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 |
|---|---|---|---|---|---|---|---|---|---|---|
equivf | := (@FracField.equivf _). | Notation | equivf | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ratio_numden (x : {ratio R}) : Ratio \n_x \d_x = x. | Proof. exact: FracField.Ratio_numden. Qed. | Lemma | Ratio_numden | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"Ratio",
"ratio"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac | := (@FracField.tofrac R). | Notation | tofrac | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [] | exporting the embedding from R to {fraction R} | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"x %:F" | := (tofrac x). | Notation | x %:F | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_is_zmod_morphism: zmod_morphism tofrac. | Proof.
move=> p q /=; unlock tofrac.
rewrite -[X in _ = _ + X]pi_opp -[RHS]pi_add.
by rewrite /addf /oppf /= !numden_Ratio ?(oner_neq0, mul1r, mulr1).
Qed. | Lemma | tofrac_is_zmod_morphism | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"addf",
"mul1r",
"mulr1",
"numden_Ratio",
"oner_neq0",
"oppf",
"pi_add",
"pi_opp",
"tofrac",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_is_additive | := tofrac_is_zmod_morphism. | Definition | tofrac_is_additive | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"tofrac_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_is_monoid_morphism: monoid_morphism tofrac. | Proof.
split=> [//|p q]; unlock tofrac; rewrite -[RHS]pi_mul.
by rewrite /mulf /= !numden_Ratio ?(oner_neq0, mul1r, mulr1).
Qed. | Lemma | tofrac_is_monoid_morphism | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"monoid_morphism",
"mul1r",
"mulf",
"mulr1",
"numden_Ratio",
"oner_neq0",
"pi_mul",
"split",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_is_multiplicative | := tofrac_is_monoid_morphism. | Definition | tofrac_is_multiplicative | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"tofrac_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac0 : 0%:F = 0. | Proof. exact: rmorph0. Qed. | Lemma | tofrac0 | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorph0"
] | tests | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tofracN : {morph tofrac: x / - x}. | Proof. exact: rmorphN. Qed. | Lemma | tofracN | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphN",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracD : {morph tofrac: x y / x + y}. | Proof. exact: rmorphD. Qed. | Lemma | tofracD | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphD",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracB : {morph tofrac: x y / x - y}. | Proof. exact: rmorphB. Qed. | Lemma | tofracB | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphB",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracMn n : {morph tofrac: x / x *+ n}. | Proof. exact: rmorphMn. Qed. | Lemma | tofracMn | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphMn",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracMNn n : {morph tofrac: x / x *- n}. | Proof. exact: rmorphMNn. Qed. | Lemma | tofracMNn | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphMNn",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac1 : 1%:F = 1. | Proof. exact: rmorph1. Qed. | Lemma | tofrac1 | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracM : {morph tofrac: x y / x * y}. | Proof. exact: rmorphM. Qed. | Lemma | tofracM | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphM",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofracXn n : {morph tofrac: x / x ^+ n}. | Proof. exact: rmorphXn. Qed. | Lemma | tofracXn | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"rmorphXn",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_eq (p q : R): (p%:F == q%:F) = (p == q). | Proof.
apply/eqP/eqP=> [|->//]; unlock tofrac=> /eqmodP /eqP /=.
by rewrite !numden_Ratio ?(oner_eq0, mul1r, mulr1).
