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 |
|---|---|---|---|---|---|---|---|---|---|---|
mulnE : muln = mult. | Proof. by []. Qed. | Lemma | mulnE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul0n : left_zero 0 muln. | Proof. by []. Qed. | Lemma | mul0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln0 : right_zero 0 muln. | Proof. by elim. Qed. | Lemma | muln0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1n : left_id 1 muln. | Proof. exact: addn0. Qed. | Lemma | mul1n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn0",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulSn m n : m.+1 * n = n + m * n. | Proof. by []. Qed. | Lemma | mulSn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulSnr m n : m.+1 * n = m * n + n. | Proof. exact: addnC. Qed. | Lemma | mulSnr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnS m n : m * n.+1 = m + m * n. | Proof. by elim: m => // m; rewrite !mulSn !addSn addnCA => ->. Qed. | Lemma | mulnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addSn",
"addnCA",
"mulSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnSr m n : m * n.+1 = m * n + m. | Proof. by rewrite addnC mulnS. Qed. | Lemma | mulnSr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"mulnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_addn m n p : iter n (addn m) p = m * n + p. | Proof. by elim: n => /= [|n ->]; rewrite ?muln0 // mulnS addnA. Qed. | Lemma | iter_addn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnA",
"iter",
"muln0",
"mulnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_addn_0 m n : iter n (addn m) 0 = m * n. | Proof. by rewrite iter_addn addn0. Qed. | Lemma | iter_addn_0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addn0",
"iter",
"iter_addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln1 : right_id 1 muln. | Proof. by move=> n; rewrite mulnSr muln0. Qed. | Lemma | muln1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln",
"muln0",
"mulnSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnC : commutative muln. | Proof.
by move=> m n; elim: m => [|m]; rewrite (muln0, mulnS) // mulSn => ->.
Qed. | Lemma | mulnC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mulSn",
"muln",
"muln0",
"mulnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnDl : left_distributive muln addn. | Proof. by move=> m1 m2 n; elim: m1 => //= m1 IHm; rewrite -addnA -IHm. Qed. | Lemma | mulnDl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnA",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnDr : right_distributive muln addn. | Proof. by move=> m n1 n2; rewrite !(mulnC m) mulnDl. Qed. | Lemma | mulnDr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"muln",
"mulnC",
"mulnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnBl : left_distributive muln subn. | Proof.
move=> m n [|p]; first by rewrite !muln0.
by elim: m n => // [m IHm] [|n] //; rewrite mulSn subnDl -IHm.
Qed. | Lemma | mulnBl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mulSn",
"muln",
"muln0",
"subn",
"subnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnBr : right_distributive muln subn. | Proof. by move=> m n p; rewrite !(mulnC m) mulnBl. Qed. | Lemma | mulnBr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln",
"mulnBl",
"mulnC",
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnA : associative muln. | Proof. by move=> m n p; elim: m => //= m; rewrite mulSn mulnDl => ->. Qed. | Lemma | mulnA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mulSn",
"muln",
"mulnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnCA : left_commutative muln. | Proof. by move=> m n1 n2; rewrite !mulnA (mulnC m). Qed. | Lemma | mulnCA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln",
"mulnA",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnAC : right_commutative muln. | Proof. by move=> m n p; rewrite -!mulnA (mulnC n). Qed. | Lemma | mulnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln",
"mulnA",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnACA : interchange muln muln. | Proof. by move=> m n p q; rewrite -!mulnA (mulnCA n). Qed. | Lemma | mulnACA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln",
"mulnA",
"mulnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_eq0 m n : (m * n == 0) = (m == 0) || (n == 0). | Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. | Lemma | muln_eq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_eq1 m n : (m * n == 1) = (m == 1) && (n == 1). | Proof. by case: m n => [|[|m]] [|[|n]] //; rewrite muln0. Qed. | Lemma | muln_eq1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_gt0 m n : (0 < m * n) = (0 < m) && (0 < n). | Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. | Lemma | muln_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pmull m n : n > 0 -> m <= n * m. | Proof. by move/prednK <-; apply: leq_addr. Qed. | Lemma | leq_pmull | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_addr",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pmulr m n : n > 0 -> m <= m * n. | Proof. by move/leq_pmull; rewrite mulnC. Qed. | Lemma | leq_pmulr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_pmull",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_mul2l m n1 n2 : (m * n1 <= m * n2) = (m == 0) || (n1 <= n2). | Proof. by rewrite [LHS]/leq -mulnBr muln_eq0. Qed. | Lemma | leq_mul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq",
"mulnBr",
"muln_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_mul2r m n1 n2 : (n1 * m <= n2 * m) = (m == 0) || (n1 <= n2). | Proof. by rewrite -!(mulnC m) leq_mul2l. Qed. | Lemma | leq_mul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_mul2l",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_mul m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 * m2 <= n1 * n2. | Proof.
