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 |
|---|---|---|---|---|---|---|---|---|---|---|
lez_divLR d m n : 0 < d -> (d %| m)%Z -> ((m %/ d)%Z <= n) = (m <= n * d). | Proof. by move=> /ler_pM2r <- /divzK->. Qed. | Lemma | lez_divLR | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzK",
"ler_pM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz_divRL d m n : 0 < d -> (d %| m)%Z -> (n < m %/ d)%Z = (n * d < m). | Proof. by move=> /ltr_pM2r/(_ n)<- /divzK->. Qed. | Lemma | ltz_divRL | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzK",
"ltr_pM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_div d m n : d != 0 -> (d %| m)%Z -> (n == m %/ d)%Z = (n * d == m). | Proof. by move=> /mulIf/inj_eq <- /divzK->. Qed. | Lemma | eqz_div | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzK",
"inj_eq",
"mulIf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_mul d m n : d != 0 -> (d %| m)%Z -> (m == n * d) = (m %/ d == n)%Z. | Proof. by move=> d_gt0 dv_d_m; rewrite eq_sym -eqz_div // eq_sym. Qed. | Lemma | eqz_mul | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"d_gt0",
"eq_sym",
"eqz_div"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divz_mulAC d m n : (d %| m)%Z -> (m %/ d)%Z * n = (m * n %/ d)%Z. | Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !divz0 mul0r.
by move/divzK=> {2} <-; rewrite mulrAC mulzK.
Qed. | Lemma | divz_mulAC | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
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"zmodp",
"matrix",
"ord... | [
"divz0",
"divzK",
"eqVneq",
"mul0r",
"mulrAC",
"mulzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_divA d m n : (d %| n)%Z -> m * (n %/ d)%Z = (m * n %/ d)%Z. | Proof. by move=> dv_d_m; rewrite !(mulrC m) divz_mulAC. Qed. | Lemma | mulz_divA | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
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"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz_mulAC",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_divCA d m n :
(d %| m)%Z -> (d %| n)%Z -> m * (n %/ d)%Z = n * (m %/ d)%Z. | Proof. by move=> dv_d_m dv_d_n; rewrite mulrC divz_mulAC ?mulz_divA. Qed. | Lemma | mulz_divCA | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
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"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz_mulAC",
"mulrC",
"mulz_divA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzA m n p : (p %| n -> n %| m * p -> m %/ (n %/ p)%Z = m * p %/ n)%Z. | Proof.
move/divzK=> p_dv_n; have [->|] := eqVneq n 0; first by rewrite div0z !divz0.
rewrite -{1 2}p_dv_n mulf_eq0 => /norP[pn_nz p_nz] /divzK; rewrite mulrA p_dv_n.
by move/mulIf=> {1} <- //; rewrite mulzK.
Qed. | Lemma | divzA | 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",
"divz0",
"divzK",
"eqVneq",
"mulIf",
"mulf_eq0",
"mulrA",
"mulzK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMA m n p : (n * p %| m -> m %/ (n * p) = (m %/ n)%Z %/ p)%Z. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite mulr0 !divz0.
have [-> | nz_n] := eqVneq n 0; first by rewrite mul0r !divz0 div0z.
by move/divzK=> {2} <-; rewrite mulrA mulrAC !mulzK.
Qed. | Lemma | divzMA | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"div0z",
"divz0",
"divzK",
"eqVneq",
"mul0r",
"mulr0",
"mulrA",
"mulrAC",
"mulzK",
"nz_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzAC m n p : (n * p %| m -> (m %/ n)%Z %/ p = (m %/ p)%Z %/ n)%Z. | Proof. by move=> np_dv_mn; rewrite -!divzMA // mulrC. Qed. | Lemma | divzAC | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divzMA",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMl p m d : p != 0 -> (d %| m -> p * m %/ (p * d) = m %/ d)%Z. | Proof.
have [-> | nz_d nz_p] := eqVneq d 0; first by rewrite mulr0 !divz0.
by move/divzK=> {1}<-; rewrite mulrCA mulzK ?mulf_neq0.
