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 |
|---|---|---|---|---|---|---|---|---|---|---|
Posz : nat >-> int. | Coercion | Posz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
NegzE (n : nat) : Negz n = - n.+1. | Proof. by []. Qed. | Lemma | NegzE | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_rect (P : int -> Type) :
P 0 -> (forall n : nat, P n -> P (n.+1))
-> (forall n : nat, P (- n) -> P (- (n.+1)))
-> forall n : int, P n. | Proof.
by move=> P0 hPp hPn []; elim=> [|n ihn]//; do ?[apply: hPn | apply: hPp].
Qed. | Lemma | int_rect | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"P0",
"apply",
"int",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_rec | := int_rect. | Definition | int_rec | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_ind | := int_rect. | Definition | int_ind | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_spec (x : int) : int -> Type | :=
| ZintNull of x = 0 : int_spec x 0
| ZintPos n of x = n.+1 : int_spec x n.+1
| ZintNeg n of x = - (n.+1)%:Z : int_spec x (- n.+1). | Variant | int_spec | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intP x : int_spec x x. | Proof. by move: x=> [] []; constructor. Qed. | Lemma | intP | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addzC : commutative addz. | Proof. by move=> [] m [] n //=; rewrite addnC. Qed. | Lemma | addzC | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addnC",
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0z : left_id 0 addz. | Proof. by do 2?case. Qed. | Lemma | add0z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppzK : involutive oppz. | Proof. by do 2?case. Qed. | Lemma | oppzK | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"oppz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppzD : {morph oppz : m n / m + n}. | Proof.
by move=> [[|n]|n] [[|m]|m] /=; rewrite ?addn0 ?subn0 ?addnS //;
rewrite !NegzE !ltnS !subSS; case: ltngtP => [?|?|->];
rewrite ?subnn // ?oppzK ?subnS ?prednK // subn_gt0.
Qed. | Lemma | oppzD | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"addn0",
"addnS",
"ltnS",
"ltngtP",
"oppz",
"oppzK",
"prednK",
"subSS",
"subn0",
"subnS",
"subn_gt0",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add1Pz (n : int) : 1 + (n - 1) = n. | Proof. by case: (intP n)=> // n' /= _; rewrite ?(subn1, addn0). Qed. | Lemma | add1Pz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addn0",
"int",
"intP",
"n'",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSz1 (n : int) : 1 + n - 1 = n. | Proof.
by apply: (inv_inj oppzK); rewrite addzC !oppzD oppzK [_ - n]addzC add1Pz.
Qed. | Lemma | subSz1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add1Pz",
"addzC",
"apply",
"int",
"oppzD",
"oppzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addSnz (m : nat) (n : int) : m.+1%N + n = 1 + (m + n). | Proof.
move: m n=> [|m] [] [|n] //=; rewrite ?add1n ?subn1 // !(ltnS, subSS).
case: ltngtP=> hnm /=; rewrite ?hnm ?subnn //.
by rewrite subnS add1n prednK ?subn_gt0.
by rewrite ltnS leqn0 subn_eq0 leqNgt hnm /= subnS subn1.
Qed. | Lemma | addSnz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add1n",
"int",
"leqNgt",
"leqn0",
"ltnS",
"ltngtP",
"nat",
"prednK",
"subSS",
"subn1",
"subnS",
"subn_eq0",
"subn_gt0",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addSz (m n : int) : (1 + m) + n = 1 + (m + n). | Proof.
case: m => [] m; first by rewrite -PoszD add1n addSnz.
rewrite !NegzE; apply: (inv_inj oppzK).
rewrite !oppzD !oppzK addSnz [-1%:Z + _]addzC addSnz add1Pz.
by rewrite [-1%:Z + _]addzC subSz1.
Qed. | Lemma | addSz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"PoszD",
"add1Pz",
"add1n",
"addSnz",
"addzC",
"apply",
"int",
"oppzD",
"oppzK",
"subSz1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addPz (m n : int) : (m - 1) + n = (m + n) - 1. | Proof.
by apply: (inv_inj oppzK); rewrite !oppzD oppzK [_ + 1]addzC addSz addzC.
