fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
ltf_nV2: {in neg &, {mono (@GRing.inv F) : x y /~ x < y}}.
Proof. exact: leW_nmono_in lef_nV2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltf_nV2
| |
ltef_pV2:= (lef_pV2, ltf_pV2).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltef_pV2
| |
ltef_nV2:= (lef_nV2, ltf_nV2).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltef_nV2
| |
invf_pgt: {in pos &, forall x y, (x < y^-1) = (y < x^-1)}.
Proof. by move=> x y *; rewrite -[x in LHS]invrK ltf_pV2// posrE invr_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_pgt
| |
invf_pge: {in pos &, forall x y, (x <= y^-1) = (y <= x^-1)}.
Proof. by move=> x y *; rewrite -[x in LHS]invrK lef_pV2// posrE invr_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_pge
| |
invf_ngt: {in neg &, forall x y, (x < y^-1) = (y < x^-1)}.
Proof. by move=> x y *; rewrite -[x in LHS]invrK ltf_nV2// negrE invr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_ngt
| |
invf_nge: {in neg &, forall x y, (x <= y^-1) = (y <= x^-1)}.
Proof. by move=> x y *; rewrite -[x in LHS]invrK lef_nV2// negrE invr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_nge
| |
invf_gt1x : 0 < x -> (1 < x^-1) = (x < 1).
Proof. by move=> x0; rewrite invf_pgt ?invr1 ?posrE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_gt1
| |
invf_ge1x : 0 < x -> (1 <= x^-1) = (x <= 1).
Proof. by move=> x0; rewrite invf_pge ?invr1 ?posrE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_ge1
| |
invf_gte1:= (invf_ge1, invf_gt1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_gte1
| |
invf_plt: {in pos &, forall x y, (x^-1 < y) = (y^-1 < x)}.
Proof. by move=> x y *; rewrite -[y in LHS]invrK ltf_pV2// posrE invr_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_plt
| |
invf_ple: {in pos &, forall x y, (x^-1 <= y) = (y^-1 <= x)}.
Proof. by move=> x y *; rewrite -[y in LHS]invrK lef_pV2// posrE invr_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_ple
| |
invf_nlt: {in neg &, forall x y, (x^-1 < y) = (y^-1 < x)}.
Proof. by move=> x y *; rewrite -[y in LHS]invrK ltf_nV2// negrE invr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_nlt
| |
invf_nle: {in neg &, forall x y, (x^-1 <= y) = (y^-1 <= x)}.
Proof. by move=> x y *; rewrite -[y in LHS]invrK lef_nV2// negrE invr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_nle
| |
invf_le1x : 0 < x -> (x^-1 <= 1) = (1 <= x).
Proof. by move=> x0; rewrite -invf_ple ?invr1 ?posrE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_le1
| |
invf_lt1x : 0 < x -> (x^-1 < 1) = (1 < x).
Proof. by move=> x0; rewrite invf_plt ?invr1 ?posrE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_lt1
| |
invf_lte1:= (invf_le1, invf_lt1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_lte1
| |
invf_cp1:= (invf_gte1, invf_lte1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invf_cp1
| |
ler_pdivlMrz x y : 0 < z -> (x <= y / z) = (x * z <= y).
Proof. by move=> z_gt0; rewrite -(@ler_pM2r _ z _ x) ?mulfVK ?gt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_pdivlMr
| |
ltr_pdivlMrz x y : 0 < z -> (x < y / z) = (x * z < y).
Proof. by move=> z_gt0; rewrite -(@ltr_pM2r _ z _ x) ?mulfVK ?gt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_pdivlMr
| |
lter_pdivlMr:= (ler_pdivlMr, ltr_pdivlMr).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_pdivlMr
| |
ler_pdivrMrz x y : 0 < z -> (y / z <= x) = (y <= x * z).
Proof. by move=> z_gt0; rewrite -(@ler_pM2r _ z) ?mulfVK ?gt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_pdivrMr
| |
ltr_pdivrMrz x y : 0 < z -> (y / z < x) = (y < x * z).
