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 |
|---|---|---|---|---|---|---|---|---|---|---|
dvdn_mul d1 d2 m1 m2 : d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2. | Proof.
by move=> /dvdnP[q1 ->] /dvdnP[q2 ->]; rewrite mulnCA -mulnA 2?dvdn_mull.
Qed. | Lemma | dvdn_mul | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdnP",
"dvdn_mull",
"mulnA",
"mulnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_trans n d m : d %| n -> n %| m -> d %| m. | Proof. by move=> d_dv_n /dvdnP[n1 ->]; apply: dvdn_mull. Qed. | Lemma | dvdn_trans | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdnP",
"dvdn_mull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_eq d m : (d %| m) = (m %/ d * d == m). | Proof.
apply/eqP/eqP=> [modm0 | <-]; last exact: modnMl.
by rewrite [RHS](divn_eq m d) modm0 addn0.
Qed. | Lemma | dvdn_eq | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addn0",
"apply",
"divn_eq",
"last",
"modnMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn2 n : (2 %| n) = ~~ odd n. | Proof. by rewrite /dvdn modn2; case (odd n). Qed. | Lemma | dvdn2 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn",
"modn2",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_odd m n : m %| n -> odd n -> odd m. | Proof. by move=> m_dv_n; apply: contraTT; rewrite -!dvdn2 => /dvdn_trans->. Qed. | Lemma | dvdn_odd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn2",
"dvdn_trans",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnK d m : d %| m -> m %/ d * d = m. | Proof. by rewrite dvdn_eq; move/eqP. Qed. | Lemma | divnK | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_divLR d m n : d %| m -> (m %/ d <= n) = (m <= n * d). | Proof. by case: d m => [|d] [|m] ///divnK=> {2}<-; rewrite leq_pmul2r. Qed. | Lemma | leq_divLR | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"leq_pmul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_divRL d m n : d %| m -> (n < m %/ d) = (n * d < m). | Proof. by move=> dv_d_m; rewrite !ltnNge leq_divLR. Qed. | Lemma | ltn_divRL | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"leq_divLR",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_div d m n : d > 0 -> d %| m -> (n == m %/ d) = (n * d == m). | Proof. by move=> d_gt0 dv_d_m; rewrite -(eqn_pmul2r d_gt0) divnK. Qed. | Lemma | eqn_div | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"d_gt0",
"divnK",
"eqn_pmul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_mul d m n : d > 0 -> d %| m -> (m == n * d) = (m %/ d == n). | Proof. by move=> d_gt0 dv_d_m; rewrite eq_sym -eqn_div // eq_sym. Qed. | Lemma | eqn_mul | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"d_gt0",
"eq_sym",
"eqn_div"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divn_mulAC d m n : d %| m -> m %/ d * n = m * n %/ d. | Proof.
case: d m => [[] //| d m] dv_d_m; apply/eqP.
by rewrite eqn_div ?dvdn_mulr // mulnAC divnK.
Qed. | Lemma | divn_mulAC | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"divnK",
"dvdn_mulr",
"eqn_div",
"mulnAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_divA d m n : d %| n -> m * (n %/ d) = m * n %/ d. | Proof. by move=> dv_d_m; rewrite !(mulnC m) divn_mulAC. Qed. | Lemma | muln_divA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn_mulAC",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_divCA d m n : d %| m -> d %| n -> m * (n %/ d) = n * (m %/ d). | Proof. by move=> dv_d_m dv_d_n; rewrite mulnC divn_mulAC ?muln_divA. Qed. | Lemma | muln_divCA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn_mulAC",
"mulnC",
"muln_divA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnA m n p : p %| n -> m %/ (n %/ p) = m * p %/ n. | Proof. by case: p => [|p] dv_n; rewrite -[in RHS](divnK dv_n) // divnMr. Qed. | Lemma | divnA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"divnMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modn_dvdm m n d : d %| m -> n %% m = n %[mod d]. | Proof.
by case/dvdnP=> q def_m; rewrite [in RHS](divn_eq n m) def_m mulnA modnMDl.
