diff options
Diffstat (limited to 'mathcomp/character/integral_char.v')
| -rw-r--r-- | mathcomp/character/integral_char.v | 28 |
1 files changed, 14 insertions, 14 deletions
diff --git a/mathcomp/character/integral_char.v b/mathcomp/character/integral_char.v index 22bd171..1022afa 100644 --- a/mathcomp/character/integral_char.v +++ b/mathcomp/character/integral_char.v @@ -1,7 +1,7 @@ (* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path. -From mathcomp Require Import div choice fintype tuple finfun bigop prime. +From mathcomp Require Import div choice fintype tuple finfun bigop prime order. From mathcomp Require Import ssralg poly finset fingroup morphism perm. From mathcomp Require Import automorphism quotient action finalg zmodp. From mathcomp Require Import commutator cyclic center pgroup sylow gseries. @@ -34,7 +34,7 @@ Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. -Import GroupScope GRing.Theory Num.Theory. +Import Order.TTheory GroupScope GRing.Theory Num.Theory. Local Open Scope ring_scope. Lemma group_num_field_exists (gT : finGroupType) (G : {group gT}) : @@ -274,7 +274,7 @@ have nz_m: m%:R != 0 :> algC by rewrite pnatr_eq0 -lt0n. pose alpha := 'chi_i g / m%:R. have a_lt1: `|alpha| < 1. rewrite normrM normfV normr_nat -{2}(divff nz_m). - rewrite ltr_def (can_eq (mulfVK nz_m)) eq_sym -{1}Dm -irr_cfcenterE // notZg. + rewrite lt_def (can_eq (mulfVK nz_m)) eq_sym -{1}Dm -irr_cfcenterE // notZg. by rewrite ler_pmul2r ?invr_gt0 ?ltr0n // -Dm char1_ge_norm ?irr_char. have Za: alpha \in Aint. have [u _ /dvdnP[v eq_uv]] := Bezoutl #|g ^: G| m_gt0. @@ -304,13 +304,13 @@ have Zbeta: beta \in Cint. have [|nz_a] := boolP (alpha == 0). by rewrite (can2_eq (divfK _) (mulfK _)) // mul0r => /eqP. have: beta != 0 by rewrite Dbeta; apply/prodf_neq0 => nu _; rewrite fmorph_eq0. -move/(norm_Cint_ge1 Zbeta); rewrite ltr_geF //; apply: ler_lt_trans a_lt1. +move/(norm_Cint_ge1 Zbeta); rewrite lt_geF //; apply: le_lt_trans a_lt1. rewrite -[`|alpha|]mulr1 Dbeta (bigD1 1%g) ?group1 //= -Da. -case: (gQnC _) => /= _ <-; rewrite gal_id normrM. -rewrite -subr_ge0 -mulrBr mulr_ge0 ?normr_ge0 // Da subr_ge0. -elim/big_rec: _ => [|nu c _]; first by rewrite normr1 lerr. -apply: ler_trans; rewrite -subr_ge0 -{1}[`|c|]mul1r normrM -mulrBl. -by rewrite mulr_ge0 ?normr_ge0 // subr_ge0 norm_a_nu. +case: (gQnC _) => /= _ <-. +rewrite gal_id normrM -subr_ge0 -mulrBr mulr_ge0 // Da subr_ge0. +elim/big_rec: _ => [|nu c _]; first by rewrite normr1 lexx. +apply: le_trans; rewrite -subr_ge0 -{1}[`|c|]mul1r normrM -mulrBl. +by rewrite mulr_ge0 // subr_ge0 norm_a_nu. Qed. End GringIrrMode. @@ -677,18 +677,18 @@ have{pi1 Zpi1} pi2_ge1: 1 <= pi2. by rewrite Cint_normK // sqr_Cint_ge1 //; apply/prodf_neq0. have Sgt0: (#|S| > 0)%N by rewrite (cardD1 g) [g \in S]Sg. rewrite -mulr_natr -ler_pdivl_mulr ?ltr0n //. -have n2chi_ge0 s: s \in S -> 0 <= `|chi s| ^+ 2 by rewrite exprn_ge0 ?normr_ge0. +have n2chi_ge0 s: s \in S -> 0 <= `|chi s| ^+ 2 by rewrite exprn_ge0. rewrite -(expr_ge1 Sgt0); last by rewrite divr_ge0 ?ler0n ?sumr_ge0. -by rewrite (ler_trans pi2_ge1) // lerif_AGM. +by rewrite (le_trans pi2_ge1) // leif_AGM. Qed. (* This is Burnside's vanishing theorem (Isaacs, Theorem (3.15)). *) Theorem nonlinear_irr_vanish gT (G : {group gT}) i : 'chi[G]_i 1%g > 1 -> exists2 x, x \in G & 'chi_i x = 0. Proof. -move=> chi1gt1; apply/exists_eq_inP; apply: contraFT (ltr_geF chi1gt1). -move/exists_inPn => -nz_chi. -rewrite -(norm_Cnat (Cnat_irr1 i)) -(@expr_le1 _ 2) ?normr_ge0 //. +move=> chi1gt1; apply/exists_eq_inP; apply: contraFT (lt_geF chi1gt1). +move=> /exists_inPn-nz_chi. +rewrite -(norm_Cnat (Cnat_irr1 i)) -(@expr_le1 _ 2)//. rewrite -(ler_add2r (#|G|%:R * '['chi_i])) {1}cfnorm_irr mulr1. rewrite (cfnormE (cfun_onG _)) mulVKf ?neq0CG // (big_setD1 1%g) //=. rewrite addrCA ler_add2l (cardsD1 1%g) group1 mulrS ler_add2l. |