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 |
|---|---|---|---|---|---|---|---|---|---|---|
oddD m n : odd (m + n) = odd m (+) odd n. | Proof. by elim: m => [|m IHn] //=; rewrite -addTb IHn addbA addTb. Qed. | Lemma | oddD | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddB m n : n <= m -> odd (m - n) = odd m (+) odd n. | Proof.
by move=> le_nm; apply: (@canRL bool) (addbK _) _; rewrite -oddD subnK.
Qed. | Lemma | oddB | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"odd",
"oddD",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddN i m : odd m = false -> i <= m -> odd (m - i) = odd i. | Proof. by move=> oddm /oddB ->; rewrite oddm. Qed. | Lemma | oddN | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd",
"oddB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddM m n : odd (m * n) = odd m && odd n. | Proof. by elim: m => //= m IHm; rewrite oddD -addTb andb_addl -IHm. Qed. | Lemma | oddM | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd",
"oddD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oddX m n : odd (m ^ n) = (n == 0) || odd m. | Proof. by elim: n => // n IHn; rewrite expnS oddM {}IHn orbC; case odd. Qed. | Lemma | oddX | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"odd",
"oddM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double n | := if n is n'.+1 then (double n').+2 else 0. | Fixpoint | double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'"
] | Doubling. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
double_rec | := double. | Definition | double_rec | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .*2" | := (double n) : nat_scope. | Notation | n .*2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleE n : double n = if n is n'.+1 then (double n').+2 else 0. | Proof. by case: n. Qed. | Lemma | doubleE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"double",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double0 : 0.*2 = 0. | Proof. by []. Qed. | Lemma | double0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleS n : n.+1.*2 = n.*2.+2. | Proof. by []. Qed. | Lemma | doubleS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double_pred n : n.-1.*2 = n.*2.-2. | Proof. by case: n. Qed. | Lemma | double_pred | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predn_doubleS n : n.+1.*2.-1 = n.*2.+1. | Proof. by []. Qed. | Lemma | predn_doubleS | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addnn n : n + n = n.*2. | Proof. by apply: eqP; elim: n => // n IHn; rewrite addnS. Qed. | Lemma | addnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnS",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul2n m : 2 * m = m.*2. | Proof. by rewrite mulSn mul1n addnn. Qed. | Lemma | mul2n | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnn",
"mul1n",
"mulSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln2 m : m * 2 = m.*2. | Proof. by rewrite mulnC mul2n. Qed. | Lemma | muln2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul2n",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleD m n : (m + n).*2 = m.*2 + n.*2. | Proof. by rewrite -!mul2n mulnDr. Qed. | Lemma | doubleD | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul2n",
"mulnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleB m n : (m - n).*2 = m.*2 - n.*2. | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | doubleB | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_double m n : (m.*2 <= n.*2) = (m <= n). | Proof. by rewrite /leq -doubleB; case (m - n). Qed. | Lemma | leq_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleB",
"leq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_double m n : (m.*2 < n.*2) = (m < n). | Proof. by rewrite 2!ltnNge leq_double. Qed. | Lemma | ltn_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_double",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_Sdouble m n : (m.*2.+1 < n.*2) = (m < n). | Proof. by rewrite -doubleS leq_double. Qed. | Lemma | ltn_Sdouble | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleS",
"leq_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_Sdouble m n : (m.*2 <= n.*2.+1) = (m <= n). | Proof. by rewrite leqNgt ltn_Sdouble -leqNgt. Qed. | Lemma | leq_Sdouble | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqNgt",
"ltn_Sdouble"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_double n : odd n.*2 = false. | Proof. by rewrite -addnn oddD addbb. Qed. | Lemma | odd_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnn",
"odd",
"oddD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double_gt0 n : (0 < n.*2) = (0 < n). | Proof. by case: n. Qed. | Lemma | double_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double_eq0 n : (n.*2 == 0) = (n == 0). | Proof. by case: n. Qed. | Lemma | double_eq0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleMl m n : (m * n).*2 = m.*2 * n. | Proof. by rewrite -!mul2n mulnA. Qed. | Lemma | doubleMl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"mul2n",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleMr m n : (m * n).*2 = m * n.*2. | Proof. by rewrite -!muln2 mulnA. Qed. | Lemma | doubleMr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"muln2",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq_doubleS_double n m : (n.*2.+1 == m.*2) = false. | Proof.
case: (leqP m n) => mn; apply/negbTE; rewrite neq_ltn ?ltn_Sdouble ?mn//.
by rewrite ltnS leq_double mn orbT.
