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 |
|---|---|---|---|---|---|---|---|---|---|---|
"\min^d_ i F" | :=
(\big[min/top]_i F) : order_scope. | Notation | \min^d_ i F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i <- r | P ) F" | :=
(\big[min/top]_(i <- r | P%B) F%O) : order_scope. | Notation | \min^d_ ( i <- r | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i < r ) F" | :=
(\big[min/top]_(i <- r) F%O) : order_scope. | Notation | \min^d_ ( i < r ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( m <= i < n | P ) F" | :=
(\big[min/top]_(m <= i < n | P%B) F%O) : order_scope. | Notation | \min^d_ ( m <= i < n | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( m <= i < n ) F" | :=
(\big[min/top]_(m <= i < n) F%O) : order_scope. | Notation | \min^d_ ( m <= i < n ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i | P ) F" | :=
(\big[min/top]_(i | P%B) F%O) : order_scope. | Notation | \min^d_ ( i | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i : t | P ) F" | :=
(\big[min/top]_(i : t | P%B) F%O) (only parsing) : order_scope. | Notation | \min^d_ ( i : t | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i : t ) F" | :=
(\big[min/top]_(i : t) F%O) (only parsing) : order_scope. | Notation | \min^d_ ( i : t ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i < n | P ) F" | :=
(\big[min/top]_(i < n | P%B) F%O) : order_scope. | Notation | \min^d_ ( i < n | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i < n ) F" | :=
(\big[min/top]_(i < n) F%O) : order_scope. | Notation | \min^d_ ( i < n ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i 'in' A | P ) F" | :=
(\big[min/top]_(i in A | P%B) F%O) : order_scope. | Notation | \min^d_ ( i 'in' A | P ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\min^d_ ( i 'in' A ) F" | :=
(\big[min/top]_(i in A) F%O) : order_scope. | Notation | \min^d_ ( i 'in' A ) F | 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",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ i F" | :=
(\big[max/bot]_i F%O) : order_scope. | Notation | \max^d_ i F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i <- r | P ) F" | :=
(\big[max/bot]_(i <- r | P%B) F%O) : order_scope. | Notation | \max^d_ ( i <- r | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i < r ) F" | :=
(\big[max/bot]_(i <- r) F%O) : order_scope. | Notation | \max^d_ ( i < r ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( m <= i < n | P ) F" | :=
(\big[max/bot]_(m <= i < n | P%B) F%O) : order_scope. | Notation | \max^d_ ( m <= i < n | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( m <= i < n ) F" | :=
(\big[max/bot]_(m <= i < n) F%O) : order_scope. | Notation | \max^d_ ( m <= i < n ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i | P ) F" | :=
(\big[max/bot]_(i | P%B) F%O) : order_scope. | Notation | \max^d_ ( i | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i : t | P ) F" | :=
(\big[max/bot]_(i : t | P%B) F%O) (only parsing) : order_scope. | Notation | \max^d_ ( i : t | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i : t ) F" | :=
(\big[max/bot]_(i : t) F%O) (only parsing) : order_scope. | Notation | \max^d_ ( i : t ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i < n | P ) F" | :=
(\big[max/bot]_(i < n | P%B) F%O) : order_scope. | Notation | \max^d_ ( i < n | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i < n ) F" | :=
(\big[max/bot]_(i < n) F%O) : order_scope. | Notation | \max^d_ ( i < n ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i 'in' A | P ) F" | :=
(\big[max/bot]_(i in A | P%B) F%O) : order_scope. | Notation | \max^d_ ( i 'in' A | P ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\max^d_ ( i 'in' A ) F" | :=
(\big[max/bot]_(i in A) F%O) : order_scope. | Notation | \max^d_ ( i 'in' A ) F | 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",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leEdual (d : disp_t) (T : preorderType d) (x y : T) :
(x <=^d y :> T^d) = (y <= x). | Proof. by []. Qed. | Lemma | leEdual | 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... | [
"disp_t"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltEdual (d : disp_t) (T : preorderType d) (x y : T) :
(x <^d y :> T^d) = (y < x). | Proof. by []. Qed. | Lemma | ltEdual | 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... | [
"disp_t"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
botEdual d (T : tPreorderType d) : (dual_bottom : T^d) = \top :> T. | Proof. by []. Qed. | Lemma | botEdual | 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... | [
"dual_bottom",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
