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 |
|---|---|---|---|---|---|---|---|---|---|---|
ler_sum_nat (m n : nat) (F G : nat -> R) :
(forall i, (m <= i < n)%N -> F i <= G i) ->
\sum_(m <= i < n) F i <= \sum_(m <= i < n) G i. | Proof. by move=> le_FG; rewrite !big_nat ler_sum. Qed. | Lemma | ler_sum_nat | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"big_nat",
"ler_sum",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_sum I (r : seq I) (P : pred I) (F G : I -> R) :
has P r -> (forall i, P i -> F i < G i) ->
\sum_(i <- r | P i) F i < \sum_(i <- r | P i) G i. | Proof.
rewrite -big_filter -[ltRHS]big_filter -size_filter_gt0.
case: filter (filter_all P r) => //= x {}r /andP[Px Pr] _ ltFG.
rewrite !big_cons ltr_leD// ?ltFG// -(all_filterP Pr) !big_filter.
by rewrite ler_sum => // i Pi; rewrite ltW ?ltFG.
Qed. | Lemma | ltr_sum | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"Px",
"all_filterP",
"big_cons",
"big_filter",
"filter",
"filter_all",
"has",
"ler_sum",
"ltRHS",
"ltW",
"ltr_leD",
"seq",
"size_filter_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_sum_nat (m n : nat) (F G : nat -> R) :
(m < n)%N -> (forall i, (m <= i < n)%N -> F i < G i) ->
\sum_(m <= i < n) F i < \sum_(m <= i < n) G i. | Proof.
move=> lt_mn i; rewrite big_nat [ltRHS]big_nat ltr_sum//.
by apply/hasP; exists m; rewrite ?mem_index_iota leqnn lt_mn.
Qed. | Lemma | ltr_sum_nat | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"big_nat",
"hasP",
"leqnn",
"ltRHS",
"ltr_sum",
"mem_index_iota",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
psumr_eq0 (I : eqType) (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) ->
(\sum_(i <- r | P i) (F i) == 0) = (all (fun i => (P i) ==> (F i == 0)) r). | Proof.
elim: r=> [|a r ihr hr] /=; rewrite (big_nil, big_cons); first by rewrite eqxx.
by case: ifP=> pa /=; rewrite ?paddr_eq0 ?ihr ?hr // sumr_ge0.
Qed. | Lemma | psumr_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"all",
"big_cons",
"big_nil",
"eqxx",
"paddr_eq0",
"seq",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
psumr_eq0P (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) -> \sum_(i | P i) F i = 0 ->
(forall i, P i -> F i = 0). | Proof.
move=> F_ge0 /eqP; rewrite psumr_eq0 // -big_all big_andE => /forallP hF i Pi.
by move: (hF i); rewrite implyTb Pi /= => /eqP.
Qed. | Lemma | psumr_eq0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"big_all",
"big_andE",
"forallP",
"psumr_eq0"
] | :TODO: Cyril : See which form to keep | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
psumr_neq0 (I : eqType) (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) ->
(\sum_(i <- r | P i) (F i) != 0) = (has (fun i => P i && (0 < F i)) r). | Proof.
move=> F_ge0; rewrite psumr_eq0// -has_predC; apply: eq_has => x /=.
by case Px: (P x); rewrite //= lt_def F_ge0 ?andbT.
Qed. | Lemma | psumr_neq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"Px",
"apply",
"eq_has",
"has",
"has_predC",
"lt_def",
"psumr_eq0",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
psumr_neq0P (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) -> \sum_(i | P i) F i <> 0 ->
(exists i, P i && (0 < F i)). | Proof. by move=> ? /eqP; rewrite psumr_neq0// => /hasP[x _ ?]; exists x. Qed. | Lemma | psumr_neq0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"hasP",
"psumr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_wpMn2r n : (0 < n)%N -> {homo (@GRing.natmul R)^~ n : x y / x < y}. | Proof.
elim: n => // -[|n] IHn _ x y ltxy//.
by rewrite mulrS [in ltRHS]mulrS ltrD// IHn.