Qed. | Lemma | modn_dvdm | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn_eq",
"dvdnP",
"modnMDl",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_leq d m : 0 < m -> d %| m -> d <= m. | Proof. by move=> m_gt0 /dvdnP[[|k] Dm]; rewrite Dm // leq_addr in m_gt0 *. Qed. | Lemma | dvdn_leq | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdnP",
"leq_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtnNdvd n d : 0 < n -> n < d -> (d %| n) = false. | Proof. by move=> n_gt0 lt_nd; rewrite /dvdn eqn0Ngt modn_small ?n_gt0. Qed. | Lemma | gtnNdvd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn",
"eqn0Ngt",
"modn_small",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_dvd m n : (m == n) = (m %| n) && (n %| m). | Proof.
case: m n => [|m] [|n] //; apply/idP/andP => [/eqP -> //| []].
by rewrite eqn_leq => Hmn Hnm; do 2 rewrite dvdn_leq //.
Qed. | Lemma | eqn_dvd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn_leq",
"eqn_leq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_pmul2l p d m : 0 < p -> (p * d %| p * m) = (d %| m). | Proof. by case: p => // p _; rewrite /dvdn -muln_modr // muln_eq0. Qed. | Lemma | dvdn_pmul2l | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn",
"muln_eq0",
"muln_modr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_pmul2r p d m : 0 < p -> (d * p %| m * p) = (d %| m). | Proof. by move=> p_gt0; rewrite -!(mulnC p) dvdn_pmul2l. Qed. | Lemma | dvdn_pmul2r | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_pmul2l",
"mulnC",
"p_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_divLR p d m : 0 < p -> p %| d -> (d %/ p %| m) = (d %| m * p). | Proof. by move=> /(@dvdn_pmul2r p _ m) <- /divnK->. Qed. | Lemma | dvdn_divLR | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"dvdn_pmul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_divRL p d m : p %| m -> (d %| m %/ p) = (d * p %| m). | Proof.
have [-> | /(@dvdn_pmul2r p d) <- /divnK-> //] := posnP p.
by rewrite divn0 muln0 dvdn0.
Qed. | Lemma | dvdn_divRL | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn0",
"divnK",
"dvdn0",
"dvdn_pmul2r",
"muln0",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_div d m : d %| m -> m %/ d %| m. | Proof. by move/divnK=> {2}<-; apply: dvdn_mulr. Qed. | Lemma | dvdn_div | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"divnK",
"dvdn_mulr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_exp2l p m n : m <= n -> p ^ m %| p ^ n. | Proof. by move/subnK <-; rewrite expnD dvdn_mull. Qed. | Lemma | dvdn_exp2l | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_mull",
"expnD",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_Pexp2l p m n : p > 1 -> (p ^ m %| p ^ n) = (m <= n). | Proof.
move=> p_gt1; case: leqP => [|gt_n_m]; first exact: dvdn_exp2l.
by rewrite gtnNdvd ?ltn_exp2l ?expn_gt0 // ltnW.
