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 |
|---|---|---|---|---|---|---|---|---|---|---|
gpredF : {in S &, forall u v, u / v \in S}. | Proof. by move=> x y xS yS; rewrite gpredM// gpredV. Qed. | Lemma | gpredF | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"gpredV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredFC u v : (u / v \in S) = (v / u \in S). | Proof. by rewrite -gpredV invgF. Qed. | Lemma | gpredFC | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"gpredV",
"invgF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredXNn n: {in S, forall u, u ^- n \in S}. | Proof. by move=> x xS; apply/gpredVr/gpredXn. Qed. | Lemma | gpredXNn | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredMr x y : x \in S -> (y * x \in S) = (y \in S). | Proof.
move=> Sx; apply/idP/idP => [Sxy|/gpredM-> //].
by rewrite -(mulgK x y) gpredF.
Qed. | Lemma | gpredMr | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredF",
"mulgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredMl x y : x \in S -> (x * y \in S) = (y \in S). | Proof.
move=> Sx; apply/idP/idP => [Sxy|/(gpredM x y Sx)//].
by rewrite -(mulKg x y) gpredM// gpredV.
Qed. | Lemma | gpredMl | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredV",
"mulKg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredFr x y : x \in S -> (y / x \in S) = (y \in S). | Proof. by rewrite -gpredV; apply: gpredMr. Qed. | Lemma | gpredFr | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredMr",
"gpredV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredFl x y : x \in S -> (x / y \in S) = (y \in S). | Proof. by rewrite -(gpredV S y); apply: gpredMl. Qed. | Lemma | gpredFl | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredMl",
"gpredV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredJ x y : x \in S -> y \in S -> x ^ y \in S. | Proof. by move=> xS yS; apply/gpredM; [apply/gpredVr|apply/gpredM]. Qed. | Lemma | gpredJ | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gpredR x y : x \in S -> y \in S -> [~ x, y] \in S. | Proof. by move=> xS yS; apply/gpredM; [apply/gpredVr|apply/gpredJ]. Qed. | Lemma | gpredR | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"gpredJ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val | := (val : H -> G). | Notation | val | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valM : {morph val : x y / x * y}. | Proof. exact: gmulfM. Qed. | Lemma | valM | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"gmulfM",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inH v Sv : H | := Sub v Sv. | Let | inH | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"Sub",
"Sv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulH (u1 u2 : H) | := inH (gpredM _ _ (valP u1) (valP u2)). | Let | mulH | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"inH",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valM : {morph (val : H -> G) : x y / x * y}. | Proof. by move=> x y; rewrite SubK. Qed. | Lemma | valM | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"SubK",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulgA : associative (@mul H). | Proof. by move=> x y z; apply/val_inj; rewrite !valM mulgA. Qed. | Lemma | mulgA | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"mul",
"valM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val1 : val 1 = 1. | Proof. exact: gmulf1. Qed. | Lemma | val1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"gmulf1",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oneH | := inH (fst umagma_closed_subproof). | Let | oneH | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"inH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val1 : (val : H -> G) 1 = 1. | Proof. exact/SubK. Qed. | Lemma | val1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"SubK",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1g : left_id 1 (@mul H). | Proof. by move=> x; apply/val_inj; rewrite valM val1 mul1g. Qed. | Lemma | mul1g | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"mul",
"val1",
"valM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulg1 : right_id 1 (@mul H). | Proof. by move=> x; apply/val_inj; rewrite valM val1 mulg1. Qed. | Lemma | mulg1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"mul",
"val1",
"valM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
umagma_closed : umagma_closed S. | Proof.
split; first exact/(fst group_closed_subproof).
exact/group_closedM/group_closed_subproof.