Qed. | Lemma | addPz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addSz",
"addzC",
"apply",
"int",
"oppzD",
"oppzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addzA : associative addz. | Proof.
elim=> [|m ihm|m ihm] n p; first by rewrite !add0z.
by rewrite -add1n PoszD !addSz ihm.
by rewrite -add1n addnC PoszD oppzD !addPz ihm.
Qed. | Lemma | addzA | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"add0z",
"add1n",
"addPz",
"addSz",
"addnC",
"addz",
"oppzD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addNz : left_inverse (0:int) oppz addz. | Proof. by do 3?elim. Qed. | Lemma | addNz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addz",
"int",
"oppz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predn_int (n : nat) : 0 < n -> n.-1%:Z = n - 1. | Proof. by case: n => //= n _; rewrite subn1. Qed. | Lemma | predn_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Mixin | := GRing.isZmodule.Build int addzA addzC add0z addNz. | Definition | Mixin | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Build",
"add0z",
"addNz",
"addzA",
"addzC",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PoszD : {morph Posz : n m / (n + m)%N >-> n + m}. | Proof. by []. Qed. | Lemma | PoszD | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NegzE (n : nat) : Negz n = -(n.+1)%:Z. | Proof. by []. Qed. | Lemma | NegzE | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_rect (P : int -> Type) :
P 0 -> (forall n : nat, P n -> P (n.+1)%N)
-> (forall n : nat, P (- (n%:Z)) -> P (- (n.+1%N%:Z)))
-> forall n : int, P n. | Proof.
by move=> P0 hPp hPn []; elim=> [|n ihn]//; do ?[apply: hPn | apply: hPp].
Qed. | Lemma | int_rect | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"P0",
"apply",
"int",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_spec (x : int) : int -> Type | :=
| ZintNull : int_spec x 0
| ZintPos n : int_spec x n.+1
| ZintNeg n : int_spec x (- (n.+1)%:Z). | Variant | int_spec | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intP x : int_spec x x. | Proof. by move: x=> [] [] *; rewrite ?NegzE; constructor. Qed. | Lemma | intP | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"int_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppzD | := @opprD int. | Definition | oppzD | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subzn (m n : nat) : (n <= m)%N -> m%:Z - n%:Z = (m - n)%N. | Proof.
elim: n=> //= [|n ihn] hmn; first by rewrite subr0 subn0.
rewrite subnS -addn1 !PoszD opprD addrA ihn 1?ltnW //.
by rewrite intZmod.predn_int // subn_gt0.
Qed. | Lemma | subzn | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"addn1",
"addrA",
"ltnW",
"nat",
"opprD",
"predn_int",
"subn0",
"subnS",
"subn_gt0",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subzSS (m n : nat) : m.+1%:Z - n.+1%:Z = m%:Z - n%:Z. | Proof. by elim: n m=> [|n ihn] m //; rewrite !subzn. Qed. | Lemma | subzSS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat",
"subzn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz (m n : int) | :=
match m, n with
| Posz m', Posz n' => (m' * n')%N%:Z
| Negz m', Negz n' => (m'.+1%N * n'.+1%N)%N%:Z
| Posz m', Negz n' => - (m' * (n'.+1%N))%N%:Z
| Negz n', Posz m' => - (m' * (n'.+1%N))%N%:Z
end. | Definition | mulz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"*%Z" | := (@mulz) : int_scope. | Notation | *%Z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x * y" | := (mulz x y) : int_scope. | Notation | x * y | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul0z : left_zero 0 *%Z. | Proof. by case=> [n|[|n]] //=; rewrite muln0. Qed. | Lemma | mul0z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzC : commutative mulz. | Proof. by move=> [] m [] n //=; rewrite mulnC. Qed. | Lemma | mulzC | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulnC",
"mulz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz0 : right_zero 0 *%Z. | Proof. by move=> x; rewrite mulzC mul0z. Qed. | Lemma | mulz0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mul0z",
"mulzC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzN (m n : int) : (m * (- n))%Z = - (m * n)%Z. | Proof.
by case: (intP m)=> {m} [|m|m]; rewrite ?mul0z //;
case: (intP n)=> {n} [|n|n]; rewrite ?mulz0 //= mulnC.