Qed. | Lemma | tofrac_eq | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"apply",
"eqmodP",
"mul1r",
"mulr1",
"numden_Ratio",
"oner_eq0",
"tofrac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tofrac_eq0 (p : R): (p%:F == 0) = (p == 0). | Proof. by rewrite tofrac_eq. Qed. | Lemma | tofrac_eq0 | algebra | algebra/fraction.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"choice",
"ssrAC",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"FracField"
] | [
"tofrac_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz (m d : int) : int | :=
let: (K, n) := match m with Posz n => (Posz, n) | Negz n => (Negz, n) end in
sgz d * K (n %/ `|d|)%N. | Definition | divz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
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"zmodp",
"matrix",
"ord... | [
"Posz",
"int",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz (m d : int) : int | := m - divz m d * d. | Definition | modz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz d m | := (`|d| %| `|m|)%N. | Definition | dvdz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz m n | := (gcdn `|m| `|n|)%:Z. | Definition | gcdz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"perm",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmz m n | := (lcmn `|m| `|n|)%:Z. | Definition | lcmz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"lcmn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdz m n : int * int | :=
if m == 0 then (0, (-1) ^+ (n < 0)%R) else
let: (u, v) := egcdn `|m| `|n| in (sgz m * u, - (-1) ^+ (n < 0)%R * v%:Z). | Definition | egcdz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"egcdn",
"int",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimez m n | := (gcdz m n == 1). | Definition | coprimez | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"d %| m" | := (m \in dvdz d) : int_scope. | Notation | d %| m | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m = n %[mod d ]" | := (modz m d = modz n d) : int_scope. | Notation | m = n %[mod d ] | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
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"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m == n %[mod d ]" | := (modz m d == modz n d) : int_scope. | Notation | m == n %[mod d ] | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m <> n %[mod d ]" | := (modz m d <> modz n d) : int_scope. | Notation | m <> n %[mod d ] | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m != n %[mod d ]" | := (modz m d != modz n d) : int_scope. | Notation | m != n %[mod d ] | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_nat (n d : nat) : (n %/ d)%Z = (n %/ d)%N. | Proof. by case: d => // d; rewrite /divz /= mul1r. Qed. | Lemma | divz_nat | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz",
"mul1r",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzN m d : (m %/ - d)%Z = - (m %/ d)%Z. | Proof. by case: m => n; rewrite /divz /= sgzN abszN mulNr. Qed. | Lemma | divzN | algebra | algebra/intdiv.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszN",
"divz",
"mulNr",
"sgzN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_abs (m d : int) : (m %/ `|d|)%Z = (-1) ^+ (d < 0)%R * (m %/ d)%Z. | Proof.
by rewrite {3}[d]intEsign !mulr_sign; case: ifP => -> //; rewrite divzN opprK.
Qed. | Lemma | divz_abs | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzN",
"int",
"intEsign",
"mulr_sign",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
div0z d : (0 %/ d)%Z = 0. | Proof.
by rewrite -(canLR (signrMK _) (divz_abs _ _)) (divz_nat 0) div0n mulr0.
Qed. | Lemma | div0z | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"div0n",
"divz_abs",
"divz_nat",
"mulr0",
"signrMK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divNz_nat m d : (d > 0)%N -> (Negz m %/ d)%Z = - (m %/ d).+1%:Z. | Proof. by case: d => // d _; apply: mul1r. Qed. | Lemma | divNz_nat | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_eq m d : m = (m %/ d)%Z * d + (m %% d)%Z. | Proof. by rewrite addrC subrK. Qed. | Lemma | divz_eq | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrC",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzN m d : (m %% - d)%Z = (m %% d)%Z. | Proof. by rewrite /modz divzN mulrNN. Qed. | Lemma | modzN | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzN",
"modz",
"mulrNN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_abs m d : (m %% `|d|%N)%Z = (m %% d)%Z. | Proof. by rewrite {2}[d]intEsign mulr_sign; case: ifP; rewrite ?modzN. Qed. | Lemma | modz_abs | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"intEsign",
"modzN",
"mulr_sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_nat (m d : nat) : (m %% d)%Z = (m %% d)%N. | Proof.
by apply: (canLR (addrK _)); rewrite addrC divz_nat {1}(divn_eq m d).