Qed. | Lemma | ltr_wpMn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltRHS",
"ltrD",
"mulrS",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wMn2r n : {homo (@GRing.natmul R)^~ n : x y / x <= y}. | Proof. by case: n => // n; exact/ltW_homo/ltr_wpMn2r. Qed. | Lemma | ler_wMn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_homo",
"ltr_wpMn2r",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_wge0 x n : 0 <= x -> 0 <= x *+ n. | Proof. by move=> /(ler_wMn2r n); rewrite mul0rn. Qed. | Lemma | mulrn_wge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler_wMn2r",
"mul0rn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_wle0 x n : x <= 0 -> x *+ n <= 0. | Proof. by move=> /(ler_wMn2r n); rewrite mul0rn. Qed. | Lemma | mulrn_wle0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler_wMn2r",
"mul0rn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpMn2l x :
0 <= x -> {homo (@GRing.natmul R x) : m n / (m <= n)%N >-> m <= n}. | Proof. by move=> xge0 m n /subnK <-; rewrite mulrnDr ler_wpDl ?mulrn_wge0. Qed. | Lemma | ler_wpMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler_wpDl",
"mulrnDr",
"mulrn_wge0",
"natmul",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wnMn2l x :
x <= 0 -> {homo (@GRing.natmul R x) : m n / (n <= m)%N >-> m <= n}. | Proof.
by move=> xle0 m n hmn /=; rewrite -lerN2 -!mulNrn ler_wpMn2l // oppr_cp0.
Qed. | Lemma | ler_wnMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lerN2",
"ler_wpMn2l",
"mulNrn",
"natmul",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_lgt0 x n : (0 < n)%N -> (0 < x) -> (0 < x *+ n). | Proof.
move=> + xgt0; elim: n => // n IHn _; rewrite mulrS (lt_le_trans xgt0)// lerDl.
by case: n IHn => // n /(_ _)/ltW->.
Qed. | Lemma | mulrn_lgt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lerDl",
"ltW",
"lt_le_trans",
"mulrS"
] | TODO negative versions | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pmulrIn x : x > 0 -> injective (GRing.natmul x). | Proof.
move=> x_neq0 m n /eqP; wlog lt_mn : m n / (m < n)%N => [hwlog|].
by case: (ltngtP m n) => // [|+ /eqP/esym/eqP] => /hwlog/[apply].
by rewrite eq_sym -subr_eq0 -mulrnBr 1?ltnW// gt_eqF// mulrn_lgt0// subn_gt0.
Qed. | Lemma | pmulrIn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"eq_sym",
"gt_eqF",
"ltnW",
"ltngtP",
"mulrnBr",
"mulrn_lgt0",
"natmul",
"subn_gt0",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pMn2l x :
0 < x -> {mono (@GRing.natmul R x) : m n / (m <= n)%N >-> m <= n}. | Proof.
move=> x_gt0; apply: le_mono; elim=> [|m IHm] [|n]//= lt_mn.
by rewrite mulr0n mulrn_lgt0.
by rewrite !mulrS ler_ltD// IHm.
Qed. | Lemma | ler_pMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"le_mono",
"ler_ltD",
"mulr0n",
"mulrS",
"mulrn_lgt0",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pMn2l x :
0 < x -> {mono (@GRing.natmul R x) : m n / (m < n)%N >-> m < n}. | Proof. by move=> x_gt0; apply: leW_mono (ler_pMn2l _). Qed. | Lemma | ltr_pMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"leW_mono",
"ler_pMn2l",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nMn2l x :
x < 0 -> {mono (@GRing.natmul R x) : m n / (n <= m)%N >-> m <= n}. | Proof. by move=> xlt0 m n /=; rewrite -lerN2 -!mulNrn ler_pMn2l// oppr_gt0. Qed. | Lemma | ler_nMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lerN2",
"ler_pMn2l",
"mulNrn",
"natmul",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nMn2l x :
x < 0 -> {mono (@GRing.natmul R x) : m n / (n < m)%N >-> m < n}. | Proof. by move=> x_lt0; apply: leW_nmono (ler_nMn2l _). Qed. | Lemma | ltr_nMn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"leW_nmono",
"ler_nMn2l",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nneg_nmod_closed : nmod_closed (@Num.nneg R). | Proof. by split; [apply: lexx | apply: addr_ge0]. Qed. | Fact | nneg_nmod_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addr_ge0",
"apply",
"lexx",
"nmod_closed",
"nneg",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_oppr_closed : oppr_closed (@Num.real R). | Proof. by move=> x; rewrite /= !realE oppr_ge0 orbC -!oppr_ge0 opprK. Qed. | Fact | real_oppr_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"opprK",
"oppr_closed",
"oppr_ge0",
"real",
"realE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real0 : 0%R \is @Num.real R. | Proof. by rewrite qualifE/= lexx. Qed. | Lemma | real0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lexx",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable0r x : (0 >=< x)%R = (x \is Num.real). | Proof. by []. Qed. | Lemma | comparable0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real"