Qed. | Lemma | divzMl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
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"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"divz0",
"divzK",
"eqVneq",
"mulf_neq0",
"mulr0",
"mulrCA",
"mulzK",
"nz_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzMr p m d : p != 0 -> (d %| m -> m * p %/ (d * p) = m %/ d)%Z. | Proof. by rewrite -!(mulrC p); apply: divzMl. Qed. | Lemma | divzMr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"divzMl",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mul2l p d m : p != 0 -> (p * d %| p * m)%Z = (d %| m)%Z. | Proof. by rewrite !dvdzE -absz_gt0 !abszM; apply: dvdn_pmul2l. Qed. | Lemma | dvdz_mul2l | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
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"tuple",
"bigop",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszM",
"absz_gt0",
"apply",
"dvdn_pmul2l",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_mul2r p d m : p != 0 -> (d * p %| m * p)%Z = (d %| m)%Z. | Proof. by rewrite !dvdzE -absz_gt0 !abszM; apply: dvdn_pmul2r. Qed. | Lemma | dvdz_mul2r | 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",
"absz_gt0",
"apply",
"dvdn_pmul2r",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_exp2l p m n : (m <= n)%N -> (p ^+ m %| p ^+ n)%Z. | Proof. by rewrite dvdzE !abszX; apply: dvdn_exp2l. Qed. | Lemma | dvdz_exp2l | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
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"tuple",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszX",
"apply",
"dvdn_exp2l",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_Pexp2l p m n : `|p| > 1 -> (p ^+ m %| p ^+ n)%Z = (m <= n)%N. | Proof. by rewrite dvdzE !abszX ltz_nat; apply: dvdn_Pexp2l. Qed. | Lemma | dvdz_Pexp2l | 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... | [
"abszX",
"apply",
"dvdn_Pexp2l",
"dvdzE",
"ltz_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_exp2r m n k : (m %| n -> m ^+ k %| n ^+ k)%Z. | Proof. by rewrite !dvdzE !abszX; apply: dvdn_exp2r. Qed. | Lemma | dvdz_exp2r | 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... | [
"abszX",
"apply",
"dvdn_exp2r",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_zmod_closed d : zmod_closed (dvdz d). | Proof.
split=> [|_ _ /dvdzP[p ->] /dvdzP[q ->]]; first exact: dvdz0.
by rewrite -mulrBl dvdz_mull.
Qed. | Fact | dvdz_zmod_closed | 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",
"dvdz0",
"dvdzP",
"dvdz_mull",
"mulrBl",
"split",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_exp k d m : (0 < k)%N -> (d %| m -> d %| m ^+ k)%Z. | Proof. by case: k => // k _ d_dv_m; rewrite exprS dvdz_mulr. Qed. | Lemma | dvdz_exp | 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_mulr",
"exprS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_mod_dvd d m n : (m == n %[mod d])%Z = (d %| m - n)%Z. | Proof.
apply/eqP/dvdz_mod0P=> eq_mn.
by rewrite -modzDml eq_mn modzDml subrr mod0z.
by rewrite -(subrK n m) -modzDml eq_mn add0r.
Qed. | Lemma | eqz_mod_dvd | 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",
"apply",
"dvdz_mod0P",
"mod0z",
"modzDml",
"subrK",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzDl m n d :
(d %| m)%Z -> ((m + n) %/ d)%Z = (m %/ d)%Z + (n %/ d)%Z. | Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !divz0.
by move/divzK=> {1}<-; rewrite divzMDl.
Qed. | Lemma | divzDl | 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... | [
"divz0",
"divzK",
"divzMDl",
"eqVneq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divzDr m n d :
(d %| n)%Z -> ((m + n) %/ d)%Z = (m %/ d)%Z + (n %/ d)%Z. | Proof. by move=> dv_n; rewrite addrC divzDl // addrC. Qed. | Lemma | divzDr | 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",
"divzDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_pcharf (R : nzRingType) p : p \in [pchar R] ->
forall n : int, (p %| n)%Z = (n%:~R == 0 :> R). | Proof.
move=> pcharRp [] n; rewrite [LHS](dvdn_pcharf pcharRp)//.
by rewrite NegzE abszN rmorphN// oppr_eq0.