move=> le_mn1 le_mn2; apply (@leq_trans (m1 * n2)).
by rewrite leq_mul2l le_mn2 orbT.
by rewrite leq_mul2r le_mn1 orbT.
Qed. | Lemma | leq_mul | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_mul2l",
"leq_mul2r",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_mul2l m n1 n2 : (m * n1 == m * n2) = (m == 0) || (n1 == n2). | Proof. by rewrite eqn_leq !leq_mul2l -orb_andr -eqn_leq. Qed. | Lemma | eqn_mul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leq_mul2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_mul2r m n1 n2 : (n1 * m == n2 * m) = (m == 0) || (n1 == n2). | Proof. by rewrite eqn_leq !leq_mul2r -orb_andr -eqn_leq. Qed. | Lemma | eqn_mul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leq_mul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pmul2l m n1 n2 : 0 < m -> (m * n1 <= m * n2) = (n1 <= n2). | Proof. by move/prednK=> <-; rewrite leq_mul2l. Qed. | Lemma | leq_pmul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_mul2l",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pmul2r m n1 n2 : 0 < m -> (n1 * m <= n2 * m) = (n1 <= n2). | Proof. by move/prednK <-; rewrite leq_mul2r. Qed. | Lemma | leq_pmul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_mul2r",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_pmul2l m n1 n2 : 0 < m -> (m * n1 == m * n2) = (n1 == n2). | Proof. by move/prednK <-; rewrite eqn_mul2l. Qed. | Lemma | eqn_pmul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_mul2l",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_pmul2r m n1 n2 : 0 < m -> (n1 * m == n2 * m) = (n1 == n2). | Proof. by move/prednK <-; rewrite eqn_mul2r. Qed. | Lemma | eqn_pmul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_mul2r",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_mul2l m n1 n2 : (m * n1 < m * n2) = (0 < m) && (n1 < n2). | Proof. by rewrite lt0n !ltnNge leq_mul2l negb_or. Qed. | Lemma | ltn_mul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_mul2l",
"lt0n",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_mul2r m n1 n2 : (n1 * m < n2 * m) = (0 < m) && (n1 < n2). | Proof. by rewrite lt0n !ltnNge leq_mul2r negb_or. Qed. | Lemma | ltn_mul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_mul2r",
"lt0n",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_pmul2l m n1 n2 : 0 < m -> (m * n1 < m * n2) = (n1 < n2). | Proof. by move/prednK <-; rewrite ltn_mul2l. Qed. | Lemma | ltn_pmul2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_mul2l",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_pmul2r m n1 n2 : 0 < m -> (n1 * m < n2 * m) = (n1 < n2). | Proof. by move/prednK <-; rewrite ltn_mul2r. Qed. | Lemma | ltn_pmul2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_mul2r",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_Pmull m n : 1 < n -> 0 < m -> m < n * m. | Proof. by move=> lt1n m_gt0; rewrite -[ltnLHS]mul1n ltn_pmul2r. Qed. | Lemma | ltn_Pmull | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnLHS",
"ltn_pmul2r",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_Pmulr m n : 1 < n -> 0 < m -> m < m * n. | Proof. by move=> lt1n m_gt0; rewrite mulnC ltn_Pmull. Qed. | Lemma | ltn_Pmulr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_Pmull",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_mull m1 m2 n1 n2 : 0 < n2 -> m1 < n1 -> m2 <= n2 -> m1 * m2 < n1 * n2. | Proof.
move=> n20 lt_mn1 le_mn2.
rewrite (@leq_ltn_trans (m1 * n2)) ?leq_mul2l ?le_mn2 ?orbT//.
by rewrite ltn_mul2r lt_mn1 n20.