Proof. by move=> z_gt0; rewrite -(@ltr_pM2r _ z) ?mulfVK ?gt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_pdivrMr
| |
lter_pdivrMr:= (ler_pdivrMr, ltr_pdivrMr).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_pdivrMr
| |
ler_pdivlMlz x y : 0 < z -> (x <= z^-1 * y) = (z * x <= y).
Proof. by move=> z_gt0; rewrite mulrC ler_pdivlMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_pdivlMl
| |
ltr_pdivlMlz x y : 0 < z -> (x < z^-1 * y) = (z * x < y).
Proof. by move=> z_gt0; rewrite mulrC ltr_pdivlMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_pdivlMl
| |
lter_pdivlMl:= (ler_pdivlMl, ltr_pdivlMl).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_pdivlMl
| |
ler_pdivrMlz x y : 0 < z -> (z^-1 * y <= x) = (y <= z * x).
Proof. by move=> z_gt0; rewrite mulrC ler_pdivrMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_pdivrMl
| |
ltr_pdivrMlz x y : 0 < z -> (z^-1 * y < x) = (y < z * x).
Proof. by move=> z_gt0; rewrite mulrC ltr_pdivrMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_pdivrMl
| |
lter_pdivrMl:= (ler_pdivrMl, ltr_pdivrMl).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_pdivrMl
| |
ler_ndivlMrz x y : z < 0 -> (x <= y / z) = (y <= x * z).
Proof. by move=> z_lt0; rewrite -(@ler_nM2r _ z) ?mulfVK ?lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_ndivlMr
| |
ltr_ndivlMrz x y : z < 0 -> (x < y / z) = (y < x * z).
Proof. by move=> z_lt0; rewrite -(@ltr_nM2r _ z) ?mulfVK ?lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_ndivlMr
| |
lter_ndivlMr:= (ler_ndivlMr, ltr_ndivlMr).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_ndivlMr
| |
ler_ndivrMrz x y : z < 0 -> (y / z <= x) = (x * z <= y).
Proof. by move=> z_lt0; rewrite -(@ler_nM2r _ z) ?mulfVK ?lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_ndivrMr
| |
ltr_ndivrMrz x y : z < 0 -> (y / z < x) = (x * z < y).
Proof. by move=> z_lt0; rewrite -(@ltr_nM2r _ z) ?mulfVK ?lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_ndivrMr
| |
lter_ndivrMr:= (ler_ndivrMr, ltr_ndivrMr).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_ndivrMr
| |
ler_ndivlMlz x y : z < 0 -> (x <= z^-1 * y) = (y <= z * x).
Proof. by move=> z_lt0; rewrite mulrC ler_ndivlMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_ndivlMl
| |
ltr_ndivlMlz x y : z < 0 -> (x < z^-1 * y) = (y < z * x).
Proof. by move=> z_lt0; rewrite mulrC ltr_ndivlMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_ndivlMl
| |
lter_ndivlMl:= (ler_ndivlMl, ltr_ndivlMl).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_ndivlMl
| |
ler_ndivrMlz x y : z < 0 -> (z^-1 * y <= x) = (z * x <= y).
Proof. by move=> z_lt0; rewrite mulrC ler_ndivrMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_ndivrMl
| |
ltr_ndivrMlz x y : z < 0 -> (z^-1 * y < x) = (z * x < y).
Proof. by move=> z_lt0; rewrite mulrC ltr_ndivrMr ?[z * _]mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_ndivrMl
| |
lter_ndivrMl:= (ler_ndivrMl, ltr_ndivrMl).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lter_ndivrMl
| |
natf_divm d : (d %| m)%N -> (m %/ d)%:R = m%:R / d%:R :> F.
Proof. by apply: pchar0_natf_div; apply: (@pchar_num F). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
natf_div
| |
normfV: {morph (norm : F -> F) : x / x ^-1}.
