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 |
|---|---|---|---|---|---|---|---|---|---|---|
"n .-2" | := n.-1.-1 (left associativity, format "n .-2") : nat_scope. | Notation | n .-2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
succnK : cancel succn predn. | Proof. by []. Qed. | Lemma | succnK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"predn",
"succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
succn_inj : injective succn. | Proof. by move=> n m []. Qed. | Lemma | succn_inj | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn m n {struct m} | :=
match m, n with
| 0, 0 => true
| m'.+1, n'.+1 => eqn m' n'
| _, _ => false
end. | Fixpoint | eqn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | Canonical comparison and eqType for nat. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eqnP : Equality.axiom eqn. | Proof.
move=> n m; apply: (iffP idP) => [|<-]; last by elim n.
by elim: n m => [|n IHn] [|m] //= /IHn->.
Qed. | Lemma | eqnP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"axiom",
"eqn",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqnE : eqn = eq_op. | Proof. by []. Qed. | Lemma | eqnE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqSS m n : (m.+1 == n.+1) = (m == n). | Proof. by []. Qed. | Lemma | eqSS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_irrelevance (x y : nat) (E E' : x = y) : E = E'. | Proof. exact: eq_irrelevance. Qed. | Lemma | nat_irrelevance | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_irrelevance",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn | := plus. | Definition | addn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | Protected addition, with a more systematic set of lemmas. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
addn_rec | := addn. | Definition | addn_rec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m + n" | := (addn m n) : nat_scope. | Notation | m + n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnE : addn = plus. | Proof. by []. Qed. | Lemma | addnE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plusE : plus = addn. | Proof. by []. Qed. | Lemma | plusE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0n : left_id 0 addn. | Proof. by []. Qed. | Lemma | add0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addSn m n : m.+1 + n = (m + n).+1. | Proof. by []. Qed. | Lemma | addSn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add1n n : 1 + n = n.+1. | Proof. by []. Qed. | Lemma | add1n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn0 : right_id 0 addn. | Proof. by move=> n; apply/eqP; elim: n. Qed. | Lemma | addn0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnS m n : m + n.+1 = (m + n).+1. | Proof. by apply/eqP; elim: m. Qed. | Lemma | addnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addSnnS m n : m.+1 + n = m + n.+1. | Proof. by rewrite addnS. Qed. | Lemma | addSnnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnCA : left_commutative addn. | Proof. by move=> m n p; elim: m => //= m; rewrite addnS => <-. Qed. | Lemma | addnCA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnC : commutative addn. | Proof. by move=> m n; rewrite -[n in LHS]addn0 addnCA addn0. Qed. | Lemma | addnC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addn0",
"addnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn1 n : n + 1 = n.+1. | Proof. by rewrite addnC. Qed. | Lemma | addn1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnA : associative addn. | Proof. by move=> m n p; rewrite (addnC n) addnCA addnC. Qed. | Lemma | addnA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnC",
"addnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnAC : right_commutative addn. | Proof. by move=> m n p; rewrite -!addnA (addnC n). Qed. | Lemma | addnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnA",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnCAC m n p : m + n + p = p + n + m. | Proof. by rewrite addnC addnA addnAC. Qed. | Lemma | addnCAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"addnAC",
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnACl m n p: m + n + p = n + (p + m). | Proof. by rewrite (addnC m) addnC addnCA. Qed. | Lemma | addnACl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"addnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnACA : interchange addn addn. | Proof. by move=> m n p q; rewrite -!addnA (addnCA n). Qed. | Lemma | addnACA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnA",
"addnCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn_eq0 m n : (m + n == 0) = (m == 0) && (n == 0). | Proof. by case: m; case: n. Qed. | Lemma | addn_eq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn_eq1 m n :