topEdual d (T : bPreorderType d) : (dual_top : T^d) = \bot :> T. | Proof. by []. Qed. | Lemma | topEdual | 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",
"dual_top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geE x y : ge x y = (y <= x). | Proof. by []. Qed. | Lemma | geE | 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... | [
"ge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtE x y : gt x y = (y < x). | Proof. by []. Qed. | Lemma | gtE | order | order/preorder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"DvdSyntax",
"DefaultSeqProdOrder",
"DefaultSeqLexiOrder",
"SetSubsetOrder.Exports",
"OrdinalOrd... | [
"gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lexx (x : T) : x <= x. | Proof. exact: le_refl. Qed. | Lemma | lexx | 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_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_refl : reflexive le | := lexx. | Definition | 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... | [
"le",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_refl : reflexive ge | := lexx. | Definition | ge_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... | [
"ge",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_trans: transitive (<=%O : rel T). | Proof. exact: le_trans. Qed. | Lemma | 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... | [
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_trans: transitive (>=%O : rel T). | Proof. by move=> ? ? ? ? /le_trans; apply. Qed. | Lemma | 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_trans",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_le_trans x y z t : z <= x -> y <= t -> x <= y -> z <= t. | Proof. by move=> + /(le_trans _)/[apply]; apply: le_trans. Qed. | Lemma | le_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... | [
"apply",
"le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_le_def x y: (x < y) = (x <= y) && ~~ (y <= x). | Proof. exact: lt_def. Qed. | Lemma | 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... | [
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltxx x: (x < x) = false. | Proof. by rewrite lt_le_def andbN. Qed. | Lemma | ltxx | 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 | |
lt_irreflexive : irreflexive lt | := ltxx. | Definition | lt_irreflexive | 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",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltexx | := (lexx, ltxx). | Definition | ltexx | 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... | [
"lexx",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_eqF x y: x < y -> (x == y) = false. | Proof. by apply: contraTF => /eqP ->; rewrite ltxx. Qed. | Lemma | lt_eqF | 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",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_eqF x y : y < x -> (x == y) = false. | Proof. by move=> /lt_eqF; rewrite eq_sym. Qed. | Lemma | gt_eqF | 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... | [
"eq_sym",
"lt_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltW x y: x < y -> x <= y. | Proof. by rewrite lt_le_def => /andP[]. Qed. | Lemma | ltW | 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 | |
lt_le_trans y x z: x < y -> y <= z -> x < z. | Proof.
rewrite !lt_le_def => /andP[] xy /negP yx yz.
apply/andP; split; first exact/(le_trans xy).
by apply/negP => /(le_trans yz).
Qed. | Lemma | lt_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... | [
"apply",
"le_trans",
"lt_le_def",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_trans: transitive (<%O : rel T). | Proof. by move=> y x z le1 /ltW le2; apply/(@lt_le_trans y). Qed. | Lemma | lt_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",
"le1",
"ltW",
"lt_le_trans",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_lt_trans y x z: x <= y -> y < z -> x < z. | Proof.
rewrite !lt_le_def => xy /andP[] yz /negP zy.
apply/andP; split; first exact/(le_trans xy).
by apply/negP => /(fun zx => le_trans zx xy).
Qed. | Lemma | le_lt_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_trans",
"lt_le_def",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_nsym x y : x < y -> y < x -> False. | Proof. by move=> xy /(lt_trans xy); rewrite ltxx. Qed. | Lemma | lt_nsym | 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... | [
"False",
"lt_trans",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_asym x y : (x < y < x) = false. | Proof. by apply/negP => /andP []; apply: lt_nsym. Qed. | Lemma | lt_asym | 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",
"lt_nsym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_gtF x y: x <= y -> (y < x) = false. | Proof.
by move=> le_xy; apply/negP => /lt_le_trans /(_ le_xy); rewrite ltxx.
