Enriched numClosedFieldType so that it factors a lot of theory from both complex and algC.

The definitions of 'i, conjC, Re, Im, n.-root, sqrtC and their theory
have been moved to the numClosedFieldType structure in ssrnum.
This covers boths the uses in algC and complex.v. To that end the
numClosedFieldType structure has been enriched with conjugation and 'i.
Note that 'i can be deduced from the property of algebraic closure and is
only here to let the user chose which definitional equality should hold
on 'i. Same thing for conjC that could be written  `|x|^+2/x, the only
nontrivial (up to my knowledge) property is the fact that conjugation
is a ring morphism.

6 files changed, 16 insertions, 15 deletions

diff --git a/mathcomp/odd_order/BGappendixC.v b/mathcomp/odd_order/BGappendixC.v
index a64c49a..16a0a3c 100644
--- a/mathcomp/odd_order/BGappendixC.v
+++ b/mathcomp/odd_order/BGappendixC.v
@@ -288,7 +288,7 @@ Proof.
 have [q_gt4 | q_le4] := ltnP 4 q.
   pose inK x := enum_rank_in (classes1 H) (x ^: H).
   have inK_E x: x \in H -> enum_val (inK x) = x ^: H.
-    by move=> Hx; rewrite enum_rankK_in ?mem_classes. 
+    by move=> Hx; rewrite enum_rankK_in ?mem_classes.
   pose j := inK s; pose k := inK (s ^+ 2)%g; pose e := gring_classM_coef j j k.
   have cPP: abelian P by rewrite -(injm_abelian inj_sigma) ?zmod_abelian.
   have Hs: s \in H by rewrite -(sdprodW defH) -[s]mulg1 mem_mulg.
@@ -355,18 +355,19 @@ have [q_gt4 | q_le4] := ltnP 4 q.
       by rewrite sub_cent1 groupX // (subsetP cPP).
     rewrite mulrnA -second_orthogonality_relation ?groupX // big_mkcond.
     by apply: ler_sum => i _; rewrite normCK; case: ifP; rewrite ?mul_conjC_ge0.
-  have sqrtP_gt0: 0 < sqrtC #|P|%:R by rewrite sqrtC_gt0 ?gt0CG.
-  have{De ub_linH'}: `|(#|P| * e)%:R - #|U|%:R ^+ 2| <= #|P|%:R * sqrtC #|P|%:R.
+  have sqrtP_gt0: 0 < sqrtC #|P|%:R :> algC by rewrite sqrtC_gt0 ?gt0CG.
+  have{De ub_linH'}:
+    `|(#|P| * e)%:R - #|U|%:R ^+ 2| <= #|P|%:R * sqrtC #|P|%:R :> algC.
     rewrite natrM De mulrCA mulrA divfK ?neq0CG // (bigID linH) /= sum_linH.
     rewrite mulrDr addrC addKr mulrC mulr_suml /chi_s2.
     rewrite (ler_trans (ler_norm_sum _ _ _)) // -ler_pdivr_mulr // mulr_suml.
     apply: ler_trans (ub_linH' 1%N isT); apply: ler_sum => i linH'i.
     rewrite ler_pdivr_mulr // degU ?divfK ?neq0CG //.
     rewrite normrM -normrX norm_conjC ler_wpmul2l ?normr_ge0 //.
-    rewrite -ler_sqr ?qualifE ?normr_ge0 ?(@ltrW _ 0) // sqrtCK. 
+    rewrite -ler_sqr ?qualifE ?normr_ge0 ?(@ltrW _ 0) // sqrtCK.
     apply: ler_trans (ub_linH' 2 isT); rewrite (bigD1 i) ?ler_paddr //=.
     by apply: sumr_ge0 => i1 _; rewrite exprn_ge0 ?normr_ge0.
-  rewrite natrM real_ler_distl ?rpredB ?rpredM ?rpred_nat // => /andP[lb_Pe _]. 
+  rewrite natrM real_ler_distl ?rpredB ?rpredM ?rpred_nat // => /andP[lb_Pe _].
   rewrite -ltC_nat -(ltr_pmul2l (gt0CG P)) {lb_Pe}(ltr_le_trans _ lb_Pe) //.
   rewrite ltr_subr_addl (@ler_lt_trans _ ((p ^ q.-1)%:R ^+ 2)) //; last first.
     rewrite  -!natrX ltC_nat ltn_sqr oU ltn_divRL ?dvdn_pred_predX //.
diff --git a/mathcomp/odd_order/PFsection11.v b/mathcomp/odd_order/PFsection11.v
index b966f25..3c4ec9f 100644
--- a/mathcomp/odd_order/PFsection11.v
+++ b/mathcomp/odd_order/PFsection11.v
@@ -232,7 +232,7 @@ Lemma bounded_proper_coherent H1 :
     (#|HU : H1| <= 2 * q * #|U : C| + 1)%N.