Qed. | Lemma | mulzN | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP",
"mul0z",
"mulnC",
"mulz0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulNz (m n : int) : ((- m) * n)%Z = - (m * n)%Z. | Proof. by rewrite mulzC mulzN mulzC. Qed. | Lemma | mulNz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"mulzC",
"mulzN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzA : associative mulz. | Proof.
by move=> [] m [] n [] p; rewrite ?NegzE ?(mulnA,mulNz,mulzN,opprK) //= ?mulnA.
Qed. | Lemma | mulzA | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"mulNz",
"mulnA",
"mulz",
"mulzN",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1z : left_id 1%Z mulz. | Proof. by case=> [[|n]|n] //=; rewrite ?mul1n// plusE addn0. Qed. | Lemma | mul1z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addn0",
"mul1n",
"mulz",
"plusE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzS (x : int) (n : nat) : (x * n.+1%:Z)%Z = x + (x * n)%Z. | Proof.
by case: (intP x)=> [|m'|m'] //=; [rewrite mulnS|rewrite mulSn -opprD].
Qed. | Lemma | mulzS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP",
"mulSn",
"mulnS",
"nat",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_addl : left_distributive mulz (+%R). | Proof.
move=> x y z; elim: z=> [|n|n]; first by rewrite !(mul0z,mulzC).
by rewrite !mulzS=> ->; rewrite addrACA.
by rewrite !mulzN !mulzS -!opprD addrACA => /oppr_inj->.
Qed. | Lemma | mulz_addl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addrACA",
"mul0z",
"mulz",
"mulzC",
"mulzN",
"mulzS",
"opprD",
"oppr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonzero1z : 1%Z != 0. | Proof. by []. Qed. | Lemma | nonzero1z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comMixin | := GRing.Zmodule_isComNzRing.Build int
mulzA mulzC mul1z mulz_addl nonzero1z. | Definition | comMixin | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Build",
"int",
"mul1z",
"mulzA",
"mulzC",
"mulz_addl",
"nonzero1z"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PoszM : {morph Posz : n m / (n * m)%N >-> n * m}. | Proof. by []. Qed. | Lemma | PoszM | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NegzS (n : nat) : Negz n.+1 = Negz n - 1. | Proof. by rewrite !NegzE -opprD -PoszD addn1. Qed. | Lemma | NegzS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"PoszD",
"addn1",
"nat",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Negz_doubleS (n : nat) : Negz n.*2.+1 = 2 * Negz n. | Proof. by rewrite !NegzE -doubleS -mul2n PoszM -mulrN. Qed. | Lemma | Negz_doubleS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"PoszM",
"doubleS",
"mul2n",
"mulrN",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intS (n : nat) : n.+1%:Z = 1 + n%:Z. | Proof. by rewrite -PoszD. Qed. | Lemma | intS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predn_int (n : nat) : (0 < n)%N -> n.-1%:Z = n%:Z - 1. | Proof. exact: intZmod.predn_int. Qed. | Lemma | predn_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitz | := [qualify a n : int | (n == 1) || (n == -1)]. | Definition | unitz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invz n : int | := n. | Definition | invz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVz : {in unitz, left_inverse 1%R invz *%R}. | Proof. by move=> n /pred2P[] ->. Qed. | Lemma | mulVz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"invz",
"pred2P",
"unitz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzn_eq1 m (n : nat) : (m * n == 1) = (m == 1) && (n == 1). | Proof. by case: m => m /=; [rewrite -PoszM [_==_]muln_eq1 | case: n]. Qed. | Lemma | mulzn_eq1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszM",
"muln_eq1",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitzPl m n : n * m = 1 -> m \is a unitz. | Proof.
rewrite qualifE => /eqP.
by case: m => m; rewrite ?NegzE ?mulrN -?mulNr mulzn_eq1 => /andP[_ /eqP->].
