fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
muln_eq0m n : (m * n == 0) = (m == 0) || (n == 0).
Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | muln_eq0 | |
muln_eq1m n : (m * n == 1) = (m == 1) && (n == 1).
Proof. by case: m n => [|[|m]] [|[|n]] //; rewrite muln0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | muln_eq1 | |
muln_gt0m n : (0 < m * n) = (0 < m) && (0 < n).
Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | muln_gt0 | |
leq_pmullm n : n > 0 -> m <= n * m.
Proof. by move/prednK <-; apply: leq_addr. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pmull | |
leq_pmulrm n : n > 0 -> m <= m * n.
Proof. by move/leq_pmull; rewrite mulnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pmulr | |
leq_mul2lm n1 n2 : (m * n1 <= m * n2) = (m == 0) || (n1 <= n2).
Proof. by rewrite [LHS]/leq -mulnBr muln_eq0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_mul2l | |
leq_mul2rm n1 n2 : (n1 * m <= n2 * m) = (m == 0) || (n1 <= n2).
Proof. by rewrite -!(mulnC m) leq_mul2l. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_mul2r | |
leq_mulm1 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_mul | |
eqn_mul2lm n1 n2 : (m * n1 == m * n2) = (m == 0) || (n1 == n2).
Proof. by rewrite eqn_leq !leq_mul2l -orb_andr -eqn_leq. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_mul2l | |
eqn_mul2rm n1 n2 : (n1 * m == n2 * m) = (m == 0) || (n1 == n2).
Proof. by rewrite eqn_leq !leq_mul2r -orb_andr -eqn_leq. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_mul2r | |
leq_pmul2lm n1 n2 : 0 < m -> (m * n1 <= m * n2) = (n1 <= n2).
Proof. by move/prednK=> <-; rewrite leq_mul2l. Qed.
Arguments leq_pmul2l [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pmul2l | |
leq_pmul2rm n1 n2 : 0 < m -> (n1 * m <= n2 * m) = (n1 <= n2).
Proof. by move/prednK <-; rewrite leq_mul2r. Qed.
Arguments leq_pmul2r [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pmul2r | |
eqn_pmul2lm n1 n2 : 0 < m -> (m * n1 == m * n2) = (n1 == n2).
Proof. by move/prednK <-; rewrite eqn_mul2l. Qed.
Arguments eqn_pmul2l [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_pmul2l | |
eqn_pmul2rm n1 n2 : 0 < m -> (n1 * m == n2 * m) = (n1 == n2).
Proof. by move/prednK <-; rewrite eqn_mul2r. Qed.
Arguments eqn_pmul2r [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_pmul2r | |
ltn_mul2lm n1 n2 : (m * n1 < m * n2) = (0 < m) && (n1 < n2).
Proof. by rewrite lt0n !ltnNge leq_mul2l negb_or. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_mul2l | |
ltn_mul2rm n1 n2 : (n1 * m < n2 * m) = (0 < m) && (n1 < n2).
Proof. by rewrite lt0n !ltnNge leq_mul2r negb_or. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_mul2r | |
ltn_pmul2lm n1 n2 : 0 < m -> (m * n1 < m * n2) = (n1 < n2).
Proof. by move/prednK <-; rewrite ltn_mul2l. Qed.
Arguments ltn_pmul2l [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_pmul2l | |
ltn_pmul2rm n1 n2 : 0 < m -> (n1 * m < n2 * m) = (n1 < n2).
Proof. by move/prednK <-; rewrite ltn_mul2r. Qed.
Arguments ltn_pmul2r [m n1 n2]. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_pmul2r | |
ltn_Pmullm n : 1 < n -> 0 < m -> m < n * m.
Proof. by move=> lt1n m_gt0; rewrite -[ltnLHS]mul1n ltn_pmul2r. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_Pmull | |
ltn_Pmulrm n : 1 < n -> 0 < m -> m < m * n.
Proof. by move=> lt1n m_gt0; rewrite mulnC ltn_Pmull. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_Pmulr | |
ltn_mullm1 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_mull | |
ltn_mulrm1 m2 n1 n2 : 0 < n1 -> m1 <= n1 -> m2 < n2 -> m1 * m2 < n1 * n2.
Proof. by move=> ? ? ?; rewrite mulnC [ltnRHS]mulnC ltn_mull. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_mulr | |
ltn_mulm1 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_mul | |
maxnMr: right_distributive muln maxn.
Proof. by case=> // m n1 n2; rewrite /maxn (fun_if (muln _)) ltn_pmul2l. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | maxnMr | |
maxnMl: left_distributive muln maxn.
Proof. by move=> m1 m2 n; rewrite -!(mulnC n) maxnMr. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | maxnMl | |
minnMr: right_distributive muln minn.