Qed. | Lemma | le_gtF | 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",
"lt_le_trans",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_geF x y : x < y -> (y <= x) = false. | Proof. by apply: contraTF => /le_gtF ->. Qed. | Lemma | lt_geF | 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_gtF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_gtF x y hxy | := le_gtF (@ltW x y hxy). | Definition | lt_gtF | 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",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_leAnge x y : (x < y) = (x <= y) && ~~ (y <= x). | Proof. exact: lt_le_def. Qed. | Lemma | lt_leAnge | 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 | |
lt_le_asym x y : (x < y <= x) = false. | Proof. by apply/negP; move=> /andP[] xy /(lt_le_trans xy); rewrite ltxx. Qed. | Lemma | lt_le_asym | 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",
"lt_le_trans",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_lt_asym x y : (x <= y < x) = false. | Proof. by rewrite andbC lt_le_asym. Qed. | Lemma | le_lt_asym | 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_asym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_leP {x y} : reflect (forall z, y <= z -> x <= z) (x <= y). | Proof. by apply: (iffP idP) => [xy z /(le_trans _)->//|]; apply. Qed. | Lemma | le_leP | 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_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_geP {x y} : reflect (forall z, z <= x -> z <= y) (x <= y). | Proof. by apply: (iffP idP) => [xy z /le_trans|]; apply. Qed. | Lemma | le_geP | 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_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_ltP {x y} : reflect (forall z, y <= z -> x < z) (x < y). | Proof. by apply: (iffP idP) => [xy z /(lt_le_trans _)|]; apply. Qed. | Lemma | lt_ltP | 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",
"lt_le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_gtP {x y} : reflect (forall z, z <= x -> z < y) (x < y). | Proof. by apply: (iffP idP) => [xy z /le_lt_trans->//|]; apply. Qed. | Lemma | lt_gtP | 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_lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_path_min x s : path <=%O x s -> all (>= x) s. | Proof. exact/order_path_min/le_trans. Qed. | Lemma | le_path_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... | [
"all",
"le_trans",
"order_path_min",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_path_min x s : path <%O x s -> all (> x) s. | Proof. exact/order_path_min/lt_trans. Qed. | Lemma | lt_path_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... | [
"all",
"lt_trans",
"order_path_min",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_path_sortedE x s : path <=%O x s = all (>= x) s && sorted <=%O s. | Proof. exact/path_sortedE/le_trans. Qed. | Lemma | le_path_sortedE | 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... | [
"all",
"le_trans",
"path",
"path_sortedE",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_path_sortedE x s : path <%O x s = all (> x) s && sorted <%O s. | Proof. exact/path_sortedE/lt_trans. Qed. | Lemma | lt_path_sortedE | 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... | [
"all",
"lt_trans",
"path",
"path_sortedE",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_sorted_pairwise s : sorted <=%O s = pairwise <=%O s. | Proof. exact/sorted_pairwise/le_trans. Qed. | Lemma | le_sorted_pairwise | 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",
"pairwise",
"sorted",
"sorted_pairwise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_pairwise s : sorted <%O s = pairwise <%O s. | Proof. exact/sorted_pairwise/lt_trans. Qed. | Lemma | lt_sorted_pairwise | 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_trans",
"pairwise",
"sorted",
"sorted_pairwise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_path_pairwise x s : path <=%O x s = pairwise <=%O (x :: s). | Proof. exact/path_pairwise/le_trans. Qed. | Lemma | le_path_pairwise | 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",
"pairwise",
"path",
"path_pairwise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_path_pairwise x s : path <%O x s = pairwise <%O (x :: s). | Proof. exact/path_pairwise/lt_trans. Qed. | Lemma | lt_path_pairwise | 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_trans",
"pairwise",
"path",
"path_pairwise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_is_uniq_le s : sorted <%O s -> uniq s && sorted <=%O s. | Proof.
rewrite le_sorted_pairwise lt_sorted_pairwise uniq_pairwise -pairwise_relI.
apply/sub_pairwise => x y/= /[dup] + /ltW ->.
by case: eqVneq => // ->; rewrite ltxx.
