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 |
|---|---|---|---|---|---|---|---|---|---|---|
eq_root_equiv | := EquivRelPack eq_root_is_equiv. | Canonical | eq_root_equiv | 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... | [
"eq_root_is_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type : Type | := {eq_quot eq_root}%qT. | Definition | type | 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... | [
"eq_root",
"qT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL (u : type) | := rootQtoL (repr u). | Definition | CtoL | 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... | [
"repr",
"rootQtoL",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_inj : injective CtoL. | Proof. by move=> u v /eqP eq_uv; rewrite -[u]reprK -[v]reprK; apply/eqmodP. Qed. | Fact | CtoL_inj | 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... | [
"CtoL",
"apply",
"eqmodP",
"reprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_P u : integralOver QtoL (CtoL u). | Proof.
rewrite /CtoL /rootQtoL; case: (repr u) => p j /=.
case: (closed_field_poly_normal _) => r Dp /=.
case: ifPn => [_ | nz_p]; first exact: integral0.
have [/(nth_default 0)-> | lt_j_r] := leqP (size r) j; first exact: integral0.
apply/integral_algebraic; exists p; rewrite // Dp -mul_polyC rootM orbC.
by rewrite ro... | Fact | CtoL_P | 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... | [
"CtoL",
"QtoL",
"apply",
"closed_field_poly_normal",
"integral0",
"integralOver",
"integral_algebraic",
"leqP",
"mem_nth",
"mul_polyC",
"nth_default",
"nz_p",
"repr",
"rootM",
"rootQtoL",
"root_prod_XsubC",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
LtoC_subproof z : integralOver QtoL z -> {u | CtoL u = z}. | Proof.
case/sig2_eqW=> p mon_p pz0; rewrite /CtoL.
pose j := index z (sval (closed_field_poly_normal (pQtoL p))).
pose u := \pi_type%qT (p, j); exists u; have /eqmodP/eqP-> := reprK u.
rewrite /rootQtoL -if_neg monic_neq0 //; apply: nth_index => /=.
case: (closed_field_poly_normal _) => r /= Dp.
by rewrite Dp (monicP _... | Fact | LtoC_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... | [
"CtoL",
"QtoL",
"apply",
"closed_field_poly_normal",
"eqmodP",
"index",
"integralOver",
"monicP",
"monic_map",
"monic_neq0",
"nth_index",
"pQtoL",
"pz0",
"qT",
"reprK",
"rootQtoL",
"root_prod_XsubC",
"scale1r",
"sig2_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
LtoC z Az | := sval (@LtoC_subproof z Az). | Definition | LtoC | 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... | [
"LtoC_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
LtoC_K z Az : CtoL (@LtoC z Az) = z. | Proof. exact: (svalP (LtoC_subproof Az)). Qed. | Fact | LtoC_K | 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... | [
"CtoL",
"LtoC",
"LtoC_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_K u : LtoC (CtoL_P u) = u. | Proof. by apply: CtoL_inj; rewrite LtoC_K. Qed. | Fact | CtoL_K | 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... | [
"CtoL_P",
"CtoL_inj",
"LtoC",
"LtoC_K",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zero | := LtoC (integral0 _). | Definition | zero | 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... | [
"LtoC",
"integral0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add u v | := LtoC (integral_add (CtoL_P u) (CtoL_P v)). | Definition | add | 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... | [
"CtoL_P",
"LtoC",
"integral_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp u | := LtoC (integral_opp (CtoL_P u)). | Definition | opp | 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... | [
"CtoL_P",
"LtoC",
"integral_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addA : associative add. | Proof. by move=> u v w; apply: CtoL_inj; rewrite !LtoC_K addrA. Qed. | Fact | addA | 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... | [
"CtoL_inj",
"LtoC_K",
"add",
"addrA",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addC : commutative add. | Proof. by move=> u v; apply: CtoL_inj; rewrite !LtoC_K addrC. Qed. | Fact | addC | 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... | [
"CtoL_inj",
"LtoC_K",
"add",
"addrC",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0 : left_id zero add. | Proof. by move=> u; apply: CtoL_inj; rewrite !LtoC_K add0r. Qed. | Fact | add0 | 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... | [
"CtoL_inj",
"LtoC_K",
"add",
"add0r",
"apply",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addN : left_inverse zero opp add. | Proof. by move=> u; apply: CtoL_inj; rewrite !LtoC_K addNr. Qed. | Fact | addN | 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... | [
