statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
dvdCP x y : reflect (exists2 z, z \in Num.int & y = z * x) (x %| y)%C. | Proof.
rewrite unfold_in; have [-> | nz_x] := eqVneq.
by apply: (iffP eqP) => [-> | [z _ ->]]; first exists 0; rewrite ?mulr0.
apply: (iffP idP) => [Zyx | [z Zz ->]]; last by rewrite mulfK.
by exists (y / x); rewrite ?divfK.
Qed. | Lemma | dvdCP | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"divfK",
"eqVneq",
"int",
"last",
"mulfK",
"mulr0"
] | Integer divisibility. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvdCP_nat x y : 0 <= x -> 0 <= y -> (x %| y)%C -> {n | y = n%:R * x}. | Proof.
move=> x_ge0 y_ge0 x_dv_y; apply: sig_eqW.
case/dvdCP: x_dv_y => z Zz -> in y_ge0 *; move: x_ge0 y_ge0 Zz.
rewrite le_eqVlt => /predU1P[<- | ]; first by exists 22%N; rewrite !mulr0.
by move=> /pmulr_lge0-> /intrEge0-> /natrP[n ->]; exists n.
Qed. | Lemma | dvdCP_nat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdCP",
"intrEge0",
"le_eqVlt",
"mulr0",
"natrP",
"pmulr_lge0",
"predU1P",
"sig_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC0 x : (x %| 0)%C. | Proof. by apply/dvdCP; exists 0; rewrite ?mul0r. Qed. | Lemma | dvdC0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdCP",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd0C x : (0 %| x)%C = (x == 0). | Proof. by rewrite unfold_in eqxx. Qed. | Lemma | dvd0C | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_mull x y z : y \in Num.int -> (x %| z)%C -> (x %| y * z)%C. | Proof.
move=> Zy /dvdCP[m Zm ->]; apply/dvdCP.
by exists (y * m); rewrite ?mulrA ?rpredM.
Qed. | Lemma | dvdC_mull | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdCP",
"int",
"mulrA",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_mulr x y z : y \in Num.int -> (x %| z)%C -> (x %| z * y)%C. | Proof. by rewrite mulrC; apply: dvdC_mull. Qed. | Lemma | dvdC_mulr | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdC_mull",
"int",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_mul2r x y z : y != 0 -> (x * y %| z * y)%C = (x %| z)%C. | Proof.
move=> nz_y; rewrite !unfold_in !(mulIr_eq0 _ (mulIf nz_y)).
by rewrite mulrAC invfM mulrA divfK.
Qed. | Lemma | dvdC_mul2r | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"divfK",
"invfM",
"mulIf",
"mulIr_eq0",
"mulrA",
"mulrAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_mul2l x y z : y != 0 -> (y * x %| y * z)%C = (x %| z)%C. | Proof. by rewrite !(mulrC y); apply: dvdC_mul2r. Qed. | Lemma | dvdC_mul2l | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdC_mul2r",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_trans x y z : (x %| y)%C -> (y %| z)%C -> (x %| z)%C. | Proof. by move=> x_dv_y /dvdCP[m Zm ->]; apply: dvdC_mull. Qed. | Lemma | dvdC_trans | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdCP",
"dvdC_mull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_refl x : (x %| x)%C. | Proof. by apply/dvdCP; exists 1; rewrite ?mul1r. Qed. | Lemma | dvdC_refl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"dvdCP",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_zmod x : zmod_closed (dvdC x). | Proof.
split=> [| _ _ /dvdCP[y Zy ->] /dvdCP[z Zz ->]]; first exact: dvdC0.
by rewrite -mulrBl dvdC_mull ?rpredB.
Qed. | Lemma | dvdC_zmod | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"dvdC",
"dvdC0",
"dvdCP",
"dvdC_mull",
"mulrBl",
"rpredB",
"split",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_nat (p n : nat) : (p %| n)%C = (p %| n)%N. | Proof.
rewrite unfold_in intrEge0 ?divr_ge0 ?invr_ge0 ?ler0n // !pnatr_eq0.
have [-> | nz_p] := eqVneq; first by rewrite dvd0n.
apply/natrP/dvdnP=> [[q def_q] | [q ->]]; exists q.
by apply/eqP; rewrite -eqC_nat natrM -def_q divfK ?pnatr_eq0.
by rewrite [num in num / _]natrM mulfK ?pnatr_eq0.