Qed. | Lemma | lt_sorted_is_uniq_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",
"eqVneq",
"le_sorted_pairwise",
"ltW",
"lt_sorted_pairwise",
"ltxx",
"pairwise_relI",
"sorted",
"sub_pairwise",
"uniq",
"uniq_pairwise"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_sorted_mask m s : sorted <=%O s -> sorted <=%O (mask m s). | Proof. exact/sorted_mask/le_trans. Qed. | Lemma | le_sorted_mask | 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",
"mask",
"sorted",
"sorted_mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_mask m s : sorted <%O s -> sorted <%O (mask m s). | Proof. exact/sorted_mask/lt_trans. Qed. | Lemma | lt_sorted_mask | 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_trans",
"mask",
"sorted",
"sorted_mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_sorted_filter a s : sorted <=%O s -> sorted <=%O (filter a s). | Proof. exact/sorted_filter/le_trans. Qed. | Lemma | le_sorted_filter | 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... | [
"filter",
"le_trans",
"sorted",
"sorted_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_filter a s : sorted <%O s -> sorted <%O (filter a s). | Proof. exact/sorted_filter/lt_trans. Qed. | Lemma | lt_sorted_filter | 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... | [
"filter",
"lt_trans",
"sorted",
"sorted_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_path_mask x m s : path <=%O x s -> path <=%O x (mask m s). | Proof. exact/path_mask/le_trans. Qed. | Lemma | le_path_mask | 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",
"mask",
"path",
"path_mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_path_mask x m s : path <%O x s -> path <%O x (mask m s). | Proof. exact/path_mask/lt_trans. Qed. | Lemma | lt_path_mask | 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_trans",
"mask",
"path",
"path_mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_path_filter x a s : path <=%O x s -> path <=%O x (filter a s). | Proof. exact/path_filter/le_trans. Qed. | Lemma | le_path_filter | 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... | [
"filter",
"le_trans",
"path",
"path_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_path_filter x a s : path <%O x s -> path <%O x (filter a s). | Proof. exact/path_filter/lt_trans. Qed. | Lemma | lt_path_filter | 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... | [
"filter",
"lt_trans",
"path",
"path_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_sorted_ltn_nth (x0 : T) (s : seq T) : sorted <=%O s ->
{in [pred n | (n < size s)%N] &,
{homo nth x0 s : i j / (i < j)%N >-> i <= j}}. | Proof. exact/sorted_ltn_nth/le_trans. Qed. | Lemma | le_sorted_ltn_nth | 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",
"nth",
"seq",
"size",
"sorted",
"sorted_ltn_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_sorted_leq_nth (x0 : T) (s : seq T) : sorted <=%O s ->
{in [pred n | (n < size s)%N] &,
{homo nth x0 s : i j / (i <= j)%N >-> i <= j}}. | Proof. exact/sorted_leq_nth/le_refl/le_trans. Qed. | Lemma | le_sorted_leq_nth | 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_refl",
"le_trans",
"nth",
"seq",
"size",
"sorted",
"sorted_leq_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_leq_nth (x0 : T) (s : seq T) : sorted <%O s ->
{in [pred n | (n < size s)%N] &,
{mono nth x0 s : i j / (i <= j)%N >-> i <= j}}. | Proof.
move=> /[dup] lt_s /lt_sorted_is_uniq_le /andP[s_uniq le_s] i j ilt jlt.
case/boolP: (i <= j)%N; first exact/le_sorted_leq_nth.
rewrite -ltnNge => /(sorted_ltn_nth lt_trans x0 lt_s j i jlt ilt).
by rewrite lt_le_def => /andP[_] /negPf.
Qed. | Lemma | lt_sorted_leq_nth | 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_sorted_leq_nth",
"lt_le_def",
"lt_sorted_is_uniq_le",
"lt_trans",
"ltnNge",
"nth",
"seq",
"size",
"sorted",
"sorted_ltn_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_ltn_nth (x0 : T) (s : seq T) : sorted <%O s ->
{in [pred n | (n < size s)%N] &,
{mono nth x0 s : i j / (i < j)%N >-> i < j}}. | Proof.
move=> ss i j ilt jlt.
rewrite lt_le_def (lt_sorted_leq_nth x0 ss)// (lt_sorted_leq_nth x0 ss)//.
by rewrite -ltnNge andbC ltn_neqAle -andbA andbb.
