diff options
Diffstat (limited to 'mathcomp/field/separable.v')
| -rw-r--r-- | mathcomp/field/separable.v | 13 |
1 files changed, 6 insertions, 7 deletions
diff --git a/mathcomp/field/separable.v b/mathcomp/field/separable.v index 6320343..7e208cd 100644 --- a/mathcomp/field/separable.v +++ b/mathcomp/field/separable.v @@ -288,8 +288,8 @@ Variables (K : {vspace L}) (D : 'End(L)). (* A deriviation only needs to be additive and satify Lebniz's law, but all *) (* the deriviations used here are going to be linear, so we only define *) (* the Derivation predicate for linear endomorphisms. *) -Definition Derivation (s := vbasis K) : bool := - all (fun u => all (fun v => D (u * v) == D u * v + u * D v) s) s. +Definition Derivation : bool := + all1rel (fun u v => D (u * v) == D u * v + u * D v) (vbasis K). Hypothesis derD : Derivation. @@ -299,7 +299,7 @@ move=> u v /coord_vbasis-> /coord_vbasis->. rewrite !(mulr_sumr, linear_sum) -big_split; apply: eq_bigr => /= j _. rewrite !mulr_suml linear_sum -big_split; apply: eq_bigr => /= i _. rewrite !(=^~ scalerAl, linearZZ) -!scalerAr linearZZ -!scalerDr !scalerA /=. -by congr (_ *: _); apply/eqP; rewrite (allP (allP derD _ _)) ?memt_nth. +by congr (_ *: _); apply/eqP/(allrelP derD); exact: memt_nth. Qed. Lemma Derivation_mul_poly (Dp := map_poly D) : @@ -314,7 +314,7 @@ End Derivation. Lemma DerivationS E K D : (K <= E)%VS -> Derivation E D -> Derivation K D. Proof. -move/subvP=> sKE derD; apply/allP=> x Kx; apply/allP=> y Ky; apply/eqP. +move/subvP=> sKE derD; apply/allrelP=> x y Kx Ky; apply/eqP. by rewrite (Derivation_mul derD) ?sKE // vbasis_mem. Qed. @@ -492,8 +492,7 @@ Qed. Lemma extendDerivationP : separable_element K x -> Derivation <<K; x>> (extendDerivation K). Proof. -move=> sep; apply/allP=> u /vbasis_mem Hu; apply/allP=> v /vbasis_mem Hv. -apply/eqP. +move=> sep; apply/allrelP=> u v /vbasis_mem Hu /vbasis_mem Hv; apply/eqP. rewrite -(Fadjoin_poly_eq Hu) -(Fadjoin_poly_eq Hv) -hornerM. rewrite !{1}extendDerivation_horner ?{1}rpredM ?Fadjoin_polyOver //. rewrite (Derivation_mul_poly derD) ?Fadjoin_polyOver //. @@ -528,7 +527,7 @@ have DK_0: (K <= lker D)%VS. apply/subvP=> v Kv; rewrite memv_ker lfunE /= Fadjoin_polyC //. by rewrite derivC horner0. have Dder: Derivation <<K; x>> D. - apply/allP=> u /vbasis_mem Kx_u; apply/allP=> v /vbasis_mem Kx_v; apply/eqP. + apply/allrelP=> u v /vbasis_mem Kx_u /vbasis_mem Kx_v; apply/eqP. rewrite !lfunE /=; set Px := Fadjoin_poly K x. set Px_u := Px u; rewrite -(Fadjoin_poly_eq Kx_u) -/Px -/Px_u. set Px_v := Px v; rewrite -(Fadjoin_poly_eq Kx_v) -/Px -/Px_v. |