Proof.
move=> x /=; have [/normrV //|Nux] := boolP (x \is a GRing.unit).
by rewrite !invr_out // unitfE normr_eq0 -unitfE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
normfV
| |
normf_div: {morph (norm : F -> F) : x y / x / y}.
Proof. by move=> x y /=; rewrite normrM normfV. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
normf_div
| |
invr_sgx : (sg x)^-1 = sgr x.
Proof. by rewrite !(fun_if GRing.inv) !(invr0, invrN, invr1). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
invr_sg
| |
sgrVx : sgr x^-1 = sgr x.
Proof. by rewrite /sgr invr_eq0 invr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sgrV
| |
splitrx : x = x / 2%:R + x / 2%:R.
Proof. by rewrite -mulr2n -[RHS]mulr_natr mulfVK //= pnatr_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
splitr
| |
lteif_pdivlMrC z x y :
0 < z -> x < y / z ?<= if C = (x * z < y ?<= if C).
Proof. by case: C => ? /=; rewrite lter_pdivlMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_pdivlMr
| |
lteif_pdivrMrC z x y :
0 < z -> y / z < x ?<= if C = (y < x * z ?<= if C).
Proof. by case: C => ? /=; rewrite lter_pdivrMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_pdivrMr
| |
lteif_pdivlMlC z x y :
0 < z -> x < z^-1 * y ?<= if C = (z * x < y ?<= if C).
Proof. by case: C => ? /=; rewrite lter_pdivlMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_pdivlMl
| |
lteif_pdivrMlC z x y :
0 < z -> z^-1 * y < x ?<= if C = (y < z * x ?<= if C).
Proof. by case: C => ? /=; rewrite lter_pdivrMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_pdivrMl
| |
lteif_ndivlMrC z x y :
z < 0 -> x < y / z ?<= if C = (y < x * z ?<= if C).
Proof. by case: C => ? /=; rewrite lter_ndivlMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_ndivlMr
| |
lteif_ndivrMrC z x y :
z < 0 -> y / z < x ?<= if C = (x * z < y ?<= if C).
Proof. by case: C => ? /=; rewrite lter_ndivrMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_ndivrMr
| |
lteif_ndivlMlC z x y :
z < 0 -> x < z^-1 * y ?<= if C = (y < z * x ?<= if C).
Proof. by case: C => ? /=; rewrite lter_ndivlMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_ndivlMl
| |
lteif_ndivrMlC z x y :
z < 0 -> z^-1 * y < x ?<= if C = (z * x < y ?<= if C).
Proof. by case: C => ? /=; rewrite lter_ndivrMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lteif_ndivrMl
| |
midf_lex y : x <= y -> (x <= mid x y) * (mid x y <= y).
Proof.
move=> lexy; rewrite ler_pdivlMr ?ler_pdivrMr ?ltr0Sn //.
by rewrite !mulrDr !mulr1 !lerD2.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
midf_le
| |
midf_ltx y : x < y -> (x < mid x y) * (mid x y < y).
Proof.
move=> ltxy; rewrite ltr_pdivlMr ?ltr_pdivrMr ?ltr0Sn //.
by rewrite !mulrDr !mulr1 !ltrD2.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
midf_lt
| |
midf_lte:= (midf_le, midf_lt).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
midf_lte
| |
ler_addgt0Prx y : reflect (forall e, e > 0 -> x <= y + e) (x <= y).
Proof.
apply/(iffP idP)=> [lexy e e_gt0 | lexye]; first by rewrite ler_wpDr// ltW.
have [||ltyx]// := comparable_leP.
rewrite (@comparabler_trans _ (y + 1))// /Order.comparable ?lexye ?ltr01//.
by rewrite lerDl ler01 orbT.
have /midf_lt [_] := ltyx; rewrite le_gtF//.
rewrite -(@addrK _ y y) (addrAC _ _ x) -addrA 2!mulrDl -splitr lexye//.
by rewrite divr_gt0// ?ltr0n// subr_gt0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_addgt0Pr
| |
ler_addgt0Plx y : reflect (forall e, e > 0 -> x <= e + y) (x <= y).
