> simple (G / H) = maxnormal H G G.
 Proof.
 move=> nsHG; have nGH := normal_norm nsHG.
-by apply/idP/idP; [exact: minnormal_maxnormal | exact: maxnormal_minnormal].
+by apply/idP/idP; [apply: minnormal_maxnormal | apply: maxnormal_minnormal].
 Qed.
 
 Lemma isog_simple gT rT (G : {group gT}) (M : {group rT}) :
@@ -496,7 +494,7 @@ Lemma chief_factor_minnormal G V U :
   chief_factor G V U -> minnormal (U / V) (G / V).
 Proof.
 case/andP=> maxV /andP[sUG nUG]; apply: maxnormal_minnormal => //.
-by have /andP[_ nVG] := maxgroupp maxV; exact: subset_trans sUG nVG.
+by have /andP[_ nVG] := maxgroupp maxV; apply: subset_trans sUG nVG.
 Qed.
 
 Lemma acts_irrQ G U V :
@@ -527,7 +525,7 @@ have /andP[ltVU nVG] := maxgroupp maxV.
 have [||s ch_s defV] := IHm V; first exact: leq_trans (proper_card ltVU) _.
   by rewrite /normal (subset_trans (proper_sub ltVU) (normal_sub nsUG)).
 exists (rcons s U); last by rewrite last_rcons.
-by rewrite rcons_path defV /= ch_s /chief_factor; exact/and3P.
+by rewrite rcons_path defV /= ch_s /chief_factor; apply/and3P.
 Qed.
 
 End Chiefs.
diff --git a/mathcomp/solvable/hall.v b/mathcomp/solvable/hall.v
index 90ba407..5e8a41b 100644
--- a/mathcomp/solvable/hall.v
+++ b/mathcomp/solvable/hall.v
@@ -1,12 +1,12 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype finset bigop prime.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient action gproduct.
-Require Import commutator center pgroup finmodule nilpotent sylow.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div fintype finset.
+From mathcomp
+Require Import prime fingroup morphism automorphism quotient action gproduct.
+From mathcomp
+Require Import gfunctor commutator center pgroup finmodule nilpotent sylow.
+From mathcomp
 Require Import abelian maximal.
 
 (*****************************************************************************)
@@ -41,7 +41,7 @@ have [-> | [p pr_p pH]] := trivgVpdiv H.
 have [P sylP] := Sylow_exists p H.
 case nPG: (P <| G); last first.
   pose N := ('N_G(P))%G; have sNG: N \subset G by rewrite subsetIl.
-  have eqHN_G: H * N = G by exact: Frattini_arg sylP.
+  have eqHN_G: H * N = G by apply: Frattini_arg sylP.
   pose H' := (H :&: N)%G.
   have nsH'N: H' <| N.
     by rewrite /normal subsetIr normsI ?normG ?(subset_trans sNG).
@@ -50,7 +50,7 @@ case nPG: (P <| G); last first.
     by rewrite -mul_cardG (mulnC #|H'|) divnMl // cardG_gt0.
   have hallH': Hall N H'.
     rewrite /Hall -divgS subsetIr //= -eq_iH.
-    by case/andP: hallH => _; apply: coprimeSg; exact: subsetIl.
+    by case/andP: hallH => _; apply: coprimeSg; apply: subsetIl.
   have: [splits N, over H'].
     apply: IHn hallH' nsH'N; apply: {n}leq_trans Gn.
     rewrite proper_card // properEneq sNG andbT; apply/eqP=> eqNG.
@@ -61,12 +61,12 @@ case nPG: (P <| G); last first.
     by rewrite /= -(setIidPr sKN) setIA tiKN.
   by rewrite eqEsubset -eqHN_G mulgS // -eqH'K mulGS mulSg ?subsetIl.
 pose Z := 'Z(P); pose Gbar := G / Z; pose Hbar := H / Z.
-have sZP: Z \subset P by exact: center_sub.
-have sZH: Z \subset H by exact: subset_trans (pHall_sub sylP).
-have sZG: Z \subset G by exact: subset_trans sHG.
-have nZG: Z <| G by apply: char_normal_trans nPG; exact: center_char.
-have nZH: Z <| H by exact: normalS nZG.
-have nHGbar: Hbar <| Gbar by exact: morphim_normal.
+have sZP: Z \subset P by apply: center_sub.
+have sZH: Z \subset H by apply: subset_trans (pHall_sub sylP).
+have sZG: Z \subset G by apply: subset_trans sHG.
+have nZG: Z <| G by apply: gFnormal_trans nPG.
+have nZH: Z <| H by apply: normalS nZG.
+have nHGbar: Hbar <| Gbar by apply: morphim_normal.
 have hallHbar: Hall Gbar Hbar by apply: morphim_Hall (normal_norm _) _.
 have: [splits Gbar, over Hbar].
   apply: IHn => //; apply: {n}leq_trans Gn; rewrite ltn_quotient //.
@@ -76,7 +76,7 @@ have: [splits Gbar, over Hbar].
 case/splitsP=> Kbar /complP[tiHKbar eqHKbar].
 have: Kbar \subset Gbar by rewrite -eqHKbar mulG_subr.
 case/inv_quotientS=> //= ZK quoZK sZZK sZKG.
-have nZZK: Z <| ZK by exact: normalS nZG.
+have nZZK: Z <| ZK by apply: normalS nZG.
 have cardZK: #|ZK| = (#|Z| * #|G : H|)%N.
   rewrite -(Lagrange sZZK); congr (_ * _)%N.
   rewrite -card_quotient -?quoZK; last by case/andP: nZZK.
@@ -86,7 +86,7 @@ have cardZK: #|ZK| = (#|Z| * #|G : H|)%N.
 have: [splits ZK, over Z].
   rewrite (Gaschutz_split nZZK _ sZZK) ?center_abelian //; last first.
     rewrite -divgS // cardZK mulKn ?cardG_gt0 //.
-    by case/andP: hallH => _; exact: coprimeSg.
+    by case/andP: hallH => _; apply: coprimeSg.
   by apply/splitsP; exists 1%G; rewrite inE -subG1 subsetIr mulg1 eqxx.
 case/splitsP=> K /complP[tiZK eqZK].
 have sKZK: K \subset ZK by rewrite -(mul1g K) -eqZK mulSg ?sub1G.
@@ -111,8 +111,8 @@ rewrite ltnS => leHn solH nHK; have [-> | ] := eqsVneq H 1.
   by apply/eqP; rewrite conjsg1 eqEcard sK1K eqK1K /=.
 pose G := (H <*> K)%G.
 have defG: G :=: H * K by rewrite -normC // -norm_joinEl // joingC.
-have sHG: H \subset G by exact: joing_subl.
-have sKG: K \subset G by exact: joing_subr.
+have sHG: H \subset G by apply: joing_subl.
+have sKG: K \subset G by apply: joing_subr.
 have nsHG: H <| G by rewrite /(H <| G) sHG join_subG normG.
 case/(solvable_norm_abelem solH nsHG)=> M [sMH nsMG ntM] /and3P[_ abelM _].
 have [sMG nMG] := andP nsMG; rewrite -defG => sK1G coHK oK1K.
@@ -134,7 +134,7 @@ case/morphimP=> x nMx Hx ->{xb} eqK1Kx; pose K2 := (K :^ x)%G.
 have{eqK1Kx} eqK12: K1 / M = K2 / M by rewrite quotientJ.
 suff [y My ->]: exists2 y, y \in M & K1 :=: K2 :^ y.
   by exists (x * y); [rewrite groupMl // (subsetP sMH) | rewrite conjsgM].
-have nMK1: K1 \subset 'N(M) by exact: nMsG.
+have nMK1: K1 \subset 'N(M) by apply: nMsG.
 have defMK: M * K1 = M <*> K1 by rewrite -normC // -norm_joinEl // joingC.
 have sMKM: M \subset M <*> K1 by rewrite joing_subl.
 have nMKM: M <| M <*> K1 by rewrite normalYl.
@@ -243,7 +243,7 @@ have{transHb} transH (K : {group gT}):
   have: y \in coset M y by rewrite val_coset (subsetP nMK, rcoset_refl).
   have: coset M y \in (H :^ x) / M.
     rewrite /quotient morphimJ //=.
-    rewrite def_x def_H in sKHxb; apply: (subsetP sKHxb); exact: mem_quotient.
+    by rewrite def_x def_H in sKHxb; apply/(subsetP sKHxb)/mem_quotient.
   case/morphimP=> z Nz Hxz ->.
   rewrite val_coset //; case/rcosetP=> t Mt ->; rewrite groupMl //.
   by rewrite mem_conjg (subsetP sMH) // -mem_conjg (normP Nx).
@@ -256,7 +256,7 @@ have [pi_p | pi'p] := boolP (p \in pi).
 have [ltHG | leGH {n IHn leGn transH}] := ltnP #|H| #|G|.
   case: (IHn _ H (leq_trans ltHG leGn)) => [|H1]; first exact: solvableS solG.
   case/and3P=> sH1H piH1 pi'H1' transH1.
-  have sH1G: H1 \subset G by exact: subset_trans sHG.
+  have sH1G: H1 \subset G by apply: subset_trans sHG.
   exists H1 => [|K sKG piK].
     apply/and3P; split => //.
     rewrite -divgS // -(Lagrange sHG) -(Lagrange sH1H) -mulnA.
@@ -283,7 +283,7 @@ have nMK: K \subset 'N(M) by apply: subset_trans sKG nMG.
 have defG1: M * K = G1 by rewrite -normC -?norm_joinEl.
 have sK1G1: K1 \subset M * K by rewrite defG1 subsetIr.
 have coMK: coprime #|M| #|K|.
-  by rewrite coprime_sym (pnat_coprime piK) //; exact: (pHall_pgroup hallM).
+  by rewrite coprime_sym (pnat_coprime piK) //; apply: (pHall_pgroup hallM).
 case: (SchurZassenhaus_trans_sol _ nMK sK1G1 coMK) => [||x Mx defK1].
 - exact: solvableS solG.
 - apply/eqP; rewrite -(eqn_pmul2l (cardG_gt0 M)) -TI_cardMg //; last first.
@@ -389,7 +389,7 @@ pose N := 'N_(A <*> G)(H)%G.
 have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
 have nGN_N: G :&: N <| N by rewrite /(_ <| N) subsetIr normsI ?normG.
 have NG_AG: G * N = A <*> G.
-  by apply: Hall_Frattini_arg hallH => //; exact/andP.
+  by apply: Hall_Frattini_arg hallH => //; apply/andP.
 have iGN_A: #|N| %/ #|G :&: N| = #|A|.
   rewrite setIC divgI -card_quotient // -quotientMidl NG_AG.
   rewrite card_quotient -?divgS //= norm_joinEl //.
@@ -426,14 +426,14 @@ have nGN_N: G :&: N <| N.
   apply/normalP; rewrite subsetIr; split=> // y Ny.
   by rewrite conjIg (normP _) // (subsetP nGN, conjGid).
 have NG_AG : G * N = A <*> G.
-  by apply: Hall_Frattini_arg hallH => //; exact/andP.
+  by apply: Hall_Frattini_arg hallH => //; apply/andP.
 have iGN_A: #|N : G :&: N| = #|A|.
   rewrite -card_quotient //; last by case/andP: nGN_N.
   rewrite (card_isog (second_isog nGN)) /= -quotientMidr (normC nGN) NG_AG.
   rewrite card_quotient // -divgS //= joingC norm_joinEr //.
   by rewrite coprime_cardMg // mulnC mulnK.
-have solGN: solvable (G :&: N) by apply: solvableS solG; exact: subsetIl.
-have oAxA: #|A :^ x^-1| = #|A| by exact: cardJg.
+have solGN: solvable (G :&: N) by apply: solvableS solG; apply: subsetIl.
+have oAxA: #|A :^ x^-1| = #|A| by apply: cardJg.
 have sAN: A \subset N by rewrite subsetI -{1}genGid genS // subsetUl.
 have nGNA: A \subset 'N(G :&: N).
   by apply/normsP=> y ?; rewrite conjIg (normsP nGA) ?(conjGid, subsetP sAN).
@@ -471,7 +471,7 @@ apply/eqP; rewrite eqEsubset subsetI morphimS ?subsetIl //=.
 rewrite (subset_trans _ (morphim_cent _ _)) ?morphimS ?subsetIr //=.
 apply/subsetP=> _ /setIP[/morphimP[x Nx Gx ->] cAHx].
 have{cAHx} cAxR y: y \in A -> [~ x, y] \in R.
-  move=> Ay; have Ny: y \in 'N(H) by exact: subsetP Ay.
+  move=> Ay; have Ny: y \in 'N(H) by apply: subsetP Ay.
   rewrite inE mem_commg // andbT coset_idr ?groupR // morphR //=.
   by apply/eqP; apply/commgP; apply: (centP cAHx); rewrite mem_quotient.
 have AxRA: A :^ x \subset R * A.
@@ -566,11 +566,11 @@ have defG: G :=: K * H.
   rewrite -{1}(mulGid G) mulgSS //= (card_Hall hallH) (card_Hall hallK).
   by rewrite mulnC partnC.
 have sGA_H: [~: G, A] \subset H.
-  rewrite gen_subG defG; apply/subsetP=> xya; case/imset2P=> xy a.
-  case/imset2P=> x y Kx Hy -> Aa -> {xya xy}.
+  rewrite gen_subG defG.
+  apply/subsetP=> _ /imset2P[_ a /imset2P[x y Kx Hy ->] Aa ->].
   rewrite commMgJ (([~ x, a] =P 1) _) ?(conj1g, mul1g).
     by rewrite groupMl ?groupV // memJ_norm ?(subsetP nHA).
-  rewrite subsetI sKG in cKA; apply/commgP; exact: (centsP cKA).
+  by rewrite subsetI sKG in cKA; apply/commgP/(centsP cKA).
 apply: pcore_max; last first.
   by rewrite /(_ <| G) /=  commg_norml commGC commg_subr nGA.
 by case/and3P: hallH => _ piH _; apply: pgroupS piH.
@@ -592,11 +592,11 @@ have [G1 | ntG] := eqsVneq G 1.
 have sG_AG: G \subset A <*> G by rewrite joing_subr.
 have sA_AG: A \subset A <*> G by rewrite joing_subl.
 have nG_AG: A <*> G \subset 'N(G) by rewrite join_subG nGA normG.
-have nsG_AG: G <| A <*> G by exact/andP.
+have nsG_AG: G <| A <*> G by apply/andP.
 case: (solvable_norm_abelem solG nsG_AG) => // M [sMG nsMAG ntM].
 have{nsMAG} [nMA nMG]: A \subset 'N(M) /\ G \subset 'N(M).
   by apply/andP; rewrite -join_subG normal_norm.
-have nMX: X \subset 'N(M) by exact: subset_trans nMG.
+have nMX: X \subset 'N(M) by apply: subset_trans nMG.
 case/is_abelemP=> p pr_p; case/and3P=> pM cMM _.
 have: #|G / M| < n by rewrite (leq_trans (ltn_quotient _ _)).
 move/(IHn _ (A / M)%G _ (X / M)%G); rewrite !(quotient_norms, quotientS) //.
@@ -609,7 +609,7 @@ have{pi'Hq' sHGq} pi'HM': pi^'.-nat #|G : HM|.
   move: pi'Hq'; rewrite -!divgS // defHM !card_quotient //.
   by rewrite -(divnMl (cardG_gt0 M)) !Lagrange.
 have{nHAq} nHMA: A \subset 'N(HM).
-  by rewrite -(quotientSGK nMA) ?normsG ?quotient_normG -?defHM //; exact/andP.
+  by rewrite -(quotientSGK nMA) ?normsG ?quotient_normG -?defHM //; apply/andP.
 case/orP: (orbN (p \in pi)) => pi_p.
   exists HM; split=> //; apply/and3P; split; rewrite /pgroup //.
   by rewrite -(Lagrange sMHM) pnat_mul -card_quotient // -defHM (pi_pnat pM).
@@ -618,7 +618,7 @@ case: (ltnP #|HM| #|G|) => [ltHG | leGHM {n IHn leGn}].
   - exact: coprimeSg coGA.
   - exact: solvableS solG.
   case/and3P: hallH => sHHM piH pi'H'.
-  have sHG: H \subset G by exact: subset_trans sHMG.
+  have sHG: H \subset G by apply: subset_trans sHMG.
   exists H; split=> //; apply/and3P; split=> //.
   rewrite -divgS // -(Lagrange sHMG) -(Lagrange sHHM) -mulnA mulKn //.
   by rewrite pnat_mul pi'H'.
@@ -640,7 +640,7 @@ have hallX: pi.-Hall(XM) X.
 have sXMG: XM \subset G by rewrite join_subG sXG.
 have hallY: pi.-Hall(XM) Y.
   have sYXM: Y \subset XM by rewrite subsetIr.
-  have piY: pi.-group Y by apply: pgroupS piH; exact: subsetIl.
+  have piY: pi.-group Y by apply: pgroupS piH; apply: subsetIl.
   rewrite /pHall sYXM piY -divgS // -(_ : Y * M = XM).
     by rewrite coprime_cardMg ?co_pi_M // mulKn //.
   rewrite /= setIC group_modr ?joing_subr //=; apply/setIidPl.
@@ -746,7 +746,7 @@ Proof.
 move=> sHG nHA coHA solH; pose N := 'N_G(H).
 have nsHN: H <| N by rewrite normal_subnorm.
 have [sHN nHn] := andP nsHN.
-have sNG: N \subset G by exact: subsetIl.
+have sNG: N \subset G by apply: subsetIl.
 have nNA: {acts A, on group N | to}.
   split; rewrite // actsEsd // injm_subnorm ?injm_sdpair1 //=.
   by rewrite normsI ?norms_norm ?im_sdpair_norm -?actsEsd.
@@ -882,12 +882,12 @@ Lemma sol_coprime_Sylow_subset A G X :
 Proof.
 move=> nGA coGA solA sXG pX nXA.
 pose nAp (Q : {group gT}) := [&& p.-group Q, Q \subset G & A \subset 'N(Q)].
-have: nAp X by exact/and3P.
+have: nAp X by apply/and3P.
 case/maxgroup_exists=> R; case/maxgroupP; case/and3P=> pR sRG nRA maxR sXR.
 have [P sylP sRP]:= Sylow_superset sRG pR.
 suffices defP: P :=: R by exists P; rewrite sylP defP.
 case/and3P: sylP => sPG pP _; apply: (nilpotent_sub_norm (pgroup_nil pP)) => //.
-pose N := 'N_G(R); have{sPG} sPN_N: 'N_P(R) \subset N by exact: setSI.
+pose N := 'N_G(R); have{sPG} sPN_N: 'N_P(R) \subset N by apply: setSI.
 apply: norm_sub_max_pgroup (pgroupS (subsetIl _ _) pP) sPN_N (subsetIr _ _).
 have nNA: A \subset 'N(N) by rewrite normsI ?norms_norm.
 have coNA: coprime #|N| #|A| by apply: coprimeSg coGA; rewrite subsetIl.
diff --git a/mathcomp/solvable/jordanholder.v b/mathcomp/solvable/jordanholder.v
index 82d9c8b..f315efa 100644
--- a/mathcomp/solvable/jordanholder.v
+++ b/mathcomp/solvable/jordanholder.v
@@ -1,11 +1,10 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path div choice fintype tuple finfun bigop finset.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient action.
+From mathcomp
+Require Import ssrfun ssrbool eqtype ssrnat seq path choice fintype.
+From mathcomp
+Require Import bigop finset fingroup morphism automorphism quotient action.
+From mathcomp
 Require Import gseries.
 