Qed. | Lemma | dvdn_Pexp2l | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_exp2l",
"expn_gt0",
"gtnNdvd",
"leqP",
"ltnW",
"ltn_exp2l",
"p_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_exp2r m n k : m %| n -> m ^ k %| n ^ k. | Proof. by case/dvdnP=> q ->; rewrite expnMn dvdn_mull. Qed. | Lemma | dvdn_exp2r | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdnP",
"dvdn_mull",
"expnMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divn_modl m n d : d %| n -> (m %% n) %/ d = (m %/ d) %% (n %/ d). | Proof. by move=> dvd_dn; rewrite modn_divl divnK. Qed. | Lemma | divn_modl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"modn_divl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_addr m d n : d %| m -> (d %| m + n) = (d %| n). | Proof. by case/dvdnP=> q ->; rewrite /dvdn modnMDl. Qed. | Lemma | dvdn_addr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn",
"dvdnP",
"modnMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_addl n d m : d %| n -> (d %| m + n) = (d %| m). | Proof. by rewrite addnC; apply: dvdn_addr. Qed. | Lemma | dvdn_addl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"apply",
"dvdn_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_add d m n : d %| m -> d %| n -> d %| m + n. | Proof. by move/dvdn_addr->. Qed. | Lemma | dvdn_add | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_add_eq d m n : d %| m + n -> (d %| m) = (d %| n). | Proof. by move=> dv_d_mn; apply/idP/idP => [/dvdn_addr | /dvdn_addl] <-. Qed. | Lemma | dvdn_add_eq | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn_addl",
"dvdn_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_subr d m n : n <= m -> d %| m -> (d %| m - n) = (d %| n). | Proof. by move=> le_n_m dv_d_m; apply: dvdn_add_eq; rewrite subnK. Qed. | Lemma | dvdn_subr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn_add_eq",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_subl d m n : n <= m -> d %| n -> (d %| m - n) = (d %| m). | Proof. by move=> le_n_m dv_d_m; rewrite -(dvdn_addl _ dv_d_m) subnK. Qed. | Lemma | dvdn_subl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_addl",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_sub d m n : d %| m -> d %| n -> d %| m - n. | Proof.
by case: (leqP n m) => [le_nm /dvdn_subr <- // | /ltnW/eqnP ->]; rewrite dvdn0.
Qed. | Lemma | dvdn_sub | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn0",
"dvdn_subr",
"eqnP",
"leqP",
"ltnW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_exp k d m : 0 < k -> d %| m -> d %| (m ^ k). | Proof. by case: k => // k _ d_dv_m; rewrite expnS dvdn_mulr. Qed. | Lemma | dvdn_exp | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_mulr",
"expnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_fact m n : 0 < m <= n -> m %| n`!. | Proof.
case: m => //= m; elim: n => //= n IHn; rewrite ltnS.
have [/IHn/dvdn_mull->||-> _] // := ltngtP m n; exact: dvdn_mulr.
Qed. | Lemma | dvdn_fact | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_mull",
"dvdn_mulr",
"ltnS",
"ltngtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_mod_dvd d m n : n <= m -> (m == n %[mod d]) = (d %| m - n). | Proof.
by move/subnK=> Dm; rewrite -[n in LHS]add0n -[in LHS]Dm eqn_modDr mod0n.
Qed. | Lemma | eqn_mod_dvd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"add0n",
"eqn_modDr",
"mod0n",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnDMl q m d : 0 < d -> (m + q * d) %/ d = (m %/ d) + q. | Proof. by move=> d_gt0; rewrite addnC divnMDl// addnC. Qed. | Lemma | divnDMl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"d_gt0",
"divnMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnMBl q m d : 0 < d -> (q * d - m) %/ d = q - (m %/ d) - (~~ (d %| m)). | Proof. by move=> d_gt0; rewrite divnB// mulnK// modnMl lt0n. Qed. | Lemma | divnMBl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"d_gt0",
"divnB",
"lt0n",
"modnMl",
"mulnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnBMl q m d : (m - q * d) %/ d = (m %/ d) - q. | Proof. by case: d => [|d]//=; rewrite divnB// mulnK// modnMl ltn0 subn0. Qed. | Lemma | divnBMl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnB",
"ltn0",
"modnMl",
"mulnK",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnDl m n d : d %| m -> (m + n) %/ d = m %/ d + n %/ d. | Proof. by case: d => // d /divnK-Dm; rewrite -[in LHS]Dm divnMDl. Qed. | Lemma | divnDl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"divnMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnDr m n d : d %| n -> (m + n) %/ d = m %/ d + n %/ d. | Proof. by move=> dv_n; rewrite addnC divnDl // addnC. Qed. | Lemma | divnDr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"divnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnBl m n d : d %| m -> (m - n) %/ d = m %/ d - (n %/ d) - (~~ (d %| n)). | Proof. by case: d => [|d] // /divnK-Dm; rewrite -[in LHS]Dm divnMBl. Qed. | Lemma | divnBl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnK",
"divnMBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnBr m n d : d %| n -> (m - n) %/ d = m %/ d - n %/ d. | Proof. by case: d => [|d]// /divnK-Dm; rewrite -[in LHS]Dm divnBMl. Qed. | Lemma | divnBr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnBMl",
"divnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivnS m d : 0 < d -> edivn m.+1 d =
if d %| m.+1 then ((m %/ d).+1, 0) else (m %/ d, (m %% d).+1). | Proof.