Proof.
by apply/(equivP (ler_addgt0Pr x y)); split=> lexy e /lexy; rewrite addrC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_addgt0Pl
| |
lt_lea b : (forall x, x < a -> x < b) -> a <= b.
Proof.
move=> ab; apply/ler_addgt0Pr => e e_gt0; rewrite -lerBDr ltW//.
by rewrite ab// ltrBlDr ltrDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lt_le
| |
gt_gea b : (forall x, b < x -> a < x) -> a <= b.
Proof.
by move=> ab; apply/ler_addgt0Pr => e e_gt0; rewrite ltW// ab// ltrDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
gt_ge
| |
real_leif_mean_squarex y :
x \is real -> y \is real -> x * y <= mid (x ^+ 2) (y ^+ 2) ?= iff (x == y).
Proof.
move=> Rx Ry; rewrite -(mono_leif (ler_pM2r (ltr_nat F 0 2))).
by rewrite divfK ?pnatr_eq0 // mulr_natr; apply: real_leif_mean_square_scaled.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
real_leif_mean_square
| |
real_leif_AGM2x y :
x \is real -> y \is real -> x * y <= mid x y ^+ 2 ?= iff (x == y).
Proof.
move=> Rx Ry; rewrite -(mono_leif (ler_pM2r (ltr_nat F 0 4))).
rewrite mulr_natr (natrX F 2 2) -exprMn divfK ?pnatr_eq0 //.
exact: real_leif_AGM2_scaled.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
real_leif_AGM2
| |
leif_AGM(I : finType) (A : {pred I}) (E : I -> F) :
let n := #|A| in let mu := (\sum_(i in A) E i) / n%:R in
{in A, forall i, 0 <= E i} ->
\prod_(i in A) E i <= mu ^+ n
?= iff [forall i in A, forall j in A, E i == E j].
Proof.
move=> n mu Ege0; have [n0 | n_gt0] := posnP n.
by rewrite n0 -big_andE !(big_pred0 _ _ _ _ (card0_eq n0)); apply/leifP.
pose E' i := E i / n%:R.
have defE' i: E' i *+ n = E i by rewrite -mulr_natr divfK ?pnatr_eq0 -?lt0n.
have /leif_AGM_scaled (i): i \in A -> 0 <= E' i *+ n by rewrite defE' => /Ege0.
rewrite -/n -mulr_suml (eq_bigr _ (in1W defE')); congr (_ <= _ ?= iff _).
by do 2![apply: eq_forallb_in => ? _]; rewrite -(eqr_pMn2r n_gt0) !defE'.
Qed.
Implicit Type p : {poly F}.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
leif_AGM
| |
Cauchy_root_boundp : p != 0 -> {b | forall x, root p x -> `|x| <= b}.
Proof.
move=> nz_p; set a := lead_coef p; set n := (size p).-1.
have [q Dp]: {q | forall x, x != 0 -> p.[x] = (a - q.[x^-1] / x) * x ^+ n}.
exists (- \poly_(i < n) p`_(n - i.+1)) => x nz_x.
rewrite hornerN mulNr opprK horner_poly mulrDl !mulr_suml addrC.
rewrite horner_coef polySpred // big_ord_recr (reindex_inj rev_ord_inj) /=.
rewrite -/n -lead_coefE; congr (_ + _); apply: eq_bigr=> i _.
by rewrite exprB ?unitfE // -exprVn mulrA mulrAC exprSr mulrA.
have [b ub_q] := poly_disk_bound q 1; exists (b / `|a| + 1) => x px0.
have b_ge0: 0 <= b by rewrite (le_trans (normr_ge0 q.[1])) ?ub_q ?normr1.
have{b_ge0} ba_ge0: 0 <= b / `|a| by rewrite divr_ge0.