 (******************************************************************************)
@@ -113,7 +112,7 @@ Qed.
 Lemma section_repr_isog s1 s2 :
   s1 \isog s2 -> section_repr s1 = section_repr s2.
 Proof.
-by move=> iso12; congr (odflt _ _); apply: eq_pick => s; exact: isog_transr.
+by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
 Qed.
 
 Definition mkfactors (G : {group gT}) (s : seq {group gT}) :=
@@ -167,7 +166,7 @@ have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqxx.
 have [N maxN] := ex_maxnormal_ntrivg ntG.
 have [|s /andP[ls cs]] := IHn N.
   by rewrite -ltnS (leq_trans _ cG) // proper_card // (maxnormal_proper maxN).
-by exists (N :: s); exact/and3P.
+by exists (N :: s); apply/and3P.
 Qed.
 
 (******************************************************************************)
@@ -234,7 +233,7 @@ have i3 : perm_eq fG1 fG2.
   rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
   exact: perm_eq_refl.
 apply: (perm_eq_trans i1); apply: (perm_eq_trans i3); rewrite perm_eq_sym.
-apply: perm_eq_trans i2; exact: perm_eq_refl.
+by apply: perm_eq_trans i2; apply: perm_eq_refl.
 Qed.
 
 End CompositionSeries.
@@ -361,7 +360,7 @@ by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP.
 Qed.
 
 Lemma maxainv_sub : maxainv K N -> N \subset K.
-Proof. move=> h; apply: proper_sub; exact: maxainv_proper. Qed.
+Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.
 
 Lemma maxainv_ainvar : maxainv K N -> A \subset 'N(N | to).
 Proof. by move/maxgroupp; case/and3P. Qed.
@@ -557,7 +556,7 @@ have nKQ1 : K <| N2 / N1.
 have sqA : qact_dom to N1 \subset A.
   by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
 have nNN2 : (N2 :&: N1) <| N2.
-  rewrite /normal subsetIl; apply: normsI => //; exact: normG.
+  by rewrite /normal subsetIl; apply: normsI => //; apply: normG.
 have aKQ1 : [acts qact_dom to N1, on K | to / N1].
   pose H':= coset (N2 :&: N1)@*^-1 H.
   have eHH' : H :=: H' / (N2 :&: N1) by rewrite cosetpreK.
@@ -653,7 +652,7 @@ have i1 : perm_eq (mksrepr G N1 :: mkfactors N1 st1)
   apply: Hi=> //; rewrite /acomps ?lst1 //= lsN csN andbT /=.
   apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
   apply: asimpleI => //.
-    apply: subset_trans (normal_norm nN2G); exact: normal_sub.
+    by apply: subset_trans (normal_norm nN2G); apply: normal_sub.
   rewrite -quotientMidl (maxainvM _ _ maxN_2) //.
     by apply: maxainv_asimple_quo.
   by move=> e; apply: neN12.
@@ -664,7 +663,7 @@ have i2 : perm_eq (mksrepr G N2 :: mkfactors N2 st2)
   apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
   have e : N1 :&: N2 :=: N2 :&: N1 by rewrite setIC.
   rewrite (group_inj (setIC N1 N2)); apply: asimpleI => //.
-    apply: subset_trans (normal_norm nN1G); exact: normal_sub.
+    by apply: subset_trans (normal_norm nN1G); apply: normal_sub.
   rewrite -quotientMidl (maxainvM _ _ maxN_1) //.
   exact: maxainv_asimple_quo.
 pose fG1 := [:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
@@ -675,7 +674,7 @@ have i3 : perm_eq fG1 fG2.
   rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
   exact: perm_eq_refl.
 apply: (perm_eq_trans i1); apply: (perm_eq_trans i3); rewrite perm_eq_sym.
-apply: perm_eq_trans i2; exact: perm_eq_refl.
+by apply: perm_eq_trans i2; apply: perm_eq_refl.
 Qed.
 
 End StrongJordanHolder.
diff --git a/mathcomp/solvable/maximal.v b/mathcomp/solvable/maximal.v
index a3cd11c..afca270 100644
--- a/mathcomp/solvable/maximal.v
+++ b/mathcomp/solvable/maximal.v
@@ -1,14 +1,14 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div choice fintype finfun bigop finset prime binomial.
-From mathcomp.fingroup
-Require Import fingroup morphism perm automorphism quotient action gproduct.
-From mathcomp.algebra
-Require Import ssralg finalg zmodp cyclic.
-Require Import pgroup center gseries commutator gfunctor.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq choice div fintype.
+From mathcomp
+Require Import finfun bigop finset prime binomial fingroup morphism perm.
+From mathcomp
+Require Import automorphism quotient action commutator gproduct gfunctor.
+From mathcomp
+Require Import ssralg finalg zmodp cyclic pgroup center gseries.
+From mathcomp
 Require Import nilpotent sylow abelian finmodule.
 
 (******************************************************************************)
@@ -65,12 +65,12 @@ Definition Fitting A := \big[dprod/1]_(p <- primes #|A|) 'O_p(A).
 
 Lemma Fitting_group_set G : group_set (Fitting G).
 Proof.
-suffices [F ->]: exists F : {group gT}, Fitting G = F by exact: groupP.
+suffices [F ->]: exists F : {group gT}, Fitting G = F by apply: groupP.
 rewrite /Fitting; elim: primes (primes_uniq #|G|) => [_|p r IHr] /=.
   by exists [1 gT]%G; rewrite big_nil.
 case/andP=> rp /IHr[F defF]; rewrite big_cons defF.
 suffices{IHr} /and3P[p'F sFG nFG]: p^'.-group F && (F <| G).
-  have nFGp: 'O_p(G) \subset 'N(F) := subset_trans (pcore_sub p G) nFG.
+  have nFGp: 'O_p(G) \subset 'N(F) := gFsub_trans _ nFG.
   have pGp: p.-group('O_p(G)) := pcore_pgroup p G.
   have{pGp} tiGpF: 'O_p(G) :&: F = 1 by rewrite coprime_TIg ?(pnat_coprime pGp).
   exists ('O_p(G) <*> F)%G; rewrite dprodEY // (sameP commG1P trivgP) -tiGpF.
@@ -78,8 +78,8 @@ suffices{IHr} /and3P[p'F sFG nFG]: p^'.-group F && (F <| G).
 move/bigdprodWY: defF => <- {F}; elim: r rp => [_|q r IHr] /=.
   by rewrite big_nil gen0 pgroup1 normal1.
 rewrite inE eq_sym big_cons -joingE -joing_idr => /norP[qp /IHr {IHr}].
-set F := <<_>> => /andP[p'F nsFG]; rewrite norm_joinEl /= -/F; last first.
-  exact: subset_trans (pcore_sub q G) (normal_norm nsFG).  
+set F := <<_>> => /andP[p'F nsFG].
+rewrite norm_joinEl /= -/F; last exact/gFsub_trans/normal_norm.
 by rewrite pgroupM p'F normalM ?pcore_normal //= (pi_pgroup (pcore_pgroup q G)).
 Qed.
 
@@ -236,7 +236,7 @@ Lemma Phi_quotient_id G : 'Phi (G / 'Phi(G)) = 1.
 Proof.
 apply/trivgP; rewrite -cosetpreSK cosetpre1 /=; apply/bigcapsP=> M maxM.
 have nPhi := Phi_normal G; have nPhiM: 'Phi(G) <| M.
-  by apply: normalS nPhi; [exact: bigcap_inf | case/maximal_eqP: maxM].
+  by apply: normalS nPhi; [apply: bigcap_inf | case/maximal_eqP: maxM].
 by rewrite sub_cosetpre_quo ?bigcap_inf // quotient_maximal_eq.
 Qed.
 
@@ -265,12 +265,12 @@ apply/eqP/idP=> [trPhi | abP].
   apply/set1gP; rewrite -trPhi; apply/bigcapP=> M.
   case/predU1P=> [-> | maxM]; first exact: groupX.
   have /andP[_ nMP] := p_maximal_normal pP maxM.
-  have nMx : x \in 'N(M) by exact: subsetP Px.
+  have nMx : x \in 'N(M) by apply: subsetP Px.
   apply: coset_idr; rewrite ?groupX ?morphX //=; apply/eqP.
   rewrite -(p_maximal_index pP maxM) -card_quotient // -order_dvdn cardSg //=.
   by rewrite cycle_subG mem_quotient.
 apply/trivgP/subsetP=> x Phi_x; rewrite -cycle_subG.
-have Px: x \in P by exact: (subsetP (Phi_sub P)).
+have Px: x \in P by apply: (subsetP (Phi_sub P)).
 have sxP: <[x]> \subset P by rewrite cycle_subG.
 case/splitsP: (abelem_splits abP sxP) => K /complP[tiKx defP].
 have [-> | nt_x] := eqVneq x 1; first by rewrite cycle1.
@@ -301,16 +301,16 @@ have [p_pr _ _] := pgroup_pdiv pP ntP.
 have [abP x1P] := abelemP p_pr Phi_quotient_abelem.
 apply/andP; split.
   have nMP: P \subset 'N(P^`(1) <*> 'Mho^1(P)) by rewrite normsY // !gFnorm. 
-  rewrite -quotient_sub1 ?(subset_trans sPhiP) //=.
-  suffices <-: 'Phi(P / (P^`(1) <*> 'Mho^1(P))) = 1 by exact: morphimF.
+  rewrite -quotient_sub1 ?gFsub_trans //=.
+  suffices <-: 'Phi(P / (P^`(1) <*> 'Mho^1(P))) = 1 by apply: morphimF.
   apply/eqP; rewrite (trivg_Phi (morphim_pgroup _ pP)) /= -quotientE.
   apply/abelemP=> //; rewrite [abelian _]quotient_cents2 ?joing_subl //.
   split=> // _ /morphimP[x Nx Px ->] /=.
   rewrite -morphX //= coset_id // (MhoE 1 pP) joing_idr expn1.
-  by rewrite mem_gen //; apply/setUP; right; exact: mem_imset.
+  by rewrite mem_gen //; apply/setUP; right; apply: mem_imset.
 rewrite -quotient_cents2 // [_ \subset 'C(_)]abP (MhoE 1 pP) gen_subG /=.
 apply/subsetP=> _ /imsetP[x Px ->]; rewrite expn1.
-have nPhi_x: x \in 'N('Phi(P)) by exact: (subsetP nPhiP).
+have nPhi_x: x \in 'N('Phi(P)) by apply: (subsetP nPhiP).
 by rewrite coset_idr ?groupX ?morphX ?x1P ?mem_morphim.
 Qed.
 
@@ -334,8 +334,8 @@ Lemma morphim_Phi rT P D (f : {morphism D >-> rT}) :
   p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).
 Proof.
 move=> pP sPD; rewrite !(@Phi_joing _ p) ?morphim_pgroup //.
-rewrite morphim_gen ?(subset_trans _ sPD) ?subUset ?der_subS ?Mho_sub //.
-by rewrite morphimU -joingE morphimR ?morphim_Mho.
+rewrite morphim_gen ?subUset ?gFsub_trans // morphimU -joingE.
+by rewrite morphimR ?morphim_Mho.
 Qed.
 
 Lemma quotient_Phi P H :
@@ -347,7 +347,7 @@ Lemma Phi_min G H :
   p.-group G -> G \subset 'N(H) -> p.-abelem (G / H) -> 'Phi(G) \subset H.
 Proof.
 move=> pG nHG; rewrite -trivg_Phi ?quotient_pgroup // -subG1 /=.
-by rewrite -(quotient_Phi pG) ?quotient_sub1 // (subset_trans (Phi_sub _)).
+by rewrite -(quotient_Phi pG) ?quotient_sub1 // gFsub_trans.
 Qed.
 
 Lemma Phi_cprod G H K :
@@ -366,17 +366,17 @@ Lemma Phi_mulg H K :
   'Phi(H * K) = 'Phi(H) * 'Phi(K).
 Proof.
 move=> pH pK cHK; have defHK := cprodEY cHK.
-have [|_ -> _] := cprodP (Phi_cprod _ defHK); rewrite /= cent_joinEr //.
-by apply: pnat_dvd (dvdn_cardMg H K) _; rewrite pnat_mul; exact/andP.
+have [|_ ->] /= := cprodP (Phi_cprod _ defHK); rewrite cent_joinEr //.
+by rewrite pgroupM pH.
 Qed.
 
 Lemma charsimpleP G :
   reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G)
           (charsimple G).
 Proof.
-apply: (iffP mingroupP); rewrite char_refl andbT => [[ntG simG]].
+apply: (iffP mingroupP); rewrite char_refl andbT => -[ntG simG].
   by split=> // K ntK chK; apply: simG; rewrite ?ntK // char_sub.
-split=> // K /andP[ntK chK] _; exact: simG.
+by split=> // K /andP[ntK chK] _; apply: simG.
 Qed.
 
 End Frattini4.
@@ -405,26 +405,22 @@ Qed.
 Lemma Fitting_max G H : H <| G -> nilpotent H -> H \subset 'F(G).
 Proof.
 move=> nsHG nilH; rewrite -(Sylow_gen H) gen_subG.
-apply/bigcupsP=> P /SylowP[p _ SylP].
+apply/bigcupsP=> P /SylowP[p _ sylP].
 case Gp: (p \in \pi(G)); last first.
-  rewrite card1_trivg ?sub1G // (card_Hall SylP).
+  rewrite card1_trivg ?sub1G // (card_Hall sylP).
   rewrite part_p'nat // (pnat_dvd (cardSg (normal_sub nsHG))) //.
   by rewrite /pnat cardG_gt0 all_predC has_pred1 Gp.
-move/nilpotent_Hall_pcore: SylP => ->{P} //.
+have{P sylP}[-> //] := nilpotent_Hall_pcore nilH sylP.
 rewrite -(bigdprodWY (erefl 'F(G))) sub_gen //.
 rewrite -(filter_pi_of (ltnSn _)) big_filter big_mkord.
 have le_pG: p < #|G|.+1.
   by rewrite ltnS dvdn_leq //; move: Gp; rewrite mem_primes => /and3P[].
 apply: (bigcup_max (Ordinal le_pG)) => //=.
-apply: pcore_max (pcore_pgroup _ _) _.
-exact: char_normal_trans (pcore_char p H) nsHG.
+by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans.
 Qed.
 