"CtoL_inj",
"LtoC_K",
"add",
"addNr",
"apply",
"opp",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_is_zmod_morphism : zmod_morphism CtoL. | Proof. by move=> u v; rewrite !LtoC_K. Qed. | Fact | CtoL_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... | [
"CtoL",
"LtoC_K",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_is_additive | := CtoL_is_zmod_morphism. | Definition | CtoL_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... | [
"CtoL_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
one | := LtoC (integral1 _). | Definition | one | 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... | [
"LtoC",
"integral1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul u v | := LtoC (integral_mul (CtoL_P u) (CtoL_P v)). | Definition | 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... | [
"CtoL_P",
"LtoC",
"integral_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv u | := LtoC (integral_inv (CtoL_P u)). | Definition | inv | 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... | [
"CtoL_P",
"LtoC",
"integral_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulA : associative mul. | Proof. by move=> u v w; apply: CtoL_inj; rewrite !LtoC_K mulrA. Qed. | Fact | mulA | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"mul",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulC : commutative mul. | Proof. by move=> u v; apply: CtoL_inj; rewrite !LtoC_K mulrC. Qed. | Fact | mulC | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"mul",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1 : left_id one mul. | Proof. by move=> u; apply: CtoL_inj; rewrite !LtoC_K mul1r. Qed. | Fact | mul1 | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"mul",
"mul1r",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulD : left_distributive mul +%R. | Proof. by move=> u v w; apply: CtoL_inj; rewrite !LtoC_K mulrDl. Qed. | Fact | mulD | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"mul",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
one_nz : one != 0 :> type. | Proof. by rewrite -(inj_eq CtoL_inj) !LtoC_K oner_eq0. Qed. | Fact | one_nz | 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... | [
"CtoL_inj",
"LtoC_K",
"inj_eq",
"one",
"oner_eq0",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_is_monoid_morphism : monoid_morphism CtoL. | Proof. by split=> [|u v]; rewrite !LtoC_K. Qed. | Fact | CtoL_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... | [
"CtoL",
"LtoC_K",
"monoid_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoL_is_multiplicative | :=
(fun g => (g.2,g.1)) CtoL_is_monoid_morphism. | Definition | CtoL_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... | [
"CtoL_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVf u : u != 0 -> inv u * u = 1. | Proof.
rewrite -(inj_eq CtoL_inj) rmorph0 => nz_u.
by apply: CtoL_inj; rewrite !LtoC_K mulVf.
Qed. | Fact | mulVf | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"inj_eq",
"inv",
"nz_u",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv0 : inv 0 = 0. | Proof. by apply: CtoL_inj; rewrite !LtoC_K invr0. Qed. | Fact | inv0 | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"inv",
"invr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closedFieldAxiom : GRing.closed_field_axiom type. | Proof.
move=> n a n_gt0; pose p := 'X^n - \poly_(i < n) CtoL (a i).
have Ap : {in p : seq L, integralRange QtoL}.
move=> _ /(nthP 0)[j _ <-]; rewrite coefB coefXn coef_poly.
apply: integral_sub; first exact: integral_nat.
by case: ifP => _; [apply: CtoL_P | apply: integral0].
have sz_p : size p = n.+1.
by rewri... | Fact | closedFieldAxiom | 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... | [
"CtoL",
"CtoL_P",
"CtoL_inj",
"LtoC",
"LtoC_K",
"QtoL",
"addNKr",
"apply",
"closed_field_axiom",
"closed_rootP",
"coefB",
"coefXn",
"coef_poly",
"eqSS",
"eq_bigr",
"hornerD",
"hornerN",
"hornerXn",
"horner_poly",
"integral0",
"integralOver",
"integralRange",
"integral_nat... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_subproof u : integralOver QtoL (conjL (CtoL u)). | Proof.
have [p mon_p pu0] := CtoL_P u; exists p => //.
rewrite -(fmorph_root conjL) conjL_K map_poly_id // => _ /(nthP 0)[j _ <-].
by rewrite coef_map fmorph_rat.
Qed. | Fact | conj_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... | [
"CtoL",
"CtoL_P",
"QtoL",
"coef_map",
"conjL",
"conjL_K",
"fmorph_rat",
"fmorph_root",
"integralOver",
"map_poly_id",
"nthP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_nmod_morphism : nmod_morphism (fun u => LtoC (conj_subproof u)). | Proof.
by split=> [|u v]; apply: CtoL_inj; rewrite LtoC_K ?raddf0// !rmorphD/= !LtoC_K.