Qed. | Lemma | dvdC_nat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"def_q",
"divfK",
"divr_ge0",
"dvd0n",
"dvdnP",
"eqC_nat",
"eqVneq",
"intrEge0",
"invr_ge0",
"ler0n",
"mulfK",
"nat",
"natrM",
"natrP",
"num",
"nz_p",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC_int (p : nat) x :
x \in Num.int -> (p %| x)%C = (p %| `|Num.floor x|)%N. | Proof.
move=> Zx; rewrite -{1}(floorK Zx) {1}[Num.floor x]intEsign.
by rewrite rmorphMsign rpredMsign dvdC_nat.
Qed. | Lemma | dvdC_int | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"dvdC_nat",
"floor",
"floorK",
"int",
"intEsign",
"nat",
"rmorphMsign",
"rpredMsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_refl e x : (x == x %[mod e])%C. | Proof. by rewrite /eqCmod subrr rpred0. Qed. | Lemma | eqCmod_refl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod",
"rpred0",
"subrr"
] | Elementary modular arithmetic. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eqCmodm0 e : (e == 0 %[mod e])%C. | Proof. by rewrite /eqCmod subr0. Qed. | Lemma | eqCmodm0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod0 e x : (x == 0 %[mod e])%C = (e %| x)%C. | Proof. by rewrite /eqCmod subr0. Qed. | Lemma | eqCmod0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_sym e x y : ((x == y %[mod e]) = (y == x %[mod e]))%C. | Proof. by rewrite /eqCmod -opprB rpredN. Qed. | Lemma | eqCmod_sym | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod",
"opprB",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_trans e y x z :
(x == y %[mod e] -> y == z %[mod e] -> x == z %[mod e])%C. | Proof.
by move=> Exy Eyz; rewrite /eqCmod -[x](subrK y) -[_ - z]addrA rpredD.
Qed. | Lemma | eqCmod_trans | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addrA",
"eqCmod",
"rpredD",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_transl e x y z :
(x == y %[mod e])%C -> (x == z %[mod e])%C = (y == z %[mod e])%C. | Proof. by move/(sym_left_transitive (eqCmod_sym e) (@eqCmod_trans e)). Qed. | Lemma | eqCmod_transl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod_sym",
"eqCmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_transr e x y z :
(x == y %[mod e])%C -> (z == x %[mod e])%C = (z == y %[mod e])%C. | Proof. by move/(sym_right_transitive (eqCmod_sym e) (@eqCmod_trans e)). Qed. | Lemma | eqCmod_transr | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmod_sym",
"eqCmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodN e x y : (- x == y %[mod e])%C = (x == - y %[mod e])%C. | Proof. by rewrite eqCmod_sym /eqCmod !opprK addrC. Qed. | Lemma | eqCmodN | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addrC",
"eqCmod",
"eqCmod_sym",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodDr e x y z : (y + x == z + x %[mod e])%C = (y == z %[mod e])%C. | Proof. by rewrite /eqCmod [z + x]addrC addrKA. Qed. | Lemma | eqCmodDr | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addrC",
"addrKA",
"eqCmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodDl e x y z : (x + y == x + z %[mod e])%C = (y == z %[mod e])%C. | Proof. by rewrite !(addrC x) eqCmodDr. Qed. | Lemma | eqCmodDl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addrC",
"eqCmodDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodD e x1 x2 y1 y2 :
(x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 + y1 == x2 + y2 %[mod e])%C. | Proof.
by rewrite -(eqCmodDl e x2 y1) -(eqCmodDr e y1); apply: eqCmod_trans.
Qed. | Lemma | eqCmodD | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"eqCmodDl",
"eqCmodDr",
"eqCmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_nat (e m n : nat) : (m == n %[mod e])%C = (m == n %[mod e]). | Proof.
without loss lenm: m n / (n <= m)%N.
by move=> IH; case/orP: (leq_total m n) => /IH //; rewrite eqCmod_sym eq_sym.
by rewrite /eqCmod -natrB // dvdC_nat eqn_mod_dvd.