Qed. | Lemma | dvdz_pcharf | 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",
"dvdn_pcharf",
"int",
"oppr_eq0",
"pchar",
"pcharRp",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_charf chRp | := (dvdz_pcharf chRp). | Notation | dvdz_charf | 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_pcharf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzz m : gcdz m m = `|m|%:Z. | Proof. by rewrite /gcdz gcdnn. Qed. | Lemma | gcdzz | 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... | [
"gcdnn",
"gcdz"
] | Greatest common divisor | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gcdzC : commutative gcdz. | Proof. by move=> m n; rewrite /gcdz gcdnC. Qed. | Lemma | gcdzC | 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... | [
"gcdnC",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd0z m : gcdz 0 m = `|m|%:Z. | Proof. by rewrite /gcdz gcd0n. Qed. | Lemma | gcd0z | 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... | [
"gcd0n",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz0 m : gcdz m 0 = `|m|%:Z. | Proof. by rewrite /gcdz gcdn0. Qed. | Lemma | gcdz0 | 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... | [
"gcdn0",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd1z : left_zero 1 gcdz. | Proof. by move=> m; rewrite /gcdz gcd1n. Qed. | Lemma | gcd1z | 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... | [
"gcd1n",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz1 : right_zero 1 gcdz. | Proof. by move=> m; rewrite /gcdz gcdn1. Qed. | Lemma | gcdz1 | 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... | [
"gcdn1",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_gcdr m n : (gcdz m n %| n)%Z. | Proof. exact: dvdn_gcdr. Qed. | Lemma | dvdz_gcdr | 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... | [
"dvdn_gcdr",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_gcdl m n : (gcdz m n %| m)%Z. | Proof. exact: dvdn_gcdl. Qed. | Lemma | dvdz_gcdl | 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... | [
"dvdn_gcdl",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz_eq0 m n : (gcdz m n == 0) = (m == 0) && (n == 0). | Proof. by rewrite -absz_eq0 eqn0Ngt gcdn_gt0 !negb_or -!eqn0Ngt !absz_eq0. Qed. | Lemma | gcdz_eq0 | 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",
"eqn0Ngt",
"gcdn_gt0",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdNz m n : gcdz (- m) n = gcdz m n. | Proof. by rewrite /gcdz abszN. Qed. | Lemma | gcdNz | 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... | [
"abszN",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzN m n : gcdz m (- n) = gcdz m n. | Proof. by rewrite /gcdz abszN. Qed. | Lemma | gcdzN | 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... | [
"abszN",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz_modr m n : gcdz m (n %% m)%Z = gcdz m n. | Proof.
rewrite -modz_abs /gcdz; move/absz: m => m.
have [-> | m_gt0] := posnP m; first by rewrite modz0.
case: n => n; first by rewrite modz_nat gcdn_modr.
rewrite modNz_nat // NegzE abszN {2}(divn_eq n m) -addnS gcdnMDl.
rewrite -addrA -opprD -intS /=; set m1 := _.+1.
have le_m1m: (m1 <= m)%N by apply: ltn_pmod.