 Lemma pcore_Fitting pi G : 'O_pi('F(G)) \subset 'O_pi(G).
-Proof.
-rewrite pcore_max ?pcore_pgroup //.
-exact: char_normal_trans (pcore_char _ _) (Fitting_normal _).
-Qed.
+Proof. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans ?Fitting_normal. Qed.
 
 Lemma p_core_Fitting p G : 'O_p('F(G)) = 'O_p(G).
 Proof.
@@ -473,7 +469,7 @@ Qed.
 Lemma FittingS gT (G H : {group gT}) : H \subset G -> H :&: 'F(G) \subset 'F(H).
 Proof.
 move=> sHG; rewrite -{2}(setIidPl sHG).
-do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; exact: morphim_Fitting.
+do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_Fitting.
 Qed.
 
 Lemma FittingJ gT (G : {group gT}) x : 'F(G :^ x) = 'F(G) :^ x.
@@ -655,7 +651,7 @@ Qed.
 Lemma charsimple_solvable G : charsimple G -> solvable G -> is_abelem G.
 Proof.
 case/charsimple_dprod=> H [sHG simH [I Aut_I defG]] solG.
-have p_pr: prime #|H| by exact: simple_sol_prime (solvableS sHG solG) simH.
+have p_pr: prime #|H| by apply: simple_sol_prime (solvableS sHG solG) simH.
 set p := #|H| in p_pr; apply/is_abelemP; exists p => //.
 elim/big_rec: _ (G) defG => [_ <-|f B If IH_B M defM]; first exact: abelem1.
 have [Af [[_ K _ defB] _ _ _]] := (subsetP Aut_I f If, dprodP defM).
@@ -686,8 +682,7 @@ Qed.
 
 Lemma trivg_Fitting G : solvable G -> ('F(G) == 1) = (G :==: 1).
 Proof.
-move=> solG; apply/idP/idP=> [F1|]; last first.
-  by rewrite -!subG1; apply: subset_trans; exact: Fitting_sub.
+move=> solG; apply/idP/idP=> [F1 | /eqP->]; last by rewrite gF1.
 apply/idPn=> /(solvable_norm_abelem solG (normal_refl _))[M [_ nsMG ntM]].
 case/is_abelemP=> p _ /and3P[pM _ _]; case/negP: ntM.
 by rewrite -subG1 -(eqP F1) Fitting_max ?(pgroup_nil pM).
@@ -697,11 +692,9 @@ Lemma Fitting_pcore pi G : 'F('O_pi(G)) = 'O_pi('F(G)).
 Proof.
 apply/eqP; rewrite eqEsubset.
 rewrite (subset_trans _ (pcoreS _ (Fitting_sub _))); last first.
-  rewrite subsetI Fitting_sub Fitting_max ?Fitting_nil //.
-  by rewrite (char_normal_trans (Fitting_char _)) ?pcore_normal.
+  by rewrite subsetI Fitting_sub Fitting_max ?Fitting_nil ?gFnormal_trans.
 rewrite (subset_trans _ (FittingS (pcore_sub _ _))) // subsetI pcore_sub.
-rewrite pcore_max ?pcore_pgroup //.
-by rewrite (char_normal_trans (pcore_char _ _)) ?Fitting_normal.
+by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans.
 Qed.
 
 End CharSimple.
@@ -722,7 +715,7 @@ Lemma sol_prime_factor_exists :
   solvable G -> G :!=: 1 -> {H : {group gT} | H <| G & prime #|G : H| }.
 Proof.
 move=> solG /ex_maxnormal_ntrivg[H maxH].
-by exists H; [exact: maxnormal_normal | exact: index_maxnormal_sol_prime].
+by exists H; [apply: maxnormal_normal | apply: index_maxnormal_sol_prime].
 Qed.
 
 End SolvablePrimeFactor.
@@ -738,7 +731,7 @@ move=> pG [defPhi defG'].
 have [-> | ntG] := eqsVneq G 1; first by rewrite center1 abelem1. 
 have [p_pr _ _] := pgroup_pdiv pG ntG.  
 have fM: {in 'Z(G) &, {morph expgn^~ p : x y / x * y}}.
-  by move=> x y /setIP[_ /centP cxG] /setIP[/cxG cxy _]; exact: expgMn.
+  by move=> x y /setIP[_ /centP cxG] /setIP[/cxG cxy _]; apply: expgMn.
 rewrite abelemE //= center_abelian; apply/exponentP=> /= z Zz.
 apply: (@kerP _ _ _ (Morphism fM)) => //; apply: subsetP z Zz.
 rewrite -{1}defG' gen_subG; apply/subsetP=> _ /imset2P[x y Gx Gy ->].
@@ -786,10 +779,10 @@ have{sZ2G'_Z} sG'Z: G^`(1) \subset 'Z(G).
   rewrite -quotient_der ?gFnorm // -quotientGI ?ucn_subS ?quotientS1 //=.
   by rewrite ucn1.
 have sCG': 'C_G(A) \subset G^`(1).
-  rewrite -quotient_sub1 //; last by rewrite subIset // char_norm ?der_char.
+  rewrite -quotient_sub1 //; last by rewrite subIset ?gFnorm.
   rewrite (subset_trans (quotient_subcent _ G A)) //= -[G in G / _]defG.
   have nGA: A \subset 'N(G) by rewrite -commg_subl defG.
-  rewrite quotientR ?(char_norm_trans (der_char _ _)) ?normG //.
+  rewrite quotientR ?gFnorm_trans ?normG //.
   rewrite coprime_abel_cent_TI ?quotient_norms ?coprime_morph //.
   exact: sub_der1_abelian.
 have defZ: 'Z(G) = G^`(1) by apply/eqP; rewrite eqEsubset (subset_trans cZA).
@@ -853,7 +846,7 @@ have pEq: p.-group (E / 'Z(E))%g by rewrite quotient_pgroup.
 have /andP[sZE nZE] := center_normal E.
 have oEq: #|E / 'Z(E)|%g = (p ^ 2)%N.
   by rewrite card_quotient -?divgS // oEp3 oZp expnS mulKn.
-have cEEq: abelian (E / 'Z(E))%g by exact: card_p2group_abelian oEq.
+have cEEq: abelian (E / 'Z(E))%g by apply: card_p2group_abelian oEq.
 have not_cEE: ~~ abelian E.
   have: #|'Z(E)| < #|E| by rewrite oEp3 oZp (ltn_exp2l 1) ?prime_gt1.
   by apply: contraL => cEE; rewrite -leqNgt subset_leq_card // subsetI subxx.
@@ -863,9 +856,9 @@ have defE': E^`(1) = 'Z(E).
   by rewrite setIC prime_TIg ?oZp.
 split; [split=> // | by rewrite oZp]; apply/eqP.
 rewrite eqEsubset andbC -{1}defE' {1}(Phi_joing pE) joing_subl.
-rewrite -quotient_sub1 ?(subset_trans (Phi_sub _)) //.
-rewrite subG1 /= (quotient_Phi pE) //= (trivg_Phi pEq); apply/abelemP=> //.
-split=> // Zx EqZx; apply/eqP; rewrite -order_dvdn /order.
+rewrite -quotient_sub1 ?gFsub_trans ?subG1 //=.
+rewrite (quotient_Phi pE) //= (trivg_Phi pEq).
+apply/abelemP=> //; split=> // Zx EqZx; apply/eqP; rewrite -order_dvdn /order.
 rewrite (card_pgroup (mem_p_elt pEq EqZx)) (@dvdn_exp2l _ _ 1) //.
 rewrite leqNgt -pfactor_dvdn // -oEq; apply: contra not_cEE => sEqZx.
 rewrite cyclic_center_factor_abelian //; apply/cyclicP.
@@ -877,7 +870,7 @@ Lemma p3group_extraspecial G :
   p.-group G -> ~~ abelian G -> logn p #|G| <= 3 -> extraspecial G.
 Proof.
 move=> pG not_cGG; have /andP[sZG nZG] := center_normal G.
-have ntG: G :!=: 1 by apply: contraNneq not_cGG => ->; exact: abelian1.
+have ntG: G :!=: 1 by apply: contraNneq not_cGG => ->; apply: abelian1.
 have ntZ: 'Z(G) != 1 by rewrite (center_nil_eq1 (pgroup_nil pG)).
 have [p_pr _ [n oG]] := pgroup_pdiv pG ntG; rewrite oG pfactorK //.
 have [_ _ [m oZ]] := pgroup_pdiv (pgroupS sZG pG) ntZ.
@@ -924,11 +917,11 @@ Qed.
 
 Lemma isog_special G (R : {group rT}) :
   G \isog R -> special G -> special R.
-Proof. by case/isogP=> f injf <-; exact: injm_special. Qed.
+Proof. by case/isogP=> f injf <-; apply: injm_special. Qed.
 
 Lemma isog_extraspecial G (R : {group rT}) :
   G \isog R -> extraspecial G -> extraspecial R.
-Proof. by case/isogP=> f injf <-; exact: injm_extraspecial. Qed.
+Proof. by case/isogP=> f injf <-; apply: injm_extraspecial. Qed.
 
 Lemma cprod_extraspecial G H K :
     p.-group G -> H \* K = G -> H :&: K = 'Z(H) ->
@@ -972,10 +965,10 @@ have fM: {in G &, {morph f : y z / y * z}}%g.
 pose fm := Morphism fM.
 have fmG: fm @* G = 'Z(G).
   have sfmG: fm @* G \subset 'Z(G).
-    by apply/subsetP=> _ /morphimP[z _ Gz ->]; exact: fZ.
+    by apply/subsetP=> _ /morphimP[z _ Gz ->]; apply: fZ.
   apply/eqP; rewrite eqEsubset sfmG; apply: contraR notZx => /(prime_TIg prZ).
   rewrite (setIidPr _) // => fmG1; rewrite inE Gx; apply/centP=> y Gy.
-  by apply/commgP; rewrite -in_set1 -[[set _]]fmG1; exact: mem_morphim. 
+  by apply/commgP; rewrite -in_set1 -[[set _]]fmG1; apply: mem_morphim. 
 have ->: 'C_G[x] = 'ker fm.
   apply/setP=> z; rewrite inE (sameP cent1P commgP) !inE.
   by rewrite -invg_comm eq_invg_mul mulg1.
@@ -995,7 +988,7 @@ apply/eqP; rewrite eqEsubset sCxH subsetI sHU /= andbT.
 apply: contraR not_sHU => not_sHCx.
 have maxCx: maximal 'C_G[x] G.
   rewrite cent1_extraspecial_maximal //; apply: contra not_cUx.
-  by rewrite inE Gx; exact: subsetP (centS sUG) _.
+  by rewrite inE Gx; apply: subsetP (centS sUG) _.
 have nsCx := p_maximal_normal pG maxCx.
 rewrite -(setIidPl sUG) -(mulg_normal_maximal nsCx maxCx sHG) ?subsetI ?sHG //.
 by rewrite -group_modr //= setIA (setIidPl sUG) mul_subG.
@@ -1014,7 +1007,7 @@ have{not_cUG} [x Gx not_cUx] := subsetPn not_cUG.
 pose W := 'C_U[x]; have sCW_G: 'C_G(W) \subset G := subsetIl G _.
 have maxW: maximal W U by rewrite subcent1_extraspecial_maximal // inE not_cUx.
 have nsWU: W <| U := p_maximal_normal (pgroupS sUG pG) maxW.
-have ltWU: W \proper U by exact: maxgroupp maxW.
+have ltWU: W \proper U by apply: maxgroupp maxW.
 have [sWU [u Uu notWu]] := properP ltWU; have sWG := subset_trans sWU sUG.
 have defU: W * <[u]> = U by rewrite (mulg_normal_maximal nsWU) ?cycle_subG.
 have iCW_CU: #|'C_G(W) : 'C_G(U)| = p.
@@ -1060,7 +1053,7 @@ have sZE: 'Z(G) \subset E.
   by rewrite /= -defG' inE !mem_commg.
 have ziER: E :&: R = 'Z(E) by rewrite setIA (setIidPl sEG).
 have cER: R \subset 'C(E) by rewrite subsetIr.
-have iCxG: #|G : 'C_G[x]| = p by exact: p_maximal_index.
+have iCxG: #|G : 'C_G[x]| = p by apply: p_maximal_index.
 have maxR: maximal R 'C_G[x].
   rewrite /R centY !cent_cycle setIA.
   rewrite subcent1_extraspecial_maximal ?subsetIl // inE Gy andbT -sub_cent1.
@@ -1171,7 +1164,7 @@ have defS: <<X>> = S.
   apply: Phi_nongen; apply/eqP; rewrite eqEsubset join_subG sPS sXS -joing_idr.
   rewrite -genM_join sub_gen // -quotientSK ?quotient_gen // -defSb genS //.
   apply/subsetP=> xb Xxb; apply/imsetP; rewrite (setIidPr nPX).
-  by exists (repr xb); rewrite /= ?coset_reprK //; exact: mem_imset.
+  by exists (repr xb); rewrite /= ?coset_reprK //; apply: mem_imset.
 pose f (a : {perm gT}) := [ffun x => if x \in X then x^-1 * a x else 1].
 have injf: {in A &, injective f}.
   move=> _ _ /morphimP[y nSy Ry ->] /morphimP[z nSz Rz ->].
@@ -1235,7 +1228,7 @@ move/IHs=> {IHs}IHs; case/cprodP=> [[_ U _ defU]]; rewrite defU => defT cEU.
 rewrite -(mulnK #|T| (cardG_gt0 (E :&: U))) -defT -mul_cardG /=.
 have ->: E :&: U = 'Z(S).
   apply/eqP; rewrite eqEsubset subsetI -{1 2}defZE subsetIl setIS //=.
-  by case/cprodP: defU => [[V _ -> _]]  <- _; exact: mulG_subr.
+  by case/cprodP: defU => [[V _ -> _]]  <- _; apply: mulG_subr.
 rewrite (IHs U) // oEp3 oZ -expnD addSn expnS mulKn ?prime_gt0 //.
 by rewrite pfactorK //= uphalf_double.
 Qed.
@@ -1251,7 +1244,7 @@ rewrite -andbA big_cons -cprodA; case/and3P; move/eqP=> oE; move/eqP=> defZE.
 move=> es_s; case/cprodP=> [[_ U _ defU]]; rewrite defU => defT cEU.
 have sUT: U \subset T by rewrite -defT mulG_subr.
 have sZU: 'Z(T) \subset U.
-  by case/cprodP: defU => [[V _ -> _] <- _]; exact: mulG_subr.
+  by case/cprodP: defU => [[V _ -> _] <- _]; apply: mulG_subr.
 have defZU: 'Z(E) = 'Z(U).
   apply/eqP; rewrite eqEsubset defZE subsetI sZU subIset ?centS ?orbT //=.
   by rewrite subsetI subIset ?sUT //= -defT centM setSI.
@@ -1259,7 +1252,7 @@ apply: (Aut_cprod_full _ defZU); rewrite ?cprodE //; last first.
   by apply: IHs; rewrite -?defZU ?defZE.
 have oZE: #|'Z(E)| = p by rewrite defZE.
 have [p2 | odd_p] := even_prime p_pr.
-  suffices <-: restr_perm 'Z(E) @* Aut E = Aut 'Z(E) by exact: Aut_in_isog.
+  suffices <-: restr_perm 'Z(E) @* Aut E = Aut 'Z(E) by apply: Aut_in_isog.
   apply/eqP; rewrite eqEcard restr_perm_Aut ?center_sub //=.
   by rewrite card_Aut_cyclic ?prime_cyclic ?oZE // {1}p2 cardG_gt0.
 have pE: p.-group E by rewrite /pgroup oE pnat_exp pnat_id.
@@ -1298,7 +1291,7 @@ have [y Ey not_cxy]: exists2 y, y \in E & y \notin 'C[x].
 have notZy: y \notin 'Z(E).
   apply: contra not_cxy; rewrite inE Ey; apply: subsetP.
   by rewrite -cent_set1 centS ?sub1set. 
-pose K := 'C_E[y]; have maxK: maximal K E by exact: cent1_extraspecial_maximal.
+pose K := 'C_E[y]; have maxK: maximal K E by apply: cent1_extraspecial_maximal.
 have nsKE: K <| E := p_maximal_normal pE maxK; have [sKE nKE] := andP nsKE.
 have oK: #|K| = (p ^ 2)%N.
   by rewrite -(divg_indexS sKE) oE (p_maximal_index pE) ?mulKn.
@@ -1403,7 +1396,7 @@ have{defB} defB : B :=: A * <[x]>.
   rewrite -quotientK ?cycle_subG ?quotient_cycle // defB quotientGK //.
   exact: normalS (normal_sub nBG) nsAG.
 apply/setIidPl; rewrite ?defB -[_ :&: _]center_prod //=.
-rewrite /center !(setIidPl _) //; exact: cycle_abelian.
+rewrite /center !(setIidPl _) //; apply: cycle_abelian.
 Qed.
 
