6767move=> p un_p; rewrite ideal_eq1P 1 :&(idI) // ; apply/negP.
6868move/fun_ext => /(_ poly1); rewrite mem_idT /I mem_idgen1.
6969case /eqT=> q /(congr1 deg); rewrite deg1 => /eq_sym.
70- case : (q = poly0) => [->|nz_q]; first by rewrite /( * ) mul0r deg0.
70+ case : (q = poly0) => [->|nz_q]; first by rewrite mul0r deg0.
7171rewrite degM_proper; 1 : by rewrite lc_polyXnDC // mulr1 lc_eq0.
7272rewrite deg_polyXnDC // -!addrA /= gtr_eqF //.
7373by rewrite (_ : 1 = 1 + 0 ) 1:// ler_lt_add // deg_ge1.
@@ -126,7 +126,7 @@ proof. by rewrite /( ** ) !scalepE -2!mulE reprK. qed.
126126
127127(* -------------------------------------------------------------------- *)
128128lemma eqv_Xn : eqv (exp X n) (-poly1).
129- proof. rewrite eqv_sym /eqv /[-] opprK /I mem_idgen1_gen. qed.
129+ proof. rewrite eqv_sym /eqv opprK /I mem_idgen1_gen. qed.
130130
131131(* -------------------------------------------------------------------- *)
132132op reduce (p : poly) : poly =
@@ -165,7 +165,7 @@ rewrite {1}(polyE p) /reduce !PCA.big_seq &(eqv_sum) /=.
165165move=> i /mem_range [rg_i _]; rewrite mulrC -scalepA.
166166pose q := exp X (i %/ n * n) * exp X (i %% n).
167167rewrite !scalepE &(eqvMl) &(eqv_trans q).
168- - rewrite (_ : q = exp X i) // /q /( * ) -exprD_nneg // -?divz_eq //.
168+ - rewrite (_ : q = exp X i) // /q -exprD_nneg // -?divz_eq //.
169169 - by rewrite mulr_ge0 // divz_ge0.
170170 - by rewrite modz_ge0 gtr_eqF.
171171apply/eqvMr; rewrite -polyCX ?divz_ge0 // (IntID.mulrC _ n).
@@ -192,7 +192,7 @@ split=> [eq_rd|eqv_pq].
192192have @/eqv @/I: eqv (reduce p) (reduce q).
193193- by rewrite !(eqv_reducel, eqv_reducer).
194194case/mem_idgen1=> r /(congr1 deg); case : (r = poly0) => [->>|nz_r].
195- - by rewrite /( * ) mul0r deg0 deg_eq0 subr_eq0 => ->.
195+ - by rewrite mul0r deg0 deg_eq0 subr_eq0 => ->.
196196have: deg (reduce q - reduce p) <= n.
197197- by rewrite &(ler_trans _ _ _ (degB _ _)) ler_maxrP !deg_reduce.
198198rewrite degM_proper.
@@ -411,8 +411,8 @@ rewrite polyDE; pose c := (bigi _ _ _ _); have ->: c.[k] = Coeff.zeror.
411411 move=> i [/mem_range [ge0_i lt_in] @/predC1 ne_ik].
412412 by rewrite polyZE polyXnE // (eq_sym k i) ne_ik /= mulr0.
413413rewrite addr0 polyZE polyXnE // = mulr1 (reducewE (2 * n)).
414- - case : (p = poly0) => [->|nz_p]; first by rewrite /( * ) mul0r deg0 /#.
415- case : (q = poly0) => [->|nz_q]; first by rewrite /( * ) mulr0 deg0 /#.
414+ - case : (p = poly0) => [->|nz_p]; first by rewrite mul0r deg0 /#.
415+ case : (q = poly0) => [->|nz_q]; first by rewrite mulr0 deg0 /#.
416416 apply: (ler_trans (deg p + deg q)); last first.
417417 - by rewrite (_ : 2 * n = n + n) 1:#ring ler_add &(deg_reduced).
418418 by rewrite &(ler_trans (deg (p * q) + 1 )) ?ler_addl // &(degM_le).
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