by re... | Lemma | gcdz_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... | [
"NegzE",
"absz",
"abszN",
"addnS",
"addrA",
"apply",
"divn_eq",
"gcdnC",
"gcdnDl",
"gcdnDr",
"gcdnMDl",
"gcdn_modr",
"gcdz",
"intS",
"ltn_pmod",
"modNz_nat",
"modz0",
"modz_abs",
"modz_nat",
"opprD",
"posnP",
"subnK",
"subzn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz_modl m n : gcdz (m %% n)%Z n = gcdz m n. | Proof. by rewrite -!(gcdzC n) gcdz_modr. Qed. | Lemma | gcdz_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... | [
"gcdz",
"gcdzC",
"gcdz_modr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzMDl q m n : gcdz m (q * m + n) = gcdz m n. | Proof. by rewrite -gcdz_modr modzMDl gcdz_modr. Qed. | Lemma | gcdzMDl | 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... | [
"gcdz",
"gcdz_modr",
"modzMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzDl m n : gcdz m (m + n) = gcdz m n. | Proof. by rewrite -{2}(mul1r m) gcdzMDl. Qed. | Lemma | gcdzDl | 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... | [
"gcdz",
"gcdzMDl",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzDr m n : gcdz m (n + m) = gcdz m n. | Proof. by rewrite addrC gcdzDl. Qed. | Lemma | gcdzDr | 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",
"gcdz",
"gcdzDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzMl n m : gcdz n (m * n) = `|n|%:Z. | Proof. by rewrite -[m * n]addr0 gcdzMDl gcdz0. Qed. | Lemma | gcdzMl | 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",
"gcdz",
"gcdz0",
"gcdzMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzMr n m : gcdz n (n * m) = `|n|%:Z. | Proof. by rewrite mulrC gcdzMl. Qed. | Lemma | gcdzMr | 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... | [
"gcdz",
"gcdzMl",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz_idPl {m n} : reflect (gcdz m n = `|m|%:Z) (m %| n)%Z. | Proof. by apply: (iffP gcdn_idPl) => [<- | []]. Qed. | Lemma | gcdz_idPl | 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",
"gcdn_idPl",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdz_idPr {m n} : reflect (gcdz m n = `|n|%:Z) (n %| m)%Z. | Proof. by rewrite gcdzC; apply: gcdz_idPl. Qed. | Lemma | gcdz_idPr | 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",
"gcdz",
"gcdzC",
"gcdz_idPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expz_min e m n : e >= 0 -> e ^+ minn m n = gcdz (e ^+ m) (e ^+ n). | Proof.
by case: e => // e _; rewrite /gcdz !abszX -expn_min -natz -natrX !natz.
Qed. | Lemma | expz_min | 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... | [
"abszX",
"expn_min",
"gcdz",
"minn",
"natrX",
"natz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_gcd p m n : (p %| gcdz m n)%Z = (p %| m)%Z && (p %| n)%Z. | Proof. exact: dvdn_gcd. Qed. | Lemma | dvdz_gcd | 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... | [
"dvdn_gcd",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzAC : right_commutative gcdz. | Proof. by move=> m n p; rewrite /gcdz gcdnAC. Qed. | Lemma | gcdzAC | 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... | [
"gcdnAC",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzA : associative gcdz. | Proof. by move=> m n p; rewrite /gcdz gcdnA. Qed. | Lemma | gcdzA | 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... | [
"gcdnA",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzCA : left_commutative gcdz. | Proof. by move=> m n p; rewrite /gcdz gcdnCA. Qed. | Lemma | gcdzCA | 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... | [
"gcdnCA",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdzACA : interchange gcdz gcdz. | Proof. by move=> m n p q; rewrite /gcdz gcdnACA. Qed. | Lemma | gcdzACA | 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... | [
"gcdnACA",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_gcdr m n p : `|m|%:Z * gcdz n p = gcdz (m * n) (m * p). | Proof. by rewrite -PoszM muln_gcdr -!abszM. Qed. | Lemma | mulz_gcdr | 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",
"abszM",
"gcdz",
"muln_gcdr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_gcdl m n p : gcdz m n * `|p|%:Z = gcdz (m * p) (n * p). | Proof. by rewrite -PoszM muln_gcdl -!abszM. Qed. | Lemma | mulz_gcdl | 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",
"abszM",
"gcdz",
"muln_gcdl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_divCA_gcd n m : n * (m %/ gcdz n m)%Z = m * (n %/ gcdz n m)%Z. | Proof. by rewrite mulz_divCA ?dvdz_gcdl ?dvdz_gcdr. Qed. | Lemma | mulz_divCA_gcd | algebra | algebra/intdiv.v | [
"HB",
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"zmodp",
"matrix",
