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 |
|---|---|---|---|---|---|---|---|---|---|---|
leq_gtF m n : m <= n -> (m > n) = false. | Proof. by rewrite (ltnNge n) => ->. Qed. | Lemma | leq_gtF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_eqVlt m n : (m <= n) = (m == n) || (m < n). | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | leq_eqVlt | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_neqAle m n : (m < n) = (m != n) && (m <= n). | Proof. by rewrite ltnNge leq_eqVlt negb_or -leqNgt eq_sym. Qed. | Lemma | ltn_neqAle | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_sym",
"leqNgt",
"leq_eqVlt",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_trans n m p : m <= n -> n <= p -> m <= p. | Proof. by elim: n m p => [|i IHn] [|m] [|p] //; apply: IHn m p. Qed. | Lemma | leq_trans | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_ltn_trans n m p : m <= n -> n < p -> m < p. | Proof. by move=> Hmn; apply: leq_trans. Qed. | Lemma | leq_ltn_trans | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_leq_trans n m p : m < n -> n <= p -> m < p. | Proof. exact: leq_trans. Qed. | Lemma | ltn_leq_trans | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnW m n : m < n -> m <= n. | Proof. exact: leq_trans. Qed. | Lemma | ltnW | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqW m n : m <= n -> m <= n.+1. | Proof. by move=> le_mn; apply: ltnW. Qed. | Lemma | leqW | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_trans n m p : m < n -> n < p -> m < p. | Proof. by move=> lt_mn /ltnW; apply: leq_trans. Qed. | Lemma | ltn_trans | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_trans",
"ltnW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_total m n : (m <= n) || (m >= n). | Proof. by rewrite -implyNb -ltnNge; apply/implyP; apply: ltnW. Qed. | Lemma | leq_total | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge",
"ltnW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_leP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | ltn_gtP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_ltn_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_geP {m n} : reflect (forall k, k <= m -> k <= n) (m <= n). | Proof. by rewrite -ltnS; apply: (iffP ltn_gtP). Qed. | Lemma | leq_geP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnS",
"ltn_gtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_ltP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | leq_gtP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltn_gtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_ltP {m n} : reflect (forall k, n <= k -> m < k) (m < n). | Proof. exact: leq_leP. Qed. | Lemma | ltn_ltP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | eqn_geP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqn_leq",
"leqnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_leP {m n} : reflect (forall k, (m <= k) = (n <= k)) (m == n). | Proof. by apply: (iffP idP) => [/eqP->//|/[dup]/[!eqn_leq]<- -> /[!leqnn]]. Qed. | Lemma | eqn_leP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqn_leq",
"leqnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_gtP {m n} : reflect (forall k, (k < m) = (k < n)) (m == n). | Proof.
apply: (iffP eqn_leP) => + k => /(_ k);
by rewrite !ltnNge => /(congr1 negb); rewrite ?negbK.
Qed. | Lemma | eqn_gtP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqn_leP",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_ltP {m n} : reflect (forall k, (m < k) = (n < k)) (m == n). | Proof.
apply: (iffP eqn_geP) => + k => /(_ k);
by rewrite !ltnNge => /(congr1 negb); rewrite ?negbK.
Qed. | Lemma | eqn_ltP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqn_geP",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubnP m : {n | m < n}. | Proof. by exists m.+1. Qed. | Lemma | ubnP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | property of a single integer (i.e., the case Mxy := x). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltnSE m n : m < n.+1 -> m <= n. | Proof. by []. Qed. | Lemma | ltnSE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubn_leq_spec m : nat -> Type | := UbnLeq n of m <= n : ubn_leq_spec m n. | Variant | ubn_leq_spec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubn_geq_spec m : nat -> Type | := UbnGeq n of m >= n : ubn_geq_spec m n. | Variant | ubn_geq_spec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubn_eq_spec m : nat -> Type | := UbnEq n of m == n : ubn_eq_spec m n. | Variant | ubn_eq_spec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubnPleq m : ubn_leq_spec m m. | Proof. by []. Qed. | Lemma | ubnPleq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ubn_leq_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubnPgeq m : ubn_geq_spec m m. | Proof. by []. Qed. | Lemma | ubnPgeq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ubn_geq_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ubnPeq m : ubn_eq_spec m m. | Proof. by []. Qed. | Lemma | ubnPeq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ubn_eq_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_ind P : (forall n, (forall m, m < n -> P m) -> P n) -> forall n, P n. | Proof.