Qed. | Fact | conj_is_nmod_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... | [
"CtoL_inj",
"LtoC",
"LtoC_K",
"apply",
"conj_subproof",
"nmod_morphism",
"raddf0",
"rmorphD",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_semi_additive | := conj_is_nmod_morphism. | Definition | conj_is_semi_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... | [
"conj_is_nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_zmod_morphism : {morph (fun u => LtoC (conj_subproof u)) : x / - x}. | Proof. by move=> u; apply: CtoL_inj; rewrite LtoC_K !raddfN /= LtoC_K. Qed. | Fact | conj_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... | [
"CtoL_inj",
"LtoC",
"LtoC_K",
"apply",
"conj_subproof",
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_additive | := conj_is_zmod_morphism. | Definition | conj_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... | [
"conj_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_monoid_morphism : monoid_morphism (fun u => LtoC (conj_subproof u)). | Proof.
split=> [|u v]; apply: CtoL_inj; first by rewrite !LtoC_K rmorph1.
by rewrite LtoC_K 3!{1}rmorphM /= !LtoC_K.
Qed. | Fact | conj_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... | [
"CtoL_inj",
"LtoC",
"LtoC_K",
"apply",
"conj_subproof",
"monoid_morphism",
"rmorph1",
"rmorphM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_is_multiplicative | :=
(fun g => (g.2,g.1)) conj_is_monoid_morphism. | Definition | conj_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... | [
"conj_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj : {rmorphism type -> type} | :=
GRing.RMorphism.Pack
(GRing.RMorphism.Class
(GRing.isNmodMorphism.Build _ _ _ conj_is_nmod_morphism)
(GRing.isMonoidMorphism.Build _ _ _ conj_is_monoid_morphism)). | Definition | conj | 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... | [
"Build",
"conj_is_monoid_morphism",
"conj_is_nmod_morphism",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjK : involutive conj. | Proof. by move=> u; apply: CtoL_inj; rewrite !LtoC_K conjL_K. Qed. | Lemma | conjK | 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... | [
"CtoL_inj",
"LtoC_K",
"apply",
"conj",
"conjL_K"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_nt : ~ conj =1 id. | Proof.
have [i i2]: exists i : type, i ^+ 2 = -1.
have [i] := @solve_monicpoly _ 2%N (nth 0 [:: -1 : type]) isT.
by rewrite !big_ord_recl big_ord0 /= mul0r mulr1 !addr0; exists i.
move/(_ i)/(congr1 CtoL); rewrite LtoC_K => iL_J.
have/lt_geF/idP[] := @ltr01 cfType.
rewrite -oppr_ge0 -(rmorphN1 CtoL).
by rewrite -i2... | Fact | conj_nt | 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... | [
"CtoL",
"LtoC_K",
"addr0",
"big_ord0",
"big_ord_recl",
"cfType",
"conj",
"expr2",
"exprn_ge0",
"id",
"lt_geF",
"ltr01",
"mul0r",
"mulr1",
"normCK",
"nth",
"oppr_ge0",
"rmorphN1",
"rmorphXn",
"solve_monicpoly",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjMixin | := Num.ClosedField.on type. | Definition | conjMixin | 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... | [
"on",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algebraic : integralRange (@ratr type). | Proof.
move=> u; have [p mon_p pu0] := CtoL_P u; exists p => {mon_p}//.
rewrite -(fmorph_root CtoL) -map_poly_comp; congr (root _ _):pu0.
by apply/esym/eq_map_poly; apply: fmorph_eq_rat.
Qed. | Lemma | algebraic | 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... | [
"CtoL",
"CtoL_P",
"apply",
"eq_map_poly",
"fmorph_eq_rat",
"fmorph_root",
"integralRange",
"map_poly_comp",
"ratr",
"root",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
archimedean : Num.archimedean_axiom type. | Proof. exact: rat_algebraic_archimedean algebraic. Qed. | Fact | archimedean | 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... | [
"algebraic",
"archimedean_axiom",
"rat_algebraic_archimedean",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isCountable | := Countable.on type. | Definition | isCountable | 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... | [
"on",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divisor | := Implementation.type. | Definition | divisor | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC | := type. | Notation | algC | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
QtoC | := (ratr : rat -> algC). | Notation | QtoC | 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",
"rat",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pQtoC | := (map_poly QtoC : {poly rat} -> {poly algC}). | Notation | pQtoC | 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",
"algC",
"map_poly",
"poly",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCi_subproof : {i : algC | i ^+ 2 = -1}. | Proof. exact: GRing.imaginary_exists. Qed. | Fact | algCi_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... | [
"algC",
"imaginary_exists"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
getCrat_spec : Type | := GetCrat_spec CtoQ of cancel QtoC CtoQ. | Variant | getCrat_spec | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
getCrat_subproof : getCrat_spec. | Proof.
have isQ := rat_algebraic_decidable algebraic.
exists (fun z => if isQ z is left Qz then sval (sig_eqW Qz) else 0) => a.
case: (isQ _) => [Qa | []]; last by exists a.
by case: (sig_eqW _) => b /= /fmorph_inj.