case: d => [|[|d]] //= _; first by rewrite edivn_def modn1 dvd1n !divn1.
rewrite -addn1 /dvdn modn_def edivnD//= (@modn_small 1)// (@divn_small 1)//.
rewrite addn1 addn0 ltnS; have [||<-] := ltngtP d.+1.
- by rewrite ltnNge -ltnS ltn_pmod.
- by rewrite addn0 mul0n subn0.
- by rewrite addn1 mul1n subnn.
Qed. | Lemma | edivnS | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addn0",
"addn1",
"divn1",
"divn_small",
"dvd1n",
"dvdn",
"edivn",
"edivnD",
"edivn_def",
"ltnNge",
"ltnS",
"ltn_pmod",
"ltngtP",
"modn1",
"modn_def",
"modn_small",
"mul0n",
"mul1n",
"subn0",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modnS m d : m.+1 %% d = if d %| m.+1 then 0 else (m %% d).+1. | Proof. by case: d => [|d]//; rewrite modn_def edivnS//; case: ifP. Qed. | Lemma | modnS | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"edivnS",
"modn_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divnS m d : 0 < d -> m.+1 %/ d = (d %| m.+1) + m %/ d. | Proof. by move=> d_gt0; rewrite /divn edivnS//; case: ifP. Qed. | Lemma | divnS | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"d_gt0",
"divn",
"edivnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divn_pred m d : m.-1 %/ d = (m %/ d) - (d %| m). | Proof.
by case: d m => [|d] [|m]; rewrite ?divn1 ?dvd1n ?subn1//= divnS// addnC addnK.
Qed. | Lemma | divn_pred | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"addnK",
"divn1",
"divnS",
"dvd1n",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modn_pred m d : d != 1 -> 0 < m ->
m.-1 %% d = if d %| m then d.-1 else (m %% d).-1. | Proof.
rewrite -subn1; case: d m => [|[|d]] [|m]//= _ _.
by rewrite ?modn1 ?dvd1n ?modn0 ?subn1.
rewrite modnB// (@modn_small 1)// [_ < _]leqn0 /dvdn mulnbl/= subn1.
by case: eqP => // ->; rewrite addn0.
Qed. | Lemma | modn_pred | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addn0",
"dvd1n",
"dvdn",
"leqn0",
"modn0",
"modn1",
"modnB",
"modn_small",
"mulnbl",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivn_pred m d : d != 1 -> 0 < m ->
edivn m.-1 d = if d %| m then ((m %/ d).-1, d.-1) else (m %/ d, (m %% d).-1). | Proof.
move=> d_neq1 m_gt0; rewrite edivn_def divn_pred modn_pred//.
by case: ifP; rewrite ?subn0 ?subn1.
Qed. | Lemma | edivn_pred | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn_pred",
"edivn",
"edivn_def",
"modn_pred",
"subn0",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn m n | :=
let n' := n %% m in if n' is 0 then m else
if m - n'.-1 is m'.+1 then gcdn (m' %% n') n' else n'. | Fixpoint | gcdn | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnE m n : gcdn m n = if m == 0 then n else gcdn (n %% m) m. | Proof.
elim/ltn_ind: m n => -[|m] IHm [|n] //=; rewrite /gcdn -/gcdn.
case def_p: (_ %% _) => // [p].
have{def_p} lt_pm: p.+1 < m.+1 by rewrite -def_p ltn_pmod.
rewrite {}IHm // subn_if_gt ltnW //=; congr gcdn.
by rewrite -(subnK (ltnW lt_pm)) modnDr.