Qed. | Lemma | modz_nat | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrC",
"addrK",
"apply",
"divn_eq",
"divz_nat",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modNz_nat m d : (d > 0)%N -> (Negz m %% d)%Z = d%:Z - 1 - (m %% d)%:Z. | Proof.
rewrite /modz => /divNz_nat->; apply: (canLR (addrK _)).
rewrite -!addrA -!opprD -!PoszD -opprB mulnSr !addnA PoszD addrK.
by rewrite addnAC -addnA mulnC -divn_eq.
Qed. | Lemma | modNz_nat | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"PoszD",
"addnA",
"addnAC",
"addrA",
"addrK",
"apply",
"divNz_nat",
"divn_eq",
"modz",
"mulnC",
"mulnSr",
"opprB",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_ge0 m d : d != 0 -> 0 <= (m %% d)%Z. | Proof.
rewrite -absz_gt0 -modz_abs => d_gt0.
case: m => n; rewrite ?modNz_nat ?modz_nat // -addrA -opprD subr_ge0.
by rewrite lez_nat ltn_mod.
Qed. | Lemma | modz_ge0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz_gt0",
"addrA",
"d_gt0",
"lez_nat",
"ltn_mod",
"modNz_nat",
"modz_abs",
"modz_nat",
"opprD",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz0 m : (m %/ 0)%Z = 0. | Proof. by case: m. Qed. | Lemma | divz0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod0z d : (0 %% d)%Z = 0. | Proof. by rewrite /modz div0z mul0r subrr. Qed. | Lemma | mod0z | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"div0z",
"modz",
"mul0r",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz0 m : (m %% 0)%Z = m. | Proof. by rewrite /modz mulr0 subr0. Qed. | Lemma | modz0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modz",
"mulr0",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_small m d : 0 <= m < `|d|%:Z -> (m %/ d)%Z = 0. | Proof.
rewrite -(canLR (signrMK _) (divz_abs _ _)); case: m => // n /divn_small.
by rewrite divz_nat => ->; rewrite mulr0.
Qed. | Lemma | divz_small | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divn_small",
"divz_abs",
"divz_nat",
"mulr0",
"signrMK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMDl q m d : d != 0 -> ((q * d + m) %/ d)%Z = q + (m %/ d)%Z. | Proof.
rewrite neq_lt -oppr_gt0 => nz_d.
wlog{nz_d} d_gt0: q d / d > 0; last case: d => // d in d_gt0 *.
move=> IH; case/orP: nz_d => /IH// /(_ (- q)).
by rewrite mulrNN !divzN -opprD => /oppr_inj.
wlog q_gt0: q m / q >= 0; last case: q q_gt0 => // q _.
move=> IH; case: q => n; first exact: IH; rewrite NegzE mul... | Lemma | divzMDl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"NegzE",
"PoszD",
"PoszM",
"addKr",
"addNKr",
"addn0",
"addnA",
"addnAC",
"addnC",
"addnS",
"addrK",
"apply",
"d_gt0",
"divNz_nat",
"divnMDl",
"divn_eq",
"divn_small",
"divzN",
"divz_nat",
"last",
"leqP",
"leq_subr",
"ltn_mod",
"ltn_pmod",
"mulNr",
"mulSnr",
"mulr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulzK m d : d != 0 -> (m * d %/ d)%Z = m. | Proof. by move=> d_nz; rewrite -[m * d]addr0 divzMDl // div0z addr0. Qed. | Lemma | mulzK | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addr0",
"div0z",
"divzMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKz m d : d != 0 -> (d * m %/ d)%Z = m. | Proof. by move=> d_nz; rewrite mulrC mulzK. Qed. | Lemma | mulKz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"mulrC",
"mulzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expzB p m n : p != 0 -> (m >= n)%N -> p ^+ (m - n) = (p ^+ m %/ p ^+ n)%Z. | Proof. by move=> p_nz /subnK{2}<-; rewrite exprD mulzK // expf_neq0. Qed. | Lemma | expzB | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"expf_neq0",
"exprD",
"mulzK",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz1 m : (m %% 1)%Z = 0. | Proof. by case: m => n; rewrite (modNz_nat, modz_nat) ?modn1. Qed. | Lemma | modz1 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modNz_nat",
"modn1",
"modz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz1 m : (m %/ 1)%Z = m. | Proof. by rewrite -{1}[m]mulr1 mulzK. Qed. | Lemma | divz1 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"mulr1",
"mulzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzz d : (d %/ d)%Z = (d != 0). | Proof. by have [-> // | d_nz] := eqVneq; rewrite -{1}[d]mul1r mulzK. Qed. | Lemma | divzz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"eqVneq",
"mul1r",
"mulzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_pmod m d : d > 0 -> (m %% d)%Z < d. | Proof.