"ord... | [
"dvdz_gcdl",
"dvdz_gcdr",
"gcdz",
"mulz_divCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_lcmr m n : (n %| lcmz m n)%Z. | Proof. exact: dvdn_lcmr. Qed. | Lemma | dvdz_lcmr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdn_lcmr",
"lcmz"
] | Least common multiple | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvdz_lcml m n : (m %| lcmz m n)%Z. | Proof. exact: dvdn_lcml. Qed. | Lemma | dvdz_lcml | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdn_lcml",
"lcmz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_lcm d1 d2 m : ((lcmn d1 d2 %| m) = (d1 %| m) && (d2 %| m))%Z. | Proof. exact: dvdn_lcm. Qed. | Lemma | dvdz_lcm | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdn_lcm",
"lcmn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmzC : commutative lcmz. | Proof. by move=> m n; rewrite /lcmz lcmnC. Qed. | Lemma | lcmzC | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"lcmnC",
"lcmz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcm0z : left_zero 0 lcmz. | Proof. by move=> x; rewrite /lcmz absz0 lcm0n. Qed. | Lemma | lcm0z | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz0",
"lcm0n",
"lcmz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmz0 : right_zero 0 lcmz. | Proof. by move=> x; rewrite /lcmz absz0 lcmn0. Qed. | Lemma | lcmz0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz0",
"lcmn0",
"lcmz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmz_ge0 m n : 0 <= lcmz m n. | Proof. by []. Qed. | Lemma | lcmz_ge0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"lcmz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmz_neq0 m n : (lcmz m n != 0) = (m != 0) && (n != 0). | Proof.
have [->|m_neq0] := eqVneq m 0; first by rewrite lcm0z.
have [->|n_neq0] := eqVneq n 0; first by rewrite lcmz0.
by rewrite gt_eqF// [0 < _]lcmn_gt0 !absz_gt0 m_neq0 n_neq0.
Qed. | Lemma | lcmz_neq0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
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"zmodp",
"matrix",
"ord... | [
"absz_gt0",
"eqVneq",
"gt_eqF",
"lcm0z",
"lcmn_gt0",
"lcmz",
"lcmz0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezE m n : coprimez m n = coprime `|m| `|n|. | Proof. by []. Qed. | Lemma | coprimezE | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"coprime",
"coprimez"
] | Coprime factors | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprimez_sym : symmetric coprimez. | Proof. by move=> m n; apply: coprime_sym. Qed. | Lemma | coprimez_sym | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"coprime_sym",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimeNz m n : coprimez (- m) n = coprimez m n. | Proof. by rewrite coprimezE abszN. Qed. | Lemma | coprimeNz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszN",
"coprimez",
"coprimezE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezN m n : coprimez m (- n) = coprimez m n. | Proof. by rewrite coprimezE abszN. Qed. | Lemma | coprimezN | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszN",
"coprimez",
"coprimezE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdz_spec m n : int * int -> Type | :=
EgcdzSpec u v of u * m + v * n = gcdz m n & coprimez u v
: egcdz_spec m n (u, v). | Variant | egcdz_spec | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"coprimez",
"gcdz",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdzP m n : egcdz_spec m n (egcdz m n). | Proof.
rewrite /egcdz; have [-> | m_nz] := eqVneq.
by split; [rewrite -abszEsign gcd0z | rewrite coprimezE absz_sign].
have m_gt0 : (`|m| > 0)%N by rewrite absz_gt0.
case: egcdnP (coprime_egcdn `|n| m_gt0) => //= u v Duv _ co_uv; split.
rewrite !mulNr -!mulrA mulrCA -abszEsg mulrCA -abszEsign.
by rewrite -!PoszM ... | Lemma | egcdzP | algebra | algebra/intdiv.v | [
"HB",
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"mathcomp",
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"egcdz",
"egcdz_spec",
"eqVneq",
"gcd0z",
"mul1n",
"mulNr",
"mulrA",
"mulrCA",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezoutz m n : {u : int & {v : int | u * m + v * n = gcdz m n}}. | Proof. by exists (egcdz m n).1, (egcdz m n).2; case: egcdzP. Qed. | Lemma | Bezoutz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"egcdz",
"egcdzP",
"gcdz",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezP m n :
reflect (exists uv, uv.1 * m + uv.2 * n = 1) (coprimez m n). | Proof.
apply: (iffP eqP) => [<-| [[u v] /= Duv]].
by exists (egcdz m n); case: egcdzP.
congr _%:Z; apply: gcdn_def; rewrite ?dvd1n // => d dv_d_n dv_d_m.
by rewrite -(dvdzE d 1) -Duv [m]intEsg [n]intEsg rpredD ?dvdz_mull.