] | Comparability in a numDomain | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
comparabler0 x : (x >=< 0)%R = (x \is Num.real). | Proof. by rewrite comparable_sym. Qed. | Lemma | comparabler0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_sym",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_comparable0 x y : (x - y >=< 0)%R = (x >=< y)%R. | Proof. by rewrite /Num.comparable subr_ge0 subr_le0. Qed. | Lemma | subr_comparable0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable",
"subr_ge0",
"subr_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparablerE x y : (x >=< y)%R = (x - y \is Num.real). | Proof. by rewrite -comparabler0 subr_comparable0. Qed. | Lemma | comparablerE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparabler0",
"real",
"subr_comparable0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_real x : 0 <= x -> x \is Num.real. | Proof. by rewrite realE => ->. Qed. | Lemma | ger0_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"realE"
] | Properties of the real subset. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ler0_real x : x <= 0 -> x \is Num.real. | Proof. by rewrite realE orbC => ->. Qed. | Lemma | ler0_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"realE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_real x : 0 < x -> x \is Num.real. | Proof. by move=> /ltW/ger0_real. Qed. | Lemma | gtr0_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ger0_real",
"ltW",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_real x : x < 0 -> x \is Num.real. | Proof. by move=> /ltW/ler0_real. Qed. | Lemma | ltr0_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler0_real",
"ltW",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_real x0 op I (P : pred I) F (s : seq I) :
{in Num.real &, forall x y, op x y \is Num.real} -> x0 \is Num.real ->
{in P, forall i, F i \is Num.real} -> \big[op/x0]_(i <- s | P i) F i \is Num.real. | Proof. exact: comparable_bigr. Qed. | Lemma | big_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_bigr",
"real",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_min_max x y : min x y + max x y = x + y. | Proof. by rewrite /min /max; case: ifP => //; rewrite addrC. Qed. | Lemma | addr_min_max | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrC",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_max_min x y : max x y + min x y = x + y. | Proof. by rewrite addrC addr_min_max. Qed. | Lemma | addr_max_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrC",
"addr_min_max",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_to_max x y : min x y = x + y - max x y. | Proof. by rewrite -[x + y]addr_min_max addrK. Qed. | Lemma | minr_to_max | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrK",
"addr_min_max",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_to_min x y : max x y = x + y - min x y. | Proof. by rewrite -[x + y]addr_max_min addrK. Qed. | Lemma | maxr_to_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrK",
"addr_max_min",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifNl C x y : (- x < y ?<= if C) = (- y < x ?<= if C). | Proof. by case: C; rewrite /= lterNl. Qed. | Lemma | lteifNl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterNl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifNr C x y : (x < - y ?<= if C) = (y < - x ?<= if C). | Proof. by case: C; rewrite /= lterNr. Qed. | Lemma | lteifNr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif0Nr C x : (0 < - x ?<= if C) = (x < 0 ?<= if C). | Proof. by case: C; rewrite /= (oppr_ge0, oppr_gt0). Qed. | Lemma | lteif0Nr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"oppr_ge0",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifNr0 C x : (- x < 0 ?<= if C) = (0 < x ?<= if C). | Proof. by case: C; rewrite /= (oppr_le0, oppr_lt0). Qed. | Lemma | lteifNr0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"oppr_le0",
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifN2 C : {mono -%R : x y /~ x < y ?<= if C :> R}. | Proof. by case: C => ? ?; rewrite /= lterN2. Qed. | Lemma | lteifN2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterN2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_oppE | := (lteif0Nr, lteifNr0, lteifN2). | Definition | lteif_oppE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lteif0Nr",
"lteifN2",
"lteifNr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifD2l C x : {mono +%R x : y z / y < z ?<= if C}. | Proof. by case: C => ? ?; rewrite /= lterD2. Qed. | Lemma | lteifD2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterD2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifD2r C x : {mono +%R^~ x : y z / y < z ?<= if C}. | Proof. by case: C => ? ?; rewrite /= lterD2. Qed. | Lemma | lteifD2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterD2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifD2 | := (lteifD2l, lteifD2r). | Definition | lteifD2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lteifD2l",
