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 |
|---|---|---|---|---|---|---|
eqSSm n : (m.+1 == n.+1) = (m == 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 | eqSS | |
nat_irrelevance(x y : nat) (E E' : x = y) : E = E'.
Proof. exact: eq_irrelevance. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | nat_irrelevance | |
addn:= plus.
Arguments addn : simpl never.
#[deprecated(since="mathcomp 2.3.0", note="Use addn instead.")] | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn | |
addn_rec:= addn. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn_rec | |
addnE: addn = plus. 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 | addnE | |
plusE: plus = addn. 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 | plusE | |
add0n: left_id 0 addn. 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 | add0n | |
addSnm n : m.+1 + n = (m + n).+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 | addSn | |
add1nn : 1 + n = n.+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 | add1n | |
addn0: right_id 0 addn. Proof. by move=> n; apply/eqP; elim: 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 | addn0 | |
addnSm n : m + n.+1 = (m + n).+1. Proof. by apply/eqP; elim: m. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnS | |
addSnnSm n : m.+1 + n = m + n.+1. Proof. by 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 | addSnnS | |
addnCA: left_commutative addn.
Proof. by move=> m n p; elim: m => //= m; 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 | addnCA | |
addnC: commutative addn.
Proof. by move=> m n; rewrite -[n in LHS]addn0 addnCA addn0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnC | |
addn1n : n + 1 = n.+1. Proof. by rewrite addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn1 | |
addnA: associative addn.
Proof. by move=> m n p; rewrite (addnC n) addnCA addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnA | |
addnAC: right_commutative addn.
Proof. by move=> m n p; rewrite -!addnA (addnC 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 | addnAC | |
addnCACm n p : m + n + p = p + n + m.
Proof. by rewrite addnC addnA addnAC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnCAC | |
addnAClm n p: m + n + p = n + (p + m).
Proof. by rewrite (addnC m) addnC addnCA. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnACl | |
addnACA: interchange addn addn.
Proof. by move=> m n p q; rewrite -!addnA (addnCA 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 | addnACA | |
addn_eq0m n : (m + n == 0) = (m == 0) && (n == 0).
Proof. by case: m; 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 | addn_eq0 | |
addn_eq1m n :
(m + n == 1) = ((m == 1) && (n == 0)) || ((m == 0) && (n == 1)).
Proof. by case: m n => [|[|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 | addn_eq1 | |
eqn_add2lp m n : (p + m == p + n) = (m == n).
Proof. by elim: p. 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_add2l | |
eqn_add2rp m n : (m + p == n + p) = (m == n).
Proof. by rewrite -!(addnC p) eqn_add2l. 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_add2r | |
addnI: right_injective addn.
Proof. by move=> p m n Heq; apply: eqP; rewrite -(eqn_add2l p) Heq eqxx. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnI | |
addIn: left_injective addn.
Proof. move=> p m n; rewrite -!(addnC p); apply addnI. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addIn | |
addn2m : m + 2 = m.+2. Proof. by rewrite addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn2 | |
add2nm : 2 + m = m.+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 | add2n | |
addn3m : m + 3 = m.+3. Proof. by rewrite addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn3 | |
add3nm : 3 + m = m.+3. 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 | add3n | |
addn4m : m + 4 = m.+4. Proof. by rewrite addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addn4 | |
add4nm : 4 + m = m.+4. 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 | add4n | |
subn:= minus.
Arguments subn : simpl never.
#[deprecated(since="mathcomp 2.3.0", note="Use subn instead.")] | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subn | |
subn_rec:= subn. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subn_rec | |
subnE: subn = minus. 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 | subnE | |
minusE: minus = subn. 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 | minusE | |
sub0n: left_zero 0 subn. 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 | sub0n | |
subn0: right_id 0 subn. Proof. by case. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subn0 | |
subnn: self_inverse 0 subn. Proof. by elim. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subnn | |
subSSn m : m.+1 - n.+1 = m - 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 | subSS | |
subn1n : n - 1 = n.-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 | subn1 | |
subn2n : (n - 2)%N = n.-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 | subn2 | |
subnDlp m n : (p + m) - (p + n) = m - n.
Proof. by elim: p. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subnDl | |
subnDrp m n : (m + p) - (n + p) = m - n.
Proof. by rewrite -!(addnC p) subnDl. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subnDr | |
addnKn : cancel (addn^~ n) (subn^~ n).
Proof. by move=> m; rewrite (subnDr n m 0) subn0. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addnK | |
addKnn : cancel (addn n) (subn^~ n).
Proof. by move=> m; rewrite addnC addnK. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | addKn | |
subSnnn : n.+1 - n = 1.
Proof. exact (addnK n 1). Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subSnn | |
subnDAm n p : n - (m + p) = (n - m) - p.
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 | subnDA | |
subnAC: right_commutative subn.
Proof. by move=> m n p; rewrite -!subnDA addnC. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subnAC | |
subnSm n : m - n.+1 = (m - n).-1.
Proof. by rewrite -addn1 subnDA subn1. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subnS | |
subSKnm n : (m.+1 - n).-1 = m - n.
Proof. by rewrite -subnS. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | subSKn | |
leqm n := m - n == 0. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leq | |
geq:= [rel m n | m >= n]. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | geq | |
ltn:= [rel m n | m < n]. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltn | |
gtn:= [rel m n | m > n]. | Definition | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | gtn | |
ltnSm n : (m < n.+1) = (m <= 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 | ltnS | |
leq0nn : 0 <= 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 | leq0n | |
ltn0Snn : 0 < n.+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 | ltn0Sn | |
ltn0n : n < 0 = false. 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 | ltn0 | |
leqnnn : n <= n. Proof. by elim: n. Qed.