Qed. | Lemma | eqCmod_nat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"dvdC_nat",
"eqCmod",
"eqCmod_sym",
"eq_sym",
"eqn_mod_dvd",
"leq_total",
"nat",
"natrB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod0_nat (e m : nat) : (m == 0 %[mod e])%C = (e %| m)%N. | Proof. by rewrite eqCmod0 dvdC_nat. Qed. | Lemma | eqCmod0_nat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"dvdC_nat",
"eqCmod0",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodMr e :
{in Num.int, forall z x y, x == y %[mod e] -> x * z == y * z %[mod e]}%C. | Proof. by move=> z Zz x y; rewrite /eqCmod -mulrBl => /dvdC_mulr->. Qed. | Lemma | eqCmodMr | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"dvdC_mulr",
"eqCmod",
"int",
"mulrBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodMl e :
{in Num.int, forall z x y, x == y %[mod e] -> z * x == z * y %[mod e]}%C. | Proof. by move=> z Zz x y Exy; rewrite !(mulrC z) eqCmodMr. Qed. | Lemma | eqCmodMl | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmodMr",
"int",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodMl0 e : {in Num.int, forall x, x * e == 0 %[mod e]}%C. | Proof. by move=> x Zx; rewrite -(mulr0 x) eqCmodMl. Qed. | Lemma | eqCmodMl0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmodMl",
"int",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodMr0 e : {in Num.int, forall x, e * x == 0 %[mod e]}%C. | Proof. by move=> x Zx; rewrite /= mulrC eqCmodMl0. Qed. | Lemma | eqCmodMr0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmodMl0",
"int",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod_addl_mul e : {in Num.int, forall x y, x * e + y == y %[mod e]}%C. | Proof. by move=> x Zx y; rewrite -{2}[y]add0r eqCmodDr eqCmodMl0. Qed. | Lemma | eqCmod_addl_mul | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"add0r",
"eqCmodDr",
"eqCmodMl0",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmodM e : {in Num.int & Num.int, forall x1 y2 x2 y1,
x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 * y1 == x2 * y2 %[mod e]}%C. | Proof.
move=> x1 y2 Zx1 Zy2 x2 y1 eq_x /(eqCmodMl Zx1)/eqCmod_trans-> //.
exact: eqCmodMr.
Qed. | Lemma | eqCmodM | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"eqCmodMl",
"eqCmodMr",
"eqCmod_trans",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratCK : cancel QtoC CtoQ. | Proof. by rewrite /getCrat; case: getCrat_subproof. Qed. | Lemma | ratCK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"CtoQ",
"QtoC",
"getCrat",
"getCrat_subproof"
] | Rational number subset. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
getCratK : {in Crat, cancel CtoQ QtoC}. | Proof. by move=> x /eqP. Qed. | Lemma | getCratK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CtoQ",
"QtoC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_rat (a : rat) : QtoC a \in Crat. | Proof. by rewrite unfold_in ratCK. Qed. | Lemma | Crat_rat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"QtoC",
"rat",
"ratCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CratP x : reflect (exists a, x = QtoC a) (x \in Crat). | Proof.
by apply: (iffP eqP) => [<- | [a ->]]; [exists (CtoQ x) | rewrite ratCK].
Qed. | Lemma | CratP | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CtoQ",
"QtoC",
"apply",
"ratCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat0 : 0 \in Crat. | Proof. by apply/CratP; exists 0; rewrite rmorph0. Qed. | Lemma | Crat0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"apply",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat1 : 1 \in Crat. | Proof. by apply/CratP; exists 1; rewrite rmorph1. Qed. | Lemma | Crat1 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"apply",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_divring_closed : divring_closed Crat. | Proof.
split=> // _ _ /CratP[x ->] /CratP[y ->].
by rewrite -rmorphB Crat_rat.
by rewrite -fmorph_div Crat_rat.