Qed. | Lemma | ltn_mull | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_ltn_trans",
"leq_mul2l",
"ltn_mul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_mulr m1 m2 n1 n2 : 0 < n1 -> m1 <= n1 -> m2 < n2 -> m1 * m2 < n1 * n2. | Proof. by move=> ? ? ?; rewrite mulnC [ltnRHS]mulnC ltn_mull. Qed. | Lemma | ltn_mulr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnRHS",
"ltn_mull",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_mul m1 m2 n1 n2 : m1 < n1 -> m2 < n2 -> m1 * m2 < n1 * n2. | Proof. by move=> ? lt2; rewrite ltn_mull ?(leq_ltn_trans _ lt2)// ltnW. Qed. | Lemma | ltn_mul | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_ltn_trans",
"ltnW",
"ltn_mull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnMr : right_distributive muln maxn. | Proof. by case=> // m n1 n2; rewrite /maxn (fun_if (muln _)) ltn_pmul2l. Qed. | Lemma | maxnMr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_pmul2l",
"maxn",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnMl : left_distributive muln maxn. | Proof. by move=> m1 m2 n; rewrite -!(mulnC n) maxnMr. Qed. | Lemma | maxnMl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"maxn",
"maxnMr",
"muln",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minnMr : right_distributive muln minn. | Proof. by case=> // m n1 n2; rewrite /minn (fun_if (muln _)) ltn_pmul2l. Qed. | Lemma | minnMr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_pmul2l",
"minn",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minnMl : left_distributive muln minn. | Proof. by move=> m1 m2 n; rewrite -!(mulnC n) minnMr. Qed. | Lemma | minnMl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"minn",
"minnMr",
"muln",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iterM (T : Type) (n m : nat) (f : T -> T) :
iter (n * m) f =1 iter n (iter m f). | Proof. by move=> x; elim: n => //= n <-; rewrite mulSn iterD. Qed. | Lemma | iterM | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"iter",
"iterD",
"mulSn",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn m n | := iterop n muln m 1. | Definition | expn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"iterop",
"muln"
] | Exponentiation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
expn_rec | := expn. | Definition | expn_rec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m ^ n" | := (expn m n) : nat_scope. | Notation | m ^ n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnE n m : expn m n = iterop n muln m 1. | Proof. by []. Qed. | Lemma | expnE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expn",
"iterop",
"muln"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn0 m : m ^ 0 = 1. | Proof. by []. Qed. | Lemma | expn0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn1 m : m ^ 1 = m. | Proof. by []. Qed. | Lemma | expn1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnS m n : m ^ n.+1 = m * m ^ n. | Proof. by case: n; rewrite ?muln1. Qed. | Lemma | expnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnSr m n : m ^ n.+1 = m ^ n * m. | Proof. by rewrite mulnC expnS. Qed. | Lemma | expnSr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_muln m n p : iter n (muln m) p = m ^ n * p. | Proof. by elim: n => /= [|n ->]; rewrite ?mul1n // expnS mulnA. Qed. | Lemma | iter_muln | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"iter",
"mul1n",
"muln",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_muln_1 m n : iter n (muln m) 1 = m ^ n. | Proof. by rewrite iter_muln muln1. Qed. | Lemma | iter_muln_1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"iter",
"iter_muln",
"muln",
"muln1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp0n n : 0 < n -> 0 ^ n = 0. | Proof. by case: n => [|[]]. Qed. | Lemma | exp0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp1n n : 1 ^ n = 1. | Proof. by elim: n => // n; rewrite expnS mul1n. Qed. | Lemma | exp1n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnD m n1 n2 : m ^ (n1 + n2) = m ^ n1 * m ^ n2. | Proof. by elim: n1 => [|n1 IHn]; rewrite !(mul1n, expnS) // IHn mulnA. Qed. | Lemma | expnD | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"mul1n",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnMn m1 m2 n : (m1 * m2) ^ n = m1 ^ n * m2 ^ n. | Proof. by elim: n => // n IHn; rewrite !expnS IHn -!mulnA (mulnCA m2). Qed. | Lemma | expnMn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"mulnA",
"mulnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnM m n1 n2 : m ^ (n1 * n2) = (m ^ n1) ^ n2. | Proof.
elim: n1 => [|n1 IHn]; first by rewrite exp1n.
by rewrite expnD expnS expnMn IHn.