Qed. | Lemma | unitzPl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"mulNr",
"mulrN",
"mulzn_eq1",
"unitz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invz_out : {in [predC unitz], invz =1 id}. | Proof. exact. Qed. | Lemma | invz_out | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"id",
"invz",
"unitz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
idomain_axiomz m n : m * n = 0 -> (m == 0) || (n == 0). | Proof. by case: m n => [[|m]|m] [[|n]|n]. Qed. | Lemma | idomain_axiomz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comMixin | := GRing.ComNzRing_hasMulInverse.Build int
mulVz unitzPl invz_out. | Definition | comMixin | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Build",
"int",
"invz_out",
"mulVz",
"unitzPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz m | := match m with Posz p => p | Negz n => n.+1 end. | Definition | absz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m - n" | := (@GRing.add int m%N (@GRing.opp int n%N)) : distn_scope. | Notation | m - n | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add",
"int",
"opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`| m |" | := (absz m) : nat_scope. | Notation | `| m | | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"absz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normz m | := (absz m)%:Z. | Notation | normz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"absz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez m n | :=
match m, n with
| Posz m', Posz n' => (m' <= n')%N
| Posz m', Negz n' => false
| Negz m', Posz n' => true
| Negz m', Negz n' => (n' <= m')%N
end. | Definition | lez | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz m n | :=
match m, n with
| Posz m', Posz n' => (m' < n')%N
| Posz m', Negz n' => false
| Negz m', Posz n' => true
| Negz m', Negz n' => (n' < m')%N
end. | Definition | ltz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_add m n : lez 0 m -> lez 0 n -> lez 0 (m + n). | Proof. by case: m n => [] m [] n. Qed. | Fact | lez_add | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_mul m n : lez 0 m -> lez 0 n -> lez 0 (m * n). | Proof. by case: m n => [] m [] n. Qed. | Fact | lez_mul | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_anti m : lez 0 m -> lez m 0 -> m = 0. | Proof. by case: m; first case. Qed. | Fact | lez_anti | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subz_ge0 m n : lez 0 (n - m) = lez m n. | Proof.
case: (intP m); case: (intP n)=> // {}m {}n /=;
rewrite ?ltnS -?opprD ?opprB ?subzSS; case: leqP=> // hmn;
by [ rewrite subzn //
| rewrite -opprB subzn ?(ltnW hmn) //;
move: hmn; rewrite -subn_gt0; case: (_ - _)%N].
Qed. | Lemma | subz_ge0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intP",
"leqP",
"lez",
"ltnS",
"ltnW",
"opprB",
"opprD",
"subn_gt0",
"subzSS",
"subzn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_total m n : lez m n || lez n m. | Proof. by move: m n => [] m [] n //=; apply: leq_total. Qed. | Fact | lez_total | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"leq_total",
"lez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normzN m : normz (- m) = normz m. | Proof. by case: m => // -[]. Qed. | Fact | normzN | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"normz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gez0_norm m : lez 0 m -> normz m = m. | Proof. by case: m. Qed. | Fact | gez0_norm | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lez",
"normz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_def m n : (ltz m n) = (n != m) && (lez m n). | Proof.
by move: m n => [] m [] n //=; rewrite (ltn_neqAle, leq_eqVlt) // eq_sym.
Qed. | Fact | ltz_def | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eq_sym",
"leq_eqVlt",
"lez",
"ltn_neqAle",
"ltz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Mixin | := Num.IntegralDomain_isLeReal.Build int
lez_add lez_mul lez_anti subz_ge0 (lez_total 0) normzN gez0_norm ltz_def. | Definition | Mixin | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Build",
"gez0_norm",
"int",
"lez_add",
"lez_anti",
"lez_mul",
"lez_total",
"ltz_def",
"normzN",
"subz_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_nat m n : (m <= n :> int) = (m <= n)%N. | Proof. by []. Qed. | Lemma | lez_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_nat m n : (m < n :> int) = (m < n)%N. | Proof. by rewrite ltnNge ltNge lez_nat. Qed. | Lemma | ltz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"lez_nat",
"ltNge",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltez_nat | := (lez_nat, ltz_nat). | Definition | ltez_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lez_nat",