 (* The two other assertions of Aschbacher 23.15 state properties of the       *)
@@ -1472,19 +1465,19 @@ have{nil_classY pY sXW sZY sZCA} defW: W = <[x]> * Z.
   congr (_ * _); apply/eqP; rewrite eqEsubset {1}[Z](OhmE 1 pA).
   rewrite subsetI setIAC subIset //; first by rewrite sZCA -[Z]Ohm_id OhmS.
   rewrite sub_gen ?setIS //; apply/subsetP=> w Ww; rewrite inE.
-  by apply/eqP; apply: exponentP w Ww; exact: exponent_Ohm1_class2.
+  by apply/eqP; apply: exponentP w Ww; apply: exponent_Ohm1_class2.
 have{sXG sAG} sXAG: XA \subset G by rewrite join_subG sXG.
 have{sXAG} nilXA: nilpotent XA := nilpotentS sXAG (pgroup_nil pG).
 have sYXA: Y \subset XA by rewrite defY defXA mulgS ?subsetIl.
 rewrite -[Y](nilpotent_sub_norm nilXA) {nilXA sYXA}//= -/Y -/XA.
-apply: subset_trans (setIS _ (char_norm_trans (Ohm_char 1 _) (subxx _))) _.
+suffices: 'N_XA('Ohm_1(Y)) \subset Y by apply/subset_trans/setIS/gFnorms.
 rewrite {XA}defXA -group_modl ?normsG /= -/W ?{W}defW ?mulG_subl //.
 rewrite {Y}defY mulgS // subsetI subsetIl {CA C}sub_astabQ subIset ?nZA //= -/Z.
 rewrite (subset_trans (quotient_subnorm _ _ _)) //= quotientMidr /= -/Z.
 rewrite -quotient_sub1 ?subIset ?cent_norm ?orbT //.
 rewrite (subset_trans (quotientI _ _ _)) ?coprime_TIg //.
-rewrite (@pnat_coprime p) // -/(pgroup p _) ?quotient_pgroup {pA}//=.
-rewrite -(setIidPr (cent_sub _)) [pnat _ _]p'group_quotient_cent_prime //.
+rewrite (@pnat_coprime p) // -/(p.-group _) ?quotient_pgroup {pA}//= -pgroupE.
+rewrite -(setIidPr (cent_sub _)) p'group_quotient_cent_prime //.
 by rewrite (dvdn_trans (dvdn_quotient _ _)) ?order_dvdn.
 Qed.
 
@@ -1497,10 +1490,8 @@ Proof.
 move=> p_odd pG; set X := 'Ohm_1('C_G(Z)).
 case/maxgroupP=> /andP[nsZG abelZ] maxZ. 
 have [sZG nZG] := andP nsZG; have [_ cZZ expZp] := and3P abelZ.
-have{nZG} nsXG: X <| G.
-  apply: (char_normal_trans (Ohm_char 1 'C_G(Z))).
-  by rewrite /normal subsetIl normsI ?normG ?norms_cent.
-have cZX : X \subset 'C(Z) := subset_trans (Ohm_sub _ _) (subsetIr _ _).
+have{nZG} nsXG: X <| G by rewrite gFnormal_trans ?norm_normalI ?norms_cent.
+have cZX : X \subset 'C(Z) by apply/gFsub_trans/subsetIr.
 have{sZG expZp} sZX: Z \subset X.
   rewrite [X](OhmE 1 (pgroupS _ pG)) ?subsetIl ?sub_gen //.
   apply/subsetP=> x Zx; rewrite !inE  ?(subsetP sZG) ?(subsetP cZZ) //=.
@@ -1513,23 +1504,23 @@ suffices{sZX} expXp: (exponent X %| p).
   case/andP=> /sub_center_normal nsZdG.
   have{nsZdG} [D defD sZD nsDG] := inv_quotientN nsZG nsZdG; rewrite defD.
   have sDG := normal_sub nsDG; have nsZD := normalS sZD sDG nsZG.
-  rewrite quotientSGK ?quotient_sub1 ?normal_norm //= -/X => sDX; case/negP.
+  rewrite quotientSGK ?quotient_sub1 ?normal_norm //= -/X => sDX /negP[].
   rewrite (maxZ D) // nsDG andbA (pgroupS sDG) ?(dvdn_trans (exponentS sDX)) //.
   have sZZD: Z \subset 'Z(D) by rewrite subsetI sZD centsC (subset_trans sDX).
   by rewrite (cyclic_factor_abelian sZZD) //= -defD cycle_cyclic.
 pose normal_abelian := [pred A : {group gT} | A <| G & abelian A].
-have{nsZG cZZ} normal_abelian_Z : normal_abelian Z by exact/andP.
+have{nsZG cZZ} normal_abelian_Z : normal_abelian Z by apply/andP.
 have{normal_abelian_Z} [A maxA sZA] := maxgroup_exists normal_abelian_Z.
 have SCN_A : A \in 'SCN(G) by apply: max_SCN pG maxA.
 move/maxgroupp: maxA => /andP[nsAG cAA] {normal_abelian}.
 have pA := pgroupS (normal_sub nsAG) pG.
 have{abelZ maxZ nsAG cAA sZA} defA1: 'Ohm_1(A) = Z.
-  apply: maxZ; last by rewrite -(Ohm1_id abelZ) OhmS.
-  by rewrite Ohm1_abelem ?(char_normal_trans (Ohm_char _ _) nsAG).
+  have: Z \subset 'Ohm_1(A) by rewrite -(Ohm1_id abelZ) OhmS.
+  by apply: maxZ; rewrite Ohm1_abelem ?gFnormal_trans.
 have{SCN_A} sX'A: X^`(1) \subset A.
   have sX_CWA1 : X \subset 'C('Ohm_1(A)) :&: 'C_G(A / 'Ohm_1(A) | 'Q).
-    rewrite subsetI /X -defA1 (Ohm1_stab_Ohm1_SCN_series _ p_odd) //= andbT.
-    exact: subset_trans (Ohm_sub _ _) (subsetIr _ _). 
+    rewrite subsetI /X -defA1 (Ohm1_stab_Ohm1_SCN_series _ p_odd) //=.
+    by rewrite gFsub_trans ?subsetIr.
   by apply: subset_trans (der1_stab_Ohm1_SCN_series SCN_A); rewrite commgSS.
 pose genXp := [pred U : {group gT} | 'Ohm_1(U) == U & ~~ (exponent U %| p)].
 apply/idPn=> expXp'; have genXp_X: genXp [group of X] by rewrite /= Ohm_id eqxx.
@@ -1582,12 +1573,11 @@ have [|K]:= @maxgroup_exists _ qcr 1 _.
 case/maxgroupP; rewrite {}/qcr subUset => /and3P[chK sPhiZ sRZ] maxK _.
 have sKG := char_sub chK; have nKG := char_normal chK.
 exists K; split=> //; apply/eqP; rewrite eqEsubset andbC setSI //=.
-have chZ: 'Z(K) \char G by [exact: subcent_char]; have nZG := char_norm chZ.
-have chC: 'C_G(K) \char G by exact: subcent_char (char_refl G) chK.
+have chZ: 'Z(K) \char G by [apply: subcent_char]; have nZG := char_norm chZ.
+have chC: 'C_G(K) \char G by apply: subcent_char chK.
 rewrite -quotient_sub1; last by rewrite subIset // char_norm.
 apply/trivgP; apply: (TI_center_nil (quotient_nil _ (pgroup_nil pG))).
-  rewrite quotient_normal // /normal subsetIl normsI ?normG ?norms_cent //.
-  exact: char_norm.
+  by rewrite quotient_normal ?norm_normalI ?norms_cent ?normal_norm.
 apply: TI_Ohm1; apply/trivgP; rewrite -trivg_quotient -sub_cosetpre_quo //.
 rewrite morphpreI quotientGK /=; last first.
   by apply: normalS (char_normal chZ); rewrite ?subsetIl ?setSI.
@@ -1598,16 +1588,16 @@ rewrite subsetI centsC cXK andbT -(mul1g K) -mulSG mul1g -(cent_joinEr cXK).
 rewrite [_ <*> K]maxK ?joing_subr //= andbC (cent_joinEr cXK).
 rewrite -center_prod // (subset_trans _ (mulG_subr _ _)).
   rewrite charM 1?charI ?(char_from_quotient (normal_cosetpre _)) //.
-  by rewrite cosetpreK (char_trans _ (center_char _)) ?Ohm_char.
+  by rewrite cosetpreK !gFchar_trans.
 rewrite (@Phi_mulg p) ?(pgroupS _ pG) // subUset commGC commMG; last first.
   by rewrite normsR ?(normsG sKG) // cents_norm // centsC.
 rewrite !mul_subG 1?commGC //.
   apply: subset_trans (commgS _ (subsetIr _ _)) _.
   rewrite -quotient_cents2 ?subsetIl // centsC // cosetpreK //.
-  by rewrite (subset_trans (Ohm_sub _ _)) // subsetIr.
-have nZX := subset_trans sXG nZG; have pX : p.-group gX by exact: pgroupS pG.
-rewrite -quotient_sub1 ?(subset_trans (Phi_sub _)) //=.
-have pXZ: p.-group (gX / 'Z(K)) by exact: morphim_pgroup.
+  exact/gFsub_trans/subsetIr.
+have nZX := subset_trans sXG nZG; have pX : p.-group gX by apply: pgroupS pG.
+rewrite -quotient_sub1 ?gFsub_trans //=.
+have pXZ: p.-group (gX / 'Z(K)) by apply: morphim_pgroup.
 rewrite (quotient_Phi pX nZX) subG1 (trivg_Phi pXZ).
 apply: (abelemS (quotientS _ (subsetIr _ _))); rewrite /= cosetpreK /=.
 have pZ: p.-group 'Z(G / 'Z(K)).
@@ -1644,7 +1634,7 @@ have Hy: y \in H.
   rewrite -(injmSK inj1) ?cycle_subG // injm_subcent // subsetI.
   rewrite morphimS ?morphim_cycle ?cycle_subG //=.
   suffices: sdpair1 [Aut G] y \in [~: G', F'].
-    by rewrite commGC; apply: subsetP; exact/commG1P.
+    by rewrite commGC; apply: subsetP; apply/commG1P.
   rewrite morphM ?groupV ?morphV //= sdpair_act // -commgEl.
   by rewrite mem_commg ?mem_morphim ?cycle_id.
 have fy: f y = y := astabP cHFP _ Hy.
diff --git a/mathcomp/solvable/nilpotent.v b/mathcomp/solvable/nilpotent.v
index e1baa3f..d9a6f8b 100644
--- a/mathcomp/solvable/nilpotent.v
+++ b/mathcomp/solvable/nilpotent.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import path fintype div bigop prime finset.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient gproduct.
-From mathcomp.algebra
-Require Import cyclic.
-Require Import commutator gfunctor center gseries.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq path fintype div.
+From mathcomp
+Require Import bigop prime finset fingroup morphism automorphism quotient.
+From mathcomp
+Require Import commutator gproduct gfunctor center gseries cyclic.
 
 (******************************************************************************)
 (*   This file defines nilpotent and solvable groups, and give some of their  *)
@@ -132,29 +129,27 @@ Lemma lcn0 A : 'L_0(A) = A. Proof. by []. Qed.
 Lemma lcn1 A : 'L_1(A) = A. Proof. by []. Qed.
 Lemma lcnSn n A : 'L_n.+2(A) = [~: 'L_n.+1(A), A]. Proof. by []. Qed.
 Lemma lcnSnS n G : [~: 'L_n(G), G] \subset 'L_n.+1(G).
-Proof. by case: n => //; exact: der1_subG. Qed.
+Proof. by case: n => //; apply: der1_subG. Qed.
 Lemma lcnE n A : 'L_n.+1(A) = lower_central_at_rec n A.
 Proof. by []. Qed.
 Lemma lcn2 A : 'L_2(A) = A^`(1). Proof. by []. Qed.
 
 Lemma lcn_group_set n G : group_set 'L_n(G).
-Proof. by case: n => [|[|n]]; exact: groupP. Qed.
+Proof. by case: n => [|[|n]]; apply: groupP. Qed.
 
 Canonical lower_central_at_group n G := Group (lcn_group_set n G).
 
 Lemma lcn_char n G : 'L_n(G) \char G.
-Proof.
-by case: n => [|n]; last elim: n => [|n IHn]; rewrite ?lcnSn ?charR ?char_refl.
-Qed.
+Proof. by case: n; last elim=> [|n IHn]; rewrite ?char_refl ?lcnSn ?charR. Qed.
 
 Lemma lcn_normal n G : 'L_n(G) <|  G.
-Proof. by apply: char_normal; exact: lcn_char. Qed.
+Proof. exact/char_normal/lcn_char. Qed.
 
 Lemma lcn_sub n G : 'L_n(G) \subset G.
-Proof. by case/andP: (lcn_normal n G). Qed.
+Proof. exact/char_sub/lcn_char. Qed.
 
 Lemma lcn_norm n G : G \subset 'N('L_n(G)).
-Proof. by case/andP: (lcn_normal n G). Qed.
+Proof. exact/char_norm/lcn_char. Qed.
 
 Lemma lcn_subS n G : 'L_n.+1(G) \subset 'L_n(G).
 Proof.
@@ -173,7 +168,7 @@ Qed.
 
 Lemma lcn_sub_leq m n G : n <= m -> 'L_m(G) \subset 'L_n(G).
 Proof.
-by move/subnK <-; elim: {m}(m - n) => // m; exact: subset_trans (lcn_subS _ _).
+by move/subnK <-; elim: {m}(m - n) => // m; apply: subset_trans (lcn_subS _ _).
 Qed.
 
 Lemma lcnS n A B : A \subset B -> 'L_n(A) \subset 'L_n(B).
@@ -268,13 +263,13 @@ Qed.
 Lemma lcnP G : reflect (exists n, 'L_n.+1(G) = 1) (nilpotent G).
 Proof.
 apply: (iffP idP) => [nilG | [n Ln1]].
-  by exists (nil_class G); exact/lcn_nil_classP.
+  by exists (nil_class G); apply/lcn_nil_classP.
 apply/forall_inP=> H /subsetIP[sHG sHR]; rewrite -subG1 -{}Ln1.
 by elim: n => // n IHn; rewrite (subset_trans sHR) ?commSg.
 Qed.
 
 Lemma abelian_nil G : abelian G -> nilpotent G.
-Proof. by move=> abG; apply/lcnP; exists 1%N; exact/commG1P. Qed.
+Proof. by move=> abG; apply/lcnP; exists 1%N; apply/commG1P. Qed.
 
 Lemma nil_class0 G : (nil_class G == 0) = (G :==: 1).
 Proof.
@@ -349,15 +344,15 @@ Qed.
 (* Now extract all the intermediate facts of the last proof. *)
 
 Lemma ucn_group_set gT (G : {group gT}) : group_set 'Z_n(G).
-Proof. by have [hZ ->] := ucn_pmap; exact: groupP. Qed.
+Proof. by have [hZ ->] := ucn_pmap; apply: groupP. Qed.
 
 Canonical upper_central_at_group gT G := Group (@ucn_group_set gT G).
 
 Lemma ucn_sub gT (G : {group gT}) : 'Z_n(G) \subset G.
-Proof. by have [hZ ->] := ucn_pmap; exact: gFsub. Qed.
+Proof. by have [hZ ->] := ucn_pmap; apply: gFsub. Qed.
 
 Lemma morphim_ucn : GFunctor.pcontinuous (upper_central_at n).
-Proof. by have [hZ ->] := ucn_pmap; exact: pmorphimF. Qed.
+Proof. by have [hZ ->] := ucn_pmap; apply: pmorphimF. Qed.
 