(m + n == 1) = ((m == 1) && (n == 0)) || ((m == 0) && (n == 1)). | Proof. by case: m n => [|[|m]] [|[|n]]. Qed. | Lemma | addn_eq1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_add2l p m n : (p + m == p + n) = (m == n). | Proof. by elim: p. Qed. | Lemma | eqn_add2l | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_add2r p m n : (m + p == n + p) = (m == n). | Proof. by rewrite -!(addnC p) eqn_add2l. Qed. | Lemma | eqn_add2r | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"eqn_add2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnI : right_injective addn. | Proof. by move=> p m n Heq; apply: eqP; rewrite -(eqn_add2l p) Heq eqxx. Qed. | Lemma | addnI | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"apply",
"eqn_add2l",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addIn : left_injective addn. | Proof. move=> p m n; rewrite -!(addnC p); apply addnI. Qed. | Lemma | addIn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnC",
"addnI",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn2 m : m + 2 = m.+2. | Proof. by rewrite addnC. Qed. | Lemma | addn2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add2n m : 2 + m = m.+2. | Proof. by []. Qed. | Lemma | add2n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn3 m : m + 3 = m.+3. | Proof. by rewrite addnC. Qed. | Lemma | addn3 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add3n m : 3 + m = m.+3. | Proof. by []. Qed. | Lemma | add3n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn4 m : m + 4 = m.+4. | Proof. by rewrite addnC. Qed. | Lemma | addn4 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add4n m : 4 + m = m.+4. | Proof. by []. Qed. | Lemma | add4n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn | := minus. | Definition | subn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | Further properties depend on ordering conditions. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
subn_rec | := subn. | Definition | subn_rec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m - n" | := (subn m n) : nat_scope. | Notation | m - n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnE : subn = minus. | Proof. by []. Qed. | Lemma | subnE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minusE : minus = subn. | Proof. by []. Qed. | Lemma | minusE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub0n : left_zero 0 subn. | Proof. by []. Qed. | Lemma | sub0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn0 : right_id 0 subn. | Proof. by case. Qed. | Lemma | subn0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnn : self_inverse 0 subn. | Proof. by elim. Qed. | Lemma | subnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSS n m : m.+1 - n.+1 = m - n. | Proof. by []. Qed. | Lemma | subSS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn1 n : n - 1 = n.-1. | Proof. by case: n => [|[]]. Qed. | Lemma | subn1 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn2 n : (n - 2)%N = n.-2. | Proof. by case: n => [|[|[]]]. Qed. | Lemma | subn2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnDl p m n : (p + m) - (p + n) = m - n. | Proof. by elim: p. Qed. | Lemma | subnDl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnDr p m n : (m + p) - (n + p) = m - n. | Proof. by rewrite -!(addnC p) subnDl. Qed. | Lemma | subnDr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnK n : cancel (addn^~ n) (subn^~ n). | Proof. by move=> m; rewrite (subnDr n m 0) subn0. Qed. | Lemma | addnK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"subn",
"subn0",
"subnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addKn n : cancel (addn n) (subn^~ n). | Proof. by move=> m; rewrite addnC addnK. Qed. | Lemma | addKn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn",
"addnC",
"addnK",
"subn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSnn n : n.+1 - n = 1. | Proof. exact (addnK n 1). Qed. | Lemma | subSnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnDA m n p : n - (m + p) = (n - m) - p. | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | subnDA | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnAC : right_commutative subn. | Proof. by move=> m n p; rewrite -!subnDA addnC. Qed. | Lemma | subnAC | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"subn",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subnS m n : m - n.+1 = (m - n).-1. | Proof. by rewrite -addn1 subnDA subn1. Qed. | Lemma | subnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addn1",
"subn1",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSKn m n : (m.+1 - n).-1 = m - n. | Proof. by rewrite -subnS. Qed. | Lemma | subSKn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"subnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq m n | := m - n == 0. | Definition | leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | Integer ordering, and its interaction with the other operations. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"m <= n" | := (leq m n) : nat_scope. | Notation | m <= n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m < n" | := (m.+1 <= n) : nat_scope. | Notation | m < n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m >= n" | := (n <= m) (only parsing) : nat_scope. | Notation | m >= n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m > n" | := (n < m) (only parsing) : nat_scope. | Notation | m > n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geq | := [rel m n | m >= n]. | Definition | geq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"rel"