Qed. | Lemma | coprimezP | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
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"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"coprimez",
"dvd1n",
"dvdzE",
"dvdz_mull",
"egcdz",
"egcdzP",
"gcdn_def",
"intEsg",
"rpredD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdz m n p :
coprimez m n -> (m * n %| p)%Z = (m %| p)%Z && (n %| p)%Z. | Proof. by move/Gauss_dvd <-; rewrite -abszM. Qed. | Lemma | Gauss_dvdz | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
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"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Gauss_dvd",
"abszM",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdzr m n p : coprimez m n -> (m %| n * p)%Z = (m %| p)%Z. | Proof. by rewrite dvdzE abszM => /Gauss_dvdr->. Qed. | Lemma | Gauss_dvdzr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Gauss_dvdr",
"abszM",
"coprimez",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdzl m n p : coprimez m p -> (m %| n * p)%Z = (m %| n)%Z. | Proof. by rewrite mulrC; apply: Gauss_dvdzr. Qed. | Lemma | Gauss_dvdzl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Gauss_dvdzr",
"apply",
"coprimez",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_gcdzr p m n : coprimez p m -> gcdz p (m * n) = gcdz p n. | Proof. by rewrite /gcdz abszM => /Gauss_gcdr->. Qed. | Lemma | Gauss_gcdzr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Gauss_gcdr",
"abszM",
"coprimez",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_gcdzl p m n : coprimez p n -> gcdz p (m * n) = gcdz p m. | Proof. by move=> co_pn; rewrite mulrC Gauss_gcdzr. Qed. | Lemma | Gauss_gcdzl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Gauss_gcdzr",
"coprimez",
"gcdz",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezMr p m n : coprimez p (m * n) = coprimez p m && coprimez p n. | Proof. by rewrite -coprimeMr -abszM. Qed. | Lemma | coprimezMr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszM",
"coprimeMr",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezMl p m n : coprimez (m * n) p = coprimez m p && coprimez n p. | Proof. by rewrite -coprimeMl -abszM. Qed. | Lemma | coprimezMl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszM",
"coprimeMl",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimez_pexpl k m n : (0 < k)%N -> coprimez (m ^+ k) n = coprimez m n. | Proof. by rewrite /coprimez /gcdz abszX; apply: coprime_pexpl. Qed. | Lemma | coprimez_pexpl | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"abszX",
"apply",
"coprime_pexpl",
"coprimez",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimez_pexpr k m n : (0 < k)%N -> coprimez m (n ^+ k) = coprimez m n. | Proof. by move=> k_gt0; rewrite !(coprimez_sym m) coprimez_pexpl. Qed. | Lemma | coprimez_pexpr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"perm",
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"divalg",
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"polydiv",
"zmodp",
"matrix",
"ord... | [
"coprimez",
"coprimez_pexpl",
"coprimez_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezXl k m n : coprimez m n -> coprimez (m ^+ k) n. | Proof. by rewrite /coprimez /gcdz abszX; apply: coprimeXl. Qed. | Lemma | coprimezXl | 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... | [
"abszX",
"apply",
"coprimeXl",
"coprimez",
"gcdz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimezXr k m n : coprimez m n -> coprimez m (n ^+ k). | Proof. by rewrite !(coprimez_sym m); apply: coprimezXl. Qed. | Lemma | coprimezXr | 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",
"coprimez",
"coprimezXl",
"coprimez_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimez_dvdl m n p : (m %| n)%N -> coprimez n p -> coprimez m p. | Proof. exact: coprime_dvdl. Qed. | Lemma | coprimez_dvdl | 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... | [
"coprime_dvdl",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimez_dvdr m n p : (m %| n)%N -> coprimez p n -> coprimez p m. | Proof. exact: coprime_dvdr. Qed. | Lemma | coprimez_dvdr | 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... | [
"coprime_dvdr",
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_pexp2r m n k : (k > 0)%N -> (m ^+ k %| n ^+ k)%Z = (m %| n)%Z. | Proof. by rewrite dvdzE !abszX; apply: dvdn_pexp2r. Qed. | Lemma | dvdz_pexp2r | 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... | [
"abszX",
"apply",
"dvdn_pexp2r",
"dvdzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
co_m12 : coprimez m1 m2. | Hypothesis | co_m12 | 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... | [
"coprimez"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
zchinese_remainder x y :
(x == y %[mod m1 * m2])%Z = (x == y %[mod m1])%Z && (x == y %[mod m2])%Z. | Proof. by rewrite !eqz_mod_dvd Gauss_dvdz. Qed. | Lemma | zchinese_remainder | 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... | [
"Gauss_dvdz",
"eqz_mod_dvd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchinese r1 r2 | :=
r1 * m2 * (egcdz m1 m2).2 + r2 * m1 * (egcdz m1 m2).1. | Definition | zchinese | 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... | [
"egcdz",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchinese_modl r1 r2 : (zchinese r1 r2 = r1 %[mod m1])%Z. | Proof.