Proof. by case=> // m n1 n2; rewrite /minn (fun_if (muln _)) ltn_pmul2l. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | minnMr | |
minnMl: left_distributive muln minn.
Proof. by move=> m1 m2 n; rewrite -!(mulnC n) minnMr. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | minnMl | |
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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | iterM | |
expnm n := iterop n muln m 1.
Arguments expn : simpl never.
#[deprecated(since="mathcomp 2.3.0", note="Use expn instead.")] | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn | |
expn_rec:= expn. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn_rec | |
expnEn m : expn m n = iterop n muln m 1. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnE | |
expn0m : m ^ 0 = 1. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn0 | |
expn1m : m ^ 1 = m. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn1 | |
expnSm n : m ^ n.+1 = m * m ^ n. Proof. by case: n; rewrite ?muln1. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnS | |
expnSrm n : m ^ n.+1 = m ^ n * m. Proof. by rewrite mulnC expnS. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnSr | |
iter_mulnm n p : iter n (muln m) p = m ^ n * p.
Proof. by elim: n => /= [|n ->]; rewrite ?mul1n // expnS mulnA. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | iter_muln | |
iter_muln_1m n : iter n (muln m) 1 = m ^ n.
Proof. by rewrite iter_muln muln1. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | iter_muln_1 | |
exp0nn : 0 < n -> 0 ^ n = 0. Proof. by case: n => [|[]]. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | exp0n | |
exp1nn : 1 ^ n = 1.
Proof. by elim: n => // n; rewrite expnS mul1n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | exp1n | |
expnDm n1 n2 : m ^ (n1 + n2) = m ^ n1 * m ^ n2.
Proof. by elim: n1 => [|n1 IHn]; rewrite !(mul1n, expnS) // IHn mulnA. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnD | |
expnMnm1 m2 n : (m1 * m2) ^ n = m1 ^ n * m2 ^ n.
Proof. by elim: n => // n IHn; rewrite !expnS IHn -!mulnA (mulnCA m2). Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnMn | |
expnMm 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnM | |
expnACm n1 n2 : (m ^ n1) ^ n2 = (m ^ n2) ^ n1.
Proof. by rewrite -!expnM mulnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnAC | |
expn_gt0m n : (0 < m ^ n) = (0 < m) || (n == 0).
Proof.
by case: m => [|m]; elim: n => //= n IHn; rewrite expnS // addn_gt0 IHn.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn_gt0 | |
expn_eq0m e : (m ^ e == 0) = (m == 0) && (e > 0).
Proof. by rewrite !eqn0Ngt expn_gt0 negb_or -lt0n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expn_eq0 | |
ltn_explm 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_expl | |
leq_exp2lm 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_exp2l | |
ltn_exp2lm n1 n2 : 1 < m -> (m ^ n1 < m ^ n2) = (n1 < n2).
Proof. by move=> m_gt1; rewrite !ltnNge leq_exp2l. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_exp2l | |
eqn_exp2lm n1 n2 : 1 < m -> (m ^ n1 == m ^ n2) = (n1 == n2).
Proof. by move=> m_gt1; rewrite !eqn_leq !leq_exp2l. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_exp2l | |
expnIm : 1 < m -> injective (expn m).
Proof. by move=> m_gt1 e1 e2 /eqP; rewrite eqn_exp2l // => /eqP. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expnI | |
leq_pexp2lm n1 n2 : 0 < m -> n1 <= n2 -> m ^ n1 <= m ^ n2.
Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite leq_exp2l]. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_pexp2l | |
ltn_pexp2lm n1 n2 : 0 < m -> m ^ n1 < m ^ n2 -> n1 < n2.
Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite ltn_exp2l]. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_pexp2l | |
ltn_exp2rm 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_exp2r | |
leq_exp2rm n e : e > 0 -> (m ^ e <= n ^ e) = (m <= n).
Proof. by move=> e_gt0; rewrite leqNgt ltn_exp2r // -leqNgt. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_exp2r | |
eqn_exp2rm n e : e > 0 -> (m ^ e == n ^ e) = (m == n).
Proof. by move=> e_gt0; rewrite !eqn_leq !leq_exp2r. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqn_exp2r | |
expIne : e > 0 -> injective (expn^~ e).
Proof. by move=> e_gt1 m n /eqP; rewrite eqn_exp2r // => /eqP. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | expIn | |
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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | iterX | |
factorialn := if n is n'.+1 then n * factorial n' else 1.
Arguments factorial : simpl never.
#[deprecated(since="mathcomp 2.3.0", note="Use factorial instead.")] | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | factorial | |
fact_rec:= factorial. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | fact_rec | |
factEn : factorial n = if n is n'.+1 then n * factorial n' else 1.