Qed. | Lemma | gcdnE | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"def_p",
"gcdn",
"ltnW",
"ltn_ind",
"ltn_pmod",
"modnDr",
"subnK",
"subn_if_gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnn : idempotent_op gcdn. | Proof. by case=> // n; rewrite gcdnE modnn. Qed. | Lemma | gcdnn | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnE",
"idempotent_op",
"modnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnC : commutative gcdn. | Proof.
move=> m n; wlog lt_nm: m n / n < m by have [? ->|? <-|-> //] := ltngtP n m.
by rewrite gcdnE -[in m == 0](ltn_predK lt_nm) modn_small.
Qed. | Lemma | gcdnC | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnE",
"ltn_predK",
"ltngtP",
"modn_small"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd0n : left_id 0 gcdn. | Proof. by case. Qed. | Lemma | gcd0n | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn0 : right_id 0 gcdn. | Proof. by case. Qed. | Lemma | gcdn0 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd1n : left_zero 1 gcdn. | Proof. by move=> n; rewrite gcdnE modn1. Qed. | Lemma | gcd1n | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnE",
"modn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn1 : right_zero 1 gcdn. | Proof. by move=> n; rewrite gcdnC gcd1n. Qed. | Lemma | gcdn1 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcd1n",
"gcdn",
"gcdnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_gcdr m n : gcdn m n %| n. | Proof.
elim/ltn_ind: m n => -[|m] IHm [|n] //=.
rewrite gcdnE; case def_p: (_ %% _) => [|p]; first by rewrite /dvdn def_p.
have lt_pm: p < m by rewrite -ltnS -def_p ltn_pmod.
rewrite /= (divn_eq n.+1 m.+1) def_p dvdn_addr ?dvdn_mull //; first exact: IHm.
by rewrite gcdnE /= IHm // (ltn_trans (ltn_pmod _ _)).
Qed. | Lemma | dvdn_gcdr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"def_p",
"divn_eq",
"dvdn",
"dvdn_addr",
"dvdn_mull",
"gcdn",
"gcdnE",
"ltnS",
"ltn_ind",
"ltn_pmod",
"ltn_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_gcdl m n : gcdn m n %| m. | Proof. by rewrite gcdnC dvdn_gcdr. Qed. | Lemma | dvdn_gcdl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_gcdr",
"gcdn",
"gcdnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_gt0 m n : (0 < gcdn m n) = (0 < m) || (0 < n). | Proof.
by case: m n => [|m] [|n] //; apply: (@dvdn_gt0 _ m.+1) => //; apply: dvdn_gcdl.
Qed. | Lemma | gcdn_gt0 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn_gcdl",
"dvdn_gt0",
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnMDl k m n : gcdn m (k * m + n) = gcdn m n. | Proof. by rewrite !(gcdnE m) modnMDl mulnC; case: m. Qed. | Lemma | gcdnMDl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnE",
"modnMDl",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnDl m n : gcdn m (m + n) = gcdn m n. | Proof. by rewrite -[m in m + n]mul1n gcdnMDl. Qed. | Lemma | gcdnDl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnMDl",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnDr m n : gcdn m (n + m) = gcdn m n. | Proof. by rewrite addnC gcdnDl. Qed. | Lemma | gcdnDr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"gcdn",
"gcdnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnMl n m : gcdn n (m * n) = n. | Proof. by case: n => [|n]; rewrite gcdnE modnMl // muln0. Qed. | Lemma | gcdnMl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnE",
"modnMl",
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnMr n m : gcdn n (n * m) = n. | Proof. by rewrite mulnC gcdnMl. Qed. | Lemma | gcdnMr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnMl",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_idPl {m n} : reflect (gcdn m n = m) (m %| n). | Proof.
by apply: (iffP idP) => [/dvdnP[q ->] | <-]; rewrite (gcdnMl, dvdn_gcdr).