move=> accP M; have [n leMn] := ubnP M; elim: n => // n IHn in M leMn *.
by apply/accP=> p /leq_trans/(_ leMn)/IHn.
Qed. | Lemma | ltn_ind | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_trans",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leP m n : reflect (m <= n)%coq_nat (m <= n). | Proof.
apply: (iffP idP); last by elim: n / => // n _ /leq_trans->.
elim: n => [|n IHn]; first by case: m.
by rewrite leq_eqVlt ltnS => /predU1P[<- // | /IHn]; right.
Qed. | Lemma | leP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"last",
"leq_eqVlt",
"leq_trans",
"ltnS",
"predU1P"
] | Link to the legacy comparison predicates. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
le_irrelevance m n le_mn1 le_mn2 : le_mn1 = le_mn2 :> (m <= n)%coq_nat. | Proof.
elim/ltn_ind: n => n IHn in le_mn1 le_mn2 *; set n1 := n in le_mn1 *.
pose def_n : n = n1 := erefl n; transitivity (eq_ind _ _ le_mn2 _ def_n) => //.
case: n1 / le_mn1 le_mn2 => [|n1 le_mn1] {n}[|n le_mn2] in (def_n) IHn *.
- by rewrite [def_n]eq_axiomK.
- by case/leP/idPn: (le_mn2); rewrite -def_n ltnn.
- by ca... | Lemma | le_irrelevance | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"def_n",
"eq_axiomK",
"leP",
"ltn_ind",
"ltnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltP m n : reflect (m < n)%coq_nat (m < n). | Proof. exact leP. Qed. | Lemma | ltP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_irrelevance m n lt_mn1 lt_mn2 : lt_mn1 = lt_mn2 :> (m < n)%coq_nat. | Proof. exact: (@le_irrelevance m.+1). Qed. | Lemma | lt_irrelevance | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"le_irrelevance"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_add2l p m n : (p + m <= p + n) = (m <= n). | Proof. by elim: p. Qed. | Lemma | leq_add2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | Monotonicity lemmas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltn_add2l p m n : (p + m < p + n) = (m < n). | Proof. by rewrite -addnS; apply: leq_add2l. Qed. | Lemma | ltn_add2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS",
"apply",
"leq_add2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_add2r p m n : (m + p <= n + p) = (m <= n). | Proof. by rewrite -!(addnC p); apply: leq_add2l. Qed. | Lemma | leq_add2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"apply",
"leq_add2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_add2r p m n : (m + p < n + p) = (m < n). | Proof. exact: leq_add2r p m.+1 n. Qed. | Lemma | ltn_add2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_add2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_add m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 + m2 <= n1 + n2. | Proof.
by move=> le_mn1 le_mn2; rewrite (@leq_trans (m1 + n2)) ?leq_add2l ?leq_add2r.