"lteifD2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBlDr C x y z : (x - y < z ?<= if C) = (x < z + y ?<= if C). | Proof. by case: C; rewrite /= lterBDr. Qed. | Lemma | lteifBlDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterBDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBrDr C x y z : (x < y - z ?<= if C) = (x + z < y ?<= if C). | Proof. by case: C; rewrite /= lterBDr. Qed. | Lemma | lteifBrDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterBDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBDr | := (lteifBlDr, lteifBrDr). | Definition | lteifBDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lteifBlDr",
"lteifBrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBlDl C x y z : (x - y < z ?<= if C) = (x < y + z ?<= if C). | Proof. by case: C; rewrite /= lterBDl. Qed. | Lemma | lteifBlDl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterBDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBrDl C x y z : (x < y - z ?<= if C) = (z + x < y ?<= if C). | Proof. by case: C; rewrite /= lterBDl. Qed. | Lemma | lteifBrDl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterBDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifBDl | := (lteifBlDl, lteifBrDl). | Definition | lteifBDl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lteifBlDl",
"lteifBrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparabler_trans : transitive (Num.comparable : rel R). | Proof. exact: comparabler_trans. Qed. | Lemma | comparabler_trans | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_leVge x y : 0 <= x -> 0 <= y -> (x <= y) || (y <= x). | Proof.
by move=> /ge_comparable + /le_comparable => /comparabler_trans/[apply].
Qed. | Lemma | ger_leVge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"comparabler_trans",
"ge_comparable",
"le_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_leVge x y : x <= 0 -> y <= 0 -> (x <= y) || (y <= x). | Proof. by rewrite -!oppr_ge0 => /(ger_leVge _) /[apply]; rewrite !lerN2. Qed. | Lemma | ler_leVge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"ger_leVge",
"lerN2",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_leVge x y : x \is Num.real -> y \is Num.real -> (x <= y) || (y <= x). | Proof. by rewrite -comparabler0 -comparable0r => /comparabler_trans P/P. Qed. | Lemma | real_leVge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable0r",
"comparabler0",
"comparabler_trans",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_comparable x y : x \is Num.real -> y \is Num.real -> x >=< y. | Proof. exact: real_leVge. Qed. | Lemma | real_comparable | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"real_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realB : {in Num.real &, forall x y, x - y \is Num.real}. | Proof. exact: rpredB. Qed. | Lemma | realB | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"rpredB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realN : {mono (@GRing.opp R) : x / x \is Num.real}. | Proof. exact: rpredN. Qed. | Lemma | realN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"opp",
"real",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realBC x y : (x - y \is Num.real) = (y - x \is Num.real). | Proof. exact: rpredBC. Qed. | Lemma | realBC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"rpredBC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realD : {in Num.real &, forall x y, x + y \is Num.real}. | Proof. exact: rpredD. Qed. | Lemma | realD | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"rpredD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_real I (P : pred I) (F : I -> R) (s : seq I) :
{in P, forall i, F i \is Num.real} -> \sum_(i <- s | P i) F i \is Num.real. | Proof. by apply/big_real; [apply: rpredD | apply: rpred0]. Qed. | Lemma | sum_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"big_real",
"real",
"rpred0",
"rpredD",
"seq"
] | #[local] Hint Resolve real0 real1 : core. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
zmod_leP x y : x \is Num.real -> y \is Num.real ->
Order.le_xor_gt x y (min y x) (min x y) (max y x) (max x y)
(x <= y) (y < x). | Proof. by move=> xR yR; apply: comparable_leP; exact: real_leVge. Qed. | Lemma | zmod_leP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"comparable_leP",
"le_xor_gt",
"max",
"min",
"real",
"real_leVge"
] | dichotomy and trichotomy | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