case: m d => n [] // d d_gt0; first by rewrite modz_nat ltz_nat ltn_pmod.
by rewrite modNz_nat // -lezD1 addrAC subrK gerDl oppr_le0.
Qed. | Lemma | ltz_pmod | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrAC",
"d_gt0",
"gerDl",
"lezD1",
"ltn_pmod",
"ltz_nat",
"modNz_nat",
"modz_nat",
"oppr_le0",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_mod m d : d != 0 -> (m %% d)%Z < `|d|. | Proof. by rewrite -absz_gt0 -modz_abs => d_gt0; apply: ltz_pmod. Qed. | Lemma | ltz_mod | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz_gt0",
"apply",
"d_gt0",
"ltz_pmod",
"modz_abs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMpl p m d : p > 0 -> (p * m %/ (p * d) = m %/ d)%Z. | Proof.
case: p => // p p_gt0; wlog d_gt0: d / d > 0; last case: d => // d in d_gt0 *.
by move=> IH; case/intP: d => [|d|d]; rewrite ?mulr0 ?divz0 ?mulrN ?divzN ?IH.
rewrite {1}(divz_eq m d) mulrDr mulrCA divzMDl ?mulf_neq0 ?gt_eqF // addrC.
rewrite divz_small ?add0r // PoszM pmulr_rge0 ?modz_ge0 ?gt_eqF //=.
by rewri... | Lemma | divzMpl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"PoszM",
"add0r",
"addrC",
"d_gt0",
"divz0",
"divzMDl",
"divzN",
"divz_eq",
"divz_small",
"gt_eqF",
"intP",
"last",
"ltr_pM2l",
"ltz_pmod",
"modz_ge0",
"mulf_neq0",
"mulr0",
"mulrCA",
"mulrDr",
"mulrN",
"p_gt0",
"pmulr_rge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMpr p m d : p > 0 -> (m * p %/ (d * p) = m %/ d)%Z. | Proof. by move=> p_gt0; rewrite -!(mulrC p) divzMpl. Qed. | Lemma | divzMpr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzMpl",
"mulrC",
"p_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_floor m d : d != 0 -> (m %/ d)%Z * d <= m. | Proof. by rewrite -subr_ge0; apply: modz_ge0. Qed. | Lemma | lez_floor | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"modz_ge0",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_div m d : (`|(m %/ d)%Z| <= `|m|)%N. | Proof.
wlog d_gt0: d / d > 0; last case: d d_gt0 => // d d_gt0.
by move=> IH; case/intP: d => [|n|n]; rewrite ?divz0 ?divzN ?abszN // IH.
case: m => n; first by rewrite divz_nat leq_div.
by rewrite divNz_nat // NegzE !abszN ltnS leq_div.
Qed. | Lemma | lez_div | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"NegzE",
"abszN",
"d_gt0",
"divNz_nat",
"divz0",
"divzN",
"divz_nat",
"intP",
"last",
"leq_div",
"ltnS"
] | leq_mod does not extend to negative m. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltz_ceil m d : d > 0 -> m < ((m %/ d)%Z + 1) * d. | Proof.
by case: d => // d d_gt0; rewrite mulrDl mul1r -ltrBlDl ltz_mod ?gt_eqF.