Qed. | Fact | getCrat_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... | [
"algebraic",
"fmorph_inj",
"getCrat_spec",
"last",
"rat_algebraic_decidable",
"sig_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpoly_subproof (x : algC) :
{p : {poly rat} | p \is monic & forall q, root (pQtoC q) x = (p %| q)%R}. | Proof.
have isQ := rat_algebraic_decidable algebraic.
have [p [mon_p px0 irr_p]] := minPoly_decidable_closure isQ (algebraic x).
exists p => // q; apply/idP/idP=> [qx0 | /dvdpP[r ->]]; last first.
by rewrite rmorphM rootM px0 orbT.
suffices /eqp_dvdl <-: gcdp p q %= p by apply: dvdp_gcdr.
rewrite irr_p ?dvdp_gcdl ?gt... | Fact | minCpoly_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... | [
"QtoC",
"algC",
"algebraic",
"apply",
"dvdpP",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp_dvdl",
"gcdp",
"gcdp_eq0",
"gcdp_map",
"gtn_eqF",
"irr_p",
"last",
"map_poly_eq0",
"minPoly_decidable_closure",
"monic",
"monic_neq0",
"pQtoC",
"poly",
"rat",
"rat_algebraic_decidable",
"rmorphM... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_divisor (x : algC) | := x : divisor. | Definition | algC_divisor | 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",
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_divisor m | := m%:~R : divisor. | Definition | int_divisor | 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... | [
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_divisor n | := n%:R : divisor. | Definition | nat_divisor | 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... | [
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCeq | := (type : eqType). | Notation | algCeq | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCzmod | := (type : zmodType). | Notation | algCzmod | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCnzRing | := (type : nzRingType). | Notation | algCnzRing | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCring | := (type : nzRingType). | Notation | algCring | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCuring | := (type : unitRingType). | Notation | algCuring | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCnum | := (type : numDomainType). | Notation | algCnum | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCfield | := (type : fieldType). | Notation | algCfield | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCnumField | := (type : numFieldType). | Notation | algCnumField | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCnumClosedField | := (type : numClosedFieldType). | Notation | algCnumClosedField | 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... | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal | := (@Num.real algCnum). | Notation | Creal | 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... | [
"algCnum",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
getCrat | := let: GetCrat_spec CtoQ _ := getCrat_subproof in CtoQ. | Definition | getCrat | 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",
"getCrat_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat : {pred algC} | := fun x => ratr (getCrat x) == x. | Definition | 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... | [
"algC",
"getCrat",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minCpoly x : {poly algC} | :=
let: exist2 p _ _ := minCpoly_subproof x in map_poly ratr p. | Definition | 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... | [
"algC",
"map_poly",
"minCpoly_subproof",
"poly",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_divisor : nat >-> divisor. | Coercion | nat_divisor | 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... | [
"divisor",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
int_divisor : int >-> divisor. | Coercion | int_divisor | 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... | [
"divisor",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
algC_divisor : algC >-> divisor. | Coercion | algC_divisor | 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",
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nCdivE (p : nat) : p = p%:R :> divisor. | Proof. by []. Qed. | Lemma | nCdivE | 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... | [
"divisor",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zCdivE (p : int) : p = p%:~R :> divisor. | Proof. by []. Qed. | Lemma | zCdivE | 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... | [
"divisor",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CdivE | := (nCdivE, zCdivE). | Definition | CdivE | 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... | [
"nCdivE",
"zCdivE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdC (x : divisor) : {pred algC} | :=
fun y => if x == 0 then y == 0 else y / x \in Num.int. | Definition | dvdC | 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",
"divisor",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x %| y" | := (y \in dvdC x) : C_expanded_scope. | Notation | x %| y | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x %| y" | := (@in_mem divisor y (mem (dvdC x))) : C_scope. | Notation | x %| y | 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... | [
"divisor",