Qed. | Lemma | gcdn_idPl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdnP",
"dvdn_gcdr",
"gcdn",
"gcdnMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_idPr {m n} : reflect (gcdn m n = n) (n %| m). | Proof. by rewrite gcdnC; apply: gcdn_idPl. Qed. | Lemma | gcdn_idPr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"gcdn",
"gcdnC",
"gcdn_idPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn_min e m n : e ^ minn m n = gcdn (e ^ m) (e ^ n). | Proof. by case: leqP => [|/ltnW] /(dvdn_exp2l e) /gcdn_idPl; rewrite gcdnC. Qed. | Lemma | expn_min | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_exp2l",
"gcdn",
"gcdnC",
"gcdn_idPl",
"leqP",
"ltnW",
"minn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_modr m n : gcdn m (n %% m) = gcdn m n. | Proof. by rewrite [in RHS](divn_eq n m) gcdnMDl. Qed. | Lemma | gcdn_modr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn_eq",
"gcdn",
"gcdnMDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_modl m n : gcdn (m %% n) n = gcdn m n. | Proof. by rewrite !(gcdnC _ n) gcdn_modr. Qed. | Lemma | gcdn_modl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnC",
"gcdn_modr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezout_rec km kn qs | :=
if qs is q :: qs' then Bezout_rec kn (NatTrec.add_mul q kn km) qs'
else (km, kn). | Fixpoint | Bezout_rec | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"add_mul"
] | Extended gcd, which computes Bezout coefficients. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
egcdn_rec m n s qs | :=
if s is s'.+1 then
let: (q, r) := edivn m n in
if r > 0 then egcdn_rec n r s' (q :: qs) else
if odd (size qs) then qs else q.-1 :: qs
else [::0]. | Fixpoint | egcdn_rec | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"edivn",
"odd",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdn m n | := Bezout_rec 0 1 (egcdn_rec m n n [::]). | Definition | egcdn | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Bezout_rec",
"egcdn_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdn_spec m n : nat * nat -> Type | :=
EgcdnSpec km kn of km * m = kn * n + gcdn m n & kn * gcdn m n < m :
egcdn_spec m n (km, kn). | Variant | egcdn_spec | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcd0n n : egcdn 0 n = (1, 0). | Proof. by case: n. Qed. | Lemma | egcd0n | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"egcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdnP m n : m > 0 -> egcdn_spec m n (egcdn m n). | Proof.
have [-> /= | n_gt0 m_gt0] := posnP n; first by split; rewrite // mul1n gcdn0.
rewrite /egcdn; set s := (s in egcdn_rec _ _ s); pose bz := Bezout_rec n m [::].
have: n < s.+1 by []; move defSpec: (egcdn_spec bz.2 bz.1) s => Spec s.
elim: s => [[]|s IHs] //= in n m (qs := [::]) bz defSpec n_gt0 m_gt0 *.
case: edi... | Lemma | egcdnP | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Bezout_rec",
"addIn",
"addn",
"addn0",
"addn1",
"addnA",
"addnACA",
"addnC",
"addn_gt0",
"addn_negb",
"addnn",
"apply",
"d_gt0",
"dvdn_gt0",
"dvdn_mulr",
"edivnP",
"egcdn",
"egcdn_rec",
"egcdn_spec",
"gcd0n",
"gcdn",
"gcdn0",
"gcdnC",
"gcdnE",
"gcdnMDl",
"gcdn_gt0"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezoutl m n : m > 0 -> {a | a < m & m %| gcdn m n + a * n}. | Proof.
move=> m_gt0; case: (egcdnP n m_gt0) => km kn def_d lt_kn_m.
exists kn; last by rewrite addnC -def_d dvdn_mull.
apply: leq_ltn_trans lt_kn_m.
by rewrite -{1}[kn]muln1 leq_mul2l gcdn_gt0 m_gt0 orbT.