rewrite real_leNgt ?rpredD ?rpred1 ?ger0_real //.
apply: contraL px0 => lb_x; rewrite rootE.
have x_ge1: 1 <= `|x| by rewrite (le_trans _ (ltW lb_x)) // ler_wpDl.
have nz_x: x != 0 by rewrite -normr_gt0 (lt_le_trans ltr01).
rewrite {}Dp // mulf_neq0 ?expf_neq0 // subr_eq0 eq_sym.
have: (b / `|a|) < `|x| by rewrite (lt_trans _ lb_x) // ltr_pwDr ?ltr01.
apply: contraTneq => /(canRL (divfK nz_x))Dax.
rewrite ltr_pdivrMr ?normr_gt0 ?lead_coef_eq0 // mulrC -normrM -{}Dax.
by rewrite le_gtF // ub_q // normfV invf_le1 ?normr_gt0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
Cauchy_root_bound
| |
natf_indexg(gT : finGroupType) (G H : {group gT}) :
H \subset G -> #|G : H|%:R = (#|G|%:R / #|H|%:R)%R :> F.
Proof. by move=> sHG; rewrite -divgS // natf_div ?cardSg. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
natf_indexg
| |
leif_mean_squarex y : x * y <= (x ^+ 2 + y ^+ 2) / 2 ?= iff (x == y).
Proof. by apply: real_leif_mean_square; apply: num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
leif_mean_square
| |
leif_AGM2x y : x * y <= ((x + y) / 2)^+ 2 ?= iff (x == y).
Proof. by apply: real_leif_AGM2; apply: num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
leif_AGM2
| |
maxr_absEx y : Num.max x y = (x + y + `|x - y|) / 2.
Proof.
apply: canRL (mulfK _) _ => //; rewrite ?pnatr_eq0//.
case: lerP => _; rewrite [2]mulr2n mulrDr mulr1.
by rewrite addrCA addrK.
by rewrite addrCA addrAC subrr add0r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
maxr_absE
| |
minr_absEx y : Num.min x y = (x + y - `|x - y|) / 2.
Proof.
apply: (addrI (Num.max x y)); rewrite addr_max_min maxr_absE.
by rewrite -mulrDl addrCA addrK mulrDl -splitr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
minr_absE
| |
poly_ivt: real_closed_axiom R. Proof. exact: poly_ivt. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
poly_ivt
| |
sqrtr_ge0a : 0 <= sqrt a.
Proof. by rewrite /sqrt; case: (sig2W _). Qed.
Hint Resolve sqrtr_ge0 : core.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr_ge0
| |
sqr_sqrtra : 0 <= a -> sqrt a ^+ 2 = a.
Proof.
by rewrite /sqrt => a_ge0; case: (sig2W _) => /= x _; rewrite a_ge0 => /eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqr_sqrtr
| |
ler0_sqrtra : a <= 0 -> sqrt a = 0.
Proof.
rewrite /sqrtr; case: (sig2W _) => x /= _.
by have [//|_ /eqP//|->] := ltrgt0P a; rewrite mulf_eq0 orbb => /eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler0_sqrtr
| |
ltr0_sqrtra : a < 0 -> sqrt a = 0.
Proof. by move=> /ltW; apply: ler0_sqrtr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr0_sqrtr
| |
sqrtr_speca : R -> bool -> bool -> R -> Type :=
| IsNoSqrtr of a < 0 : sqrtr_spec a a false true 0
| IsSqrtr b of 0 <= b : sqrtr_spec a (b ^+ 2) true false b.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr_spec
| |
sqrtrPa : sqrtr_spec a a (0 <= a) (a < 0) (sqrt a).
Proof.
have [a_ge0|a_lt0] := ger0P a.
by rewrite -{1 2}[a]sqr_sqrtr //; constructor.
by rewrite ltr0_sqrtr //; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtrP
| |
sqrtr_sqra : sqrt (a ^+ 2) = `|a|.