Qed. | Lemma | neq_doubleS_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"leqP",
"leq_double",
"ltnS",
"ltn_Sdouble",
"neq_ltn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
half (n : nat) : nat | := if n is n'.+1 then uphalf n' else n
with uphalf (n : nat) : nat := if n is n'.+1 then n'./2.+1 else n
where "n ./2" := (half n) : nat_scope. | Fixpoint | half | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"n'",
"nat"
] | Halving. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
uphalfE n : uphalf n = n.+1./2. | Proof. by []. Qed. | Lemma | uphalfE | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
doubleK : cancel double half. | Proof. by elim=> //= n ->. Qed. | Lemma | doubleK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"double",
"half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
half_double | := doubleK. | Definition | half_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double_inj | := can_inj doubleK. | Definition | double_inj | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uphalf_double n : uphalf n.*2 = n. | Proof. by elim: n => //= n ->. Qed. | Lemma | uphalf_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uphalf_half n : uphalf n = odd n + n./2. | Proof. by elim: n => //= n ->; rewrite addnA addn_negb. Qed. | Lemma | uphalf_half | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"addn_negb",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_double_half n : odd n + n./2.*2 = n. | Proof.
by elim: n => //= n {3}<-; rewrite uphalf_half doubleD; case (odd n).
Qed. | Lemma | odd_double_half | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleD",
"odd",
"uphalf_half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
halfK n : n./2.*2 = n - odd n. | Proof. by rewrite -[n in n - _]odd_double_half addnC addnK. Qed. | Lemma | halfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"addnK",
"odd",
"odd_double_half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uphalfK n : (uphalf n).*2 = odd n + n. | Proof. by rewrite uphalfE halfK/=; case: odd; rewrite ?subn1. Qed. | Lemma | uphalfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"halfK",
"odd",
"subn1",
"uphalfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_halfK n : odd n -> n./2.*2 = n.-1. | Proof. by rewrite halfK => ->; rewrite subn1. Qed. | Lemma | odd_halfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"halfK",
"odd",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
even_halfK n : ~~ odd n -> n./2.*2 = n. | Proof. by rewrite halfK => /negbTE->; rewrite subn0. Qed. | Lemma | even_halfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"halfK",
"odd",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_uphalfK n : odd n -> (uphalf n).*2 = n.+1. | Proof. by rewrite uphalfK => ->. Qed. | Lemma | odd_uphalfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd",
"uphalfK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
even_uphalfK n : ~~ odd n -> (uphalf n).*2 = n. | Proof. by rewrite uphalfK => /negbTE->. Qed. | Lemma | even_uphalfK | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd",
"uphalfK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
half_bit_double n (b : bool) : (b + n.*2)./2 = n. | Proof. by case: b; rewrite /= (half_double, uphalf_double). Qed. | Lemma | half_bit_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"half_double",
"uphalf_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
halfD m n : (m + n)./2 = (odd m && odd n) + (m./2 + n./2). | Proof.
rewrite -[n in LHS]odd_double_half addnCA.
rewrite -[m in LHS]odd_double_half -addnA -doubleD.
by do 2!case: odd; rewrite /= ?add0n ?half_double ?uphalf_double.
Qed. | Lemma | halfD | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"add0n",
"addnA",
"addnCA",
"doubleD",
"half_double",
"odd",
"odd_double_half",
"uphalf_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
half_leq m n : m <= n -> m./2 <= n./2. | Proof. by move/subnK <-; rewrite halfD addnA leq_addl. Qed. | Lemma | half_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"halfD",
"leq_addl",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geq_half_double m n : (m <= n./2) = (m.*2 <= n). | Proof.
rewrite -[X in _.*2 <= X]odd_double_half.
case: odd; last by rewrite leq_double.
by case: m => // m; rewrite doubleS ltnS ltn_double.
Qed. | Lemma | geq_half_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"doubleS",
"last",
"leq_double",
"ltnS",
"ltn_double",
"odd",
"odd_double_half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_half_double m n : (m./2 < n) = (m < n.*2). | Proof. by rewrite ltnNge geq_half_double -ltnNge. Qed. | Lemma | ltn_half_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"geq_half_double",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_half_double m n : (m./2 <= n) = (m <= n.*2.+1). | Proof. by case: m => [|[|m]] //; rewrite ltnS ltn_half_double. Qed. | Lemma | leq_half_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnS",
"ltn_half_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn_half_double m n : (n < m./2) = (n.*2.+1 < m). | Proof. by rewrite ltnNge leq_half_double -ltnNge. Qed. | Lemma | gtn_half_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_half_double",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
half_gt0 n : (0 < n./2) = (1 < n). | Proof. by case: n => [|[]]. Qed. | Lemma | half_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uphalf_leq m n : m <= n -> uphalf m <= uphalf n. | Proof.
move/subnK <-; rewrite !uphalf_half oddD halfD !addnA.
by do 2 case: odd; apply: leq_addl.