Qed. | Fact | Crat_divring_closed | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"Crat_rat",
"divring_closed",
"fmorph_div",
"rmorphB",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_Crat (S : divringClosed algC) : {subset Crat <= S}. | Proof. by move=> _ /CratP[a ->]; apply: rpred_rat. Qed. | Lemma | rpred_Crat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"algC",
"apply",
"rpred_rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_Crat z : z \in Crat -> z^* = z. | Proof. by move/getCratK <-; rewrite fmorph_div !rmorph_int. Qed. | Lemma | conj_Crat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"fmorph_div",
"getCratK",
"rmorph_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal_Crat : {subset Crat <= Creal}. | Proof. by move=> x /conj_Crat/CrealP. Qed. | Lemma | Creal_Crat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"Creal",
"CrealP",
"conj_Crat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_rat a : (QtoC a \in Num.int) = (a \in Num.int). | Proof.
apply/idP/idP=> [Za | /numqK <-]; last by rewrite rmorph_int.
apply/intrP; exists (Num.floor (QtoC a)); apply: (can_inj ratCK).
by rewrite rmorph_int floorK.
Qed. | Lemma | Cint_rat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"QtoC",
"apply",
"floor",
"floorK",
"int",
"intrP",
"last",
"numqK",
"ratCK",
"rmorph_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpolyP x :
{p : {poly rat} | minCpoly x = pQtoC p /\ p \is monic
& forall q, root (pQtoC q) x = (p %| q)%R}. | Proof. by rewrite /minCpoly; case: (minCpoly_subproof x) => p; exists p. Qed. | Lemma | minCpolyP | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"minCpoly",
"minCpoly_subproof",
"monic",
"pQtoC",
"poly",
"rat",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpoly_monic x : minCpoly x \is monic. | Proof. by have [p [-> mon_p] _] := minCpolyP x; rewrite map_monic. Qed. | Lemma | minCpoly_monic | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"map_monic",
"minCpoly",
"minCpolyP",
"monic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpoly_eq0 x : (minCpoly x == 0) = false. | Proof. exact/negbTE/monic_neq0/minCpoly_monic. Qed. | Lemma | minCpoly_eq0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"minCpoly",
"minCpoly_monic",
"monic_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_minCpoly x : root (minCpoly x) x. | Proof. by have [p [-> _] ->] := minCpolyP x. Qed. | Lemma | root_minCpoly | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"minCpoly",
"minCpolyP",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_minCpoly x : (1 < size (minCpoly x))%N. | Proof. by apply: root_size_gt1 (root_minCpoly x); rewrite ?minCpoly_eq0. Qed. | Lemma | size_minCpoly | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"minCpoly",
"minCpoly_eq0",
"root_minCpoly",
"root_size_gt1",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_Crat nu : {in Crat, nu =1 id}. | Proof. by move=> _ /CratP[a ->]; apply: fmorph_rat. Qed. | Lemma | aut_Crat | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"apply",
"fmorph_rat",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_aut nu x : (nu x \in Crat) = (x \in Crat). | Proof.
apply/idP/idP=> /CratP[a] => [|->]; last by rewrite fmorph_rat Crat_rat.
by rewrite -(fmorph_rat nu) => /fmorph_inj->; apply: Crat_rat.
Qed. | Lemma | Crat_aut | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Crat",
"CratP",
"Crat_rat",
"apply",
"fmorph_inj",
"fmorph_rat",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut_subproof nu x : {y | nu y = x}. | Proof.
have [r Dp] := closed_field_poly_normal (minCpoly x).
suffices /mapP/sig2_eqW[y _ ->]: x \in map nu r by exists y.
rewrite -root_prod_XsubC; congr (root _ x): (root_minCpoly x).
have [q [Dq _] _] := minCpolyP x; rewrite Dq -(eq_map_poly (fmorph_rat nu)).