zmod_ltP x y : x \is Num.real -> y \is Num.real ->
Order.lt_xor_ge x y (min y x) (min x y) (max y x) (max x y)
(y <= x) (x < y). | Proof. by move=> xR yR; apply: comparable_ltP; exact: real_leVge. Qed. | Lemma | zmod_ltP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"comparable_ltP",
"lt_xor_ge",
"max",
"min",
"real",
"real_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_ltgtP x y : x \is Num.real -> y \is Num.real ->
Order.compare x y (min y x) (min x y) (max y x) (max x y)
(y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y). | Proof. by move=> xR yR; apply: comparable_ltgtP; exact: real_leVge. Qed. | Lemma | zmod_ltgtP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"comparable_ltgtP",
"compare",
"max",
"min",
"real",
"real_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltNge : {in real &, forall x y, (x < y) = ~~ (y <= x)}. | Proof. by move=> x y xR yR /=; case: zmod_leP. Qed. | Lemma | real_ltNge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_leNgt : {in real &, forall x y, (x <= y) = ~~ (y < x)}. | Proof. by move=> x y xR yR /=; case: zmod_leP. Qed. | Lemma | real_leNgt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_xor_lt0 (x : R) : R -> R -> R -> R ->
bool -> bool -> Set | :=
| Ger0NotLt0 of 0 <= x : ger0_xor_lt0 x 0 0 x x false true
| Ltr0NotGe0 of x < 0 : ger0_xor_lt0 x x x 0 0 true false. | Variant | ger0_xor_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_xor_gt0 (x : R) : R -> R -> R -> R ->
bool -> bool -> Set | :=
| Ler0NotLe0 of x <= 0 : ler0_xor_gt0 x x x 0 0 false true
| Gtr0NotGt0 of 0 < x : ler0_xor_gt0 x 0 0 x x true false. | Variant | ler0_xor_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparer0 x : R -> R -> R -> R ->
bool -> bool -> bool -> bool -> bool -> bool -> Set | :=
| ComparerGt0 of 0 < x : comparer0 x 0 0 x x false false false true false true
| ComparerLt0 of x < 0 : comparer0 x x x 0 0 false false true false true false
| ComparerEq0 of x = 0 : comparer0 x 0 0 0 0 true true true true false false. | Variant | comparer0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_ge0P x : x \is Num.real -> ger0_xor_lt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0) (x < 0) (0 <= x). | Proof.
move=> hx; case: comparable_leP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | zmod_ge0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_leP",
"ger0_xor_lt0",
"max",
"min",
"real",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_le0P x : x \is Num.real -> ler0_xor_gt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0)
(0 < x) (x <= 0). | Proof.
move=> hx; case: comparable_ltP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | zmod_le0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_ltP",
"ler0_xor_gt0",
"max",
"min",
"real",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_ltgt0P x : x \is Num.real ->
comparer0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
(0 == x) (x == 0) (x <= 0) (0 <= x) (x < 0) (x > 0). | Proof.
move=> hx; case: (@comparable_ltgtP _ _ 0 x);
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | zmod_ltgt0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_ltgtP",
"comparer0",
"max",
"min",
"real",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_real : {in real &, forall x y, max x y \is Num.real}. | Proof. exact: comparable_maxr. Qed. | Lemma | max_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_maxr",
"max",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min_real : {in real &, forall x y, min x y \is Num.real}. | Proof. exact: comparable_minr. Qed. | Lemma | min_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"comparable_minr",
"min",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigmax_real I x0 (r : seq I) (P : pred I) (f : I -> R):
x0 \is Num.real -> {in P, forall i : I, f i \is Num.real} ->
\big[max/x0]_(i <- r | P i) f i \is Num.real. | Proof. exact/big_real/max_real. Qed. | Lemma | bigmax_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"big_real",
"max",
"max_real",
"real",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigmin_real I x0 (r : seq I) (P : pred I) (f : I -> R):
x0 \is Num.real -> {in P, forall i : I, f i \is Num.real} ->
\big[min/x0]_(i <- r | P i) f i \is Num.real. | Proof. exact/big_real/min_real. Qed. | Lemma | bigmin_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"big_real",
"min",
"min_real",
"real",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_neqr_lt : {in real &, forall x y, (x != y) = (x < y) || (y < x)}. | Proof. by move=> * /=; case: zmod_ltgtP. Qed. | Lemma | real_neqr_lt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"real",