 Canonical ucn_igFun := [igFun by ucn_sub & morphim_ucn].
 Canonical ucn_gFun := [gFun by morphim_ucn].
@@ -432,7 +427,7 @@ have <-: 'Z(H / Z) * 'Z(K / Z) = 'Z(G / Z).
   by rewrite -mulHK quotientMl // center_prod ?quotient_cents.
 have ZquoZ (B A : {group gT}):
   B \subset 'C(A) -> 'Z_n(A) * 'Z_n(B) = Z -> 'Z(A / Z) = 'Z_n.+1(A) / Z.
-- move=> cAB {defZ}defZ; have cAZnB := subset_trans (ucn_sub n B) cAB. 
+- move=> cAB {defZ}defZ; have cAZnB: 'Z_n(B) \subset 'C(A) := gFsub_trans _ cAB.
   have /second_isom[/=]: A \subset 'N(Z).
     by rewrite -defZ normsM ?gFnorm ?cents_norm // centsC.
   suffices ->: Z :&: A = 'Z_n(A).
@@ -442,7 +437,7 @@ have ZquoZ (B A : {group gT}):
   by apply: subset_trans (ucn_sub_geq A n_gt0); rewrite /= setIC ucn1 setIS.
 rewrite (ZquoZ H K) 1?centsC 1?(centC cZn) // {ZquoZ}(ZquoZ K H) //.
 have cZn1: 'Z_n.+1(K) \subset 'C('Z_n.+1(H)) by apply: centSS cHK; apply: gFsub.
-rewrite -quotientMl ?quotientK ?mul_subG ?(subset_trans (gFsub _ _)) //=.
+rewrite -quotientMl ?quotientK ?mul_subG ?gFsub_trans //=.
 rewrite cprodE // -cent_joinEr ?mulSGid //= cent_joinEr //= -/Z.
 by rewrite -defZ mulgSS ?ucn_subS.
 Qed.
@@ -475,25 +470,24 @@ Proof.
 rewrite !eqEsubset sub1G ucn_sub /= andbT -(ucn0 G).
 elim: {1 3}n 0 (addn0 n) => [j <- //|i IHi j].
 rewrite addSnnS => /IHi <- {IHi}; rewrite ucnSn lcnSn.
-have nZG := normal_norm (ucn_normal j G).
-have nZL := subset_trans (lcn_sub _ _) nZG.
-by rewrite -sub_morphim_pre // subsetI morphimS ?lcn_sub // quotient_cents2.
+rewrite -sub_morphim_pre ?gFsub_trans ?gFnorm_trans // subsetI.
+by rewrite morphimS ?gFsub // quotient_cents2 ?gFsub_trans ?gFnorm_trans.
 Qed.
 
 Lemma ucnP G : reflect (exists n, 'Z_n(G) = G) (nilpotent G).
 Proof.
-apply: (iffP (lcnP G)) => [] [n /eqP clGn];
+apply: (iffP (lcnP G)) => -[n /eqP-clGn];
   by exists n; apply/eqP; rewrite ucn_lcnP in clGn *.
 Qed.
 
 Lemma ucn_nil_classP n G :
   nilpotent G -> reflect ('Z_n(G) = G) (nil_class G <= n).
 Proof.
-move=> nilG; rewrite (sameP (lcn_nil_classP n nilG) eqP) ucn_lcnP; exact: eqP.
+move=> nilG; rewrite (sameP (lcn_nil_classP n nilG) eqP) ucn_lcnP; apply: eqP.
 Qed.
 
 Lemma ucn_id n G : 'Z_n('Z_n(G)) = 'Z_n(G).
-Proof. by rewrite -{2}['Z_n(G)]gFid. Qed.
+Proof. exact: gFid. Qed.
 
 Lemma ucn_nilpotent n G : nilpotent 'Z_n(G). 
 Proof. by apply/ucnP; exists n; rewrite ucn_id. Qed.
@@ -527,7 +521,7 @@ Lemma injm_nil G : 'injm f -> G \subset D -> nilpotent (f @* G) = nilpotent G.
 Proof.
 move=> injf sGD; apply/idP/idP; last exact: morphim_nil.
 case/ucnP=> n; rewrite -injm_ucn // => /injm_morphim_inj defZ.
-by apply/ucnP; exists n; rewrite defZ ?(subset_trans (ucn_sub n G)).
+by apply/ucnP; exists n; rewrite defZ ?gFsub_trans.
 Qed.
 
 Lemma nil_class_morphim G : nilpotent G -> nil_class (f @* G) <= nil_class G.
@@ -542,7 +536,7 @@ Proof.
 move=> injf sGD; case nilG: (nilpotent G).
   apply/eqP; rewrite eqn_leq nil_class_morphim //.
   rewrite (sameP (lcn_nil_classP _ nilG) eqP) -subG1.
-  rewrite -(injmSK injf) ?(subset_trans (lcn_sub _ _)) // morphim1.
+  rewrite -(injmSK injf) ?gFsub_trans // morphim1.
   by rewrite morphim_lcn // (lcn_nil_classP _ _ (leqnn _)) //= injm_nil.
 transitivity #|G|; apply/eqP; rewrite eqn_leq.
   rewrite -(card_injm injf sGD) (leq_trans (index_size _ _)) ?size_mkseq //.
@@ -608,21 +602,21 @@ have{nsGH} [i sZH []]: exists2 i, 'Z_i(G) \subset H & ~ 'Z_i.+1(G) \subset H.
   elim: n => [|i IHi] in nsGH *; first by rewrite sub1G in nsGH.
   by case sZH: ('Z_i(G) \subset H); [exists i | apply: IHi; rewrite sZH].
 apply: subset_trans sNH; rewrite subsetI ucn_sub -commg_subr.
-by apply: subset_trans sZH; apply: subset_trans (ucn_comm i G); exact: commgS.
+by apply: subset_trans sZH; apply: subset_trans (ucn_comm i G); apply: commgS.
 Qed.
 
 Lemma nilpotent_proper_norm G H :
   nilpotent G -> H \proper G -> H \proper 'N_G(H).
 Proof.
 move=> nilG; rewrite properEneq properE subsetI normG => /andP[neHG sHG].
-by rewrite sHG; apply: contra neHG; move/(nilpotent_sub_norm nilG)->.
+by rewrite sHG; apply: contra neHG => /(nilpotent_sub_norm nilG)->.
 Qed.
 
 Lemma nilpotent_subnormal G H : nilpotent G -> H \subset G -> H <|<| G.
 Proof.
 move=> nilG; elim: {H}_.+1 {-2}H (ltnSn (#|G| - #|H|)) => // m IHm H.
-rewrite ltnS => leGHm sHG; have:= sHG; rewrite subEproper.
-case/predU1P=> [->|]; first exact: subnormal_refl.
+rewrite ltnS => leGHm sHG.
+have [->|] := eqVproper sHG; first exact: subnormal_refl.
 move/(nilpotent_proper_norm nilG); set K := 'N_G(H) => prHK.
 have snHK: H <|<| K by rewrite normal_subnormal ?normalSG.
 have sKG: K \subset G by rewrite subsetIl.
@@ -676,7 +670,7 @@ by rewrite (forall_inP nilG) // subsetI sHG (subset_trans sHH') ?commgS.
 Qed.
 
 Lemma abelian_sol G : abelian G -> solvable G.
-Proof. by move/abelian_nil; exact: nilpotent_sol. Qed.
+Proof. by move/abelian_nil/nilpotent_sol. Qed.
 
 Lemma solvable1 : solvable [1 gT]. Proof. exact: abelian_sol (abelian1 gT). Qed.
 
@@ -705,7 +699,7 @@ elim: n => [|n IHn]; first by rewrite subn0 prednK.
 rewrite dergSn subnS -ltnS.
 have [-> | ntGn] := eqVneq G^`(n) 1; first by rewrite commG1 cards1.
 case: (_ - _) IHn => [|n']; first by rewrite leqNgt cardG_gt1 ntGn.
-by apply: leq_trans (proper_card _); exact: sol_der1_proper (der_sub _ _) _.
+by apply: leq_trans (proper_card _); apply: sol_der1_proper (der_sub _ _) _.
 Qed.
 
 End Solvable.
@@ -725,8 +719,7 @@ Lemma injm_sol : 'injm f -> G \subset D -> solvable (f @* G) = solvable G.
 Proof.
 move=> injf sGD; apply/idP/idP; last exact: morphim_sol.
 case/derivedP=> n Gn1; apply/derivedP; exists n; apply/trivgP.
-rewrite -(injmSK injf) ?(subset_trans (der_sub _ _)) ?morphim_der //.
-by rewrite Gn1 morphim1.
+by rewrite -(injmSK injf) ?gFsub_trans ?morphim_der // Gn1 morphim1.
 Qed.
 
 End MorphSol.
diff --git a/mathcomp/solvable/pgroup.v b/mathcomp/solvable/pgroup.v
index b1d2bf6..3d48a8a 100644
--- a/mathcomp/solvable/pgroup.v
+++ b/mathcomp/solvable/pgroup.v
@@ -1,14 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div fintype bigop finset prime.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient action gproduct.
-From mathcomp.algebra
-Require Import cyclic.
-Require Import gfunctor.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div.
+From mathcomp
+Require Import fintype bigop finset prime fingroup morphism.
+From mathcomp
+Require Import gfunctor automorphism quotient action gproduct cyclic.
 
 (******************************************************************************)
 (* Standard group notions and constructions based on the prime decomposition  *)
@@ -125,7 +122,7 @@ Implicit Types (x y z : gT) (A B C D : {set gT}) (G H K P Q R : {group gT}).
 
 Lemma trivgVpdiv G : G :=: 1 \/ (exists2 p, prime p & p %| #|G|).
 Proof.
-have [leG1|lt1G] := leqP #|G| 1; first by left; exact: card_le1_trivg.
+have [leG1|lt1G] := leqP #|G| 1; first by left; apply: card_le1_trivg.
 by right; exists (pdiv #|G|); rewrite ?pdiv_dvd ?pdiv_prime.
 Qed.
 
@@ -140,13 +137,13 @@ Qed.
 Lemma pgroupE pi A : pi.-group A = pi.-nat #|A|. Proof. by []. Qed.
 
 Lemma sub_pgroup pi rho A : {subset pi <= rho} -> pi.-group A -> rho.-group A.
-Proof. by move=> pi_sub_rho; exact: sub_in_pnat (in1W pi_sub_rho). Qed.
+Proof. by move=> pi_sub_rho; apply: sub_in_pnat (in1W pi_sub_rho). Qed.
 
 Lemma eq_pgroup pi rho A : pi =i rho -> pi.-group A = rho.-group A.
 Proof. exact: eq_pnat. Qed.
 
 Lemma eq_p'group pi rho A : pi =i rho -> pi^'.-group A = rho^'.-group A.
-Proof. by move/eq_negn; exact: eq_pnat. Qed.
+Proof. by move/eq_negn; apply: eq_pnat. Qed.
 
 Lemma pgroupNK pi A : pi^'^'.-group A = pi.-group A.
 Proof. exact: pnatNK. Qed.
@@ -164,7 +161,7 @@ Lemma p'groupEpi p G : p^'.-group G = (p \notin \pi(G)).
 Proof. exact: p'natEpi (cardG_gt0 G). Qed.
 
 Lemma pgroup_pi G : \pi(G).-group G.
-Proof. by rewrite /=; exact: pnat_pi. Qed.
+Proof. by rewrite /=; apply: pnat_pi. Qed.
 
 Lemma partG_eq1 pi G : (#|G|`_pi == 1%N) = pi^'.-group G.
 Proof. exact: partn_eq1 (cardG_gt0 G). Qed.
@@ -178,10 +175,10 @@ Lemma pgroup1 pi : pi.-group [1 gT].
 Proof. by rewrite /pgroup cards1. Qed.
 
 Lemma pgroupS pi G H : H \subset G -> pi.-group G -> pi.-group H.
-Proof. by move=> sHG; exact: pnat_dvd (cardSg sHG). Qed.
+Proof. by move=> sHG; apply: pnat_dvd (cardSg sHG). Qed.
 
 Lemma oddSg G H : H \subset G -> odd #|G| -> odd #|H|.
-Proof. by rewrite !odd_2'nat; exact: pgroupS. Qed.
+Proof. by rewrite !odd_2'nat; apply: pgroupS. Qed.
 
 Lemma odd_pgroup_odd p G : odd p -> p.-group G -> odd #|G|.
 Proof.
@@ -212,7 +209,7 @@ Proof. by rewrite /pgroup cardJg. Qed.
 
 Lemma pgroup_p p P : p.-group P -> p_group P.
 Proof.
-case: (leqP #|P| 1); first by move=> /card_le1_trivg-> _; exact: pgroup1.
+case: (leqP #|P| 1); first by move=> /card_le1_trivg-> _; apply: pgroup1.
 move/pdiv_prime=> pr_q pgP; have:= pgroupP pgP _ pr_q (pdiv_dvd _).
 by rewrite /p_group => /eqnP->.
 Qed.
@@ -220,7 +217,7 @@ Qed.
 Lemma p_groupP P : p_group P -> exists2 p, prime p & p.-group P.
 Proof.
 case: (ltnP 1 #|P|); first by move/pdiv_prime; exists (pdiv #|P|).
-move/card_le1_trivg=> -> _; exists 2 => //; exact: pgroup1.
+by move/card_le1_trivg=> -> _; exists 2 => //; apply: pgroup1.
 Qed.
 
 Lemma pgroup_pdiv p G :
@@ -254,7 +251,7 @@ Proof. by case/and3P. Qed.
 Lemma pHallP pi G H : reflect (H \subset G /\ #|H| = #|G|`_pi) (pi.-Hall(G) H).
 Proof.
 apply: (iffP idP) => [piH | [sHG oH]].
-  split; [exact: pHall_sub piH | exact: card_Hall].
+  by split; [apply: pHall_sub piH | apply: card_Hall].
 rewrite /pHall sHG -divgS // /pgroup oH.
 by rewrite -{2}(@partnC pi #|G|) ?mulKn ?part_pnat.
 Qed.
@@ -268,7 +265,7 @@ Lemma coprime_mulpG_Hall pi G K R :
 Proof.
 move=> defG piK pi'R; apply/andP.
 rewrite /pHall piK -!divgS /= -defG ?mulG_subl ?mulg_subr //= pnatNK.
-by rewrite coprime_cardMg ?(pnat_coprime piK) // mulKn ?mulnK //; exact/and3P.
+by rewrite coprime_cardMg ?(pnat_coprime piK) // mulKn ?mulnK //; apply/and3P.
 Qed.
 
 Lemma coprime_mulGp_Hall pi G K R :
@@ -284,15 +281,15 @@ Lemma eq_in_pHall pi rho G H :
 Proof.
 move=> eq_pi_rho; apply: andb_id2l => sHG.
 congr (_ && _); apply: eq_in_pnat => p piHp. 
-  by apply: eq_pi_rho; exact: (piSg sHG).
+  by apply: eq_pi_rho; apply: (piSg sHG).
 by congr (~~ _); apply: eq_pi_rho; apply: (pi_of_dvd (dvdn_indexg G H)).
 Qed.
 
 Lemma eq_pHall pi rho G H : pi =i rho -> pi.-Hall(G) H = rho.-Hall(G) H.
-Proof. by move=> eq_pi_rho; exact: eq_in_pHall (in1W eq_pi_rho). Qed.
+Proof. by move=> eq_pi_rho; apply: eq_in_pHall (in1W eq_pi_rho). Qed.
 
 Lemma eq_p'Hall pi rho G H : pi =i rho -> pi^'.-Hall(G) H = rho^'.-Hall(G) H.
-Proof. by move=> eq_pi_rho; exact: eq_pHall (eq_negn _). Qed.
+Proof. by move=> eq_pi_rho; apply: eq_pHall (eq_negn _). Qed.
 
 Lemma pHallNK pi G H : pi^'^'.-Hall(G) H = pi.-Hall(G) H.
 Proof. exact: eq_pHall (negnK _). Qed.
@@ -322,7 +319,7 @@ by case/andP=> sHG coHG /=; rewrite /pHall sHG /pgroup pnat_pi -?coprime_pi'.
 Qed.
 
 Lemma HallP G H : Hall G H -> exists pi, pi.-Hall(G) H.
-Proof. by exists \pi(H); exact: Hall_pi. Qed.
+Proof. by exists \pi(H); apply: Hall_pi. Qed.
 
 Lemma sdprod_Hall G K H : K ><| H = G -> Hall G K = Hall G H.
 Proof.
@@ -355,7 +352,7 @@ Qed.
 
 Lemma compl_p'Hall pi K H G :
   pi^'.-Hall(G) K -> (H \in [complements to K in G]) = pi.-Hall(G) H.
-Proof. by move/compl_pHall->; exact: eq_pHall (negnK pi). Qed.
+Proof. by move/compl_pHall->; apply: eq_pHall (negnK pi). Qed.
 