rewrite (map_poly_comp nu) -{q}Dq Dp (monicP (minCpoly_mon... | Lemma | algC_invaut_subproof | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"apply",
"big_map",
"closed_field_poly_normal",
"eq_bigr",
"eq_map_poly",
"fmorph_rat",
"map",
"mapP",
"map_polyC",
"map_polyX",
"map_poly_comp",
"minCpoly",
"minCpolyP",
"minCpoly_monic",
"monicP",
"rmorphB",
"rmorph_prod",
"root",
"root_minCpoly",
"root_prod_XsubC",
"scale1... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut nu x | := sval (algC_invaut_subproof nu x). | Definition | algC_invaut | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_invaut_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invautK nu : cancel (algC_invaut nu) nu. | Proof. by move=> x; rewrite /algC_invaut; case: algC_invaut_subproof. Qed. | Lemma | algC_invautK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_invaut",
"algC_invaut_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_autK nu : cancel nu (algC_invaut nu). | Proof. exact: inj_can_sym (algC_invautK nu) (fmorph_inj nu). Qed. | Lemma | algC_autK | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_invaut",
"algC_invautK",
"fmorph_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut_is_zmod_morphism nu : zmod_morphism (algC_invaut nu). | Proof. exact: can2_zmod_morphism (algC_autK nu) (algC_invautK nu). Qed. | Fact | algC_invaut_is_zmod_morphism | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_autK",
"algC_invaut",
"algC_invautK",
"can2_zmod_morphism",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut_is_additive | := algC_invaut_is_zmod_morphism. | Definition | algC_invaut_is_additive | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_invaut_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut_is_monoid_morphism nu : monoid_morphism (algC_invaut nu). | Proof. exact: can2_monoid_morphism (algC_autK nu) (algC_invautK nu). Qed. | Fact | algC_invaut_is_monoid_morphism | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_autK",
"algC_invaut",
"algC_invautK",
"can2_monoid_morphism",
"monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_invaut_is_multiplicative nu | :=
(fun g => (g.2,g.1)) (algC_invaut_is_monoid_morphism nu). | Definition | algC_invaut_is_multiplicative | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_invaut_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpoly_aut nu x : minCpoly (nu x) = minCpoly x. | Proof.
wlog suffices dvd_nu: nu x / (minCpoly x %| minCpoly (nu x))%R.
apply/eqP; rewrite -eqp_monic ?minCpoly_monic //; apply/andP; split=> //.
by rewrite -{2}(algC_autK nu x) dvd_nu.
have [[q [Dq _] min_q] [q1 [Dq1 _] _]] := (minCpolyP x, minCpolyP (nu x)).
rewrite Dq Dq1 dvdp_map -min_q -(fmorph_root nu) -map_po... | Lemma | minCpoly_aut | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_autK",
"apply",
"dvdp_map",
"eq_map_poly",
"eqp_monic",
"fmorph_rat",
"fmorph_root",
"map_poly_comp",
"minCpoly",
"minCpolyP",
"minCpoly_monic",
"root_minCpoly",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cchar | := (Cpchar) (only parsing). | Notation | Cchar | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Cpchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^^ f" | := (map_poly f p)
(at level 30, f at level 30, format "p ^^ f"). | Notation | p ^^ f | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR | := in_algR {algRval :> algC; algRvalP : algRval \is Creal}. | Record | algR | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
total_algR : total (<=%O : rel (algR : porderType _)). | Proof. by move=> x y; apply/real_leVge/valP/valP. Qed. | Lemma | total_algR | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR",
"apply",
"real_leVge",
"rel",
"total",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algRval_is_zmod_morphism : zmod_morphism algRval. | Proof. by []. Qed. | Lemma | algRval_is_zmod_morphism | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algRval_is_additive | := algRval_is_zmod_morphism. | Definition | algRval_is_additive | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algRval_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algRval_is_monoid_morphism : monoid_morphism algRval. | Proof. by []. Qed. | Lemma | algRval_is_monoid_morphism | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algRval_is_multiplicative | :=
(fun g => (g.2,g.1)) algRval_is_monoid_morphism. | Definition | algRval_is_multiplicative | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algRval_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_norm (x : algR) : algR | := in_algR (normr_real (val x)). | Definition | algR_norm | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR",
"normr_real",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_ler_normD x y : algR_norm (x + y) <= (algR_norm x + algR_norm y). | Proof. exact: ler_normD. Qed. | Lemma | algR_ler_normD | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_norm",
"ler_normD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_normr0_eq0 x : algR_norm x = 0 -> x = 0. | Proof. by move=> /(congr1 val)/normr0_eq0 ?; apply/val_inj. Qed. | Lemma | algR_normr0_eq0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_norm",
"apply",
"normr0_eq0",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_normrMn x n : algR_norm (x *+ n) = algR_norm x *+ n. | Proof. by apply/val_inj; rewrite /= !rmorphMn/= normrMn. Qed. | Lemma | algR_normrMn | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_norm",
"apply",
"normrMn",
"rmorphMn",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_normrN x : algR_norm (- x) = algR_norm x. | Proof. by apply/val_inj; apply: normrN. Qed. | Lemma | algR_normrN | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_norm",
"apply",
"normrN",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
z : algR | := 0. | Let | z | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_addr_gt0 (x y : algR) : z < x -> z < y -> z < x + y. | Proof. exact: addr_gt0. Qed. | Lemma | algR_addr_gt0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"addr_gt0",
"algR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_ger_leVge (x y : algR) : z <= x -> z <= y -> (x <= y) || (y <= x). | Proof. exact: ger_leVge. Qed. | Lemma | algR_ger_leVge | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR",
"ger_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_normrM : {morph algR_norm : x y / x * y}. | Proof. by move=> *; apply/val_inj; apply: normrM. Qed. | Lemma | algR_normrM | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_norm",
"apply",
"normrM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_ler_def (x y : algR) : (x <= y) = (algR_norm (y - x) == y - x). | Proof. by apply: ler_def. Qed. | Lemma | algR_ler_def | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR",
"algR_norm",
"apply",
"ler_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_archiFieldMixin : Num.archimedean_axiom algR. | Proof.
