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 |
|---|---|---|---|---|---|---|---|---|---|---|
incomparable_leF x y : (x >< y) -> (x <= y) = false. | Proof. by apply: contraNF; rewrite /comparable => ->. Qed. | Lemma | incomparable_leF | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incomparable_ltF x y : (x >< y) -> (x < y) = false. | Proof. by rewrite lt_le_def => /incomparable_leF ->. Qed. | Lemma | incomparable_ltF | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"incomparable_leF",
"lt_le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_comparable (x y : T) : x <= y -> x >=< y. | Proof. by rewrite /comparable => ->. Qed. | Lemma | le_comparable | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_comparable (x y : T) : x < y -> x >=< y. | Proof. by rewrite /comparable => /ltW ->. Qed. | Lemma | lt_comparable | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_comparable (x y : T) : y <= x -> x >=< y. | Proof. by rewrite /comparable orbC => ->. Qed. | Lemma | ge_comparable | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_comparable (x y : T) : y < x -> x >=< y. | Proof. by rewrite /comparable orbC => /ltW ->. Qed. | Lemma | gt_comparable | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_refl x C : reflect (x <= x ?= iff C) C. | Proof. by apply: (iffP idP) => [-> | <-] //; split; rewrite ?eqxx. Qed. | Lemma | leif_refl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"eqxx",
"split"
] | leif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_leif x y C : x <= y ?= iff C -> (x == y) = C. | Proof. by move=> []. Qed. | Lemma | eq_leif | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqTleif x y C : x <= y ?= iff C -> C -> x = y. | Proof. by move=> [] _ <- /eqP. Qed. | Lemma | eqTleif | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_trans x y z C1 C2 :
x < y ?<= if C1 -> y < z ?<= if C2 -> x < z ?<= if C1 && C2. | Proof.
case: C1 C2 => [][];
[exact: le_trans | exact: le_lt_trans | exact: lt_le_trans | exact: lt_trans].
Qed. | Lemma | lteif_trans | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_lt_trans",
"le_trans",
"lt_le_trans",
"lt_trans"
] | lteif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lteifxx x C : (x < x ?<= if C) = C. | Proof. by case: C; rewrite /= ltexx. Qed. | Lemma | lteifxx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifNF x y C : y < x ?<= if ~~ C -> (x < y ?<= if C) = false. | Proof. by case: C => [/lt_geF|/le_gtF]. Qed. | Lemma | lteifNF | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_gtF",
"lt_geF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifS x y C : x < y -> x < y ?<= if C. | Proof. by case: C => //= /ltW. Qed. | Lemma | lteifS | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifT x y : (x < y ?<= if true) = (x <= y). | Proof. by []. Qed. | Lemma | lteifT | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifF x y : (x < y ?<= if false) = (x < y). | Proof. by []. Qed. | Lemma | lteifF | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_orb x y : {morph lteif x y : p q / p || q}. | Proof.
case=> [][] /=.
- by rewrite orbb.
- by case/boolP: (x < y) => [/ltW -> //|_]; rewrite orbF.
- by case/boolP: (x < y) => [/ltW ->|].
- by rewrite orbb.
Qed. | Lemma | lteif_orb | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltW",
"lteif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_andb x y : {morph lteif x y : p q / p && q}. | Proof.
case=> [][] /=.
- by rewrite andbb.
- by rewrite lt_le_def andbA andbb.
- by rewrite andbC lt_le_def andbA andbb.
- by rewrite andbb.