"zmod_ltgtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerB_real x y : x <= y -> y - x \is Num.real. | Proof. by move=> le_xy; rewrite ger0_real // subr_ge0. Qed. | Lemma | lerB_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ger0_real",
"real",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gerB_real x y : x <= y -> x - y \is Num.real. | Proof. by move=> le_xy; rewrite ler0_real // subr_le0. Qed. | Lemma | gerB_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler0_real",
"real",
"subr_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_real y x : x <= y -> (x \is Num.real) = (y \is Num.real). | Proof. by move=> le_xy; rewrite -(addrNK x y) rpredDl ?lerB_real. Qed. | Lemma | ler_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrNK",
"lerB_real",
"real",
"rpredDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_real x y : y <= x -> (x \is Num.real) = (y \is Num.real). | Proof. by move=> le_yx; rewrite -(ler_real le_yx). Qed. | Lemma | ger_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ler_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nreal_leF x y : y \is Num.real -> x \notin real -> (x <= y) = false. | Proof. by move=> yR; apply: contraNF=> /ler_real->. Qed. | Lemma | Nreal_leF | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"ler_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nreal_geF x y : y \is Num.real -> x \notin real -> (y <= x) = false. | Proof. by move=> yR; apply: contraNF=> /ger_real->. Qed. | Lemma | Nreal_geF | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"ger_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nreal_ltF x y : y \is Num.real -> x \notin real -> (x < y) = false. | Proof. by move=> yR xNR; rewrite lt_def Nreal_leF ?andbF. Qed. | Lemma | Nreal_ltF | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"Nreal_leF",
"lt_def",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nreal_gtF x y : y \is Num.real -> x \notin real -> (y < x) = false. | Proof. by move=> yR xNR; rewrite lt_def Nreal_geF ?andbF. Qed. | Lemma | Nreal_gtF | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"Nreal_geF",
"lt_def",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_wlog_ler P :
(forall a b, P b a -> P a b) -> (forall a b, a <= b -> P a b) ->
forall a b : R, a \is Num.real -> b \is Num.real -> P a b. | Proof.
move=> sP hP a b ha hb; wlog: a b ha hb / a <= b => [hwlog|]; last exact: hP.
by case: (@zmod_leP a b)=> // [/hP //|/ltW hba]; apply/sP/hP.
Qed. | Lemma | real_wlog_ler | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"last",
"ltW",
"real",
"zmod_leP"
] | real wlog | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_wlog_ltr P :
(forall a, P a a) -> (forall a b, (P b a -> P a b)) ->
(forall a b, a < b -> P a b) ->
forall a b : R, a \is Num.real -> b \is Num.real -> P a b. | Proof.
move=> rP sP hP; apply: real_wlog_ler=> // a b.
by rewrite le_eqVlt; case: eqVneq => [->|] //= _ /hP.
Qed. | Lemma | real_wlog_ltr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"apply",
"eqVneq",
"le_eqVlt",
"real",
"real_wlog_ler"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_oppr_max : {in real &, {morph -%R : x y / max x y >-> min x y : R}}. | Proof.
by move=> x y xr yr; rewrite !(fun_if, if_arg) ltrN2; case: zmod_ltgtP => // ->.
Qed. | Lemma | real_oppr_max | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltrN2",
"max",
"min",
"real",
"zmod_ltgtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_oppr_min : {in real &, {morph -%R : x y / min x y >-> max x y : R}}. | Proof.
by move=> x y xr yr; rewrite -[RHS]opprK real_oppr_max ?realN// !opprK.
Qed. | Lemma | real_oppr_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"max",
"min",
"opprK",
"real",
"realN",
"real_oppr_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_addr_minl : {in real & real & real, @left_distributive R R +%R min}. | Proof.
by move=> x y z xr yr zr; case: (@zmod_leP (_ + _)); rewrite ?rpredD//;
rewrite lterD2; case: zmod_leP.
Qed. | Lemma | real_addr_minl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterD2",
"min",
"real",
"rpredD",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_addr_minr : {in real & real & real, @right_distributive R R +%R min}. | Proof. by move=> x y z xr yr zr; rewrite !(addrC x) real_addr_minl. Qed. | Lemma | real_addr_minr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrC",
"min",
"real",
"real_addr_minl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_addr_maxl : {in real & real & real, @left_distributive R R +%R max}. | Proof.
by move=> x y z xr yr zr; case: (@zmod_leP (_ + _)); rewrite ?realD//;
rewrite lterD2; case: zmod_leP.