#[global] Hint Resolve leqnn : core. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leqnn | |
ltnSnn : n < n.+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 | ltnSn | |
eq_leqm n : m = n -> m <= n. Proof. by move->. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | eq_leq | |
leqnSnn : n <= n.+1. Proof. by elim: n. Qed.
#[global] Hint Resolve leqnSn : core. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leqnSn | |
leq_predn : n.-1 <= 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 | leq_pred | |
leqSpredn : n <= n.-1.+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 | leqSpred | |
ltn_predLn : (n.-1 < n) = (0 < n).
Proof. by case: n => [//|n]; rewrite ltnSn. 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_predL | |
ltn_predRLm n : (m < n.-1) = (m.+1 < n).
Proof. by case: n => [//|n]; rewrite succnK. 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_predRL | |
ltn_predKm n : m < n -> n.-1.+1 = 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 | ltn_predK | |
prednKn : 0 < n -> n.-1.+1 = n.
Proof. exact: ltn_predK. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | prednK | |
leqNgtm n : (m <= n) = ~~ (n < m).
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 | leqNgt | |
leqVgtm n : (m <= n) || (n < m). Proof. by rewrite leqNgt orNb. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | leqVgt | |
ltnNgem n : (m < n) = ~~ (n <= m).
Proof. by rewrite 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 | ltnNge | |
ltnnn : n < n = false.
Proof. by rewrite ltnNge leqnn. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltnn | |
leqn0n : (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 | leqn0 | |
lt0nn : (0 < n) = (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 | lt0n | |
lt0n_neq0n : 0 < n -> 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 | lt0n_neq0 | |
eqn0Ngtn : (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 | eqn0Ngt | |
neq0_lt0nn : (n == 0) = false -> 0 < n. Proof. by case: n. Qed.
#[global] Hint Resolve lt0n_neq0 neq0_lt0n : core. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | neq0_lt0n | |
eqn_leqm n : (m == n) = (m <= n <= m).
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 | eqn_leq | |
anti_leq: antisymmetric leq.
Proof. by move=> m n; rewrite -eqn_leq => /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 | anti_leq | |
neq_ltnm n : (m != n) = (m < n) || (n < m).
Proof. by rewrite eqn_leq negb_and orbC -!ltnNge. Qed. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | neq_ltn | |
gtn_eqFm n : m < n -> n == m = false.
Proof. by rewrite eqn_leq (leqNgt 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 | gtn_eqF | |
ltn_eqFm n : m < n -> m == n = false.
Proof. by move/gtn_eqF; rewrite eq_sym. 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_eqF | |
ltn_geFm n : m < n -> m >= n = false.
Proof. by rewrite (leqNgt 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 | ltn_geF | |
leq_gtFm n : m <= n -> m > n = false.
Proof. by rewrite (ltnNge 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_gtF | |
leq_eqVltm n : (m <= n) = (m == n) || (m < n).
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 | leq_eqVlt | |
ltn_neqAlem n : (m < n) = (m != n) && (m <= n).
Proof. by rewrite ltnNge leq_eqVlt negb_or -leqNgt eq_sym. 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_neqAle | |
leq_transn m p : m <= n -> n <= p -> m <= p.
Proof. by elim: n m p => [|i IHn] [|m] [|p] //; apply: IHn m p. 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_trans | |
leq_ltn_transn m p : m <= n -> n < p -> m < p.
Proof. by move=> Hmn; apply: leq_trans. 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_ltn_trans | |
ltnWm n : m < n -> m <= n.
Proof. exact: leq_trans. Qed.
#[global] Hint Resolve ltnW : core. | Lemma | boot | [
"From Corelib Require Import PosDef",
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype"
] | boot/ssrnat.v | ltnW | |
leqWm n : m <= n -> m <= n.+1.
Proof. by move=> le_mn; apply: 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 | leqW | |
ltn_transn m p : m < n -> n < p -> m < p.
Proof. by move=> lt_mn /ltnW; apply: leq_trans. 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_trans | |
leq_totalm n : (m <= n) || (m >= n).
Proof. by rewrite -implyNb -ltnNge; apply/implyP; apply: 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_total | |
leq_leP{m n} : reflect (forall k, n <= k -> m <= k) (m <= n).
Proof. by apply: (iffP idP) => [mn k /(leq_trans _)->//|]; apply. 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_leP | |
ltn_gtP{m n} : reflect (forall k, k <= m -> k < n) (m < n).
Proof. by apply: (iffP idP) => [mn k /leq_ltn_trans->//|]; apply. 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_gtP | |
leq_geP{m n} : reflect (forall k, k <= m -> k <= n) (m <= n).
Proof. by rewrite -ltnS; apply: (iffP ltn_gtP). 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_geP | |
leq_ltP{m n} : reflect (forall k, n < k -> m < k) (m <= n).
Proof. by apply: (iffP idP) => [mn k|/(_ n.+1)]; [exact: leq_trans|exact]. 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_ltP | |
leq_gtP{m n} : reflect (forall k, k < m -> k < n) (m <= n).
Proof. by case: m => [|m]; [constructor|apply: (iffP ltn_gtP)]. 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_gtP | |
ltn_ltP{m n} : reflect (forall k, n <= k -> m < k) (m < n).
Proof. exact: leq_leP. 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_ltP | |
eqn_geP{m n} : reflect (forall k, (k <= m) = (k <= n)) (m == n).
Proof. by apply: (iffP idP) => [/eqP->//|/[dup]/[!eqn_leq]-> <- /[!leqnn]]. 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_geP |
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