Qed. | Lemma | lteif_andb | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lt_le_def",
"lteif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_imply C1 C2 x y : C1 ==> C2 -> x < y ?<= if C1 -> x < y ?<= if C2. | Proof. by case: C1 C2 => [][] //= _ /ltW. Qed. | Lemma | lteif_imply | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteifW C x y : x < y ?<= if C -> x <= y. | Proof. by case: C => // /ltW. Qed. | Lemma | lteifW | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrW_lteif | := lteifS. | Notation | ltrW_lteif | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lteifS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minElt x y : min x y = if x < y then x else y. | Proof. by []. Qed. | Lemma | minElt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"min"
] | min and max | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
maxElt x y : max x y = if x < y then y else x. | Proof. by []. Qed. | Lemma | maxElt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minxx : idempotent_op (min : T -> T -> T). | Proof. by rewrite /min => x; rewrite ltxx. Qed. | Lemma | minxx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"idempotent_op",
"ltxx",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxxx : idempotent_op (max : T -> T -> T). | Proof. by rewrite /max => x; rewrite ltxx. Qed. | Lemma | maxxx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"idempotent_op",
"ltxx",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min_minKx x y : min (min x y) y = min x y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | min_minKx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min_minxK x y : min x (min x y) = min x y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | min_minxK | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_maxKx x y : max (max x y) y = max x y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | max_maxKx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_maxxK x y : max x (max x y) = max x y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | max_maxxK | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_minl z : {in >=< z &, forall x y, min x y >=< z}. | Proof. by move=> x y cmp_xz cmp_yz; rewrite /min; case: ifP. Qed. | Lemma | comparable_minl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_minr z : {in >=<%O z &, forall x y, z >=< min x y}. | Proof. by move=> x y cmp_xz cmp_yz; rewrite /min; case: ifP. Qed. | Lemma | comparable_minr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_maxl z : {in >=< z &, forall x y, max x y >=< z}. | Proof. by move=> x y cmp_xz cmp_yz; rewrite /max; case: ifP. Qed. | Lemma | comparable_maxl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_maxr z : {in >=<%O z &, forall x y, z >=< max x y}. | Proof. by move=> x y cmp_xz cmp_yz; rewrite /max; case: ifP. Qed. | Lemma | comparable_maxr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_le_min : (z <= min x y) = (z <= x) && (z <= y). | Proof.
move: cmp_xy; rewrite /min /comparable lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- by rewrite andbC; case/boolP: (z <= y) => zy //=; apply/esym/(le_trans zy).
- by case/boolP: (z <= x) => zx //=; apply/esym/(le_trans zx).
- by rewrite andbC; case/boolP: (z <= y) => zy //=; apply/e... | Lemma | comparable_le_min | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_trans",
"lt_le_def",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_ge_min : (min x y <= z) = (x <= z) || (y <= z). | Proof.
move: cmp_xy; rewrite /min /comparable lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- rewrite orbC; case/boolP: (y <= z) => //= /negP yz.
by apply/esym/negP => /(le_trans yx).
- by case/boolP: (x <= z) => //= /negP xz; apply/esym/negP => /(le_trans xy).
- rewrite orbC; case/boolP: ... | Lemma | comparable_ge_min | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_trans",
"lt_le_def",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lt_min : (z < min x y) = (z < x) && (z < y). | Proof.
move: cmp_xy; rewrite /min /comparable !lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- rewrite -!lt_le_def; case/boolP: (z < x) => //= /negP zx.
by apply/negP => zy; apply/zx/(lt_le_trans zy).
- rewrite -!lt_le_def; case/boolP: (z < x) => //= zx; apply/esym/(lt_trans zx).
by rewr... | Lemma | comparable_lt_min | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"lt_le_def",
"lt_le_trans",
"lt_trans",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_gt_min : (min x y < z) = (x < z) || (y < z). | Proof.
move: cmp_xy; rewrite /min /comparable !lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- rewrite -!lt_le_def orbC; case/boolP: (y < z) => //= /negP yz.
by apply/esym/negP => xz; apply/yz/(le_lt_trans yx).
- rewrite -!lt_le_def; case/boolP: (x < z) => //= /negP xz.
by apply/esym/neg... | Lemma | comparable_gt_min | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_lt_trans",
"lt_le_def",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_le_max : (z <= max x y) = (z <= x) || (z <= y). | Proof.
move: cmp_xy; rewrite /max /comparable lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- case/boolP: (z <= x) => //= /negP zx.
by apply/esym/negP => zy; apply/zx/(le_trans zy).
- rewrite orbC; case/boolP: (z <= y) => //= /negP zy.
by apply/esym/negP => zx; apply/zy/(le_trans zx).
- ... | Lemma | comparable_le_max | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_trans",
"lt_le_def",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_ge_max : (max x y <= z) = (x <= z) && (y <= z). | Proof.
move: cmp_xy; rewrite /max /comparable lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- case/boolP: (x <= z) => //= xz.
by apply/esym/(le_trans yx).