] | For sorting, etc. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltn | := [rel m n | m < n]. | Definition | ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn | := [rel m n | m > n]. | Definition | gtn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m <= n <= p" | := ((m <= n) && (n <= p)) : nat_scope. | Notation | m <= n <= p | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m < n <= p" | := ((m < n) && (n <= p)) : nat_scope. | Notation | m < n <= p | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m <= n < p" | := ((m <= n) && (n < p)) : nat_scope. | Notation | m <= n < p | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m < n < p" | := ((m < n) && (n < p)) : nat_scope. | Notation | m < n < p | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnS m n : (m < n.+1) = (m <= n). | Proof. by []. Qed. | Lemma | ltnS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq0n n : 0 <= n. | Proof. by []. Qed. | Lemma | leq0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn0Sn n : 0 < n.+1. | Proof. by []. Qed. | Lemma | ltn0Sn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn0 n : (n < 0) = false. | Proof. by []. Qed. | Lemma | ltn0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqnn n : n <= n. | Proof. by elim: n. Qed. | Lemma | leqnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnSn n : n < n.+1. | Proof. by []. Qed. | Lemma | ltnSn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_leq m n : m = n -> m <= n. | Proof. by move->. Qed. | Lemma | eq_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqnSn n : n <= n.+1. | Proof. by elim: n. Qed. | Lemma | leqnSn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_pred n : n.-1 <= n. | Proof. by case: n => /=. Qed. | Lemma | leq_pred | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqSpred n : n <= n.-1.+1. | Proof. by case: n => /=. Qed. | Lemma | leqSpred | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_predL n : (n.-1 < n) = (0 < n). | Proof. by case: n => [//|n]; rewrite ltnSn. Qed. | Lemma | ltn_predL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_predRL m n : (m < n.-1) = (m.+1 < n). | Proof. by case: n => [//|n]; rewrite succnK. Qed. | Lemma | ltn_predRL | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"succnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_predK m n : m < n -> n.-1.+1 = n. | Proof. by case: n. Qed. | Lemma | ltn_predK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prednK n : 0 < n -> n.-1.+1 = n. | Proof. exact: ltn_predK. Qed. | Lemma | prednK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_predK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqNgt m n : (m <= n) = ~~ (n < m). | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | leqNgt | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqVgt m n : (m <= n) || (n < m). | Proof. by rewrite leqNgt orNb. Qed. | Lemma | leqVgt | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnNge m n : (m < n) = ~~ (n <= m). | Proof. by rewrite leqNgt. Qed. | Lemma | ltnNge | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnn n : (n < n) = false. | Proof. by rewrite ltnNge leqnn. Qed. | Lemma | ltnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqnn",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqn0 n : (n <= 0) = (n == 0). | Proof. by case: n. Qed. | Lemma | leqn0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0n n : (0 < n) = (n != 0). | Proof. by case: n. Qed. | Lemma | lt0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0n_neq0 n : 0 < n -> n != 0. | Proof. by case: n. Qed. | Lemma | lt0n_neq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn0Ngt n : (n == 0) = ~~ (n > 0). | Proof. by case: n. Qed. | Lemma | eqn0Ngt | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq0_lt0n n : (n == 0) = false -> 0 < n. | Proof. by case: n. Qed. | Lemma | neq0_lt0n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_leq m n : (m == n) = (m <= n <= m). | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | eqn_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
anti_leq : antisymmetric leq. | Proof. by move=> m n; rewrite -eqn_leq => /eqP. Qed. | Lemma | anti_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq_ltn m n : (m != n) = (m < n) || (n < m). | Proof. by rewrite eqn_leq negb_and orbC -!ltnNge. Qed. | Lemma | neq_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn_eqF m n : m < n -> (n == m) = false. | Proof. by rewrite eqn_leq (leqNgt n) => ->. Qed. | Lemma | gtn_eqF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"leqNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_eqF m n : m < n -> (m == n) = false. | Proof. by move/gtn_eqF; rewrite eq_sym. Qed. | Lemma | ltn_eqF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_sym",
"gtn_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_geF m n : m < n -> (m >= n) = false. | Proof. by rewrite (leqNgt n) => ->. Qed. | Lemma | ltn_geF | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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