Proof.
have /eqP : sqrt (a ^+ 2) ^+ 2 = `|a| ^+ 2.
by rewrite -normrX ger0_norm ?sqr_sqrtr ?sqr_ge0.
rewrite eqf_sqr => /predU1P[-> //|ha].
have := sqrtr_ge0 (a ^+ 2); rewrite (eqP ha) oppr_ge0 normr_le0 => /eqP ->.
by rewrite normr0 oppr0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr_sqr
| |
sqrtrMa b : 0 <= a -> sqrt (a * b) = sqrt a * sqrt b.
Proof.
case: (sqrtrP a) => // {}a a_ge0 _; case: (sqrtrP b) => [b_lt0 | {}b b_ge0].
by rewrite mulr0 ler0_sqrtr // nmulr_lle0 ?mulr_ge0.
by rewrite mulrACA sqrtr_sqr ger0_norm ?mulr_ge0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtrM
| |
sqrtr0: sqrt 0 = 0 :> R.
Proof. by move: (sqrtr_sqr 0); rewrite exprS mul0r => ->; rewrite normr0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr0
| |
sqrtr1: sqrt 1 = 1 :> R.
Proof. by move: (sqrtr_sqr 1); rewrite expr1n => ->; rewrite normr1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr1
| |
sqrtr_eq0a : (sqrt a == 0) = (a <= 0).
Proof.
case: sqrtrP => [/ltW ->|b]; first by rewrite eqxx.
case: ltrgt0P => [b_gt0|//|->]; last by rewrite exprS mul0r lexx.
by rewrite lt_geF ?pmulr_rgt0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr_eq0
| |
sqrtr_gt0a : (0 < sqrt a) = (0 < a).
Proof. by rewrite lt0r sqrtr_ge0 sqrtr_eq0 -ltNge andbT. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr_gt0
| |
eqr_sqrta b : 0 <= a -> 0 <= b -> (sqrt a == sqrt b) = (a == b).
Proof.
move=> a_ge0 b_ge0; apply/eqP/eqP=> [HS|->] //.
by move: (sqr_sqrtr a_ge0); rewrite HS (sqr_sqrtr b_ge0).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
eqr_sqrt
| |
ler_wsqrtr: {homo @sqrt R : a b / a <= b}.
Proof.
move=> a b /= le_ab; case: (boolP (0 <= a))=> [pa|]; last first.
by rewrite -ltNge; move/ltW; rewrite -sqrtr_eq0; move/eqP->.
rewrite -(@ler_pXn2r R 2) ?nnegrE ?sqrtr_ge0 //.
by rewrite !sqr_sqrtr // (le_trans pa).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_wsqrtr
| |
ler_psqrt: {in @nneg R &, {mono sqrt : a b / a <= b}}.
Proof.
apply: le_mono_in => x y x_gt0 y_gt0.
rewrite !lt_neqAle => /andP[neq_xy le_xy].
by rewrite ler_wsqrtr // eqr_sqrt // neq_xy.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_psqrt
| |
ler_sqrta b : 0 <= b -> (sqrt a <= sqrt b) = (a <= b).
Proof.
move=> b_ge0; have [a_le0|a_gt0] := ler0P a; last first.
by rewrite ler_psqrt // nnegrE ltW.
by rewrite ler0_sqrtr // sqrtr_ge0 (le_trans a_le0).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ler_sqrt
| |
ltr_sqrta b : 0 < b -> (sqrt a < sqrt b) = (a < b).
Proof.
move=> b_gt0; have [a_le0|a_gt0] := ler0P a; last first.
by rewrite (leW_mono_in ler_psqrt)//; apply: ltW.
by rewrite ler0_sqrtr // sqrtr_gt0 b_gt0 (le_lt_trans a_le0).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltr_sqrt
| |
sqrtrVx : 0 <= x -> sqrt (x^-1) = (sqrt x)^-1.