- rewrite andbC; case/boolP: (y <= z) => //= yz.
by apply/esym/(le_trans xy).
- case/boolP: (x <= z) => //= xz.
by apply/esym/(le_t... | Lemma | comparable_ge_max | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_trans",
"lt_le_def",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lt_max : (z < max x y) = (z < x) || (z < y). | Proof.
move: cmp_xy; rewrite /max /comparable !lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- rewrite -!lt_le_def; case/boolP: (z < x) => //= /negP zx.
by apply/esym/negP => zy; apply/zx/(lt_le_trans zy).
- rewrite -!lt_le_def orbC; case/boolP: (z < y) => //= /negP zy.
by apply/esym/neg... | Lemma | comparable_lt_max | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"lt_le_def",
"lt_le_trans",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_gt_max : (max x y < z) = (x < z) && (y < z). | Proof.
move: cmp_xy; rewrite /max /comparable !lt_le_def.
case/boolP: (x <= y) => xy/=; case/boolP: (y <= x) => yx//= _.
- rewrite -!lt_le_def; case/boolP: (x < z) => //= xz.
by apply/esym/(le_lt_trans yx).
- rewrite -!lt_le_def andbC; case/boolP: (y < z) => //= yz.
by apply/esym/(le_lt_trans xy).
- rewrite -!lt_le... | Lemma | comparable_gt_max | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable",
"le_lt_trans",
"lt_le_def",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_minxK : max (min x y) y = y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | comparable_minxK | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_minKx : max x (min x y) = x. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | comparable_minKx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_maxxK : min (max x y) y = y. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | comparable_maxxK | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_maxKx : min x (max x y) = x. | Proof. by rewrite !(fun_if, if_arg) ltxx/=; case: (x < y). Qed. | Lemma | comparable_maxKx | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltxx",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lteif_minr C :
(z < min x y ?<= if C) = (z < x ?<= if C) && (z < y ?<= if C). | Proof. by case: C; rewrite /= (comparable_le_min, comparable_lt_min). Qed. | Lemma | comparable_lteif_minr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_le_min",
"comparable_lt_min",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lteif_minl C :
(min x y < z ?<= if C) = (x < z ?<= if C) || (y < z ?<= if C). | Proof. by case: C; rewrite /= (comparable_ge_min, comparable_gt_min). Qed. | Lemma | comparable_lteif_minl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_ge_min",
"comparable_gt_min",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lteif_maxr C :
(z < max x y ?<= if C) = (z < x ?<= if C) || (z < y ?<= if C). | Proof. by case: C; rewrite /= (comparable_le_max, comparable_lt_max). Qed. | Lemma | comparable_lteif_maxr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_le_max",
"comparable_lt_max",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_lteif_maxl C :
(max x y < z ?<= if C) = (x < z ?<= if C) && (y < z ?<= if C). | Proof. by case: C; rewrite /= (comparable_ge_max, comparable_gt_max). Qed. | Lemma | comparable_lteif_maxl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_ge_max",
"comparable_gt_max",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_minA : min x (min y z) = min (min x y) z. | Proof.
move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/= !lt_le_def.
case/boolP: (x <= y) => xy;
case/boolP: (y <= x) => yx;
case/boolP: (x <= z) => xz;
case/boolP: (z <= x) => zx;
case/boolP: (y <= z) => yz;
case/boolP: (z <= y) => zy //= _ _ _.
- by move: zx; rewrite (le_trans zy yx).
- by move: zx; r... | Lemma | comparable_minA | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_trans",
"lt_le_def",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_maxA : max x (max y z) = max (max x y) z. | Proof.
move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/= !lt_le_def.
case/boolP: (x <= y) => xy;
case/boolP: (y <= x) => yx;
case/boolP: (x <= z) => xz;
case/boolP: (z <= x) => zx;
case/boolP: (y <= z) => yz;
case/boolP: (z <= y) => zy //= _ _ _.
- by move: zx; rewrite (le_trans zy yx).