"ltz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leNz_nat m n : (- m%:Z <= n). | Proof. by case: m. Qed. | Lemma | leNz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltNz_nat m n : (- m%:Z < n) = (m != 0) || (n != 0). | Proof. by move: m n=> [|?] []. Qed. | Lemma | ltNz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteNz_nat | := (leNz_nat, ltNz_nat). | Definition | lteNz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"leNz_nat",
"ltNz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lezN_nat m n : (m%:Z <= - n%:Z) = (m == 0) && (n == 0). | Proof. by move: m n=> [|?] []. Qed. | Lemma | lezN_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltzN_nat m n : (m%:Z < - n%:Z) = false. | Proof. by move: m n=> [|?] []. Qed. | Lemma | ltzN_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le0z_nat n : 0 <= n :> int. | Proof. by []. Qed. | Lemma | le0z_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez0_nat n : (n <= 0 :> int) = (n == 0 :> nat). | Proof. by elim: n. Qed. | Lemma | lez0_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltezN_nat | := (lezN_nat, ltzN_nat). | Definition | ltezN_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lezN_nat",
"ltzN_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltez_natE | := (ltez_nat, lteNz_nat, ltezN_nat, le0z_nat, lez0_nat). | Definition | ltez_natE | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"le0z_nat",
"lez0_nat",
"lteNz_nat",
"ltezN_nat",
"ltez_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtz0_ge1 x : (0 < x) = (1 <= x). | Proof. by case: (intP x). Qed. | Lemma | gtz0_ge1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez1D x y : (1 + x <= y) = (x < y). | Proof. by rewrite -subr_gt0 gtz0_ge1 lterBDr. Qed. | Lemma | lez1D | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"gtz0_ge1",
"lterBDr",
"subr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lezD1 x y : (x + 1 <= y) = (x < y). | Proof. by rewrite addrC lez1D. Qed. | Lemma | lezD1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addrC",
"lez1D"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz1D x y : (x < 1 + y) = (x <= y). | Proof. by rewrite -lez1D lerD2l. Qed. | Lemma | ltz1D | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lerD2l",
"lez1D"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltzD1 x y : (x < y + 1) = (x <= y). | Proof. by rewrite -lezD1 lerD2r. Qed. | Lemma | ltzD1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"lerD2r",
"lezD1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intmul (R : zmodType) (x : R) (n : int) | :=
match n with
| Posz n => (x *+ n)%R
| Negz n => (x *- (n.+1))%R
end. | Definition | intmul | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int"
] | definition of intmul | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"*~%R" | := (@intmul _) : function_scope. | Notation | *~%R | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x *~ n" | := (intmul x n) : ring_scope. | Notation | x *~ n | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr | := ( *~%R 1). | Notation | intr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n %:~R" | := (1 *~ n)%R : ring_scope. | Notation | n %:~R | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn (R : zmodType) (x : R) (n : nat) : x *+ n = x *~ n%:Z. | Proof. by []. Qed. | Lemma | pmulrn | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulrn (R : zmodType) (x : R) (n : nat) : x *- n = x *~ - n%:Z. | Proof. by case: n; rewrite // oppr0. Qed. | Lemma | nmulrn | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmodule (M : Type) : Type | := M. | Definition | zmodule | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"M ^z" | := (zmodule M) (format "M ^z") : type_scope. | Notation | M ^z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"zmodule"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrzA_C m n x : (x *~ n) *~ m = x *~ (m * n). | Proof.
elim: m=> [|m _|m _]; elim: n=> [|n _|n _]; rewrite /intmul //=;
rewrite ?(muln0, mulr0n, mul0rn, oppr0, mulNrn, opprK) //;
do ?by rewrite mulnC mulrnA.
* by rewrite -mulrnA mulnC.
* by rewrite -mulrnA.
Qed. | Fact | mulrzA_C | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intmul",
"mul0rn",
"mulNrn",
"muln0",
"mulnC",
"mulr0n",
"mulrnA",
"oppr0",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrzAC m n x : (x *~ n) *~ m = (x *~ m) *~ n. | Proof. by rewrite !mulrzA_C mulrC. Qed. | Fact | mulrzAC | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulrC",
"mulrzA_C"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr1z (x : M) : x *~ 1 = x. | Proof. by []. Qed. | Fact | mulr1z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrzDl m : {morph ( *~%R^~ m : M -> M) : x y / x + y}. | Proof. by case: m => m x y; rewrite /intmul mulrnDl // opprD. Qed. | Fact | mulrzDl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intmul",
"mulrnDl",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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