Qed. | Lemma | uphalf_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"apply",
"halfD",
"leq_addl",
"odd",
"oddD",
"subnK",
"uphalf_half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_uphalf_double m n : (uphalf m <= n) = (m <= n.*2). | Proof. by rewrite uphalfE leq_half_double. Qed. | Lemma | leq_uphalf_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_half_double",
"uphalfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geq_uphalf_double m n : (m <= uphalf n) = (m.*2 <= n.+1). | Proof. by rewrite uphalfE geq_half_double. Qed. | Lemma | geq_uphalf_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"geq_half_double",
"uphalfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtn_uphalf_double m n : (n < uphalf m) = (n.*2 < m). | Proof. by rewrite uphalfE gtn_half_double. Qed. | Lemma | gtn_uphalf_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"gtn_half_double",
"uphalfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_uphalf_double m n : (uphalf m < n) = (m.+1 < n.*2). | Proof. by rewrite uphalfE ltn_half_double. Qed. | Lemma | ltn_uphalf_double | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_half_double",
"uphalfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uphalf_gt0 n : (0 < uphalf n) = (0 < n). | Proof. by case: n. Qed. | Lemma | uphalf_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_geq m n : odd n -> (m <= n) = (m./2.*2 <= n). | Proof.
move=> odd_n; rewrite -[m in LHS]odd_double_half -[n]odd_double_half odd_n.
by case: (odd m); rewrite // leq_Sdouble ltnS leq_double.
Qed. | Lemma | odd_geq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_Sdouble",
"leq_double",
"ltnS",
"odd",
"odd_double_half"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_ltn m n : odd n -> (n < m) = (n < m./2.*2). | Proof. by move=> odd_n; rewrite !ltnNge odd_geq. Qed. | Lemma | odd_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltnNge",
"odd",
"odd_geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_gt0 n : odd n -> n > 0. | Proof. by case: n. Qed. | Lemma | odd_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_gt2 n : odd n -> n > 1 -> n > 2. | Proof. by move=> odd_n n_gt1; rewrite odd_geq. Qed. | Lemma | odd_gt2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"odd",
"odd_geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulnn m : m * m = m ^ 2. | Proof. by rewrite !expnS muln1. Qed. | Lemma | mulnn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expnS",
"muln1"
] | Squares and square identities. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sqrnD m n : (m + n) ^ 2 = m ^ 2 + n ^ 2 + 2 * (m * n). | Proof.
rewrite -!mulnn mul2n mulnDr !mulnDl (mulnC n) -!addnA.
by congr (_ + _); rewrite addnA addnn addnC.
Qed. | Lemma | sqrnD | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"addnC",
"addnn",
"mul2n",
"mulnC",
"mulnDl",
"mulnDr",
"mulnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrnB m n : n <= m -> (m - n) ^ 2 = m ^ 2 + n ^ 2 - 2 * (m * n). | Proof.
move/subnK <-; rewrite addnK sqrnD -addnA -addnACA -addnA.
by rewrite addnn -mul2n -mulnDr -mulnDl addnK.
Qed. | Lemma | sqrnB | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnA",
"addnACA",
"addnK",
"addnn",
"mul2n",
"mulnDl",
"mulnDr",
"sqrnD",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrnD_sub m n : n <= m -> (m + n) ^ 2 - 4 * (m * n) = (m - n) ^ 2. | Proof.
move=> le_nm; rewrite -[4]/(2 * 2) -mulnA mul2n -addnn subnDA.
by rewrite sqrnD addnK sqrnB.