- by move: zx; r... | Lemma | comparable_maxA | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_trans",
"lt_le_def",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_min_maxl : min (max x y) z = max (min x z) (min y z). | Proof.
move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/= !lt_le_def.
case/boolP: (x <= y) => xy;
case/boolP: (y <= x) => yx;
case/boolP: (x <= z) => xz;
case/boolP: (z <= x) => zx;
case/boolP: (y <= z) => yz;
case/boolP: (z <= y) => zy;
rewrite ?lexx => //= _ _ _.
- by move: zx; rewrite (le_trans zy y... | Lemma | comparable_min_maxl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_trans",
"lexx",
"lt_le_def",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_max_minr :
max x (min y z) = min (max x y) (max x z). | Proof.
move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/= !lt_le_def.
case/boolP: (x <= y) => xy;
case/boolP: (y <= x) => yx;
case/boolP: (x <= z) => xz;
case/boolP: (z <= x) => zx;
case/boolP: (y <= z) => yz;
case/boolP: (z <= y) => zy;
rewrite ?lexx => //= _ _ _.
- by move: zx; rewrite (le_trans zy y... | Lemma | comparable_max_minr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le_trans",
"lexx",
"lt_le_def",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_comparable : {in P &, forall i j, F i >=< F j}. | Hypothesis | F_comparable | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
comparable_arg_minP: extremum_spec <=%O P F (arg_min i0 P F). | Proof.
by apply: extremum_inP => // [x _|y x z _ _ _]; [apply: lexx|apply: le_trans].
Qed. | Lemma | comparable_arg_minP | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"arg_min",
"extremum_inP",
"extremum_spec",
"i0",
"le_trans",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_arg_maxP: extremum_spec >=%O P F (arg_max i0 P F). | Proof.
apply: extremum_inP => // [x _|y x z _ _ _|]; [exact: lexx|exact: ge_trans|].
by move=> x y xP yP; rewrite orbC [_ || _]F_comparable.
Qed. | Lemma | comparable_arg_maxP | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"F_comparable",
"apply",
"arg_max",
"extremum_inP",
"extremum_spec",
"ge_trans",
"i0",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_bigl x x0 op I (P : pred I) F (s : seq I) :
{in >=< x &, forall y z, op y z >=< x} -> x0 >=< x ->
{in P, forall i, F i >=< x} -> \big[op/x0]_(i <- s | P i) F i >=< x. | Proof. by move=> *; elim/big_ind : _. Qed. | Lemma | comparable_bigl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"big_ind",
"seq"
] | monotonicity | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
comparable_bigr x x0 op I (P : pred I) F (s : seq I) :
{in >=<%O x &, forall y z, x >=< op y z} -> x >=< x0 ->
{in P, forall i, x >=< F i} -> x >=< \big[op/x0]_(i <- s | P i) F i. | Proof. by move=> *; elim/big_ind : _. Qed. | Lemma | comparable_bigr | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"big_ind",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigmax_lt : x0 < x -> (forall i, P i -> f i < x) ->
\big[max/x0]_(i <- r | P i) f i < x. | Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite maxElt; case: ifPn. Qed. | Lemma | bigmax_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"big_ind",
"max",
"maxElt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_bigmin : x < x0 -> (forall i, P i -> x < f i) ->
x < \big[min/x0]_(i <- r | P i) f i. | Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite minElt; case: ifPn. Qed. | Lemma | lt_bigmin | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"big_ind",
"min",