Qed. | Lemma | real_addr_maxl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"lterD2",
"max",
"real",
"realD",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_addr_maxr : {in real & real & real, @right_distributive R R +%R max}. | Proof. by move=> x y z xr yr zr; rewrite !(addrC x) real_addr_maxl. Qed. | Lemma | real_addr_maxr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"addrC",
"max",
"real",
"real_addr_maxl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_mono :
{homo f : x y / x < y} -> {in real &, {mono f : x y / x <= y}}. | Proof.
move=> mf x y xR yR /=; have [lt_xy | le_yx] := zmod_leP xR yR.
by rewrite ltW_homo.
by rewrite lt_geF ?mf.
Qed. | Lemma | real_mono | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_homo",
"lt_geF",
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_nmono :
{homo f : x y /~ x < y} -> {in real &, {mono f : x y /~ x <= y}}. | Proof.
move=> mf x y xR yR /=; have [lt_xy|le_yx] := zmod_ltP xR yR.
by rewrite lt_geF ?mf.
by rewrite ltW_nhomo.
Qed. | Lemma | real_nmono | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_nhomo",
"lt_geF",
"real",
"zmod_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_mono_in :
{in D &, {homo f : x y / x < y}} ->
{in [pred x in D | x \is real] &, {mono f : x y / x <= y}}. | Proof.
move=> Dmf x y /andP[hx xR] /andP[hy yR] /=.
have [lt_xy|le_yx] := zmod_leP xR yR; first by rewrite (ltW_homo_in Dmf).
by rewrite lt_geF ?Dmf.
Qed. | Lemma | real_mono_in | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_homo_in",
"lt_geF",
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_nmono_in :
{in D &, {homo f : x y /~ x < y}} ->
{in [pred x in D | x \is real] &, {mono f : x y /~ x <= y}}. | Proof.
move=> Dmf x y /andP[hx xR] /andP[hy yR] /=.
have [lt_xy|le_yx] := zmod_ltP xR yR; last by rewrite (ltW_nhomo_in Dmf).
by rewrite lt_geF ?Dmf.
Qed. | Lemma | real_nmono_in | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"last",
"ltW_nhomo_in",
"lt_geF",
"real",
"zmod_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realn_mono : {homo f' : x y / x < y >-> (x < y)} ->
{in real &, {mono f' : x y / x <= y >-> (x <= y)}}. | Proof.
move=> mf x y xR yR /=; have [lt_xy | le_yx] := zmod_leP xR yR.
by rewrite ltW_homo.
by rewrite lt_geF ?mf.
Qed. | Lemma | realn_mono | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_homo",
"lt_geF",
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realn_nmono : {homo f' : x y / y < x >-> (x < y)} ->
{in real &, {mono f' : x y / y <= x >-> (x <= y)}}. | Proof.
move=> mf x y xR yR /=; have [lt_xy|le_yx] := zmod_ltP xR yR.
by rewrite lt_geF ?mf.
by rewrite ltW_nhomo.
Qed. | Lemma | realn_nmono | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_nhomo",
"lt_geF",
"real",
"zmod_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realn_mono_in : {in D &, {homo f' : x y / x < y >-> (x < y)}} ->
{in [pred x in D | x \is real] &, {mono f' : x y / x <= y >-> (x <= y)}}. | Proof.
move=> Dmf x y /andP[hx xR] /andP[hy yR] /=.
have [lt_xy|le_yx] := zmod_leP xR yR; first by rewrite (ltW_homo_in Dmf).
by rewrite lt_geF ?Dmf.
Qed. | Lemma | realn_mono_in | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"ltW_homo_in",
"lt_geF",
"real",
"zmod_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realn_nmono_in : {in D &, {homo f' : x y / y < x >-> (x < y)}} ->
{in [pred x in D | x \is real] &, {mono f' : x y / y <= x >-> (x <= y)}}. | Proof.
move=> Dmf x y /andP[hx xR] /andP[hy yR] /=.
have [lt_xy|le_yx] := zmod_ltP xR yR; last by rewrite (ltW_nhomo_in Dmf).
by rewrite lt_geF ?Dmf.
Qed. | Lemma | realn_nmono_in | algebra.numeric_hierarchy | algebra/numeric_hierarchy/orderedzmod.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"Order.TTheory",
"GRing.Theory",
"Order.PreOCoercions",
"Num.Syntax",
"Num.Exports"
] | [
"last",
"ltW_nhomo_in",
"lt_geF",
"real",
"zmod_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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