move=> /= x; have := real_floorD1_gt (valP `|x|).
set n := Num.floor _ + 1 => x_lt.
exists (`|(n + 1)%R|%N); apply: (lt_le_trans x_lt _).
by rewrite /= rmorphMn/= pmulrn ler_int (le_trans _ (lez_abs _))// lerDl.
Qed. | Definition | algR_archiFieldMixin | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR",
"apply",
"archimedean_axiom",
"floor",
"le_trans",
"lerDl",
"ler_int",
"lez_abs",
"lt_le_trans",
"pmulrn",
"real_floorD1_gt",
"rmorphMn",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_pfactor (x : algC) : {poly algR} | :=
if (x \is Creal) =P true is ReflectT xR then 'X - (in_algR xR)%:P else
'X^2 - (in_algR (Creal_Re x) *+ 2) *: 'X + ((in_algR (normr_real x))^+2)%:P. | Definition | algR_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"Creal_Re",
"algC",
"algR",
"normr_real",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactor x | := (algR_pfactor x ^^ algRval). | Notation | algC_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_pfactor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_pfactorRE (x : algC) (xR : x \is Creal) :
algR_pfactor x = 'X - (in_algR xR)%:P. | Proof.
rewrite /algR_pfactor; case: eqP xR => //= p1 p2.
by rewrite (bool_irrelevance p1 p2).
Qed. | Lemma | algR_pfactorRE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC",
"algR_pfactor",
"bool_irrelevance"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactorRE (x : algC) : x \is Creal ->
algC_pfactor x = 'X - x%:P. | Proof. by move=> xR; rewrite algR_pfactorRE map_polyXsubC. Qed. | Lemma | algC_pfactorRE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC",
"algC_pfactor",
"algR_pfactorRE",
"map_polyXsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_pfactorCE (x : algC) : x \isn't Creal ->
algR_pfactor x =
'X^2 - (in_algR (Creal_Re x) *+ 2) *: 'X + ((in_algR (normr_real x))^+2)%:P. | Proof. by rewrite /algR_pfactor; case: eqP => // p; rewrite p. Qed. | Lemma | algR_pfactorCE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"Creal_Re",
"algC",
"algR_pfactor",
"normr_real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactorCE (x : algC) : x \isn't Creal ->
algC_pfactor x = ('X - x%:P) * ('X - x^*%:P). | Proof.
move=> xNR; rewrite mulrBl !mulrBr -rmorphM/= -normCK opprB addrACA -opprD.
rewrite algR_pfactorCE// rmorphD rmorphB/= linearZ map_polyC map_polyXn/=.
(* TODO: Remove the pattern below once we require Rocq >= 9.2 *)
rewrite map_polyX [LHS]addrAC; congr (_ - _).
rewrite mulrC !mul_polyC -scalerDl.
by rewrite -mul... | Lemma | algC_pfactorCE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"ReE",
"addrAC",
"addrACA",
"addrC",
"algC",
"algC_pfactor",
"algR_pfactorCE",
"divfK",
"linearZ",
"map_polyC",
"map_polyX",
"map_polyXn",
"mul_polyC",
"mulr2n",
"mulrBl",
"mulrBr",
"mulrC",
"mulr_natr",
"normCK",
"opprB",
"opprD",
"pnatr_eq0",
"rmorphB",
"rm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactorE x :
algC_pfactor x = ('X - x%:P) * ('X - x^*%:P) ^+ (x \isn't Creal). | Proof.
by have [/algC_pfactorRE|/algC_pfactorCE] := boolP (_ \is _); rewrite ?mulr1.