Qed. | Lemma | expnM | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"exp1n",
"expnD",
"expnMn",
"expnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnAC m n1 n2 : (m ^ n1) ^ n2 = (m ^ n2) ^ n1. | Proof. by rewrite -!expnM mulnC. Qed. | Lemma | expnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnM",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn_gt0 m n : (0 < m ^ n) = (0 < m) || (n == 0). | Proof.
by case: m => [|m]; elim: n => //= n IHn; rewrite expnS // addn_gt0 IHn.
Qed. | Lemma | expn_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn_gt0",
"expnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn_eq0 m e : (m ^ e == 0) = (m == 0) && (e > 0). | Proof. by rewrite !eqn0Ngt expn_gt0 negb_or -lt0n. Qed. | Lemma | expn_eq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn0Ngt",
"expn_gt0",
"lt0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_expl m n : 1 < m -> n < m ^ n. | Proof.
move=> m_gt1; elim: n => //= n; rewrite -(leq_pmul2l (ltnW m_gt1)) expnS.
by apply: leq_trans; apply: ltn_Pmull.
Qed. | Lemma | ltn_expl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"expnS",
"leq_pmul2l",
"leq_trans",
"ltnW",
"ltn_Pmull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_exp2l m n1 n2 : 1 < m -> (m ^ n1 <= m ^ n2) = (n1 <= n2). | Proof.
move=> m_gt1; elim: n1 n2 => [|n1 IHn] [|n2] //; last 1 first.
- by rewrite !expnS leq_pmul2l ?IHn // ltnW.
- by rewrite expn_gt0 ltnW.
by rewrite leqNgt (leq_trans m_gt1) // expnS leq_pmulr // expn_gt0 ltnW.
Qed. | Lemma | leq_exp2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"expn_gt0",
"last",
"leqNgt",
"leq_pmul2l",
"leq_pmulr",
"leq_trans",
"ltnW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_exp2l m n1 n2 : 1 < m -> (m ^ n1 < m ^ n2) = (n1 < n2). | Proof. by move=> m_gt1; rewrite !ltnNge leq_exp2l. Qed. | Lemma | ltn_exp2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_exp2l",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_exp2l m n1 n2 : 1 < m -> (m ^ n1 == m ^ n2) = (n1 == n2). | Proof. by move=> m_gt1; rewrite !eqn_leq !leq_exp2l. Qed. | Lemma | eqn_exp2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leq_exp2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expnI m : 1 < m -> injective (expn m). | Proof. by move=> m_gt1 e1 e2 /eqP; rewrite eqn_exp2l // => /eqP. Qed. | Lemma | expnI | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_exp2l",
"expn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pexp2l m n1 n2 : 0 < m -> n1 <= n2 -> m ^ n1 <= m ^ n2. | Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite leq_exp2l]. Qed. | Lemma | leq_pexp2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"exp1n",
"leq_exp2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_pexp2l m n1 n2 : 0 < m -> m ^ n1 < m ^ n2 -> n1 < n2. | Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite ltn_exp2l]. Qed. | Lemma | ltn_pexp2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"exp1n",
"ltn_exp2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_exp2r m n e : e > 0 -> (m ^ e < n ^ e) = (m < n). | Proof.
move=> e_gt0; apply/idP/idP=> [|ltmn].
rewrite !ltnNge; apply: contra => lemn.
by elim: e {e_gt0} => // e IHe; rewrite !expnS leq_mul.
by elim: e e_gt0 => // [[|e] IHe] _; rewrite ?expn1 // ltn_mul // IHe.