Qed. | Lemma | leq_add | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_add2l",
"leq_add2r",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_addl m n : n <= m + n. | Proof. exact: (leq_add2r n 0). Qed. | Lemma | leq_addl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_add2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_addr m n : n <= n + m. | Proof. by rewrite addnC leq_addl. Qed. | Lemma | leq_addr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"leq_addl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_addl m n p : m < n -> m < p + n. | Proof. by move/leq_trans=> -> //; apply: leq_addl. Qed. | Lemma | ltn_addl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_addl",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_addr m n p : m < n -> m < n + p. | Proof. by move/leq_trans=> -> //; apply: leq_addr. Qed. | Lemma | ltn_addr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_addr",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn_gt0 m n : (0 < m + n) = (0 < m) || (0 < n). | Proof. by rewrite !lt0n -negb_and addn_eq0. Qed. | Lemma | addn_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn_eq0",
"lt0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn_gt0 m n : (0 < n - m) = (m < n). | Proof. by elim: m n => [|m IHm] [|n] //; apply: IHm n. Qed. | Lemma | subn_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn_eq0 m n : (m - n == 0) = (m <= n). | Proof. by []. Qed. | Lemma | subn_eq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subLR m n p : (m - n <= p) = (m <= n + p). | Proof. by rewrite -subn_eq0 -subnDA. Qed. | Lemma | leq_subLR | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subnDA",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subr m n : n - m <= n. | Proof. by rewrite leq_subLR leq_addl. Qed. | Lemma | leq_subr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_addl",
"leq_subLR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subrR m n : (n < n - m) = false. | Proof. by rewrite ltnNge leq_subr. Qed. | Lemma | ltn_subrR | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subr",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subrR m n : (n <= n - m) = (m == 0) || (n == 0). | Proof. by case: m n => [|m] [|n]; rewrite ?subn0 ?leqnn ?ltn_subrR. Qed. | Lemma | leq_subrR | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqnn",
"ltn_subrR",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subrL m n : (n - m < n) = (0 < m) && (0 < n). | Proof. by rewrite ltnNge leq_subrR negb_or !lt0n. Qed. | Lemma | ltn_subrL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subrR",
"lt0n",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnKC m n : m <= n -> m + (n - m) = n. | Proof. by elim: m n => [|m IHm] [|n] // /(IHm n) {2}<-. Qed. | Lemma | subnKC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBn m n : m + (n - m) = m - n + n. | Proof. by elim: m n => [|m IHm] [|n] //; rewrite addSn addnS IHm. Qed. | Lemma | addnBn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addSn",
"addnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnK m n : m <= n -> (n - m) + m = n. | Proof. by rewrite addnC; apply: subnKC. Qed. | Lemma | subnK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"apply",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBA m n p : p <= n -> m + (n - p) = m + n - p. | Proof. by move=> le_pn; rewrite -[in RHS](subnK le_pn) addnA addnK. Qed. | Lemma | addnBA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"addnK",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBAC m n p : n <= m -> m - n + p = m + p - n. | Proof. by move=> le_nm; rewrite addnC addnBA // addnC. Qed. | Lemma | addnBAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBCA m n p : p <= m -> p <= n -> m + (n - p) = n + (m - p). | Proof. by move=> le_pm le_pn; rewrite !addnBA // addnC. Qed. | Lemma | addnBCA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnABC m n p : p <= m -> p <= n -> m + (n - p) = m - p + n. | Proof. by move=> le_pm le_pn; rewrite addnBA // addnBAC. Qed. | Lemma | addnABC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnBAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnBA m n p : p <= n -> m - (n - p) = m + p - n. | Proof. by move=> le_pn; rewrite -[in RHS](subnK le_pn) subnDr. Qed. | Lemma | subnBA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subnDr",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnA m n p : p <= n -> n <= m -> m - (n - p) = m - n + p. | Proof. by move=> le_pn lr_nm; rewrite addnBAC // subnBA. Qed. | Lemma | subnA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBAC",
"subnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subKn m n : m <= n -> n - (n - m) = m. | Proof. by move/subnBA->; rewrite addKn. Qed. | Lemma | subKn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addKn",
"subnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSn m n : m <= n -> n.+1 - m = (n - m).+1. | Proof. by rewrite -add1n => /addnBA <-. Qed. | Lemma | subSn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add1n",
"addnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnSK m n : m < n -> (n - m.+1).+1 = n - m. | Proof. by move/subSn. Qed. | Lemma | subnSK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnCBA m n p : p <= n -> m + (n - p) = n + m - p. | Proof. by move=> pn; rewrite (addnC n m) addnBA. Qed. | Lemma | addnCBA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBr_leq n p m : n <= p -> m + (n - p) = m. | Proof. by rewrite -subn_eq0 => /eqP->; rewrite addn0. Qed. | Lemma | addnBr_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn0",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBl_leq m n p : m <= n -> m - n + p = p. | Proof. by rewrite -subn_eq0; move/eqP => ->; rewrite add0n. Qed. | Lemma | addnBl_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add0n",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnDAC m n p : m - (n + p) = m - p - n. | Proof. by rewrite addnC subnDA. Qed. | Lemma | subnDAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnCBA m n p : p <= n -> m - (n - p) = p + m - n. | Proof. by move=> pn; rewrite addnC subnBA. Qed. | Lemma | subnCBA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnBr_leq n p m : n <= p -> m - (n - p) = m. | Proof. by rewrite -subn_eq0 => /eqP->; rewrite subn0. Qed. | Lemma | subnBr_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn0",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnBl_leq m n p : m <= n -> (m - n) - p = 0. | Proof. by rewrite -subn_eq0 => /eqP->. Qed. | Lemma | subnBl_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnBAC m n p : p <= n -> n <= m -> m - (n - p) = p + (m - n). | Proof. by move=> pn nm; rewrite subnA // addnC. Qed. | Lemma | subnBAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subDnAC m n p : p <= n -> m + n - p = n - p + m. | Proof. by move=> pn; rewrite addnC -addnBAC. Qed. | Lemma | subDnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBAC",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subDnCA m n p : p <= m -> m + n - p = n + (m - p). | Proof. by move=> pm; rewrite addnC -addnBA. Qed. | Lemma | subDnCA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subDnCAC m n p : m <= p -> m + n - p = n - (p - m). | Proof. by move=> mp; rewrite addnC -subnBA. Qed. | Lemma | subDnCAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnBC m n : m - n + n = n - m + m. | Proof. by rewrite -[in RHS]addnBn addnC. Qed. | Lemma | addnBC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBn",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnCB m n : m - n + n = m + (n - m). | Proof. by rewrite addnBC addnC. Qed. | Lemma | addnCB | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBC",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addBnAC m n p : n <= m -> m - n + p = p + m - n. | Proof. by move=> nm; rewrite [p + m]addnC addnBAC. Qed. | Lemma | addBnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBAC",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addBnCAC m n p : n <= m -> n <= p -> m - n + p = p - n + m. | Proof. by move=> nm np; rewrite addnC addnBA // subDnCA // addnC. Qed. | Lemma | addBnCAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnBA",
"addnC",
"subDnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addBnA m n p : n <= m -> p <= n -> m - n + p = m - (n - p). | Proof. by move=> nm pn; rewrite subnBA // -subDnAC // addnC. Qed. | Lemma | addBnA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subDnAC",
"subnBA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subBnAC m n p : m - n - p = m - (p + n). | Proof. by rewrite addnC -subnDA. Qed. | Lemma | subBnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predn_sub m n : (m - n).-1 = (m.-1 - n). | Proof. by case: m => // m; rewrite subSKn. Qed. | Lemma | predn_sub | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subSKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sub2r p m n : m <= n -> m - p <= n - p. | Proof. by move=> le_mn; rewrite leq_subLR (leq_trans le_mn) // -leq_subLR. Qed. | Lemma | leq_sub2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subLR",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sub2l p m n : m <= n -> p - n <= p - m. | Proof.
rewrite -(leq_add2r (p - m)) leq_subLR.
by apply: leq_trans; rewrite -leq_subLR.