Qed. | Lemma | ltz_ceil | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"d_gt0",
"gt_eqF",
"ltrBlDl",
"ltz_mod",
"mul1r",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_divLR m n d : d > 0 -> ((m %/ d)%Z < n) = (m < n * d). | Proof.
move=> d_gt0; apply/idP/idP.
by rewrite -[_ < n]lezD1 -(ler_pM2r d_gt0); exact/lt_le_trans/ltz_ceil.
by rewrite -(ltr_pM2r d_gt0 _ n); apply/le_lt_trans/lez_floor; rewrite gt_eqF.
Qed. | Lemma | ltz_divLR | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"d_gt0",
"gt_eqF",
"le_lt_trans",
"ler_pM2r",
"lezD1",
"lez_floor",
"lt_le_trans",
"ltr_pM2r",
"ltz_ceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_divRL m n d : d > 0 -> (m <= (n %/ d)%Z) = (m * d <= n). | Proof. by move=> d_gt0; rewrite !leNgt ltz_divLR. Qed. | Lemma | lez_divRL | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"d_gt0",
"leNgt",
"ltz_divLR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_pdiv2r d : 0 <= d -> {homo divz^~ d : m n / m <= n}. | Proof.
by case: d => [[|d]|]// _ [] m [] n //; rewrite /divz !mul1r; apply: leq_div2r.
Qed. | Lemma | lez_pdiv2r | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"divz",
"leq_div2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_ge0 m d : d > 0 -> ((m %/ d)%Z >= 0) = (m >= 0). | Proof. by case: d m => // d [] n d_gt0; rewrite (divz_nat, divNz_nat). Qed. | Lemma | divz_ge0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"d_gt0",
"divNz_nat",
"divz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMA_ge0 m n p : n >= 0 -> (m %/ (n * p) = (m %/ n)%Z %/ p)%Z. | Proof.
case: n => // [[|n]] _; first by rewrite mul0r !divz0 div0z.
wlog p_gt0: p / p > 0; last case: p => // p in p_gt0 *.
by case/intP: p => [|p|p] IH; rewrite ?mulr0 ?divz0 ?mulrN ?divzN // IH.
rewrite {2}(divz_eq m (n.+1%:Z * p)) mulrA mulrAC !divzMDl // ?gt_eqF //.
rewrite [rhs in _ + rhs]divz_small ?addr0 // lt... | Lemma | divzMA_ge0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addr0",
"div0z",
"divz0",
"divzMDl",
"divzN",
"divz_eq",
"divz_ge0",
"divz_small",
"gt_eqF",
"intP",
"last",
"ltz_divLR",
"ltz_pmod",
"modz_ge0",
"mul0r",
"mulr0",
"mulrA",
"mulrAC",
"mulrC",
"mulrN",
"p_gt0",
"pmulr_lgt0",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_small m d : 0 <= m < d -> (m %% d)%Z = m. | Proof. by case: m d => //= m [] // d; rewrite modz_nat => /modn_small->. Qed. | Lemma | modz_small | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modn_small",
"modz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_mod m d : ((m %% d)%Z = m %[mod d])%Z. | Proof.
rewrite -!(modz_abs _ d); case: {d}`|d|%N => [|d]; first by rewrite !modz0.
by rewrite modz_small ?modz_ge0 ?ltz_mod.
Qed. | Lemma | modz_mod | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"ltz_mod",
"modz0",
"modz_abs",
"modz_ge0",
"modz_small"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzMDl p m d : (p * d + m = m %[mod d])%Z. | Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite mulr0 add0r.
by rewrite /modz divzMDl // mulrDl [_ + m]addrC addrKA.