Qed. | Lemma | algC_pfactorE | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC_pfactor",
"algC_pfactorCE",
"algC_pfactorRE",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_algC_pfactor x : size (algC_pfactor x) = (x \isn't Creal).+2. | Proof.
have [xR|xNR] := boolP (_ \is _); first by rewrite algC_pfactorRE// size_XsubC.
by rewrite algC_pfactorCE// size_mul ?size_XsubC ?polyXsubC_eq0.
Qed. | Lemma | size_algC_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC_pfactor",
"algC_pfactorCE",
"algC_pfactorRE",
"polyXsubC_eq0",
"size",
"size_XsubC",
"size_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_algR_pfactor x : size (algR_pfactor x) = (x \isn't Creal).+2. | Proof. by have := size_algC_pfactor x; rewrite size_map_poly. Qed. | Lemma | size_algR_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algR_pfactor",
"size",
"size_algC_pfactor",
"size_map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactor_eq0 x : (algC_pfactor x == 0) = false. | Proof. by rewrite -size_poly_eq0 size_algC_pfactor. Qed. | Lemma | algC_pfactor_eq0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_pfactor",
"size_algC_pfactor",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_pfactor_eq0 x : (algR_pfactor x == 0) = false. | Proof. by rewrite -size_poly_eq0 size_algR_pfactor. Qed. | Lemma | algR_pfactor_eq0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_pfactor",
"size_algR_pfactor",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_pfactorCgt0 x y : x \isn't Creal -> y \is Creal ->
(algC_pfactor x).[y] > 0. | Proof.
move=> xNR yR; rewrite algC_pfactorCE// hornerM !hornerXsubC.
rewrite [x]algCrect conjC_rect ?Creal_Re ?Creal_Im// !opprD !addrA opprK.
rewrite -subr_sqr exprMn sqrCi mulN1r opprK ltr_wpDl//.
- by rewrite real_exprn_even_ge0// ?rpredB// ?Creal_Re.
by rewrite real_exprn_even_gt0 ?Creal_Im ?orTb//=; apply/eqP/Crea... | Lemma | algC_pfactorCgt0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"Creal_Im",
"Creal_ImP",
"Creal_Re",
"addrA",
"algC_pfactor",
"algC_pfactorCE",
"algCrect",
"apply",
"conjC_rect",
"exprMn",
"hornerM",
"hornerXsubC",
"ltr_wpDl",
"mulN1r",
"opprD",
"opprK",
"real_exprn_even_ge0",
"real_exprn_even_gt0",
"rpredB",
"sqrCi",
"subr_sqr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_pfactorR_mul_gt0 (x a b : algC) :
x \is Creal -> a \is Creal -> b \is Creal ->
a <= b ->
((algC_pfactor x).[a] * (algC_pfactor x).[b] <= 0) =
(a <= x <= b). | Proof.
move=> xR aR bR ab; rewrite !algC_pfactorRE// !hornerXsubC.
have [lt_xa|lt_ax|->]/= := real_ltgtP xR aR; last first.
- by rewrite subrr mul0r lexx ab.
- by rewrite nmulr_rle0 ?subr_lt0 ?subr_ge0.
rewrite pmulr_rle0 ?subr_gt0// subr_le0.
by apply: negbTE; rewrite -real_ltNge// (lt_le_trans lt_xa).