 Lemma sdprod_normal_p'HallP pi K H G :
   K <| G -> pi^'.-Hall(G) H -> reflect (K ><| H = G) (pi.-Hall(G) K).
@@ -402,8 +399,7 @@ Qed.
 Lemma pHall_subl pi G K H :
   H \subset K -> K \subset G -> pi.-Hall(G) H -> pi.-Hall(K) H.
 Proof.
-move=> sHK sKG; rewrite /pHall sHK; case/and3P=> _ ->.
-by apply: pnat_dvd; exact: indexSg.
+by move=> sHK sKG; rewrite /pHall sHK => /and3P[_ ->]; apply/pnat_dvd/indexSg.
 Qed.
 
 Lemma Hall1 G : Hall G 1.
@@ -446,14 +442,14 @@ Lemma p_eltM_norm pi x y :
   x \in 'N(<[y]>) -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).
 Proof.
 move=> nyx pi_x pi_y; apply: (@mem_p_elt pi _ (<[x]> <*> <[y]>)%G).
-  by rewrite /= norm_joinEl ?cycle_subG // pgroupM; exact/andP.
+  by rewrite /= norm_joinEl ?cycle_subG // pgroupM; apply/andP.
 by rewrite groupM // mem_gen // inE cycle_id ?orbT.
 Qed.
 
 Lemma p_eltM pi x y : commute x y -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).
 Proof.
 move=> cxy; apply: p_eltM_norm; apply: (subsetP (cent_sub _)).
-by rewrite cent_gen cent_set1; exact/cent1P.
+by rewrite cent_gen cent_set1; apply/cent1P.
 Qed.
 
 Lemma p_elt1 pi : pi.-elt (1 : gT).
@@ -469,10 +465,10 @@ Lemma p_eltJ pi x y : pi.-elt (x ^ y) = pi.-elt x.
 Proof. by congr pnat; rewrite orderJ. Qed.
 
 Lemma sub_p_elt pi1 pi2 x : {subset pi1 <= pi2} -> pi1.-elt x -> pi2.-elt x.
-Proof. by move=> pi12; apply: sub_in_pnat => q _; exact: pi12. Qed.
+Proof. by move=> pi12; apply: sub_in_pnat => q _; apply: pi12. Qed.
 
 Lemma eq_p_elt pi1 pi2 x : pi1 =i pi2 -> pi1.-elt x = pi2.-elt x.
-Proof. by move=> pi12; exact: eq_pnat. Qed.
+Proof. by move=> pi12; apply: eq_pnat. Qed.
 
 Lemma p_eltNK pi x : pi^'^'.-elt x = pi.-elt x.
 Proof. exact: pnatNK. Qed.
@@ -480,7 +476,7 @@ Proof. exact: pnatNK. Qed.
 Lemma eq_constt pi1 pi2 x : pi1 =i pi2 -> x.`_pi1 = x.`_pi2.
 Proof.
 move=> pi12; congr (x ^+ (chinese _ _ 1 0)); apply: eq_partn => // a.
-by congr (~~ _); exact: pi12.
+by congr (~~ _); apply: pi12.
 Qed.
 
 Lemma consttNK pi x : x.`_pi^'^' = x.`_pi.
@@ -514,8 +510,8 @@ Proof. by rewrite -{1}(consttC pi^' x) consttNK mulgK p_elt_constt. Qed.
 Lemma order_constt pi (x : gT) : #[x.`_pi] = #[x]`_pi.
 Proof.
 rewrite -{2}(consttC pi x) orderM; [|exact: commuteX2|]; last first.
-  by apply: (@pnat_coprime pi); exact: p_elt_constt.
-by rewrite partnM // part_pnat_id ?part_p'nat ?muln1 //; exact: p_elt_constt.
+  by apply: (@pnat_coprime pi); apply: p_elt_constt.
+by rewrite partnM // part_pnat_id ?part_p'nat ?muln1 //; apply: p_elt_constt.
 Qed.
 
 Lemma consttM pi x y : commute x y -> (x * y).`_pi = x.`_pi * y.`_pi.
@@ -535,7 +531,7 @@ Qed.
 Lemma consttX pi x n : (x ^+ n).`_pi = x.`_pi ^+ n.
 Proof.
 elim: n => [|n IHn]; first exact: constt1.
-rewrite !expgS consttM ?IHn //; exact: commuteX.
+by rewrite !expgS consttM ?IHn //; apply: commuteX.
 Qed.
 
 Lemma constt1P pi x : reflect (x.`_pi = 1) (pi^'.-elt x).
@@ -562,12 +558,12 @@ Qed.
 Lemma prod_constt x : \prod_(0 <= p < #[x].+1) x.`_p = x.
 Proof.
 pose lp n := [pred p | p < n].
-have: (lp #[x].+1).-elt x by apply/pnatP=> // p _; exact: dvdn_leq.
+have: (lp #[x].+1).-elt x by apply/pnatP=> // p _; apply: dvdn_leq.
 move/constt_p_elt=> def_x; symmetry; rewrite -{1}def_x {def_x}.
 elim: _.+1 => [|p IHp].
-  by rewrite big_nil; apply/constt1P; exact/pgroupP.
+  by rewrite big_nil; apply/constt1P; apply/pgroupP.
 rewrite big_nat_recr //= -{}IHp -(consttC (lp p) x.`__); congr (_ * _).
-  rewrite sub_in_constt // => q _; exact: leqW.
+  by rewrite sub_in_constt // => q _; apply: leqW.
 set y := _.`__; rewrite -(consttC p y) (constt1P p^' _ _) ?mulg1.
   by rewrite 2?sub_in_constt // => q _; move/eqnP->; rewrite !inE ?ltnn.
 rewrite /p_elt pnatNK !order_constt -partnI.
@@ -604,7 +600,7 @@ Qed.
 Lemma norm_sub_max_pgroup pi H M G :
     [max M | pi.-subgroup(G) M] -> pi.-group H -> H \subset G ->
   H \subset 'N(M) -> H \subset M.
-Proof. by move=> maxM piH sHG /normC; exact: comm_sub_max_pgroup piH sHG. Qed.
+Proof. by move=> maxM piH sHG /normC; apply: comm_sub_max_pgroup piH sHG. Qed.
 
 Lemma sub_pHall pi H G K :
   pi.-Hall(G) H -> pi.-group K -> H \subset K -> K \subset G -> K :=: H.
@@ -615,9 +611,9 @@ Qed.
 
 Lemma Hall_max pi H G : pi.-Hall(G) H -> [max H | pi.-subgroup(G) H].
 Proof.
-move=> hallH; apply/maxgroupP; split=> [|K].
+move=> hallH; apply/maxgroupP; split=> [|K /andP[sKG piK] sHK].
   by rewrite /psubgroup; case/and3P: hallH => ->.
-case/andP=> sKG piK sHK; exact: (sub_pHall hallH).
+exact: (sub_pHall hallH).
 Qed.
 
 Lemma pHall_id pi H G : pi.-Hall(G) H -> pi.-group G -> H :=: G.
@@ -644,7 +640,7 @@ have{pG n leGn IHn} pZ: p %| #|'C_G(G)|.
     by have /andP[] := no_x y.
   by apply/implyP; rewrite -index_cent1 indexgI implyNb -Euclid_dvdM ?LagrangeI.
 have [Q maxQ _]: {Q | [max Q | p^'.-subgroup('C_G(G)) Q] & 1%G \subset Q}.
-  apply: maxgroup_exists; exact: psubgroup1.
+  by apply: maxgroup_exists; apply: psubgroup1.
 case/andP: (maxgroupp maxQ) => sQC; rewrite /pgroup p'natE // => /negP[].
 apply: dvdn_trans pZ (cardSg _); apply/subsetP=> x /setIP[Gx Cx].
 rewrite -sub1set -gen_subG (normal_sub_max_pgroup maxQ) //; last first.
@@ -670,7 +666,7 @@ Qed.
 
 Lemma mem_normal_Hall pi H G x :
   pi.-Hall(G) H -> H <| G -> x \in G -> (x \in H) = pi.-elt x.
-Proof. by rewrite -!cycle_subG; exact: sub_normal_Hall. Qed.
+Proof. by rewrite -!cycle_subG; apply: sub_normal_Hall. Qed.
 
 Lemma uniq_normal_Hall pi H G K :
   pi.-Hall(G) H -> H <| G -> [max K | pi.-subgroup(G) K] -> K :=: H.
@@ -723,17 +719,17 @@ Variables (aT rT : finGroupType) (D : {group aT}) (f : {morphism D >-> rT}).
 Implicit Types (pi : nat_pred) (G H P : {group aT}).
 
 Lemma morphim_pgroup pi G : pi.-group G -> pi.-group (f @* G).
-Proof. by apply: pnat_dvd; exact: dvdn_morphim. Qed.
+Proof. by apply: pnat_dvd; apply: dvdn_morphim. Qed.
 
 Lemma morphim_odd G : odd #|G| -> odd #|f @* G|.
-Proof. by rewrite !odd_2'nat; exact: morphim_pgroup. Qed.
+Proof. by rewrite !odd_2'nat; apply: morphim_pgroup. Qed.
 
 Lemma pmorphim_pgroup pi G :
    pi.-group ('ker f) -> G \subset D -> pi.-group (f @* G) = pi.-group G.
 Proof.
 move=> piker sGD; apply/idP/idP=> [pifG|]; last exact: morphim_pgroup.
 apply: (@pgroupS _ _ (f @*^-1 (f @* G))); first by rewrite -sub_morphim_pre.
-by rewrite /pgroup card_morphpre ?morphimS // pnat_mul; exact/andP.
+by rewrite /pgroup card_morphpre ?morphimS // pnat_mul; apply/andP.
 Qed.
 
 Lemma morphim_p_index pi G H :
@@ -760,7 +756,7 @@ Qed.
 
 Lemma morphim_Hall G H : H \subset D -> Hall G H -> Hall (f @* G) (f @* H).
 Proof.
-by move=> sHD /HallP[pi piH]; apply: (@pHall_Hall _ pi); exact: morphim_pHall.
+by move=> sHD /HallP[pi piH]; apply: (@pHall_Hall _ pi); apply: morphim_pHall.
 Qed.
 
 Lemma morphim_pSylow p G P :
@@ -768,7 +764,7 @@ Lemma morphim_pSylow p G P :
 Proof. exact: morphim_pHall. Qed.
 
 Lemma morphim_p_group P : p_group P -> p_group (f @* P).
-Proof. by move/morphim_pgroup; exact: pgroup_p. Qed.
+Proof. by move/morphim_pgroup; apply: pgroup_p. Qed.
 
 Lemma morphim_Sylow G P : P \subset D -> Sylow G P -> Sylow (f @* G) (f @* P).
 Proof.
@@ -776,12 +772,12 @@ by move=> sPD /andP[pP hallP]; rewrite /Sylow morphim_p_group // morphim_Hall.
 Qed.
 
 Lemma morph_p_elt pi x : x \in D -> pi.-elt x -> pi.-elt (f x).
-Proof. by move=> Dx; apply: pnat_dvd; exact: morph_order. Qed.
+Proof. by move=> Dx; apply: pnat_dvd; apply: morph_order. Qed.
 
 Lemma morph_constt pi x : x \in D -> f x.`_pi = (f x).`_pi.
 Proof.
 move=> Dx; rewrite -{2}(consttC pi x) morphM ?groupX //.
-rewrite consttM; last by rewrite !morphX //; exact: commuteX2.
+rewrite consttM; last by rewrite !morphX //; apply: commuteX2.
 have: pi.-elt (f x.`_pi) by rewrite morph_p_elt ?groupX ?p_elt_constt //.
 have: pi^'.-elt (f x.`_pi^') by rewrite morph_p_elt ?groupX ?p_elt_constt //.
 by move/constt1P->; move/constt_p_elt->; rewrite mulg1.
@@ -817,7 +813,7 @@ Lemma ltn_log_quotient :
 Proof.
 move=> pG ntH sHG; apply: contraLR (ltn_quotient ntH sHG); rewrite -!leqNgt.
 rewrite {2}(card_pgroup pG) {2}(card_pgroup (morphim_pgroup _ pG)).
-by case: (posnP p) => [-> //|]; exact: leq_pexp2l.
+by case: (posnP p) => [-> //|]; apply: leq_pexp2l.
 Qed.
 
 End Pquotient.
@@ -883,7 +879,7 @@ Canonical pcore_mod_group pi B : {group gT} :=
 Definition pseries := foldr pcore_mod 1 (rev pis).
 
 Lemma pseries_group_set : group_set pseries.
-Proof. rewrite /pseries; case: rev => [|pi1 pi1']; exact: groupP. Qed.
+Proof. by rewrite /pseries; case: rev => [|pi1 pi1']; apply: groupP. Qed.
 
 Canonical pseries_group : {group gT} := group pseries_group_set.
 
@@ -906,7 +902,7 @@ Lemma pcore_psubgroup G : pi.-subgroup(G) 'O_pi(G).
 Proof.
 have [M maxM _]: {M | [max M | pi.-subgroup(G) M] & 1%G \subset M}.
   by apply: maxgroup_exists; rewrite /psubgroup sub1G pgroup1.
-have sOM: 'O_pi(G) \subset M by exact: bigcap_inf.
+have sOM: 'O_pi(G) \subset M by apply: bigcap_inf.
 have /andP[piM sMG] := maxgroupp maxM. 
 by rewrite /psubgroup (pgroupS sOM) // (subset_trans sOM).
 Qed.
@@ -918,7 +914,7 @@ Lemma pcore_sub G : 'O_pi(G) \subset G.
 Proof. by case/andP: (pcore_psubgroup G). Qed.
 
 Lemma pcore_sub_Hall G H : pi.-Hall(G) H -> 'O_pi(G) \subset H.
-Proof. by move/Hall_max=> maxH; exact: bigcap_inf. Qed.
+Proof. by move/Hall_max=> maxH; apply: bigcap_inf. Qed.
 
 Lemma pcore_max G H : pi.-group H -> H <| G -> H \subset 'O_pi(G).
 Proof.
@@ -952,11 +948,11 @@ Qed.
 
 Lemma sub_Hall_pcore G K :
   pi.-Hall(G) 'O_pi(G) -> K \subset G -> (K \subset 'O_pi(G)) = pi.-group K.
-Proof. by move=> hallGpi; exact: sub_normal_Hall (pcore_normal G). Qed.
+Proof. by move=> hallGpi; apply: sub_normal_Hall (pcore_normal G). Qed.
 
 Lemma mem_Hall_pcore G x :
   pi.-Hall(G) 'O_pi(G) -> x \in G -> (x \in 'O_pi(G)) = pi.-elt x.
-Proof. move=> hallGpi; exact: mem_normal_Hall (pcore_normal G). Qed.
+Proof. by move=> hallGpi; apply: mem_normal_Hall (pcore_normal G). Qed.
 
 Lemma sdprod_Hall_pcoreP H G :
   pi.-Hall(G) 'O_pi(G) -> reflect ('O_pi(G) ><| H = G) (pi^'.-Hall(G) H).
@@ -986,14 +982,14 @@ Lemma morphim_pcore pi : GFunctor.pcontinuous (pcore pi).
 Proof.
 move=> gT rT D G f; apply/bigcapsP=> M /normal_sub_max_pgroup; apply.
   by rewrite morphim_pgroup ?pcore_pgroup.
-apply: morphim_normal; exact: pcore_normal.
+by apply: morphim_normal; apply: pcore_normal.
 Qed.
 
 Lemma pcoreS pi gT (G H : {group gT}) :
   H \subset G -> H :&: 'O_pi(G) \subset 'O_pi(H).
 Proof.
 move=> sHG; rewrite -{2}(setIidPl sHG).
-do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; exact: morphim_pcore.
+by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_pcore.
 Qed.
 
 Canonical pcore_igFun pi := [igFun by pcore_sub pi & morphim_pcore pi].
@@ -1009,17 +1005,12 @@ Variable F : GFunctor.pmap.
 
 Lemma pcore_mod_sub pi gT (G : {group gT}) : pcore_mod G pi (F _ G) \subset G.
 Proof.
-have nFD := gFnorm F G; rewrite sub_morphpre_im ?pcore_sub //=.
-  by rewrite ker_coset_prim subIset // gen_subG gFsub.
-by apply: subset_trans (pcore_sub _ _) _; apply: morphimS.
+by rewrite sub_morphpre_im ?gFsub_trans ?morphimS ?gFnorm //= ker_coset gFsub.
 Qed.
 
 Lemma quotient_pcore_mod pi gT (G : {group gT}) (B : {set gT}) :
   pcore_mod G pi B / B = 'O_pi(G / B).
-Proof.
-apply: morphpreK; apply: subset_trans (pcore_sub _ _) _.
-by rewrite /= /quotient -morphimIdom  morphimS ?subsetIl.
-Qed.
+Proof. exact/morphpreK/gFsub_trans/morphim_sub. Qed.
 