Qed. | Lemma | sqrnD_sub | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnK",
"addnn",
"mul2n",
"mulnA",
"sqrnB",
"sqrnD",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subn_sqr m n : m ^ 2 - n ^ 2 = (m - n) * (m + n). | Proof. by rewrite mulnBl !mulnDr addnC (mulnC m) subnDl. Qed. | Lemma | subn_sqr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"mulnBl",
"mulnC",
"mulnDr",
"subnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_sqr m n : (m ^ 2 < n ^ 2) = (m < n). | Proof. by rewrite ltn_exp2r. Qed. | Lemma | ltn_sqr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_exp2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_sqr m n : (m ^ 2 <= n ^ 2) = (m <= n). | Proof. by rewrite leq_exp2r. Qed. | Lemma | leq_sqr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_exp2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrn_gt0 n : (0 < n ^ 2) = (0 < n). | Proof. exact: (ltn_sqr 0). Qed. | Lemma | sqrn_gt0 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqn_sqr m n : (m ^ 2 == n ^ 2) = (m == n). | Proof. by rewrite eqn_exp2r. Qed. | Lemma | eqn_sqr | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_exp2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrn_inj : injective (expn ^~ 2). | Proof. exact: expIn. Qed. | Lemma | sqrn_inj | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"expIn",
"expn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif m n C | := ((m <= n) * ((m == n) = C))%type. | Definition | leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"type"
] | non-strict one. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"m <= n ?= 'iff' C" | := (leqif m n C) : nat_scope. | Notation | m <= n ?= 'iff' C | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_of_leqif m n C (H : m <= n ?= iff C) | := H.1 : m <= n. | Coercion | leq_of_leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqifP m n C : reflect (m <= n ?= iff C) (if C then m == n else m < n). | Proof.
rewrite ltn_neqAle; apply: (iffP idP) => [|lte]; last by rewrite !lte; case C.
by case C => [/eqP-> | /andP[/negPf]]; split=> //; apply: eqxx.
Qed. | Lemma | leqifP | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqxx",
"last",
"ltn_neqAle",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_refl m C : reflect (m <= m ?= iff C) C. | Proof. by apply: (iffP idP) => [-> | <-] //; split; rewrite ?eqxx. Qed. | Lemma | leqif_refl | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqxx",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_trans m1 m2 m3 C12 C23 :
m1 <= m2 ?= iff C12 -> m2 <= m3 ?= iff C23 -> m1 <= m3 ?= iff C12 && C23. | Proof.
move=> ltm12 ltm23; apply/leqifP; rewrite -ltm12.
have [->|eqm12] := eqVneq; first by rewrite ltn_neqAle !ltm23 andbT; case C23.
by rewrite (@leq_trans m2) ?ltm23 // ltn_neqAle eqm12 ltm12.
Qed. | Lemma | leqif_trans | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"eqVneq",
"leq_trans",
"leqifP",
"ltn_neqAle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mono_leqif f : {mono f : m n / m <= n} ->
forall m n C, (f m <= f n ?= iff C) = (m <= n ?= iff C). | Proof. by move=> f_mono m n C; rewrite /leqif !eqn_leq !f_mono. Qed. | Lemma | mono_leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"f_mono",
"leqif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_geq m n : m <= n -> m <= n ?= iff (m >= n). | Proof. by move=> lemn; split=> //; rewrite eqn_leq lemn. Qed. | Lemma | leqif_geq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_eq m n : m <= n -> m <= n ?= iff (m == n). | Proof. by []. Qed. | Lemma | leqif_eq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geq_leqif a b C : a <= b ?= iff C -> (b <= a) = C. | Proof. by case=> le_ab; rewrite eqn_leq le_ab. Qed. | Lemma | geq_leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eqn_leq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_leqif a b C : a <= b ?= iff C -> (a < b) = ~~ C. | Proof. by move=> le_ab; rewrite ltnNge (geq_leqif le_ab). Qed. | Lemma | ltn_leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"geq_leqif",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltnNleqif x y C : x <= y ?= iff ~~ C -> (x < y) = C. | Proof. by move=> /ltn_leqif; rewrite negbK. Qed. | Lemma | ltnNleqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"ltn_leqif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_leqif x y C : x <= y ?= iff C -> (x == y) = C. | Proof. by move=> /leqifP; case: C ltngtP => [] []. Qed. | Lemma | eq_leqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leqifP",
"ltngtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqTleqif x y C : x <= y ?= iff C -> C -> x = y. | Proof. by move=> /eq_leqif<-/eqP. Qed. | Lemma | eqTleqif | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"eq_leqif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_add m1 n1 C1 m2 n2 C2 :
m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 ->
m1 + m2 <= n1 + n2 ?= iff C1 && C2. | Proof.
rewrite -(mono_leqif (leq_add2r m2)) -(mono_leqif (leq_add2l n1) m2).
exact: leqif_trans.