Qed. | Lemma | algR_pfactorR_mul_gt0 | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC",
"algC_pfactor",
"algC_pfactorRE",
"apply",
"hornerXsubC",
"last",
"lexx",
"lt_le_trans",
"mul0r",
"nmulr_rle0",
"pmulr_rle0",
"real_ltNge",
"real_ltgtP",
"subr_ge0",
"subr_gt0",
"subr_le0",
"subr_lt0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monic_algC_pfactor x : algC_pfactor x \is monic. | Proof. by rewrite algC_pfactorE rpredM ?rpredX ?monicXsubC. Qed. | Lemma | monic_algC_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algC_pfactor",
"algC_pfactorE",
"monic",
"monicXsubC",
"rpredM",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monic_algR_pfactor x : algR_pfactor x \is monic. | Proof. by have := monic_algC_pfactor x; rewrite map_monic. Qed. | Lemma | monic_algR_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"algR_pfactor",
"map_monic",
"monic",
"monic_algC_pfactor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_algR_pfactor (p : {poly algR}) :
{ r : seq algC |
p ^^ algRval = val (lead_coef p) *: \prod_(z <- r) algC_pfactor z }. | Proof.
wlog p_monic : p / p \is monic => [hwlog|].
have [->|pN0] := eqVneq p 0.
by exists [::]; rewrite lead_coef0/= rmorph0 scale0r.
have [|r] := hwlog ((lead_coef p)^-1 *: p).
by rewrite monicE lead_coefZ mulVf ?lead_coef_eq0//.
rewrite !lead_coefZ mulVf ?lead_coef_eq0//= scale1r.
rewrite map_polyZ/= ... | Lemma | poly_algR_pfactor | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"CrealE",
"CrealP",
"algC",
"algC_pfactor",
"algC_pfactorE",
"algC_pfactor_eq0",
"algR",
"algR_pfactor",
"big_cons",
"big_nil",
"big_rem",
"closed_field_poly_normal",
"conj",
"divp1",
"divp_pmul2l",
"dvd1p",
"dvdp_XsubCl",
"dvdp_eq_mul",
"dvdp_mul2l",
"eqVneq",
"eq... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algR_rcfMixin : Num.real_closed_axiom algR. | Proof.
move=> p a b le_ab /andP[pa_le0 pb_ge0]/=.
case: ltgtP pa_le0 => //= pa0 _; last first.
by exists a; rewrite ?lexx// rootE pa0.
case: ltgtP pb_ge0 => //= pb0 _; last first.
by exists b; rewrite ?lexx ?andbT// rootE -pb0.
have p_neq0 : p != 0 by apply: contraTneq pa0 => ->; rewrite horner0 ltxx.
have {pa0 pb0... | Definition | algR_rcfMixin | field | field/algC.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"generic_quotient",
"countalg",
"ssrnum",
"closed_field",
"ss... | [
"Creal",
"algC_pfactorCgt0",
"algC_pfactorRE",
"algR",
"algR_pfactorR_mul_gt0",
"all",
"allP",
"apply",
"bigID",
"big_cons",
"big_filter",
"big_nil",
"big_split",
"contraTneq",
"eq_bigr",
"eqxx",
"expr2",
"exprn_even_gt0",
"filter",
"filter_all",
"ger0_real",
"horner0",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^@" | := (p ^ in_alg _) (format "p ^@"): ring_scope. | Notation | p ^@ | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"in_alg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"<< E ; u >>" | := <<E; u>>%VS. | Notation | << E ; u >> | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Qmorphism C | := {rmorphism rat -> C}. | Notation | Qmorphism | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_algebraic_archimedean (C : numFieldType) (QtoC : Qmorphism C) :
integralRange QtoC -> Num.archimedean_axiom C. | Proof.
move=> algC x.
without loss x_ge0: x / 0 <= x by rewrite -normr_id; apply.
have [-> | nz_x] := eqVneq x 0; first by exists 1; rewrite normr0.
have [p mon_p px0] := algC x; exists (\sum_(j < size p) `|numq p`_j|)%N.
rewrite ger0_norm // real_ltNge ?rpred_nat ?ger0_real //.
apply: contraL px0 => lb_x; rewrite root... | Lemma | rat_algebraic_archimedean | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"Qmorphism",
"QtoC",
"abszE",
"absz_gt0",
"add0r",
"algC",
"apply",
"archimedean_axiom",
"big_ord0",
"big_ord_recr",
"coef_map",
"eqVneq",
"exprS",
"exprn_gt0",
"fmorph_eq_rat",
"ger0_norm",
"ger0_real",
"gt_eqF",
"horner_coef",
"integralRange",
"intr_norm",
"invf_le1",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
decidable_embedding sT T (f : sT -> T) | :=
forall y, decidable (exists x, y = f x). | Definition | decidable_embedding | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"decidable",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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