"dvdC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCmod (e x y : divisor) | := (e %| x - y)%C. | Definition | eqCmod | 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... | [
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x == y %[mod e ]" | := (eqCmod e x y) : C_scope. | Notation | x == y %[mod e ] | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x != y %[mod e ]" | := (~~ (x == y %[mod e])%C) : C_scope. | Notation | x != y %[mod e ] | 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... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZtoQ | := (intr : int -> rat). | Notation | ZtoQ | 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... | [
"int",
"intr",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZtoC | := (intr : int -> algC). | Notation | ZtoC | 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",
"int",
"intr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CtoQ | := getCrat. | Notation | CtoQ | 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... | [
"getCrat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrp | := (map_poly intr). | Notation | intrp | 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... | [
"intr",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pZtoQ | := (map_poly ZtoQ). | Notation | pZtoQ | 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... | [
"ZtoQ",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pZtoC | := (map_poly ZtoC). | Notation | pZtoC | 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... | [
"ZtoC",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pQtoC | := (map_poly ratr). | Notation | pQtoC | 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",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_inj_ZtoC | := (intr_inj : injective ZtoC). | Let | intr_inj_ZtoC | 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... | [
"ZtoC",
"intr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqC_nat n p : (n%:R == p%:R :> algC) = (n == p) | := eqr_nat _ n p. | Definition | eqC_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... | [
"algC",
"eqr_nat"
] | Specialization of a few basic ssrnum order lemmas. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
leC_nat n p : (n%:R <= p%:R :> algC) = (n <= p)%N | := ler_nat _ n p. | Definition | leC_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... | [
"algC",
"ler_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltC_nat n p : (n%:R < p%:R :> algC) = (n < p)%N | := ltr_nat _ n p. | Definition | ltC_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... | [
"algC",
"ltr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cpchar : [pchar algC] =i pred0 | := @pchar_num _. | Definition | Cpchar | 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",
"pchar",
"pchar_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CratrE | :=
let CnF : numClosedFieldType := algC in
let QtoCm : {rmorphism _ -> _} := @ratr CnF in
((rmorph0 QtoCm, rmorph1 QtoCm, rmorphMn QtoCm, rmorphN QtoCm, rmorphD QtoCm),
(rmorphM QtoCm, rmorphXn QtoCm, fmorphV QtoCm),
(rmorphMz QtoCm, rmorphXz QtoCm, @ratr_norm CnF, @ratr_sg CnF),
=^~ (@ler_rat CnF, @ltr_... | Definition | CratrE | 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",
"fmorphV",
"fmorph_inj",
"inj_eq",
"ler_rat",
"ltr_rat",
"ratr",
"ratr_norm",
"ratr_sg",
"rmorph0",
"rmorph1",
"rmorphD",
"rmorphM",
"rmorphMn",
"rmorphMz",
"rmorphN",
"rmorphXn",
"rmorphXz"
] | Missing norm and integer exponent, due to gaps in ssrint and rat. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
CintrE | :=
let CnF : numClosedFieldType := algC in
let ZtoCm : {rmorphism _ -> _} := *~%R (1 : CnF) in
((rmorph0 ZtoCm, rmorph1 ZtoCm, rmorphMn ZtoCm, rmorphN ZtoCm, rmorphD ZtoCm),
(rmorphM ZtoCm, rmorphXn ZtoCm),
(rmorphMz ZtoCm, @intr_norm CnF, @intr_sg CnF),
=^~ (@ler_int CnF, @ltr_int CnF, (inj_eq (@intr_in... | Definition | CintrE | 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",
"inj_eq",
"intr_inj",
"intr_norm",
"intr_sg",
"ler_int",
"ltr_int",
"rmorph0",
"rmorph1",
"rmorphD",
"rmorphM",
"rmorphMn",
"rmorphMz",
"rmorphN",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz2 : 2 != 0 :> algC. | Proof. by rewrite pnatr_eq0. Qed. | Let | nz2 | 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",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_algebraic x | := Algebraics.Implementation.algebraic x. | Definition | algC_algebraic | 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... | [
"algebraic"
] | Conjugation and norm. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
algCrect x : x = 'Re x + 'i * 'Im x. | Proof. by rewrite [LHS]Crect. Qed. | Lemma | algCrect | 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... | [
"Crect",
"Im",
"Re"
] | Real number subset. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
algCreal_Re x : 'Re x \is Creal. | Proof. by rewrite Creal_Re. Qed. | Lemma | algCreal_Re | 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",
"Re"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algCreal_Im x : 'Im x \is Creal. | Proof. by rewrite Creal_Im. Qed. | Lemma | algCreal_Im | 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",
"Im"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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