 Lemma morphim_pcore_mod pi gT rT (D G : {group gT}) (f : {morphism D >-> rT}) :
   f @* pcore_mod G pi (F _ G) \subset pcore_mod (f @* G) pi (F _ (f @* G)).
@@ -1111,7 +1102,7 @@ Qed.
 
 Lemma pseries_pop2 pi1 pi2 gT (G : {group gT}) :
   'O_pi1(G) = 1 -> 'O_{pi1, pi2}(G) = 'O_pi2(G).
-Proof. by move/pseries_pop->; exact: pseries1. Qed.
+Proof. by move/pseries_pop->; apply: pseries1. Qed.
 
 Lemma pseries_sub_catl pi1s pi2s gT (G : {group gT}) :
   pseries pi1s G \subset pseries (pi1s ++ pi2s) G.
@@ -1126,9 +1117,7 @@ Proof. by rewrite pseries_rcons quotient_pcore_mod. Qed.
 
 Lemma pseries_norm2 pi1s pi2s gT (G : {group gT}) :
   pseries pi2s G \subset 'N(pseries pi1s G).
-Proof.
-apply: subset_trans (normal_norm (pseries_normal pi1s G)); exact: pseries_sub.
-Qed.
+Proof. by rewrite gFsub_trans ?gFnorm. Qed.
 
 Lemma pseries_sub_catr pi1s pi2s gT (G : {group gT}) :
   pseries pi2s G \subset pseries (pi1s ++ pi2s) G.
@@ -1177,7 +1166,7 @@ rewrite /= cat_rcons -(IHpi (pi :: pi2s)) {1}quotient_pseries IHpi.
 apply/eqP; rewrite quotient_pseries eqEsubset !pcore_max ?pcore_pgroup //=.
   rewrite -quotient_pseries morphim_normal // /(_ <| _) pseries_norm2.
   by rewrite -cat_rcons pseries_sub_catl.
-by rewrite (char_normal_trans (pcore_char _ _)) ?quotient_normal ?gFnormal. 
+by rewrite gFnormal_trans ?quotient_normal ?gFnormal.
 Qed.
 
 Lemma pseries_char_catl pi1s pi2s gT (G : {group gT}) :
@@ -1197,8 +1186,7 @@ rewrite /= -Epis {1}quotient_pseries Epis quotient_pseries.
 apply/eqP; rewrite eqEsubset !pcore_max ?pcore_pgroup //=.
   rewrite -quotient_pseries morphim_normal // /(_ <| _) pseries_norm2.
   by rewrite pseries_sub_catr.
-apply: char_normal_trans (pcore_char pi _) (morphim_normal _ _).
-exact: pseries_normal.
+by rewrite gFnormal_trans ?morphim_normal ?gFnormal.
 Qed.
 
 Lemma pseries_char_catr pi1s pi2s gT (G : {group gT}) :
@@ -1208,12 +1196,11 @@ Proof. by rewrite -(pseries_catr_id pi1s pi2s G) pseries_char. Qed.
 Lemma pcore_modp pi gT (G H : {group gT}) :
   H <| G -> pi.-group H -> pcore_mod G pi H = 'O_pi(G).
 Proof.
-move=> nsHG piH; apply/eqP; rewrite eqEsubset andbC.
-have nHG := normal_norm nsHG; have sOG := subset_trans (pcore_sub pi _).
-rewrite -sub_morphim_pre ?(sOG, morphim_pcore) // pcore_max //.
-  rewrite -(pquotient_pgroup piH) ?subsetIl //.
-  by rewrite quotient_pcore_mod pcore_pgroup.
-by rewrite -{2}(quotientGK nsHG) morphpre_normal ?pcore_normal ?sOG ?morphimS.
+move=> nsHG piH; have nHG := normal_norm nsHG; apply/eqP.
+rewrite eqEsubset andbC -sub_morphim_pre ?(gFsub_trans, morphim_pcore) //=.
+rewrite -[G in 'O_pi(G)](quotientGK nsHG) pcore_max //.
+  by rewrite -(pquotient_pgroup piH) ?subsetIl // cosetpreK pcore_pgroup.
+by rewrite morphpre_normal ?gFnormal ?gFsub_trans ?morphim_sub.
 Qed.
 
 Lemma pquotient_pcore pi gT (G H : {group gT}) :
@@ -1221,9 +1208,7 @@ Lemma pquotient_pcore pi gT (G H : {group gT}) :
 Proof. by move=> nsHG piH; rewrite -quotient_pcore_mod pcore_modp. Qed.
 
 Lemma trivg_pcore_quotient pi gT (G : {group gT}) : 'O_pi(G / 'O_pi(G)) = 1.
-Proof.
-by rewrite pquotient_pcore ?pcore_normal ?pcore_pgroup // trivg_quotient.
-Qed.
+Proof. by rewrite pquotient_pcore ?gFnormal ?pcore_pgroup ?trivg_quotient. Qed.
 
 Lemma pseries_rcons_id pis pi gT (G : {group gT}) :
   pseries (rcons (rcons pis pi) pi) G = pseries (rcons pis pi) G.
@@ -1246,11 +1231,11 @@ Lemma sub_in_pcore pi rho G :
 Proof.
 move=> pi_sub_rho; rewrite pcore_max ?pcore_normal //.
 apply: sub_in_pnat (pcore_pgroup _ _) => p.
-move/(piSg (pcore_sub _ _)); exact: pi_sub_rho.
+by move/(piSg (pcore_sub _ _)); apply: pi_sub_rho.
 Qed.
 
 Lemma sub_pcore pi rho G : {subset pi <= rho} -> 'O_pi(G) \subset 'O_rho(G).
-Proof. by move=> pi_sub_rho; exact: sub_in_pcore (in1W pi_sub_rho). Qed.
+Proof. by move=> pi_sub_rho; apply: sub_in_pcore (in1W pi_sub_rho). Qed.
 
 Lemma eq_in_pcore pi rho G : {in \pi(G), pi =i rho} -> 'O_pi(G) = 'O_rho(G).
 Proof.
@@ -1259,31 +1244,30 @@ by rewrite !sub_in_pcore // => p /eq_pi_rho->.
 Qed.
 
 Lemma eq_pcore pi rho G : pi =i rho -> 'O_pi(G) = 'O_rho(G).
-Proof. by move=> eq_pi_rho; exact: eq_in_pcore (in1W eq_pi_rho). Qed.
+Proof. by move=> eq_pi_rho; apply: eq_in_pcore (in1W eq_pi_rho). Qed.
 
 Lemma pcoreNK pi G : 'O_pi^'^'(G) = 'O_pi(G).
-Proof. by apply: eq_pcore; exact: negnK. Qed.
+Proof. by apply: eq_pcore; apply: negnK. Qed.
 
 Lemma eq_p'core pi rho G : pi =i rho -> 'O_pi^'(G) = 'O_rho^'(G).
-Proof. by move/eq_negn; exact: eq_pcore. Qed.
+Proof. by move/eq_negn; apply: eq_pcore. Qed.
 
 Lemma sdprod_Hall_p'coreP pi H G :
   pi^'.-Hall(G) 'O_pi^'(G) -> reflect ('O_pi^'(G) ><| H = G) (pi.-Hall(G) H).
-Proof. by rewrite -(pHallNK pi G H); exact: sdprod_Hall_pcoreP. Qed.
+Proof. by rewrite -(pHallNK pi G H); apply: sdprod_Hall_pcoreP. Qed.
 
 Lemma sdprod_p'core_HallP pi H G :
   pi.-Hall(G) H -> reflect ('O_pi^'(G) ><| H = G) (pi^'.-Hall(G) 'O_pi^'(G)).
-Proof. by rewrite -(pHallNK pi G H); exact: sdprod_pcore_HallP. Qed.
+Proof. by rewrite -(pHallNK pi G H); apply: sdprod_pcore_HallP. Qed.
 
 Lemma pcoreI pi rho G : 'O_[predI pi & rho](G) = 'O_pi('O_rho(G)).
 Proof.
 apply/eqP; rewrite eqEsubset !pcore_max //.
-- rewrite /pgroup pnatI [pnat _ _]pcore_pgroup.
-  exact: pgroupS (pcore_sub _ _) (pcore_pgroup _ _).
-- exact: char_normal_trans (pcore_char _ _) (pcore_normal _ _).
-- by apply: sub_in_pnat (pcore_pgroup _ _) => p _ /andP[].
+- by rewrite /pgroup pnatI -!pgroupE !(pcore_pgroup, pgroupS (pcore_sub pi _)).
+- by rewrite !gFnormal_trans.
+- by apply: sub_pgroup (pcore_pgroup _ _) => p /andP[].
 apply/andP; split; first by apply: sub_pcore => p /andP[].
-by rewrite (subset_trans (pcore_sub _ _)) ?gFnorm.
+by rewrite gFnorm_trans ?normsG ?gFsub.
 Qed.
 
 Lemma bigcap_p'core pi G :
@@ -1298,7 +1282,7 @@ apply/eqP; rewrite eqEsubset subsetI pcore_sub pcore_max /=.
   rewrite (dvdn_trans qGpi') ?cardSg ?subIset //= orbC.
   by rewrite (bigcap_inf (Ordinal ltqG)).
 rewrite /normal subsetIl normsI ?normG // norms_bigcap //.
-by apply/bigcapsP => p _; exact: gFnorm.
+by apply/bigcapsP => p _; apply: gFnorm.
 Qed.
 
 Lemma coprime_pcoreC (rT : finGroupType) pi G (R : {group rT}) :
@@ -1310,10 +1294,9 @@ Proof. by rewrite coprime_TIg ?coprime_pcoreC. Qed.
 
 Lemma pcore_setI_normal pi G H : H <| G -> 'O_pi(G) :&: H = 'O_pi(H).
 Proof.
-move=> nsHG; apply/eqP; rewrite eqEsubset subsetI pcore_sub.
-rewrite !pcore_max ?(pgroupS (subsetIl _ H)) ?pcore_pgroup //=.
-  exact: char_normal_trans (pcore_char pi H) nsHG.
-by rewrite setIC (normalGI (normal_sub nsHG)) ?pcore_normal.
+move=> nsHG; apply/eqP; rewrite eqEsubset subsetI pcore_sub setIC.
+rewrite !pcore_max ?(pgroupS (subsetIr H _)) ?pcore_pgroup ?gFnormal_trans //=.
+by rewrite norm_normalI ?gFnorm_trans ?normsG ?normal_sub.
 Qed.
 
 End EqPcore.
diff --git a/mathcomp/solvable/primitive_action.v b/mathcomp/solvable/primitive_action.v
index 06ab30e..2d07bba 100644
--- a/mathcomp/solvable/primitive_action.v
+++ b/mathcomp/solvable/primitive_action.v
@@ -1,12 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div fintype tuple finset.
-From mathcomp.fingroup
-Require Import fingroup action.
-Require Import gseries.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat.
+From mathcomp
+Require Import div seq fintype tuple finset.
+From mathcomp
+Require Import fingroup action gseries.
 
 (******************************************************************************)
 (* n-transitive and primitive actions:                                        *)
@@ -65,7 +64,7 @@ move=> Sx trG; rewrite /primitive trG negb_exists.
 apply/forallP/maximal_eqP=> /= [primG | [_ maxCx] Q].
   split=> [|H sCH sHG]; first exact: subsetIl.
   pose X := orbit to H x; pose Q := orbit (to^*)%act G X.
-  have Xx: x \in X by exact: orbit_refl.
+  have Xx: x \in X by apply: orbit_refl.
   have defH: 'N_(G)(X | to) = H.
     have trH: [transitive H, on X | to] by apply/imsetP; exists x.
     have sHN: H \subset 'N_G(X | to) by rewrite subsetI sHG atrans_acts.
@@ -79,13 +78,13 @@ apply/forallP/maximal_eqP=> /= [primG | [_ maxCx] Q].
   - rewrite card_orbit astab1_set defH -(@ltn_pmul2l #|H|) ?Lagrange // muln1.
     rewrite oHG -(@ltn_pmul2l #|H|) ?Lagrange // -(card_orbit_stab to G x).
     by rewrite -(atransP trG x Sx) mulnC card_orbit ltn_pmul2r.
-  - by apply/actsP=> a Ga Y; apply: orbit_transr; exact: mem_orbit.
+  - by apply/actsP=> a Ga Y; apply/orbit_transl/mem_orbit.
   apply/and3P; split; last 1 first.
   - rewrite orbit_sym; apply/imsetP=> [[a _]] /= defX.
     by rewrite defX /setact imset0 inE in Xx.
   - apply/eqP/setP=> y; apply/bigcupP/idP=> [[_ /imsetP[a Ga ->]] | Sy].
       case/imsetP=> _ /imsetP[b Hb ->] ->.
-      by rewrite !(actsP (atrans_acts trG)) //; exact: subsetP Hb.
+      by rewrite !(actsP (atrans_acts trG)) //; apply: subsetP Hb.
     case: (atransP2 trG Sx Sy) => a Ga ->.
     by exists ((to^*)%act X a); apply: mem_imset; rewrite // orbit_refl.
   apply/trivIsetP=> _ _ /imsetP[a Ga ->] /imsetP[b Gb ->].
@@ -179,7 +178,7 @@ Lemma n_act_dtuple t a :
 Proof.
 move/astabsP=> toSa /dtuple_onP[t_inj St]; apply/dtuple_onP.
 split=> [i j | i]; rewrite !tnth_map ?[_ \in S]toSa //.
-by move/act_inj; exact: t_inj.
+by move/act_inj; apply: t_inj.
 Qed.
 
 End NTransitive.
@@ -202,7 +201,7 @@ Variables (gT : finGroupType) (sT : finType).
 Variables (to : {action gT &-> sT}) (G : {group gT}) (S : {set sT}).
 
 Lemma card_uniq_tuple n (t : n.-tuple sT) : uniq t -> #|t| = n.
-Proof. by move/card_uniqP->; exact: size_tuple. Qed.
+Proof. by move/card_uniqP->; apply: size_tuple. Qed.
 
 Lemma n_act0 (t : 0.-tuple sT) a : n_act to t a = [tuple].
 Proof. exact: tuple0. Qed.
@@ -223,7 +222,7 @@ Qed.
 
 Lemma dtuple_on_subset n (S1 S2 : {set sT}) t :
   S1 \subset S2 -> t \in n.-dtuple(S1) -> t \in n.-dtuple(S2).
-Proof. by move=> sS12; rewrite !inE => /andP[-> /subset_trans]; exact. Qed.
+Proof. by move=> sS12; rewrite !inE => /andP[-> /subset_trans]; apply. Qed.
 
 Lemma n_act_add n x (t : n.-tuple sT) a :
   n_act to [tuple of x :: t] a = [tuple of to x a :: n_act to t a].
@@ -254,7 +253,7 @@ case/and3P=> Sx ntx dt; set xt := [tuple of _] => tr_xt.
 apply/imsetP; exists t => //.
 apply/setP=> u; apply/idP/imsetP=> [du | [a Ga ->{u}]].
   case: (ext_t u du) => y; rewrite tr_xt.
-  by case/imsetP=> a Ga [_ def_u]; exists a => //; exact: val_inj.
+  by case/imsetP=> a Ga [_ def_u]; exists a => //; apply: val_inj.
 have: n_act to xt a \in dtuple_on _ S by rewrite tr_xt mem_imset.
 by rewrite n_act_add dtuple_on_add; case/and3P.
 Qed.
@@ -267,14 +266,14 @@ have trdom1 x: ([tuple x] \in 1.-dtuple(S)) = (x \in S).
 move=> m_gt0 /(ntransitive_weak m_gt0) {m m_gt0}.
 case/imsetP; case/tupleP=> x t0; rewrite {t0}(tuple0 t0) trdom1 => Sx trx.
 apply/imsetP; exists x => //; apply/setP=> y; rewrite -trdom1 trx.
-apply/imsetP/imsetP=> [[a ? [->]]|[a ? ->]]; exists a => //; exact: val_inj.
+by apply/imsetP/imsetP=> [[a ? [->]]|[a ? ->]]; exists a => //; apply: val_inj.
 Qed.
 