Qed. | Lemma | leq_sub2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_add2r",
"leq_subLR",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sub m1 m2 n1 n2 : m1 <= m2 -> n2 <= n1 -> m1 - n1 <= m2 - n2. | Proof. by move/(leq_sub2r n1)=> le_m12 /(leq_sub2l m2); apply: leq_trans. Qed. | Lemma | leq_sub | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_sub2l",
"leq_sub2r",
"leq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_sub2r p m n : p < n -> m < n -> m - p < n - p. | Proof. by move/subnSK <-; apply: (@leq_sub2r p.+1). Qed. | Lemma | ltn_sub2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_sub2r",
"subnSK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_sub2l p m n : m < p -> m < n -> p - n < p - m. | Proof. by move/subnSK <-; apply: leq_sub2l. Qed. | Lemma | ltn_sub2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leq_sub2l",
"subnSK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subRL m n p : (n < p - m) = (m + n < p). | Proof. by rewrite !ltnNge leq_subLR. Qed. | Lemma | ltn_subRL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subLR",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_psubRL m n p : 0 < n -> (n <= p - m) = (m + n <= p). | Proof. by move=> /prednK<-; rewrite ltn_subRL addnS. Qed. | Lemma | leq_psubRL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS",
"ltn_subRL",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_psubLR m n p : 0 < p -> (m - n < p) = (m < n + p). | Proof. by move=> /prednK<-; rewrite ltnS leq_subLR addnS. Qed. | Lemma | ltn_psubLR | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS",
"leq_subLR",
"ltnS",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subRL m n p : m <= p -> (n <= p - m) = (m + n <= p). | Proof. by move=> /subnKC{2}<-; rewrite leq_add2l. Qed. | Lemma | leq_subRL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_add2l",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subLR m n p : n <= m -> (m - n < p) = (m < n + p). | Proof. by move=> /subnKC{2}<-; rewrite ltn_add2l. Qed. | Lemma | ltn_subLR | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_add2l",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subCl m n p : (m - n <= p) = (m - p <= n). | Proof. by rewrite !leq_subLR // addnC. Qed. | Lemma | leq_subCl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"leq_subLR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subCr m n p : (p < m - n) = (n < m - p). | Proof. by rewrite !ltn_subRL // addnC. Qed. | Lemma | ltn_subCr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"ltn_subRL"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_psubCr m n p : 0 < p -> 0 < n -> (p <= m - n) = (n <= m - p). | Proof. by move=> p_gt0 n_gt0; rewrite !leq_psubRL // addnC. Qed. | Lemma | leq_psubCr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"leq_psubRL",
"n_gt0",
"p_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_psubCl m n p : 0 < p -> 0 < n -> (m - n < p) = (m - p < n). | Proof. by move=> p_gt0 n_gt0; rewrite !ltn_psubLR // addnC. Qed. | Lemma | ltn_psubCl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"ltn_psubLR",
"n_gt0",
"p_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_subCr m n p : n <= m -> p <= m -> (p <= m - n) = (n <= m - p). | Proof. by move=> np pm; rewrite !leq_subRL // addnC. Qed. | Lemma | leq_subCr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"leq_subRL"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_subCl m n p : n <= m -> p <= m -> (m - n < p) = (m - p < n). | Proof. by move=> nm pm; rewrite !ltn_subLR // addnC. Qed. | Lemma | ltn_subCl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"ltn_subLR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sub2rE p m n : p <= n -> (m - p <= n - p) = (m <= n). | Proof. by move=> pn; rewrite leq_subLR subnKC. Qed. | Lemma | leq_sub2rE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subLR",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sub2lE m n p : n <= m -> (m - p <= m - n) = (n <= p). | Proof. by move=> nm; rewrite leq_subCl subKn. Qed. | Lemma | leq_sub2lE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_subCl",
"subKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_sub2rE p m n : p <= m -> (m - p < n - p) = (m < n). | Proof. by move=> pn; rewrite ltn_subRL addnC subnK. Qed. | Lemma | ltn_sub2rE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"ltn_subRL",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_sub2lE m n p : p <= m -> (m - p < m - n) = (n < p). | Proof. by move=> pm; rewrite ltn_subCr subKn. Qed. | Lemma | ltn_sub2lE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_subCr",
"subKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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