Qed. | Lemma | Bezoutl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"addnC",
"apply",
"dvdn_mull",
"egcdnP",
"gcdn",
"gcdn_gt0",
"last",
"leq_ltn_trans",
"leq_mul2l",
"muln1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezoutr m n : n > 0 -> {a | a < n & n %| gcdn m n + a * m}. | Proof. by rewrite gcdnC; apply: Bezoutl. Qed. | Lemma | Bezoutr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Bezoutl",
"apply",
"gcdn",
"gcdnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_gcd p m n : (p %| gcdn m n) = (p %| m) && (p %| n). | Proof.
apply/idP/andP=> [dv_pmn | [dv_pm dv_pn]].
by rewrite !(dvdn_trans dv_pmn) ?dvdn_gcdl ?dvdn_gcdr.
have [->|n_gt0] := posnP n; first by rewrite gcdn0.
case: (Bezoutr m n_gt0) => // km _ /(dvdn_trans dv_pn).
by rewrite dvdn_addl // dvdn_mull.
Qed. | Lemma | dvdn_gcd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Bezoutr",
"apply",
"dvdn_addl",
"dvdn_gcdl",
"dvdn_gcdr",
"dvdn_mull",
"dvdn_trans",
"gcdn",
"gcdn0",
"n_gt0",
"posnP"
] | Back to the gcd. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gcdnAC : right_commutative gcdn. | Proof.
suffices dvd m n p: gcdn (gcdn m n) p %| gcdn (gcdn m p) n.
by move=> m n p; apply/eqP; rewrite eqn_dvd !dvd.
rewrite !dvdn_gcd dvdn_gcdr.
by rewrite !(dvdn_trans (dvdn_gcdl _ p)) ?dvdn_gcdl ?dvdn_gcdr.
Qed. | Lemma | gcdnAC | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvd",
"dvdn_gcd",
"dvdn_gcdl",
"dvdn_gcdr",
"dvdn_trans",
"eqn_dvd",
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnA : associative gcdn. | Proof. by move=> m n p; rewrite !(gcdnC m) gcdnAC. Qed. | Lemma | gcdnA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnAC",
"gcdnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnCA : left_commutative gcdn. | Proof. by move=> m n p; rewrite !gcdnA (gcdnC m). Qed. | Lemma | gcdnCA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnA",
"gcdnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdnACA : interchange gcdn gcdn. | Proof. by move=> m n p q; rewrite -!gcdnA (gcdnCA n). Qed. | Lemma | gcdnACA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"gcdnA",
"gcdnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_gcdr : right_distributive muln gcdn. | Proof.
move=> p m n; have [-> //|p_gt0] := posnP p.
elim/ltn_ind: m n => m IHm n; rewrite gcdnE [RHS]gcdnE muln_eq0 (gtn_eqF p_gt0).
by case: posnP => // m_gt0; rewrite -muln_modr //=; apply/IHm/ltn_pmod.
Qed. | Lemma | muln_gcdr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"gcdn",
"gcdnE",
"gtn_eqF",
"ltn_ind",
"ltn_pmod",
"muln",
"muln_eq0",
"muln_modr",
"p_gt0",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_gcdl : left_distributive muln gcdn. | Proof. by move=> m n p; rewrite -!(mulnC p) muln_gcdr. Qed. | Lemma | muln_gcdl | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn",
"muln",
"mulnC",
"muln_gcdr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdn_def d m n :
d %| m -> d %| n -> (forall d', d' %| m -> d' %| n -> d' %| d) ->
gcdn m n = d. | Proof.
move=> dv_dm dv_dn gdv_d; apply/eqP.
by rewrite eqn_dvd dvdn_gcd dv_dm dv_dn gdv_d ?dvdn_gcdl ?dvdn_gcdr.