rewrite /zchinese; have [u v /= Duv _] := egcdzP m1 m2.
rewrite -[r1 in RHS]mulr1 -((gcdz _ _ =P 1) co_m12) -Duv.
by rewrite mulrDr addrC mulrAC 2!mulrA 2!modzMDl mulrAC.
Qed. | Lemma | zchinese_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... | [
"addrC",
"co_m12",
"egcdzP",
"gcdz",
"modzMDl",
"mulr1",
"mulrA",
"mulrAC",
"mulrDr",
"r1",
"r2",
"zchinese"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchinese_modr r1 r2 : (zchinese r1 r2 = r2 %[mod m2])%Z. | Proof.
rewrite /zchinese; have [u v /= Duv _] := egcdzP m1 m2.
rewrite -[r2 in RHS]mulr1 -((gcdz _ _ =P 1) co_m12) -Duv.
by rewrite mulrAC modzMDl mulrAC addrC mulrDr !mulrA modzMDl.
Qed. | Lemma | zchinese_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... | [
"addrC",
"co_m12",
"egcdzP",
"gcdz",
"modzMDl",
"mulr1",
"mulrA",
"mulrAC",
"mulrDr",
"r1",
"r2",
"zchinese"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchinese_mod x : (x = zchinese (x %% m1)%Z (x %% m2)%Z %[mod m1 * m2])%Z. | Proof.
apply/eqP; rewrite zchinese_remainder //.
by rewrite zchinese_modl zchinese_modr !modz_mod !eqxx.
Qed. | Lemma | zchinese_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... | [
"apply",
"eqxx",
"modz_mod",
"zchinese",
"zchinese_modl",
"zchinese_modr",
"zchinese_remainder"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontents (p : {poly int}) : int | :=
sgz (lead_coef p) * \big[gcdn/0]_(i < size p) `|(p`_i)%R|%N. | Definition | zcontents | 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... | [
"gcdn",
"int",
"lead_coef",
"poly",
"sgz",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_contents p : sgz (zcontents p) = sgz (lead_coef p). | Proof.
rewrite /zcontents mulrC sgzM sgz_id; set d := _%:Z.
have [-> | nz_p] := eqVneq p 0; first by rewrite lead_coef0 mulr0.
rewrite gtr0_sgz ?mul1r // ltz_nat polySpred ?big_ord_recr //= -lead_coefE.
by rewrite gcdn_gt0 orbC absz_gt0 lead_coef_eq0 nz_p.