 Lemma ntransitive_primitive m :
   1 < m -> [transitive^m G, on S | to] -> [primitive G, on S | to].
 Proof.
 move=> lt1m /(ntransitive_weak lt1m) {m lt1m}tr2G.
-have trG: [transitive G, on S | to] by exact: ntransitive1 tr2G.
+have trG: [transitive G, on S | to] by apply: ntransitive1 tr2G.
 have [x Sx _]:= imsetP trG; rewrite (trans_prim_astab Sx trG).
 apply/maximal_eqP; split=> [|H]; first exact: subsetIl; rewrite subEproper.
 case/predU1P; first by [left]; case/andP=> sCH /subsetPn[a Ha nCa] sHG.
@@ -287,7 +286,7 @@ rewrite eqEsubset acts_sub_orbit // Sy andbT; apply/subsetP=> z Sz.
 have [-> | zx] := eqVneq z x; first by rewrite orbit_sym mem_orbit.
 pose ty := [tuple y; x]; pose tz := [tuple z; x].
 have [Sty Stz]: ty \in 2.-dtuple(S) /\ tz \in 2.-dtuple(S).
-  rewrite !inE !memtE !subset_all /= !mem_seq1 !andbT; split; exact/and3P.
+  by rewrite !inE !memtE !subset_all /= !mem_seq1 !andbT; split; apply/and3P.
 case: (atransP2 tr2G Sty Stz) => b Gb [->] /esym/astab1P cxb.
 by rewrite mem_orbit // (subsetP sCH) // inE Gb.
 Qed.
@@ -310,12 +309,12 @@ case/and3P=> Sx1 nt1x1 dt1 trt1; have Gtr1 := ntransitive1 (ltn0Sn _) Gtr.
 case: (atransP2 Gtr1 Sx1 Sx) => // a Ga x1ax.
 pose t := n_act to t1 a.
 have dxt: [tuple of x :: t] \in m.+1.-dtuple(S).
-  rewrite trt1 x1ax; apply/imsetP; exists a => //; exact: val_inj.
+  by rewrite trt1 x1ax; apply/imsetP; exists a => //; apply: val_inj.
 apply/imsetP; exists t; first by rewrite dtuple_on_add_D1 Sx in dxt.
 apply/setP=> t2; apply/idP/imsetP => [dt2|[b]].
   have: [tuple of x :: t2] \in dtuple_on _ S by rewrite dtuple_on_add_D1 Sx.
   case/(atransP2 Gtr dxt)=> b Gb [xbx tbt2].
-  exists b; [rewrite inE Gb; exact/astab1P | exact: val_inj].
+  by exists b; [rewrite inE Gb; apply/astab1P | apply: val_inj].
 case/setIP=> Gb /astab1P xbx ->{t2}.
 rewrite n_act_dtuple //; last by rewrite dtuple_on_add_D1 Sx in dxt.
 apply/astabsP=> y; rewrite !inE -{1}xbx (inj_eq (act_inj _ _)).
diff --git a/mathcomp/solvable/sylow.v b/mathcomp/solvable/sylow.v
index abaa2a5..e347a74 100644
--- a/mathcomp/solvable/sylow.v
+++ b/mathcomp/solvable/sylow.v
@@ -1,16 +1,11 @@
 (* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
 Require Import mathcomp.ssreflect.ssreflect.
-From mathcomp.ssreflect
-Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq.
-From mathcomp.discrete
-Require Import div fintype finset bigop prime.
-From mathcomp.fingroup
-Require Import fingroup morphism automorphism quotient action gproduct.
-From mathcomp.algebra
-Require Import ssralg poly ssrnum ssrint rat.
-From mathcomp.algebra
-Require Import polydiv finalg zmodp matrix mxalgebra vector cyclic.
-Require Import commutator pgroup center nilpotent.
+From mathcomp
+Require Import ssrbool ssrfun eqtype ssrnat seq div fintype prime.
+From mathcomp
+Require Import bigop finset fingroup morphism automorphism quotient action.
+From mathcomp
+Require Import cyclic gproduct gfunctor commutator pgroup center nilpotent.
 
 (******************************************************************************)
 (*   The Sylow theorem and its consequences, including the Frattini argument, *)
@@ -68,11 +63,11 @@ Lemma pcore_sub_astab_irr G M :
   'O_p(G) \subset 'C_G(M | to).
 Proof.
 move=> pM sMR /mingroupP[/andP[ntM nMG] minM].
-have [sGpG nGpG]:= andP (pcore_normal p G).
-have sGD := acts_dom nMG; have sGpD := subset_trans sGpG sGD.
+have /andP[sGpG nGpG]: 'O_p(G) <| G := gFnormal _ G.
+have sGD := acts_dom nMG; have sGpD: 'O_p(G) \subset D := gFsub_trans _ sGD.
 rewrite subsetI sGpG -gacentC //=; apply/setIidPl; apply: minM (subsetIl _ _).
 rewrite nontrivial_gacent_pgroup ?pcore_pgroup //=; last first.
-  by split; rewrite ?(subset_trans sGpG).
+  by split; rewrite ?gFsub_trans.
 by apply: subset_trans (acts_subnorm_subgacent sGpD nMG); rewrite subsetI subxx.
 Qed.
 
@@ -108,7 +103,7 @@ have S_pG P: P \in S -> P \subset G /\ p.-group P.
 have SmaxN P Q: Q \in S -> Q \subset 'N(P) -> maxp 'N_G(P) Q.
   rewrite inE => /maxgroupP[/andP[sQG pQ] maxQ] nPQ.
   apply/maxgroupP; rewrite /psubgroup subsetI sQG nPQ.
-  by split=> // R; rewrite subsetI -andbA andbCA => /andP[_]; exact: maxQ.
+  by split=> // R; rewrite subsetI -andbA andbCA => /andP[_]; apply: maxQ.
 have nrmG P: P \subset G -> P <| 'N_G(P).
   by move=> sPG; rewrite /normal subsetIr subsetI sPG normG.
 have sylS P: P \in S -> p.-Sylow('N_G(P)) P.
@@ -123,7 +118,7 @@ have{defCS} oG_mod: {in S &, forall P Q, #|oG P| = (Q \in oG P) %[mod p]}.
   move=> P Q S_P S_Q; have [sQG pQ] := S_pG _ S_Q.
   have soP_S: oG P \subset S by rewrite acts_sub_orbit.
   have /pgroup_fix_mod-> //: [acts Q, on oG P | 'JG].
-    apply/actsP=> x /(subsetP sQG) Gx R; apply: orbit_transr.
+    apply/actsP=> x /(subsetP sQG) Gx R; apply: orbit_transl.
     exact: mem_orbit.
   rewrite -{1}(setIidPl soP_S) -setIA defCS // (cardsD1 Q) setDE.
   by rewrite -setIA setICr setI0 cards0 addn0 inE set11 andbT.
@@ -206,7 +201,7 @@ Lemma card_Syl P : p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|.
 Proof. by case: Sylow's_theorem P. Qed.
 
 Lemma card_Syl_dvd : #|'Syl_p(G)| %| #|G|.
-Proof. by case Sylow_exists => P /card_Syl->; exact: dvdn_indexg. Qed.
+Proof. by case Sylow_exists => P /card_Syl->; apply: dvdn_indexg. Qed.
 
 Lemma card_Syl_mod : prime p -> #|'Syl_p(G)| %% p = 1%N.
 Proof. by case Sylow's_theorem. Qed.
@@ -262,13 +257,13 @@ Qed.
 Lemma p2group_abelian P : p.-group P -> logn p #|P| <= 2 -> abelian P.
 Proof.
 move=> pP lePp2; pose Z := 'Z(P); have sZP: Z \subset P := center_sub P.
-case: (eqVneq Z 1); first by move/(trivg_center_pgroup pP)->; exact: abelian1.
+case: (eqVneq Z 1); first by move/(trivg_center_pgroup pP)->; apply: abelian1.
 case/(pgroup_pdiv (pgroupS sZP pP)) => p_pr _ [k oZ].
 apply: cyclic_center_factor_abelian.
-case: (eqVneq (P / Z) 1) => [-> |]; first exact: cyclic1.
+have [->|] := eqVneq (P / Z) 1; first exact: cyclic1.
 have pPq := quotient_pgroup 'Z(P) pP; case/(pgroup_pdiv pPq) => _ _ [j oPq].
 rewrite prime_cyclic // oPq; case: j oPq lePp2 => //= j.
-rewrite card_quotient ?gfunctor.gFnorm //.
+rewrite card_quotient ?gFnorm //.
 by rewrite -(Lagrange sZP) lognM // => ->; rewrite oZ !pfactorK ?addnS.
 Qed.
 
@@ -343,7 +338,7 @@ Proof.
 move=> defG nHG coKR; apply/eqP; rewrite eqEcard mulG_subG /= -defG.
 rewrite !setSI ?mulG_subl ?mulG_subr //=.
 rewrite coprime_cardMg ?(coKR, coprimeSg (subsetIl _ _), coprime_sym) //=.
-pose pi := \pi(K); have piK: pi.-group K by exact: pgroup_pi.
+pose pi := \pi(K); have piK: pi.-group K by apply: pgroup_pi.
 have pi'R: pi^'.-group R by rewrite /pgroup -coprime_pi' /=.
 have [hallK hallR] := coprime_mulpG_Hall defG piK pi'R.
 have nsHG: H :&: G <| G by rewrite /normal subsetIr normsI ?normG.
@@ -372,7 +367,7 @@ by rewrite ltn_Pdiv // ltnNge -trivg_card_le1.
 Qed.
 
 Lemma pgroup_sol p P : p.-group P -> solvable P.
-Proof. by move/pgroup_nil; exact: nilpotent_sol. Qed.
+Proof. by move/pgroup_nil; apply: nilpotent_sol. Qed.
 
 Lemma small_nil_class G : nil_class G <= 5 -> nilpotent G.
 Proof.
@@ -404,10 +399,10 @@ have nHN: H <| 'N_G(H) by rewrite normal_subnorm.
 have{maxH} hallH: pi.-Hall('N_G(H)) H.
   apply: normal_max_pgroup_Hall => //; apply/maxgroupP.
   rewrite /psubgroup normal_sub // piH; split=> // K.
-  by rewrite subsetI -andbA andbCA => /andP[_]; exact: maxH.
-rewrite /normal sHG; apply/setIidPl; symmetry.
-apply: nilpotent_sub_norm; rewrite ?subsetIl ?setIS //=.
-by rewrite char_norms // -{1}(normal_Hall_pcore hallH) // pcore_char.
+  by rewrite subsetI -andbA andbCA => /andP[_ /maxH].
+rewrite /normal sHG; apply/setIidPl/esym.
+apply: nilpotent_sub_norm; rewrite ?subsetIl ?setIS //= char_norms //.
+by congr (_ \char _): (pcore_char pi 'N_G(H)); apply: normal_Hall_pcore.
 Qed.
 
 Lemma nilpotent_Hall_pcore pi G H :
@@ -430,12 +425,12 @@ Qed.
 Lemma nilpotent_pcoreC pi G : nilpotent G -> 'O_pi(G) \x 'O_pi^'(G) = G.
 Proof.
 move=> nilG; have trO: 'O_pi(G) :&: 'O_pi^'(G) = 1.
-  by apply: coprime_TIg; apply: (@pnat_coprime pi); exact: pcore_pgroup.
+  by apply: coprime_TIg; apply: (@pnat_coprime pi); apply: pcore_pgroup.
 rewrite dprodE //.
   apply/eqP; rewrite eqEcard mul_subG ?pcore_sub // (TI_cardMg trO).
   by rewrite !(card_Hall (nilpotent_pcore_Hall _ _)) // partnC ?leqnn.
 rewrite (sameP commG1P trivgP) -trO subsetI commg_subl commg_subr.
-by rewrite !(subset_trans (pcore_sub _ _)) ?normal_norm ?pcore_normal.
+by rewrite !gFsub_trans ?gFnorm.
 Qed.
 
 Lemma sub_nilpotent_cent2 H K G :
@@ -485,7 +480,7 @@ elim: c => // c IHc in gT P def_c pP *; set e := logn p _.
 have nilP := pgroup_nil pP; have sZP := center_sub P.
 have [e_le2 | e_gt2] := leqP e 2.
   by rewrite -def_c leq_max nil_class1 (p2group_abelian pP).
-have pPq: p.-group (P / 'Z(P)) by exact: quotient_pgroup.
+have pPq: p.-group (P / 'Z(P)) by apply: quotient_pgroup.
 rewrite -(subnKC e_gt2) ltnS (leq_trans (IHc _ _ _ pPq)) //.
   by rewrite nil_class_quotient_center ?def_c.
 rewrite geq_max /= -add1n -leq_subLR -subn1 -subnDA -subSS leq_sub2r //.
@@ -534,7 +529,7 @@ have /cyclicP[x' def_p']: cyclic 'O_p^'(G).
   by have:= pcore_pgroup p^' G; rewrite def_p' /pgroup p'natE ?pG.
 apply/cyclicP; exists (x * x'); rewrite -{}defG def_p def_p' cycleM //.
   by red; rewrite -(centsP Cpp') // (def_p, def_p') cycle_id.
-rewrite /order -def_p -def_p' (@pnat_coprime p) //; exact: pcore_pgroup.
+by rewrite /order -def_p -def_p' (@pnat_coprime p) //; apply: pcore_pgroup.
 Qed.
 
 End Zgroups.
@@ -582,8 +577,8 @@ have{pE} pE: {in E &, forall x1 x2, p.-group <<[set x1; x2]>>}.
   rewrite -(mulgKV y1 y2) conjgM -2!conjg_set1 -conjUg genJ pgroupJ.
   by rewrite pE // groupMl ?groupV.
 have sEG: <<E>> \subset G by rewrite gen_subG class_subG.
-have nEG: G \subset 'N(E) by exact: class_norm.
-have Ex: x \in E by exact: class_refl.
+have nEG: G \subset 'N(E) by apply: class_norm.
+have Ex: x \in E by apply: class_refl.
 have [P Px sylP]: exists2 P : {group gT}, x \in P & p.-Sylow(<<E>>) P.
   have sxxE: <<[set x; x]>> \subset <<E>> by rewrite genS // setUid sub1set.
   have{sxxE} [P sylP sxxP] := Sylow_superset sxxE (pE _ _ Ex Ex).
@@ -600,14 +595,14 @@ have{Ex Px}: P_D [set x].
 case/(@maxset_exists _ P_D)=> D /maxsetP[]; rewrite {P_yD P_D}/=.
 rewrite subsetI sub1set -andbA => /and3P[sDP sDE /existsP[y0]].
 set B := _ |: D; rewrite inE -andbA => /and3P[Py0 Ey0 pB] maxD Dx.
-have sDgE: D \subset <<E>> by exact: sub_gen.
-have sDG: D \subset G by exact: subset_trans sEG.
+have sDgE: D \subset <<E>> by apply: sub_gen.
+have sDG: D \subset G by apply: subset_trans sEG.
 have sBE: B \subset E by rewrite subUset sub1set Ey0.
-have sBG: <<B>> \subset G by exact: subset_trans (genS _) sEG.
+have sBG: <<B>> \subset G by apply: subset_trans (genS _) sEG.
 have sDB: D \subset B by rewrite subsetUr.
 have defD: D :=: P :&: <<B>> :&: E.
   apply/eqP; rewrite eqEsubset ?subsetI sDP sDE sub_gen //=.
-  apply/setUidPl; apply: maxD; last exact: subsetUl.
+  apply/setUidPl; apply: maxD; last apply: subsetUl.
   rewrite subUset subsetI sDP sDE setIAC subsetIl.
   apply/existsP; exists y0; rewrite inE Py0 Ey0 /= setUA -/B.
   by rewrite -[<<_>>]joing_idl joingE setKI genGid.
@@ -645,7 +640,7 @@ have [y1 Ny1 Py1]: exists2 y1, y1 \in 'N_E(D) & y1 \notin P.
 have [y2 Ny2 Dy2]: exists2 y2, y2 \in 'N_(P :&: E)(D) & y2 \notin D.
   case sNN: ('N_P('N_P(D)) \subset 'N_P(D)).
     have [z /= Ez sEzP] := Sylow_Jsub sylP (genS sBE) pB.
-    have Gz: z \in G by exact: subsetP Ez.
+    have Gz: z \in G by apply: subsetP Ez.
     have /subsetPn[y Bzy Dy]: ~~ (B :^ z \subset D).
       apply/negP; move/subset_leq_card; rewrite cardJg cardsU1.
       by rewrite {1}defD 2!inE (negPf Py0) ltnn.
@@ -657,7 +652,7 @@ have [y2 Ny2 Dy2]: exists2 y2, y2 \in 'N_(P :&: E)(D) & y2 \notin D.
   case/subsetPn=> y Dzy Dy; exists y => //; apply: subsetP Dzy.
   rewrite -setIA setICA subsetI sub_conjg (normsP nEG) ?groupV //.
     by rewrite sDE -(normP NNz); rewrite conjSg subsetI sDP.
-  apply: subsetP Pz; exact: (subset_trans (pHall_sub sylP)).
+  by apply: subsetP Pz; apply: (subset_trans (pHall_sub sylP)).
 suff{Dy2} Dy2D: y2 |: D = D by rewrite -Dy2D setU11 in Dy2.
 apply: maxD; last by rewrite subsetUr.
 case/setIP: Ny2 => PEy2 Ny2; case/setIP: Ny1 => Ey1 Ny1.