"minElt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraTle b x y : x >=< y -> (y < x -> ~~ b) -> (b -> x <= y). | Proof. by move=> /comparable_leNgt ->; case: (y < x); case: b. Qed. | Lemma | comparable_contraTle | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraTlt b x y : x >=< y -> (y <= x -> ~~ b) -> (b -> x < y). | Proof. by move=> /comparable_ltNge ->; case: (y <= x); case: b. Qed. | Lemma | comparable_contraTlt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraPle P x y : x >=< y -> (y < x -> ~ P) -> (P -> x <= y). | Proof. by move=> /comparable_leNgt -> np p; apply/negP => /np. Qed. | Lemma | comparable_contraPle | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraPlt P x y : x >=< y -> (y <= x -> ~ P) -> (P -> x < y). | Proof. by move=> /comparable_ltNge -> np p; apply/negP => /np. Qed. | Lemma | comparable_contraPlt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraNle b x y : x >=< y -> (y < x -> b) -> (~~ b -> x <= y). | Proof. by move=> /comparable_leNgt ->; case: (y < x); case: b. Qed. | Lemma | comparable_contraNle | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraNlt b x y : x >=< y -> (y <= x -> b) -> (~~ b -> x < y). | Proof. by move=> /comparable_ltNge ->; case: (y <= x); case: b. Qed. | Lemma | comparable_contraNlt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_not_le P x y : x >=< y -> (y < x -> P) -> (~ P -> x <= y). | Proof. by move=> /comparable_leNgt -> np p; apply/negP => /np. Qed. | Lemma | comparable_contra_not_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_not_lt P x y : x >=< y -> (y <= x -> P) -> (~ P -> x < y). | Proof. by move=> /comparable_ltNge -> np p; apply/negP => /np. Qed. | Lemma | comparable_contra_not_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraFle b x y : x >=< y -> (y < x -> b) -> (b = false -> x <= y). | Proof. by move=> /comparable_leNgt -> np /negP p; apply/negP => /np. Qed. | Lemma | comparable_contraFle | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contraFlt b x y : x >=< y -> (y <= x -> b) -> (b = false -> x < y). | Proof. by move=> /comparable_ltNge -> np /negP p; apply/negP => /np. Qed. | Lemma | comparable_contraFlt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_leq_le m n x y : x >=< y ->
(y < x -> (n < m)%N) -> ((m <= n)%N -> x <= y). | Proof. by rewrite ltnNge; apply/comparable_contraTle. Qed. | Lemma | comparable_contra_leq_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_contraTle",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_leq_lt m n x y : x >=< y ->
(y <= x -> (n < m)%N) -> ((m <= n)%N -> x < y). | Proof. by rewrite ltnNge; apply/comparable_contraTlt. Qed. | Lemma | comparable_contra_leq_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_contraTlt",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_ltn_le m n x y : x >=< y ->
(y < x -> (n <= m)%N) -> ((m < n)%N -> x <= y). | Proof. by rewrite ltnNge; apply/comparable_contraNle. Qed. | Lemma | comparable_contra_ltn_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_contraNle",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_ltn_lt m n x y : x >=< y ->
(y <= x -> (n <= m)%N) -> ((m < n)%N -> x < y). | Proof. by rewrite ltnNge; apply/comparable_contraNlt. Qed. | Lemma | comparable_contra_ltn_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"comparable_contraNlt",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_le x y z t : z >=< t ->
(t < z -> y < x) -> (x <= y -> z <= t). | Proof.
rewrite /comparable lt_le_def; case: (z <= t) => //= -> /(_ erefl) yx.
by move=> /(lt_le_trans yx); rewrite ltxx.
Qed. | Lemma | comparable_contra_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"lt_le_def",
"lt_le_trans",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_le_lt x y z t : z >=< t ->
(t <= z -> y < x) -> (x <= y -> z < t). | Proof.
rewrite /comparable [z < t]lt_le_def orbC; case: (t <= z) => /= [_|-> //].
by move=> /(_ erefl) yx /(lt_le_trans yx); rewrite ltxx.
Qed. | Lemma | comparable_contra_le_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"lt_le_def",
"lt_le_trans",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_lt_le x y z t : z >=< t ->
(t < z -> y <= x) -> (x < y -> z <= t). | Proof.
rewrite /comparable lt_le_def; case: (z <= t) => //= -> /(_ erefl) yx.
by move=> /(le_lt_trans yx); rewrite ltxx.
Qed. | Lemma | comparable_contra_lt_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"le_lt_trans",
"lt_le_def",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_contra_lt x y z t : z >=< t ->
(t <= z -> y <= x) -> (x < y -> z < t). | Proof.
rewrite /comparable [z < t]lt_le_def orbC; case: (t <= z) => /= [_|-> //].
by move=> /(_ erefl) yx /(le_lt_trans yx); rewrite ltxx.