Qed. | Lemma | umagma_closed | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"group_closedM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invH (u : H) | := inH (gpredVr _ (valP u)). | Let | invH | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"inH",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVg : left_inverse 1%g invH *%g. | Proof. by move=> x; apply/val_inj; rewrite valM SubK mulVg val1. Qed. | Lemma | mulVg | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"SubK",
"apply",
"invH",
"val1",
"valM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulgV : right_inverse 1%g invH *%g. | Proof. by move=> x; apply/val_inj; rewrite valM SubK mulgV val1. Qed. | Lemma | mulgV | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"SubK",
"apply",
"invH",
"val1",
"valM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i <- r | P ) F" | :=
(\big[*%g/1]_(i <- r | P%B) F%g) : group_scope. | Notation | \prod_ ( i <- r | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i <- r ) F" | :=
(\big[*%g/1]_(i <- r) F%g) : group_scope. | Notation | \prod_ ( i <- r ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( m <= i < n | P ) F" | :=
(\big[*%g/1]_(m <= i < n | P%B) F%g) : group_scope. | Notation | \prod_ ( m <= i < n | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( m <= i < n ) F" | :=
(\big[*%g/1]_(m <= i < n) F%g) : group_scope. | Notation | \prod_ ( m <= i < n ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i | P ) F" | :=
(\big[*%g/1]_(i | P%B) F%g) : group_scope. | Notation | \prod_ ( i | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ i F" | :=
(\big[*%g/1]_i F%g) : group_scope. | Notation | \prod_ i F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i : t | P ) F" | :=
(\big[*%g/1]_(i : t | P%B) F%g) (only parsing) : group_scope. | Notation | \prod_ ( i : t | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i : t ) F" | :=
(\big[*%g/1]_(i : t) F%g) (only parsing) : group_scope. | Notation | \prod_ ( i : t ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i < n | P ) F" | :=
(\big[*%g/1]_(i < n | P%B) F%g) : group_scope. | Notation | \prod_ ( i < n | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i < n ) F" | :=
(\big[*%g/1]_(i < n) F%g) : group_scope. | Notation | \prod_ ( i < n ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i 'in' A | P ) F" | :=
(\big[*%g/1]_(i in A | P%B) F%g) : group_scope. | Notation | \prod_ ( i 'in' A | P ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i 'in' A ) F" | :=
(\big[*%g/1]_(i in A) F%g) : group_scope. | Notation | \prod_ ( i 'in' A ) F | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mul f g | := [ffun a => f a * g a]. | Definition | ffun_mul | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mulgA : associative (@ffun_mul aT rT). | Proof. by move=> f1 f2 f3; apply/ffunP=> a; rewrite !ffunE mulgA. Qed. | Fact | ffun_mulgA | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT",
"apply",
"f1",
"f2",
"f3",
"ffunE",
"ffunP",
"ffun_mul",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_one | := [ffun a : aT => (1 : rT)]. | Definition | ffun_one | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mul1g : left_id (@ffun_one aT rT) *%g. | Proof. by move=> f; apply/ffunP => a; rewrite !ffunE mul1g. Qed. | Fact | ffun_mul1g | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT",
"apply",
"ffunE",
"ffunP",
"ffun_one",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mulg1 : right_id (@ffun_one aT rT) *%g. | Proof. by move=> f; apply/ffunP => a; rewrite !ffunE mulg1. Qed. | Fact | ffun_mulg1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT",
"apply",
"ffunE",
"ffunP",
"ffun_one",
"mulg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_inv f | := [ffun a => (f a)^-1]. | Definition | ffun_inv | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mulVg :
left_inverse (@ffun_one aT rT) (@ffun_inv _ _) (@ffun_mul _ _). | Proof. by move=> f; apply/ffunP=> a; rewrite !ffunE mulVg. Qed. | Fact | ffun_mulVg | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT",
"apply",
"ffunE",
"ffunP",
"ffun_inv",
"ffun_mul",
"ffun_one",
"mulVg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_mulgV :
right_inverse (@ffun_one aT rT) (@ffun_inv _ _) (@ffun_mul _ _). | Proof. by move=> f; apply/ffunP=> a; rewrite !ffunE mulgV. Qed. | Fact | ffun_mulgV | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"aT",
"apply",
"ffunE",
"ffunP",
"ffun_inv",
"ffun_mul",
"ffun_one",
"mulgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_pair (x y : G * H) | := (x.1 * y.1, x.2 * y.2). | Definition | mul_pair | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fst_is_multiplicative : {morph fst : x y / x * y}. | Proof. by []. Qed. | Fact | fst_is_multiplicative | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