 Proof.
 move=> nsH1_M psH1_M' cohH1; have [nsHHU _ _ _ _] := sdprod_context defHU.
-suffices: #|HU : H1|%:R - 1 <= 2%:R * #|M : HC|%:R * sqrtC #|HC : HC|%:R.
+suffices: #|HU : H1|%:R - 1 <= 2%:R * #|M : HC|%:R * sqrtC #|HC : HC|%:R :> algC.
   rewrite  indexgg sqrtC1 mulr1 -leC_nat natrD -ler_subl_addr -mulnA natrM.
   congr (_ <= _ * _%:R); apply/eqP; rewrite -(eqn_pmul2l (cardG_gt0 HC)).
   rewrite Lagrange ?normal_sub // mulnCA -(dprod_card defHC) -mulnA mulnC.
diff --git a/mathcomp/odd_order/PFsection3.v b/mathcomp/odd_order/PFsection3.v
index cb55ae4..9011122 100644
--- a/mathcomp/odd_order/PFsection3.v
+++ b/mathcomp/odd_order/PFsection3.v
@@ -1360,7 +1360,7 @@ have{oxi_00} oxi_i0 i j i0: '[xi_ i j, xi_ i0 0] = ((i == i0) && (j == 0))%:R.
   by rewrite cfdotC Xi0_X0j // conjC0.
 have [-> | nzi2] := altP (i2 =P 0); first exact: oxi_0j.
 have [-> | nzj2] := altP (j2 =P 0); first exact: oxi_i0.
-rewrite cfdotC eq_sym; apply: canLR conjCK _; rewrite rmorph_nat.
+rewrite cfdotC eq_sym; apply: canLR (@conjCK _) _; rewrite rmorph_nat.
 have [-> | nzi1] := altP (i1 =P 0); first exact: oxi_0j.
 have [-> | nzj1] := altP (j1 =P 0); first exact: oxi_i0.
 have ->: xi_ i1 j1 = beta i1 j1 + xi_ i1 0 + xi_ 0 j1 by rewrite /xi_ !ifN.
diff --git a/mathcomp/odd_order/PFsection5.v b/mathcomp/odd_order/PFsection5.v
index d318f5f..3f90da7 100644
--- a/mathcomp/odd_order/PFsection5.v
+++ b/mathcomp/odd_order/PFsection5.v
@@ -492,7 +492,7 @@ Definition subcoherent S tau R :=
       (*c*) pairwise_orthogonal S,
       (*d*) {in S, forall xi : 'CF(L : {set gT}),
               [/\ {subset R xi <= 'Z[irr G]}, orthonormal (R xi)
-                & tau (xi - xi^*)%CF = \sum_(alpha <- R xi) alpha]}
+                & tau (xi - xi^*%CF) = \sum_(alpha <- R xi) alpha]}
     & (*e*) {in S &, forall xi phi : 'CF(L),
               orthogonal phi (xi :: xi^*%CF) -> orthogonal (R phi) (R xi)}].
 
@@ -621,7 +621,7 @@ have isoS1: {in S1, isometry [eta tau with eta1 |-> zeta1], to 'Z[irr G]}.
   split=> [xi eta | eta]; rewrite !in_cons /=; last first.
     by case: eqP => [-> | _  /isoS[/Ztau/zcharW]].
   do 2!case: eqP => [-> _|_  /isoS[? ?]] //; last exact: Itau.
-  by apply/(can_inj conjCK); rewrite -!cfdotC.
+  by apply/(can_inj (@conjCK _)); rewrite -!cfdotC.
 have [nu Dnu IZnu] := Zisometry_of_iso freeS1 isoS1.
 exists nu; split=> // phi; rewrite zcharD1E => /andP[].
 case/(zchar_expansion (free_uniq freeS1)) => b Zb {phi}-> phi1_0.
@@ -646,7 +646,7 @@ have N_S: {subset S <= character} by move=> _ /irrS/irrP[i ->]; apply: irr_char.
 have Z_S: {subset S <= 'Z[irr L]} by move=> chi /N_S/char_vchar.
 have o1S: orthonormal S by apply: sub_orthonormal (irr_orthonormal L).
 have [[_ dotSS] oS] := (orthonormalP o1S, orthonormal_orthogonal o1S).
-pose beta chi := tau (chi - chi^*)%CF; pose eqBP := _ =P beta _.
+pose beta chi := tau (chi - chi^*%CF); pose eqBP := _ =P beta _.
 have Zbeta: {in S, forall chi, chi - (chi^*)%CF \in 'Z[S, L^#]}.
   move=> chi Schi; rewrite /= zcharD1E rpredB ?mem_zchar ?ccS //= !cfunE.
   by rewrite subr_eq0 conj_Cnat // Cnat_char1 ?N_S.