Qed. | Lemma | gcdn_def | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"dvdn_gcd",
"dvdn_gcdl",
"dvdn_gcdr",
"eqn_dvd",
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_divCA_gcd n m : n * (m %/ gcdn n m) = m * (n %/ gcdn n m). | Proof. by rewrite muln_divCA ?dvdn_gcdl ?dvdn_gcdr. Qed. | Lemma | muln_divCA_gcd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_gcdl",
"dvdn_gcdr",
"gcdn",
"muln_divCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmn m n | := m * n %/ gcdn m n. | Definition | lcmn | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdn"
] | We derive the lcm directly. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lcmnC : commutative lcmn. | Proof. by move=> m n; rewrite /lcmn mulnC gcdnC. Qed. | Lemma | lcmnC | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"gcdnC",
"lcmn",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcm0n : left_zero 0 lcmn. | Proof. by move=> n; apply: div0n. Qed. | Lemma | lcm0n | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"div0n",
"lcmn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmn0 : right_zero 0 lcmn. | Proof. by move=> n; rewrite lcmnC lcm0n. Qed. | Lemma | lcmn0 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"lcm0n",
"lcmn",
"lcmnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcm1n : left_id 1 lcmn. | Proof. by move=> n; rewrite /lcmn gcd1n mul1n divn1. Qed. | Lemma | lcm1n | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divn1",
"gcd1n",
"lcmn",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmn1 : right_id 1 lcmn. | Proof. by move=> n; rewrite lcmnC lcm1n. Qed. | Lemma | lcmn1 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"lcm1n",
"lcmn",
"lcmnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_lcm_gcd m n : lcmn m n * gcdn m n = m * n. | Proof. by apply/eqP; rewrite divnK ?dvdn_mull ?dvdn_gcdr. Qed. | Lemma | muln_lcm_gcd | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"divnK",
"dvdn_gcdr",
"dvdn_mull",
"gcdn",
"lcmn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmn_gt0 m n : (0 < lcmn m n) = (0 < m) && (0 < n). | Proof. by rewrite -muln_gt0 ltn_divRL ?dvdn_mull ?dvdn_gcdr. Qed. | Lemma | lcmn_gt0 | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"dvdn_gcdr",
"dvdn_mull",
"lcmn",
"ltn_divRL",
"muln_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_lcmr : right_distributive muln lcmn. | Proof.
case=> // m n p; rewrite /lcmn -muln_gcdr -!mulnA divnMl // mulnCA.
by rewrite muln_divA ?dvdn_mull ?dvdn_gcdr.
Qed. | Lemma | muln_lcmr | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnMl",
"dvdn_gcdr",
"dvdn_mull",
"lcmn",
"muln",
"mulnA",
"mulnCA",
"muln_divA",
"muln_gcdr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_lcml : left_distributive muln lcmn. | Proof. by move=> m n p; rewrite -!(mulnC p) muln_lcmr. Qed. | Lemma | muln_lcml | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"lcmn",
"muln",
"mulnC",
"muln_lcmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmnA : associative lcmn. | Proof.
move=> m n p; rewrite [LHS]/lcmn [RHS]/lcmn mulnC.
rewrite !divn_mulAC ?dvdn_mull ?dvdn_gcdr // -!divnMA ?dvdn_mulr ?dvdn_gcdl //.
rewrite mulnC mulnA !muln_gcdr; congr (_ %/ _).
by rewrite ![_ * lcmn _ _]mulnC !muln_lcm_gcd !muln_gcdl -!(mulnC m) gcdnA.
Qed. | Lemma | lcmnA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"divnMA",
"divn_mulAC",
"dvdn_gcdl",
"dvdn_gcdr",
"dvdn_mull",
"dvdn_mulr",
"gcdnA",
"lcmn",
"mulnA",
"mulnC",
"muln_gcdl",
"muln_gcdr",
"muln_lcm_gcd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcmnCA : left_commutative lcmn. | Proof. by move=> m n p; rewrite !lcmnA (lcmnC m). Qed. | Lemma | lcmnCA | boot | boot/div.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"lcmn",
"lcmnA",
"lcmnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.