Qed. | Lemma | modzMDl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"add0r",
"addrC",
"addrKA",
"divzMDl",
"eqVneq",
"modz",
"mulr0",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_modr {p m d} : 0 < p -> p * (m %% d)%Z = ((p * m) %% (p * d))%Z. | Proof.
case: p => // p p_gt0; rewrite mulrBr; apply: canLR (addrK _) _.
by rewrite mulrCA -(divzMpl p_gt0) subrK.
Qed. | Lemma | mulz_modr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrK",
"apply",
"divzMpl",
"mulrBr",
"mulrCA",
"p_gt0",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_modl {p m d} : 0 < p -> (m %% d)%Z * p = ((m * p) %% (d * p))%Z. | Proof. by rewrite -!(mulrC p); apply: mulz_modr. Qed. | Lemma | mulz_modl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"mulrC",
"mulz_modr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzDl m d : (d + m = m %[mod d])%Z. | Proof. by rewrite -{1}[d]mul1r modzMDl. Qed. | Lemma | modzDl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzMDl",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzDr m d : (m + d = m %[mod d])%Z. | Proof. by rewrite addrC modzDl. Qed. | Lemma | modzDr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrC",
"modzDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzz d : (d %% d)%Z = 0. | Proof. by rewrite -{1}[d]addr0 modzDl mod0z. Qed. | Lemma | modzz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addr0",
"mod0z",
"modzDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzMl p d : (p * d %% d)%Z = 0. | Proof. by rewrite -[p * d]addr0 modzMDl mod0z. Qed. | Lemma | modzMl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addr0",
"mod0z",
"modzMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzMr p d : (d * p %% d)%Z = 0. | Proof. by rewrite mulrC modzMl. Qed. | Lemma | modzMr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzMl",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzDml m n d : ((m %% d)%Z + n = m + n %[mod d])%Z. | Proof. by rewrite {2}(divz_eq m d) -[_ * d + _ + n]addrA modzMDl. Qed. | Lemma | modzDml | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrA",
"divz_eq",
"modzMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzDmr m n d : (m + (n %% d)%Z = m + n %[mod d])%Z. | Proof. by rewrite !(addrC m) modzDml. Qed. | Lemma | modzDmr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrC",
"modzDml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzDm m n d : ((m %% d)%Z + (n %% d)%Z = m + n %[mod d])%Z. | Proof. by rewrite modzDml modzDmr. Qed. | Lemma | modzDm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzDml",
"modzDmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_modDl p m n d : (p + m == p + n %[mod d])%Z = (m == n %[mod d])%Z. | Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !modz0 (inj_eq (addrI p)).
apply/eqP/eqP=> eq_mn; last by rewrite -modzDmr eq_mn modzDmr.
by rewrite -(addKr p m) -modzDmr eq_mn modzDmr addKr.
Qed. | Lemma | eqz_modDl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addKr",
"addrI",
"apply",
"eqVneq",
"inj_eq",
"last",
"modz0",
"modzDmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_modDr p m n d : (m + p == n + p %[mod d])%Z = (m == n %[mod d])%Z. | Proof. by rewrite -!(addrC p) eqz_modDl. Qed. | Lemma | eqz_modDr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addrC",
"eqz_modDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzMml m n d : ((m %% d)%Z * n = m * n %[mod d])%Z. | Proof. by rewrite {2}(divz_eq m d) [in RHS]mulrDl mulrAC modzMDl. Qed. | Lemma | modzMml | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz_eq",
"modzMDl",
"mulrAC",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzMmr m n d : (m * (n %% d)%Z = m * n %[mod d])%Z. | Proof. by rewrite !(mulrC m) modzMml. Qed. | Lemma | modzMmr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzMml",
"mulrC"
] | FIXME: rewrite pattern | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
modzMm m n d : ((m %% d)%Z * (n %% d)%Z = m * n %[mod d])%Z. | Proof. by rewrite modzMml modzMmr. Qed. | Lemma | modzMm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzMml",
"modzMmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzXm k m d : ((m %% d)%Z ^+ k = m ^+ k %[mod d])%Z. | Proof. by elim: k => // k IHk; rewrite !exprS -modzMmr IHk modzMm. Qed. | Lemma | modzXm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"exprS",
"modzMm",
"modzMmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modzNm m d : (- (m %% d)%Z = - m %[mod d])%Z. | Proof. by rewrite -mulN1r modzMmr mulN1r. Qed. | Lemma | modzNm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"modzMmr",
"mulN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modz_absm m d : ((-1) ^+ (m < 0)%R * (m %% d)%Z = `|m|%:Z %[mod d])%Z. | Proof. by rewrite modzMmr -abszEsign. Qed. | Lemma | modz_absm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszEsign",
"modzMmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdzE d m : (d %| m)%Z = (`|d| %| `|m|)%N. | Proof. by []. Qed. | Lemma | dvdzE | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [] | Divisibility * | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvdz0 d : (d %| 0)%Z. | Proof. exact: dvdn0. Qed. | Lemma | dvdz0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd0z n : (0 %| n)%Z = (n == 0). | Proof. by rewrite -absz_eq0 -dvd0n. Qed. | Lemma | dvd0z | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz_eq0",
"dvd0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz1 d : (d %| 1)%Z = (`|d|%N == 1). | Proof. exact: dvdn1. Qed. | Lemma | dvdz1 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd1z m : (1 %| m)%Z. | Proof. exact: dvd1n. Qed. | Lemma | dvd1z | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvd1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdzz m : (m %| m)%Z. | Proof. exact: dvdnn. Qed. | Lemma | dvdzz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mull d m n : (d %| n)%Z -> (d %| m * n)%Z. | Proof. by rewrite !dvdzE abszM; apply: dvdn_mull. Qed. | Lemma | dvdz_mull | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszM",
"apply",
"dvdn_mull",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mulr d m n : (d %| m)%Z -> (d %| m * n)%Z. | Proof. by move=> d_m; rewrite mulrC dvdz_mull. Qed. | Lemma | dvdz_mulr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdz_mull",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mul d1 d2 m1 m2 : (d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2)%Z. | Proof. by rewrite !dvdzE !abszM; apply: dvdn_mul. Qed. | Lemma | dvdz_mul | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszM",
"apply",
"dvdn_mul",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_trans n d m : (d %| n -> n %| m -> d %| m)%Z. | Proof. by rewrite !dvdzE; apply: dvdn_trans. Qed. | Lemma | dvdz_trans | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"dvdn_trans",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdzP d m : reflect (exists q, m = q * d) (d %| m)%Z. | Proof.
apply: (iffP dvdnP) => [] [q Dm]; last by exists `|q|%N; rewrite Dm abszM.
exists ((-1) ^+ (m < 0)%R * q%:Z * (-1) ^+ (d < 0)%R).
by rewrite -!mulrA -abszEsign -PoszM -Dm -intEsign.
Qed. | Lemma | dvdzP | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"PoszM",
"abszEsign",
"abszM",
"apply",
"dvdnP",
"intEsign",
"last",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mod0P d m : reflect (m %% d = 0)%Z (d %| m)%Z. | Proof.
apply: (iffP dvdzP) => [[q ->] | md0]; first by rewrite modzMl.
by rewrite (divz_eq m d) md0 addr0; exists (m %/ d)%Z.
Qed. | Lemma | dvdz_mod0P | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"addr0",
"apply",
"divz_eq",
"dvdzP",
"modzMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_eq d m : (d %| m)%Z = ((m %/ d)%Z * d == m). | Proof. by rewrite (sameP dvdz_mod0P eqP) subr_eq0 eq_sym. Qed. | Lemma | dvdz_eq | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdz_mod0P",
"eq_sym",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzK d m : (d %| m)%Z -> (m %/ d)%Z * d = m. | Proof. by rewrite dvdz_eq => /eqP. Qed. | Lemma | divzK | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdz_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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