@@ -885,7 +885,7 @@ Lemma subcoherent_norm chi psi (tau1 : {additive 'CF(L) -> 'CF(G)}) X Y :
     [/\ chi \in S, psi \in 'Z[irr L] & orthogonal (chi :: chi^*)%CF psi] ->
     let S0 := chi - psi :: chi - chi^*%CF in
     {in 'Z[S0], isometry tau1, to 'Z[irr G]} ->
-    tau1 (chi - chi^*)%CF = tau (chi - chi^*)%CF ->
+    tau1 (chi - chi^*%CF) = tau (chi - chi^*%CF) ->
     [/\ tau1 (chi - psi) = X - Y, '[X, Y] = 0 & orthogonal Y (R chi)] ->
  [/\ (*a*) '[chi] <= '[X]
    & (*b*) '[psi] <= '[Y] ->
diff --git a/mathcomp/odd_order/PFsection6.v b/mathcomp/odd_order/PFsection6.v
index 6d9ecfc..cbde798 100644
--- a/mathcomp/odd_order/PFsection6.v
+++ b/mathcomp/odd_order/PFsection6.v
@@ -83,13 +83,13 @@ Lemma coherent_seqIndD_bound (A B C D : {group gT}) :
   (*a*) [/\ A \proper K, B \subset D, D \subset C, C \subset K
           & D / B \subset 'Z(C / B)]%g ->
   (*b*) coherent (S A) L^# tau -> \unless coherent (S B) L^# tau,
-  #|K : A|%:R - 1 <= 2%:R * #|L : C|%:R * sqrtC #|C : D|%:R.
+  #|K : A|%:R - 1 <= 2%:R * #|L : C|%:R * sqrtC #|C : D|%:R :> algC.
 Proof.
 move=> [nsAL nsBL nsCL nsDL] [ltAK sBD sDC sCK sDbZC] cohA.
 have sBC := subset_trans sBD sDC; have sBK := subset_trans sBC sCK.
 have [sAK nsBK] := (proper_sub ltAK, normalS sBK sKL nsBL).
 have{sBC} [nsAK nsBC] := (normalS sAK sKL nsAL, normalS sBC sCK nsBK).
-rewrite real_lerNgt ?rpredB ?ger0_real ?mulr_ge0 ?sqrtC_ge0 ?ler0n //.
+rewrite real_lerNgt ?rpredB ?ger0_real ?mulr_ge0 ?sqrtC_ge0 ?ler0n ?ler01 //.
 apply/unless_contra; rewrite negbK -(Lagrange_index sKL sCK) natrM => lb_KA.
 pose S2 : seq 'CF(L) := [::]; pose S1 := S2 ++ S A; rewrite -[S A]/S1 in cohA.
 have ccsS1S: cfConjC_subset S1 calS by apply: seqInd_conjC_subset1.
@@ -153,7 +153,7 @@ have sAbZH: (A / B \subset 'Z(H / B))%g.
   by apply: homg_quotientS; rewrite ?(subset_trans sHL) ?normal_norm.
 have ltAH: A \proper H.
   by rewrite properEneq sAH (contraTneq _ lbHA) // => ->; rewrite indexgg addn1.
-set x := sqrtC #|H : A|%:R.
+set x : algC := sqrtC #|H : A|%:R.
 have [nz_x x_gt0]: x != 0 /\ 0 < x by rewrite gtr_eqF sqrtC_gt0 gt0CiG.
 without loss{cohA} ubKA: / #|K : A|%:R - 1 <= 2%:R * #|L : H|%:R * x.
   have [sAK ltAK] := (subset_trans sAH sHK, proper_sub_trans ltAH sHK).
diff --git a/mathcomp/odd_order/PFsection7.v b/mathcomp/odd_order/PFsection7.v
index cea9319..4610829 100644
--- a/mathcomp/odd_order/PFsection7.v
+++ b/mathcomp/odd_order/PFsection7.v
@@ -324,7 +324,7 @@ transitivity (\sum_(x in A) \sum_(xi <- S) \sum_(mu <- S) F xi mu x).
   apply: eq_bigr => x Ax; rewrite part_a // sum_cfunE -mulrA mulr_suml.
   apply: eq_bigr => xi _; rewrite mulrA -mulr_suml rmorph_sum; congr (_ * _).
   rewrite mulr_sumr; apply: eq_bigr => mu _; rewrite !cfunE (cfdotC mu).
-  rewrite -{1}[mu x]conjCK -fmorph_div -rmorphM conjCK -4!mulrA 2!(mulrCA _^-1).
+  rewrite -{1}[mu x]conjCK -fmorph_div -rmorphM conjCK -3!mulrA 2!(mulrCA _^-1).
   by rewrite (mulrA _^-1) -invfM 2!(mulrCA (xi x)) mulrA 2!(mulrA _^*).
 rewrite exchange_big; apply: eq_bigr => xi _; rewrite exchange_big /=.
 apply: eq_big_seq => mu Smu; have Tmu := sST mu Smu.