snd_is_multiplicative : {morph snd : x y / x * y}. | Proof. by []. Qed. | Fact | snd_is_multiplicative | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_mulgA : associative (@mul (G * H)%type). | Proof. by move=> x y z; congr (_, _); apply/mulgA. Qed. | Lemma | pair_mulgA | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"mul",
"mulgA",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
one_pair : G * H | := (1, 1). | Definition | one_pair | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fst_is_umagma_morphism : fst (1 : G * H) = 1. | Proof. by []. Qed. | Fact | fst_is_umagma_morphism | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
snd_is_umagma_morphism : snd (1 : G * H) = 1. | Proof. by []. Qed. | Fact | snd_is_umagma_morphism | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_mul1g : left_id (@one_pair G H) *%g. | Proof. by move=> [x y]; congr (_, _); rewrite mul1g. Qed. | Lemma | pair_mul1g | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mul1g",
"one_pair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_mulg1 : right_id (@one_pair G H) *%g. | Proof. by move=> [x y]; congr (_, _); rewrite mulg1. Qed. | Lemma | pair_mulg1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mulg1",
"one_pair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_pair (u : G * H) | := (u.1 ^-1, u.2 ^-1). | Definition | inv_pair | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_mulVg : left_inverse one (@inv_pair G H) mul. | Proof. by move=> x; congr (_, _); apply/mulVg. Qed. | Lemma | pair_mulVg | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"inv_pair",
"mul",
"mulVg",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_mulgV : right_inverse one (@inv_pair G H) mul. | Proof. by move=> x; congr (_, _); apply/mulgV. Qed. | Lemma | pair_mulgV | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"inv_pair",
"mul",
"mulgV",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
to_multiplicative | := @id Type. | Definition | to_multiplicative | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_closed | := {in S &, forall u v, u + v \in S}. | Definition | addr_closed | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commuteT x y : @commute (to_multiplicative V) x y. | Proof. exact/addrC. Qed. | Lemma | commuteT | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addrC",
"commute",
"to_multiplicative"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrCA : @left_commutative V V +%R. | Proof. exact: SemiGroup.mulmCA. Qed. | Lemma | addrCA | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulmCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrAC : @right_commutative V V +%R. | Proof. exact: SemiGroup.mulmAC. Qed. | Lemma | addrAC | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulmAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrACA : @interchange V +%R +%R. | Proof. exact: SemiGroup.mulmACA. Qed. | Lemma | addrACA | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulmACA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natmul (V : baseAddUMagmaType) (x : V) n : V | := iterop n +%R x 0. | Definition | natmul | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"iterop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr0n x : x *+ 0 = 0. | Proof. by []. Qed. | Lemma | mulr0n | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr1n x : x *+ 1 = x. | Proof. by []. Qed. | Lemma | mulr1n | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr2n x : x *+ 2 = x + x. | Proof. by []. Qed. | Lemma | mulr2n | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrb x (b : bool) : x *+ b = (if b then x else 0). | Proof. exact: (@expgb (to_multiplicative V)). Qed. | Lemma | mulrb | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgb",
"to_multiplicative"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrSS x n : x *+ n.+2 = x + x *+ n.+1. | Proof. by []. Qed. | Lemma | mulrSS | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addumagma_closed | := 0 \in S /\ addr_closed S. | Definition | addumagma_closed | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addr_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr0 (V : addUMagmaType) : right_id (@zero V) add. | Proof. by move=> x; rewrite addrC add0r. Qed. | Lemma | addr0 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"add",
"add0r",
"addrC",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i <- r | P ) F" | := (\big[+%R/0]_(i <- r | P) F). | Notation | \sum_ ( i <- r | P ) F | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( m <= i < n ) F" | := (\big[+%R/0]_(m <= i < n) F). | Notation | \sum_ ( m <= i < n ) F | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i < n ) F" | := (\big[+%R/0]_(i < n) F). | Notation | \sum_ ( i < n ) F | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i 'in' A ) F" | := (\big[+%R/0]_(i in A) F). | Notation | \sum_ ( i 'in' A ) F | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G | := to_multiplicative V. | Let | G | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"to_multiplicative"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrS x n : x *+ n.+1 = x + (x *+ n). | Proof. exact: (@expgS G). Qed. | Lemma | mulrS | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgS"