Qed. | Lemma | sgz_contents | 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",
"big_ord_recr",
"eqVneq",
"gcdn_gt0",
"gtr0_sgz",
"lead_coef",
"lead_coef0",
"lead_coefE",
"lead_coef_eq0",
"ltz_nat",
"mul1r",
"mulr0",
"mulrC",
"nz_p",
"polySpred",
"sgz",
"sgzM",
"sgz_id",
"zcontents"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontents_eq0 p : (zcontents p == 0) = (p == 0). | Proof. by rewrite -sgz_eq0 sgz_contents sgz_eq0 lead_coef_eq0. Qed. | Lemma | zcontents_eq0 | 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... | [
"lead_coef_eq0",
"sgz_contents",
"sgz_eq0",
"zcontents"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontents0 : zcontents 0 = 0. | Proof. by apply/eqP; rewrite zcontents_eq0. Qed. | Lemma | zcontents0 | 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",
"zcontents",
"zcontents_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontentsZ a p : zcontents (a *: p) = a * zcontents p. | Proof.
have [-> | nz_a] := eqVneq a 0; first by rewrite scale0r mul0r zcontents0.
rewrite {2}[a]intEsg mulrCA -mulrA -PoszM big_distrr /= mulrCA mulrA -sgzM.
rewrite -lead_coefZ; congr (_ * _%:Z); rewrite size_scale //.
by apply: eq_bigr => i _; rewrite coefZ abszM.
Qed. | Lemma | zcontentsZ | 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",
"abszM",
"apply",
"big_distrr",
"coefZ",
"eqVneq",
"eq_bigr",
"intEsg",
"lead_coefZ",
"mul0r",
"mulrA",
"mulrCA",
"scale0r",
"sgzM",
"size_scale",
"zcontents",
"zcontents0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontents_monic p : p \is monic -> zcontents p = 1. | Proof.
move=> mon_p; rewrite /zcontents polySpred ?monic_neq0 //.
by rewrite big_ord_recr /= -lead_coefE (monicP mon_p) gcdn1.
Qed. | Lemma | zcontents_monic | 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... | [
"big_ord_recr",
"gcdn1",
"lead_coefE",
"monic",
"monicP",
"monic_neq0",
"polySpred",
"zcontents"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdz_contents a p : (a %| zcontents p)%Z = (p \is a polyOver (dvdz a)). | Proof.
rewrite dvdzE abszM absz_sg lead_coef_eq0.
have [-> | nz_p] := eqVneq; first by rewrite mul0n dvdn0 rpred0.
rewrite mul1n; apply/dvdn_biggcdP/(all_nthP 0)=> a_dv_p i ltip /=.
exact: (a_dv_p (Ordinal ltip)).
exact: a_dv_p.
Qed. | Lemma | dvdz_contents | 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",
"absz_sg",
"all_nthP",
"apply",
"dvdn0",
"dvdn_biggcdP",
"dvdz",
"dvdzE",
"eqVneq",
"lead_coef_eq0",
"mul0n",
"mul1n",
"nz_p",
"polyOver",
"rpred0",
"zcontents"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_divzK {a} p :
p \is a polyOver (dvdz a) -> a *: map_poly (divz^~ a) p = p. | Proof.
move/polyOverP=> a_dv_p; apply/polyP=> i.
by rewrite coefZ coef_map_id0 ?div0z // mulrC divzK.
Qed. | Lemma | map_poly_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... | [
"apply",
"coefZ",
"coef_map_id0",
"div0z",
"divz",
"divzK",
"dvdz",
"map_poly",
"mulrC",
"polyOver",
"polyOverP",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyOver_dvdzP a p :
reflect (exists q, p = a *: q) (p \is a polyOver (dvdz a)). | Proof.
apply: (iffP idP) => [/map_poly_divzK | [q ->]].
by exists (map_poly (divz^~ a) p).
by apply/polyOverP=> i; rewrite coefZ dvdz_mulr.
Qed. | Lemma | polyOver_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... | [
"apply",
"coefZ",
"divz",
"dvdz",
"dvdz_mulr",
"map_poly",
"map_poly_divzK",
"polyOver",
"polyOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive p | := map_poly (divz^~ (zcontents p)) p. | Definition | zprimitive | 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",
"map_poly",
"zcontents"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zpolyEprim p : p = zcontents p *: zprimitive p. | Proof. by rewrite map_poly_divzK // -dvdz_contents. Qed. | Lemma | zpolyEprim | 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_contents",
"map_poly_divzK",
"zcontents",
"zprimitive"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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