Proof. by case: n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | factE | |
fact0: 0`! = 1. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | fact0 | |
factSn : (n.+1)`! = n.+1 * n`!. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | factS | |
fact_gt0n : n`! > 0.
Proof. by elim: n => //= n IHn; rewrite muln_gt0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | fact_gt0 | |
fact_geqn : n <= n`!.
Proof. by case: n => // n; rewrite factS -(addn1 n) leq_pmulr ?fact_gt0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | fact_geq | |
ltn_factm 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 | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_fact | |
nat_of_bool(b : bool) := if b then 1 else 0. | Coercion | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_of_bool | |
leq_b1(b : bool) : b <= 1. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_b1 | |
addn_negb(b : bool) : ~~ b + b = 1. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn_negb | |
eqb0(b : bool) : (b == 0 :> nat) = ~~ b. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqb0 | |
eqb1(b : bool) : (b == 1 :> nat) = b. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eqb1 | |
lt0b(b : bool) : (b > 0) = b. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | lt0b | |
sub1b(b : bool) : 1 - b = ~~ b. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | sub1b | |
mulnb(b1 b2 : bool) : b1 * b2 = b1 && b2.
Proof. by case: b1; case: b2. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mulnb | |
mulnbl(b : bool) n : b * n = (if b then n else 0).
Proof. by case: b; rewrite ?mul1n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mulnbl | |
mulnbr(b : bool) n : n * b = (if b then n else 0).
Proof. by rewrite mulnC mulnbl. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mulnbr | |
oddn := if n is n'.+1 then ~~ odd n' else false. | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | odd | |
oddSn : odd n.+1 = ~~ odd n. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddS | |
oddb(b : bool) : odd b = b. Proof. by case: b. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddb | |
oddDm n : odd (m + n) = odd m (+) odd n.
Proof. by elim: m => [|m IHn] //=; rewrite -addTb IHn addbA addTb. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddD | |
oddBm n : n <= m -> odd (m - n) = odd m (+) odd n.
Proof.
by move=> le_nm; apply: (@canRL bool) (addbK _) _; rewrite -oddD subnK.
Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddB | |
oddNi m : odd m = false -> i <= m -> odd (m - i) = odd i.
Proof. by move=> oddm /oddB ->; rewrite oddm. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddN | |
oddMm n : odd (m * n) = odd m && odd n.
Proof. by elim: m => //= m IHm; rewrite oddD -addTb andb_addl -IHm. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddM | |
oddXm n : odd (m ^ n) = (n == 0) || odd m.
Proof. by elim: n => // n IHn; rewrite expnS oddM {}IHn orbC; case odd. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | oddX | |
doublen := if n is n'.+1 then (double n').+2 else 0.
Arguments double : simpl never.
#[deprecated(since="mathcomp 2.3.0", note="Use double instead.")] | Fixpoint | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double | |
double_rec:= double. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double_rec | |
doubleEn : double n = if n is n'.+1 then (double n').+2 else 0.
Proof. by case: n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | doubleE | |
double0: 0.*2 = 0. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double0 | |
doubleSn : n.+1.*2 = n.*2.+2. Proof. by []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | doubleS | |
double_predn : n.-1.*2 = n.*2.-2. Proof. by case: n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double_pred | |
addnnn : n + n = n.*2.
Proof. by apply: eqP; elim: n => // n IHn; rewrite addnS. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnn | |
mul2nm : 2 * m = m.*2.
Proof. by rewrite mulSn mul1n addnn. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | mul2n | |
muln2m : m * 2 = m.*2.
Proof. by rewrite mulnC mul2n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | muln2 | |
doubleDm n : (m + n).*2 = m.*2 + n.*2.
Proof. by rewrite -!mul2n mulnDr. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | doubleD | |
doubleBm n : (m - n).*2 = m.*2 - n.*2.
Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | doubleB | |
leq_doublem n : (m.*2 <= n.*2) = (m <= n).
Proof. by rewrite /leq -doubleB; case (m - n). Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_double | |
ltn_doublem n : (m.*2 < n.*2) = (m < n).
Proof. by rewrite 2!ltnNge leq_double. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_double | |
ltn_Sdoublem n : (m.*2.+1 < n.*2) = (m < n).
Proof. by rewrite -doubleS leq_double. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn_Sdouble | |
leq_Sdoublem n : (m.*2 <= n.*2.+1) = (m <= n).
Proof. by rewrite leqNgt ltn_Sdouble -leqNgt. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq_Sdouble | |
odd_doublen : odd n.*2 = false.
Proof. by rewrite -addnn oddD addbb. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | odd_double | |
double_gt0n : (0 < n.*2) = (0 < n).
Proof. by case: n. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | double_gt0 |
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