] | addr0 proved above | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulrSr x n : x *+ n.+1 = x *+ n + x. | Proof. exact: (@expgSr G). Qed. | Lemma | mulrSr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul0rn n : 0 *+ n = 0 :> V. | Proof. exact: (@expg1n G). Qed. | Lemma | mul0rn | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expg1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrnDl n : {morph (fun x => x *+ n) : x y / x + y}. | Proof. by move=> x y; apply/(@expgMn G)/commuteT. Qed. | Lemma | mulrnDl | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"apply",
"commuteT",
"expgMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrnDr x m n : x *+ (m + n) = x *+ m + x *+ n. | Proof. exact: (@expgnDr G). Qed. | Lemma | mulrnDr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrnA x m n : x *+ (m * n) = x *+ m *+ n. | Proof. exact: (@expgnA G). Qed. | Lemma | mulrnA | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrnAC x m n : x *+ m *+ n = x *+ n *+ m. | Proof. exact: (@expgnAC G). Qed. | Lemma | mulrnAC | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"expgnAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_addr n x y : iter n (+%R x) y = x *+ n + y. | Proof. exact: (@iter_mulg G). Qed. | Lemma | iter_addr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"iter",
"iter_mulg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_addr_0 n x : iter n (+%R x) 0 = x *+ n. | Proof. exact: (@iter_mulg_1 G). Qed. | Lemma | iter_addr_0 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"iter",
"iter_mulg_1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumrMnl I r P (F : I -> V) n :
\sum_(i <- r | P i) F i *+ n = (\sum_(i <- r | P i) F i) *+ n. | Proof. by rewrite (big_morph _ (mulrnDl n) (mul0rn _)). Qed. | Lemma | sumrMnl | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"big_morph",
"mul0rn",
"mulrnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumrMnr x I r P (F : I -> nat) :
\sum_(i <- r | P i) x *+ F i = x *+ (\sum_(i <- r | P i) F i). | Proof. by rewrite (big_morph _ (mulrnDr x) (erefl _)). Qed. | Lemma | sumrMnr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"big_morph",
"mulrnDr",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumr_const (I : finType) (A : pred I) x : \sum_(i in A) x = x *+ #|A|. | Proof. by rewrite big_const -Monoid.iteropE. Qed. | Lemma | sumr_const | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"big_const",
"iteropE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumr_const_nat m n x : \sum_(n <= i < m) x = x *+ (m - n). | Proof. by rewrite big_const_nat iter_addr_0. Qed. | Lemma | sumr_const_nat | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"big_const_nat",
"iter_addr_0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmod_closed | := addumagma_closed. | Notation | nmod_closed | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addumagma_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x - y" | := (x + - y) : ring_scope. | Notation | x - y | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x *- n" | := (- (x *+ n)) : ring_scope. | Notation | x *- n | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_closed | := {in S, forall u, - u \in S}. | Definition | oppr_closed | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_closed | := {in S &, forall u v, u - v \in S}. | Definition | subr_closed | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrN (V : zmodType) : @right_inverse V V V 0 -%R +%R. | Proof. by move=> x; rewrite addrC addNr. Qed. | Lemma | addrN | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addNr",
"addrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subrr | := addrN. | Definition | subrr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addKr : @left_loop V V -%R +%R. | Proof. exact: (@mulKg G). Qed. | Lemma | addKr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulKg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addNKr : @rev_left_loop V V -%R +%R. | Proof. exact: (@mulVKg G). Qed. | Lemma | addNKr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulVKg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrK : @right_loop V V -%R +%R. | Proof. exact: (@mulgK G). Qed. | Lemma | addrK | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addrNK : @rev_right_loop V V -%R +%R. | Proof. exact: (@mulgVK G). Qed. | Lemma | addrNK | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"mulgVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subrK | := addrNK. | Definition | subrK | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addrNK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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