Qed. | Lemma | leqif_add | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"leq_add2l",
"leq_add2r",
"leqif_trans",
"mono_leqif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqif_mul m1 n1 C1 m2 n2 C2 :
m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 ->
m1 * m2 <= n1 * n2 ?= iff (n1 * n2 == 0) || (C1 && C2). | Proof.
case: n1 => [|n1] le1; first by case: m1 le1 => [|m1] [_ <-] //.
case: n2 m2 => [|n2] [|m2] /=; try by case=> // _ <-; rewrite !muln0 ?andbF.
have /leq_pmul2l-/mono_leqif<-: 0 < n1.+1 by [].
by apply: leqif_trans; have /leq_pmul2r-/mono_leqif->: 0 < m2.+1.
Qed. | Lemma | leqif_mul | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"le1",
"leq_pmul2l",
"leq_pmul2r",
"leqif_trans",
"mono_leqif",
"muln0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_Cauchy m n : 2 * (m * n) <= m ^ 2 + n ^ 2 ?= iff (m == n). | Proof.
without loss le_nm: m n / n <= m.
by have [?|/ltnW ?] := leqP n m; last rewrite eq_sym addnC (mulnC m); apply.
apply/leqifP; have [-> | ne_mn] := eqVneq; first by rewrite addnn mul2n.
by rewrite -subn_gt0 -sqrnB // sqrn_gt0 subn_gt0 ltn_neqAle eq_sym ne_mn.
Qed. | Lemma | nat_Cauchy | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnC",
"addnn",
"apply",
"eqVneq",
"eq_sym",
"last",
"leqP",
"leqifP",
"ltnW",
"ltn_neqAle",
"mul2n",
"mulnC",
"sqrnB",
"sqrn_gt0",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_AGM2 m n : 4 * (m * n) <= (m + n) ^ 2 ?= iff (m == n). | Proof.
rewrite -[4]/(2 * 2) -mulnA mul2n -addnn sqrnD; apply/leqifP.
by rewrite ltn_add2r eqn_add2r ltn_neqAle !nat_Cauchy; case: eqVneq.
Qed. | Lemma | nat_AGM2 | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"addnn",
"apply",
"eqVneq",
"eqn_add2r",
"leqifP",
"ltn_add2r",
"ltn_neqAle",
"mul2n",
"mulnA",
"nat_Cauchy",
"sqrnD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraTleq b m n : (n < m -> ~~ b) -> (b -> m <= n). | Proof. by rewrite ltnNge; apply: contraTT. Qed. | Lemma | contraTleq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraTltn b m n : (n <= m -> ~~ b) -> (b -> m < n). | Proof. by rewrite ltnNge; apply: contraTN. Qed. | Lemma | contraTltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraPleq P m n : (n < m -> ~ P) -> (P -> m <= n). | Proof. by rewrite ltnNge; apply: contraPT. Qed. | Lemma | contraPleq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraPltn P m n : (n <= m -> ~ P) -> (P -> m < n). | Proof. by rewrite ltnNge; apply: contraPN. Qed. | Lemma | contraPltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraNleq b m n : (n < m -> b) -> (~~ b -> m <= n). | Proof. by rewrite ltnNge; apply: contraNT. Qed. | Lemma | contraNleq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraNltn b m n : (n <= m -> b) -> (~~ b -> m < n). | Proof. by rewrite ltnNge; apply: contraNN. Qed. | Lemma | contraNltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_not_leq P m n : (n < m -> P) -> (~ P -> m <= n). | Proof. by rewrite ltnNge; apply: contra_notT. Qed. | Lemma | contra_not_leq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_not_ltn P m n : (n <= m -> P) -> (~ P -> m < n). | Proof. by rewrite ltnNge; apply: contra_notN. Qed. | Lemma | contra_not_ltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraFleq b m n : (n < m -> b) -> (b = false -> m <= n). | Proof. by rewrite ltnNge; apply: contraFT. Qed. | Lemma | contraFleq | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contraFltn b m n : (n <= m -> b) -> (b = false -> m < n). | Proof. by rewrite ltnNge; apply: contraFN. Qed. | Lemma | contraFltn | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_leqT b m n : (~~ b -> m < n) -> (n <= m -> b). | Proof. by rewrite ltnNge; apply: contraTT. Qed. | Lemma | contra_leqT | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
contra_ltnT b m n : (~~ b -> m <= n) -> (n < m -> b). | Proof. by rewrite ltnNge; apply: contraNT. Qed. | Lemma | contra_ltnT | boot | boot/ssrnat.v | [
"Corelib",
"PosDef",
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"NatTrec"
] | [
"apply",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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