Qed. | Lemma | comparable_contra_lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"comparable",
"le_lt_trans",
"lt_le_def",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leW_mono : {mono f : x y / x <= y} -> {mono f : x y / x < y}. | Proof. by move=> fmono x y; rewrite !lt_le_def !fmono. Qed. | Lemma | leW_mono | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lt_le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leW_nmono : {mono f : x y /~ x <= y} -> {mono f : x y /~ x < y}. | Proof. by move=> fmono x y; rewrite !lt_le_def !fmono. Qed. | Lemma | leW_nmono | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lt_le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leW_mono_in :
{in D &, {mono f : x y / x <= y}} -> {in D &, {mono f : x y / x < y}}. | Proof. by move=> fmono x y xD yD; rewrite !lt_le_def !fmono. Qed. | Lemma | leW_mono_in | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lt_le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leW_nmono_in :
{in D &, {mono f : x y /~ x <= y}} -> {in D &, {mono f : x y /~ x < y}}. | Proof. by move=> fmono x y xD yD; rewrite !lt_le_def !fmono. Qed. | Lemma | leW_nmono_in | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"lt_le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le0x x : \bot <= x. | Proof. exact: le0x. Qed. | Lemma | le0x | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"bot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltx0 x : (x < \bot) = false. | Proof. exact/le_gtF/le0x. Qed. | Lemma | ltx0 | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"bot",
"le0x",
"le_gtF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lex1 x : x <= \top. | Proof. exact: lex1. Qed. | Lemma | lex1 | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt1x x : (\top < x) = false. | Proof. exact: (@ltx0 _ T^d). Qed. | Lemma | lt1x | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"ltx0",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_trans : transitive (fun x y => le y x). | Proof. by move=> x y z /[swap]; apply: le_trans. Qed. | Let | ge_trans | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"le",
"le_trans"
] | TODO: print nice error message when keyed type is not provided | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
le_refl : reflexive le. | Proof. by move=> x; rewrite le_def eqxx. Qed. | Let | le_refl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"eqxx",
"le",
"le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_trans : transitive le. | Proof.
move=> y x z; rewrite !le_def; case: (eqVneq x y) => [->|]//= neq_xy.
by case: (eqVneq y z) => /= [<- ->|_ /lt_trans yx /yx ->]; rewrite orbT.
Qed. | Let | le_trans | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"eqVneq",
"le",
"le_def",
"lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_le_def x y : lt x y = (le x y) && ~~ (le y x). | Proof.
rewrite !le_def eq_sym; have [->|_ /=] := eqVneq x y; first by rewrite lt_irr.
case/boolP: (lt x y) => //= xy; apply/esym/negP => /(lt_trans xy).
by rewrite lt_irr.
Qed. | Let | lt_le_def | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"eqVneq",
"eq_sym",
"le",
"le_def",
"lt",
"lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le (x y : T) | := f x <= f y. | Definition | le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt (x y : T) | := f x < f y. | Definition | lt | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
refl : reflexive le. | Proof. by move=> ?; apply: lexx. Qed. | Fact | refl | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"le",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trans : transitive le. | Proof. by move=> ? ? ?; apply: le_trans. Qed. | Fact | trans | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"le",
"le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_trans : transitive (fun x y => le y x). | Proof. by move=> ? ? ?; apply: ge_trans. Qed. | Fact | ge_trans | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"apply",
"le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_le_def x y : lt x y = le x y && ~~ le y x. | Proof. exact: lt_le_def. Qed. | Fact | lt_le_def | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"le",
"lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PrePcan | := isPreorder.Build disp T lt_le_def refl trans. | Definition | PrePcan | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"Build",
"lt_le_def",
"refl",
"trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_morphism d (T : preorderType d) d' (T' : preorderType d')
(f : T -> T') : Prop | := {mono f : x y / x <= y}. | Definition | order_morphism | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"T'"
] | Morphism hierarchy. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"{ 'omorphism' T -> T' }" | :=
(@OrderMorphism.type _ T%type _ T'%type) : type_scope. | Notation | { 'omorphism' T -> T' } | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"T'",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
omorph_le (d : disp_t) (T : preorderType d) (d' : disp_t) (T' : preorderType d')
(f : {omorphism T -> T'}) : {homo f : x y / x <= y}. | Proof. exact: omorph_le_subproof. Qed. | Lemma | omorph_le | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"T'",
"disp_t"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
idfun_is_nondecreasing : nondecreasing (@idfun T). | Proof. by []. Qed. | Fact | idfun_is_nondecreasing | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"nondecreasing"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.