Proof.
case: ltrgt0P => // [x_gt0 _|->]; last by rewrite !(invr0, sqrtr0).
have sx_neq0 : sqrt x != 0 by rewrite sqrtr_eq0 -ltNge.
apply: (mulfI sx_neq0).
by rewrite -sqrtrM !(divff, ltW, sqrtr1) // lt0r_neq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtrV
| |
normCK: forall x, `|x| ^+ 2 = x * x^* := normCK_subdef.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
normCK
| |
sqrCi: 'i ^+ 2 = -1 :> C := sqrCi.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrCi
| |
mulCii: 'i * 'i = -1 :> C. Proof. exact: sqrCi. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
mulCii
| |
conjCK: involutive (@conj C).
Proof.
have JE x : x^* = `|x|^+2 / x.
have [->|x_neq0] := eqVneq x 0; first by rewrite rmorph0 invr0 mulr0.
by apply: (canRL (mulfK _)) => //; rewrite mulrC -normCK.
move=> x; have [->|x_neq0] := eqVneq x 0; first by rewrite !rmorph0.
rewrite !JE normrM normfV exprMn normrX normr_id.
rewrite invfM exprVn (AC (2*2) (1*(2*3)*4))/= -invfM -exprMn.
by rewrite divff ?mul1r ?invrK // !expf_eq0 normr_eq0 //.
Qed.
Let Re2 z := z + z^*.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
conjCK
| |
nnegImz := (0 <= 'i * (z^* - z)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
nnegIm
| |
argCley z := nnegIm z ==> nnegIm y && (Re2 z <= Re2 y).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
argCle
| |
rootC_specn (x : C) : Type :=
RootCspec (y : C) of if (n > 0)%N then y ^+ n = x else y = 0
& forall z, (n > 0)%N -> z ^+ n = x -> argCle y z.
Fact rootC_subproof n x : rootC_spec n x.
Proof.
have realRe2 u : Re2 u \is Num.real by
rewrite realEsqr expr2 {2}/Re2 -{2}[u]conjCK addrC -rmorphD -normCK exprn_ge0.
have argCle_total : total argCle.
move=> u v; rewrite /total /argCle.
by do 2!case: (nnegIm _) => //; rewrite ?orbT //= real_leVge.
have argCle_trans : transitive argCle.
move=> u v w /implyP geZuv /implyP geZvw; apply/implyP.
by case/geZvw/andP=> /geZuv/andP[-> geRuv] /le_trans->.
pose p := 'X^n - (x *+ (n > 0))%:P; have [r0 Dp] := closed_field_poly_normal p.
have sz_p : size p = n.+1.
rewrite size_polyDl ?size_polyXn // ltnS size_polyN size_polyC mulrn_eq0.
by case: posnP => //; case: negP.
pose r := sort argCle r0; have r_arg: sorted argCle r by apply: sort_sorted.
have{} Dp: p = \prod_(z <- r) ('X - z%:P).
rewrite Dp lead_coefE sz_p coefB coefXn coefC -mulrb -mulrnA mulnb lt0n andNb.
by rewrite subr0 eqxx scale1r; apply/esym/perm_big; rewrite perm_sort.
have mem_rP z: (n > 0)%N -> reflect (z ^+ n = x) (z \in r).
move=> n_gt0; rewrite -root_prod_XsubC -Dp rootE !hornerE n_gt0.
by rewrite subr_eq0; apply: eqP.
exists r`_0 => [|z n_gt0 /(mem_rP z n_gt0) r_z].
have sz_r: size r = n by apply: succn_inj; rewrite -sz_p Dp size_prod_XsubC.
case: posnP => [n0 | n_gt0]; first by rewrite nth_default // sz_r n0.
by apply/mem_rP=> //; rew
...
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
rootC_spec
| |
nthrootn x := let: RootCspec y _ _ := rootC_subproof n x in y.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
nthroot
| |
sqrtC:= 2.-root.
Fact Re_lock : unit. Proof. exact: tt. Qed.
Fact Im_lock : unit. Proof. exact: tt. Qed.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtC
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.