Qed. | Lemma | ltn_exp2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"expn1",
"expnS",
"leq_mul",
"ltnNge",
"ltn_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_exp2r m n e : e > 0 -> (m ^ e <= n ^ e) = (m <= n). | Proof. by move=> e_gt0; rewrite leqNgt ltn_exp2r // -leqNgt. Qed. | Lemma | leq_exp2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqNgt",
"ltn_exp2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_exp2r m n e : e > 0 -> (m ^ e == n ^ e) = (m == n). | Proof. by move=> e_gt0; rewrite !eqn_leq !leq_exp2r. Qed. | Lemma | eqn_exp2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leq_exp2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expIn e : e > 0 -> injective (expn^~ e). | Proof. by move=> e_gt1 m n /eqP; rewrite eqn_exp2r // => /eqP. Qed. | Lemma | expIn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_exp2r",
"expn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iterX (T : Type) (n m : nat) (f : T -> T) :
iter (n ^ m) f =1 iter m (iter n) f. | Proof. elim: m => //= m ihm x; rewrite expnS iterM; exact/eq_iter. Qed. | Lemma | iterX | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_iter",
"expnS",
"iter",
"iterM",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factorial n | := if n is n'.+1 then n * factorial n' else 1. | Fixpoint | factorial | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | Factorial. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
fact_rec | := factorial. | Definition | fact_rec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"factorial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n `!" | := (factorial n) (at level 1, format "n `!") : nat_scope. | Notation | n `! | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"factorial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factE n : factorial n = if n is n'.+1 then n * factorial n' else 1. | Proof. by case: n. Qed. | Lemma | factE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"factorial",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fact0 : 0`! = 1. | Proof. by []. Qed. | Lemma | fact0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factS n : (n.+1)`! = n.+1 * n`!. | Proof. by []. Qed. | Lemma | factS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fact_gt0 n : n`! > 0. | Proof. by elim: n => //= n IHn; rewrite muln_gt0. Qed. | Lemma | fact_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fact_geq n : n <= n`!. | Proof. by case: n => // n; rewrite factS -(addn1 n) leq_pmulr ?fact_gt0. Qed. | Lemma | fact_geq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn1",
"factS",
"fact_gt0",
"leq_pmulr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_fact m n : 0 < m -> m < n -> m`! < n`!. | Proof.
case: m n => // m n _; elim: n m => // n ih [|m] ?; last by rewrite ltn_mul ?ih.
by rewrite -[_.+1]muln1 leq_mul ?fact_gt0.
Qed. | Lemma | ltn_fact | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"fact_gt0",
"last",
"leq_mul",
"ltn_mul",
"muln1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_of_bool (b : bool) | := if b then 1 else 0. | Coercion | nat_of_bool | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | Parity and bits. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
leq_b1 (b : bool) : b <= 1. | Proof. by case: b. Qed. | Lemma | leq_b1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn_negb (b : bool) : ~~ b + b = 1. | Proof. by case: b. Qed. | Lemma | addn_negb | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqb0 (b : bool) : (b == 0 :> nat) = ~~ b. | Proof. by case: b. Qed. | Lemma | eqb0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqb1 (b : bool) : (b == 1 :> nat) = b. | Proof. by case: b. Qed. | Lemma | eqb1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0b (b : bool) : (b > 0) = b. | Proof. by case: b. Qed. | Lemma | lt0b | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub1b (b : bool) : 1 - b = ~~ b. | Proof. by case: b. Qed. | Lemma | sub1b | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnb (b1 b2 : bool) : b1 * b2 = b1 && b2. | Proof. by case: b1; case: b2. Qed. | Lemma | mulnb | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnbl (b : bool) n : b * n = (if b then n else 0). | Proof. by case: b; rewrite ?mul1n. Qed. | Lemma | mulnbl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnbr (b : bool) n : n * b = (if b then n else 0). | Proof. by rewrite mulnC mulnbl. Qed. | Lemma | mulnbr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mulnC",
"mulnbl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd n | := if n is n'.+1 then ~~ odd n' else false. | Fixpoint | odd | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddS n : odd n.+1 = ~~ odd n. | Proof. by []. Qed. | Lemma | oddS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddb (b : bool) : odd b = b. | Proof. by case: b. Qed. | Lemma | oddb | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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