Qed. | Lemma | lt_sorted_ltn_nth | 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",
"lt_sorted_leq_nth",
"ltnNge",
"ltn_neqAle",
"nth",
"seq",
"size",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_le_path x s1 s2 : subseq s1 s2 -> path <=%O x s2 -> path <=%O x s1. | Proof. exact/subseq_path/le_trans. Qed. | Lemma | subseq_le_path | 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",
"path",
"s1",
"s2",
"subseq",
"subseq_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_lt_path x s1 s2 : subseq s1 s2 -> path <%O x s2 -> path <%O x s1. | Proof. exact/subseq_path/lt_trans. Qed. | Lemma | subseq_lt_path | 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_trans",
"path",
"s1",
"s2",
"subseq",
"subseq_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_le_sorted s1 s2 : subseq s1 s2 -> sorted <=%O s2 -> sorted <=%O s1. | Proof. exact/subseq_sorted/le_trans. Qed. | Lemma | subseq_le_sorted | 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",
"s1",
"s2",
"sorted",
"subseq",
"subseq_sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_lt_sorted s1 s2 : subseq s1 s2 -> sorted <%O s2 -> sorted <%O s1. | Proof. exact/subseq_sorted/lt_trans. Qed. | Lemma | subseq_lt_sorted | 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_trans",
"s1",
"s2",
"sorted",
"subseq",
"subseq_sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_uniq s : sorted <%O s -> uniq s. | Proof. exact/sorted_uniq/ltxx/lt_trans. Qed. | Lemma | lt_sorted_uniq | 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_trans",
"ltxx",
"sorted",
"sorted_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_sorted_eq s1 s2 :
sorted <%O s1 -> sorted <%O s2 -> s1 =i s2 -> s1 = s2. | Proof. exact/irr_sorted_eq/ltxx/lt_trans. Qed. | Lemma | lt_sorted_eq | 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... | [
"irr_sorted_eq",
"lt_trans",
"ltxx",
"s1",
"s2",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_lt_nth x0 s i : sorted <%O s -> (i < size s)%N ->
[seq x <- s | x < nth x0 s i] = take i s. | Proof.
move=> ss i_lt/=; rewrite -[X in filter _ X](mkseq_nth x0) filter_map.
under eq_in_filter => j do
[rewrite ?mem_iota => j_s /=; rewrite lt_sorted_ltn_nth//].
by rewrite (filter_iota_ltn 0) ?map_nth_iota0 // ltnW.
Qed. | Lemma | filter_lt_nth | 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... | [
"eq_in_filter",
"filter",
"filter_iota_ltn",
"filter_map",
"lt_sorted_ltn_nth",
"ltnW",
"map_nth_iota0",
"mem_iota",
"mkseq_nth",
"nth",
"seq",
"size",
"sorted",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_lt_nth x0 s i : sorted <%O s -> (i < size s)%N ->
count (< nth x0 s i) s = i. | Proof.
by move=> ss i_lt; rewrite -size_filter/= filter_lt_nth// size_take i_lt.
Qed. | Lemma | count_lt_nth | 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... | [
"count",
"filter_lt_nth",
"nth",
"size",
"size_filter",
"size_take",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_le_nth x0 s i : sorted <%O s -> (i < size s)%N ->
[seq x <- s | x <= nth x0 s i] = take i.+1 s. | Proof.
move=> ss i_lt/=; rewrite -[X in filter _ X](mkseq_nth x0) filter_map.
under eq_in_filter => j do
[rewrite ?mem_iota => j_s /=; rewrite lt_sorted_leq_nth//].
by rewrite (filter_iota_leq 0)// map_nth_iota0.
Qed. | Lemma | filter_le_nth | 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... | [
"eq_in_filter",
"filter",
"filter_iota_leq",
"filter_map",
"lt_sorted_leq_nth",
"map_nth_iota0",
"mem_iota",
"mkseq_nth",
"nth",
"seq",
"size",
"sorted",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_le_nth x0 s i : sorted <%O s -> (i < size s)%N ->
count (<= nth x0 s i) s = i.+1. | Proof.
by move=> ss i_lt; rewrite -size_filter/= filter_le_nth// size_takel.
Qed. | Lemma | count_le_nth | 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... | [
"count",
"filter_le_nth",
"nth",
"size",
"size_filter",
"size_takel",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_filter_lt x s :
sorted <=%O s -> [seq y <- s | y < x] = take (count (< x) s) s. | Proof.
elim: s => [//|y s IHs]/=; rewrite (path_sortedE le_trans) => /andP[le_y_s ss].
case: ifP => [|ltyxF]; rewrite IHs//.
rewrite (@eq_in_count _ _ pred0) ?count_pred0/= ?take0// => z.
by move=> /(allP le_y_s) yz; apply: contraFF ltyxF; apply: le_lt_trans.
Qed. | Lemma | sorted_filter_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... | [
"allP",
"apply",
"count",
"count_pred0",
"eq_in_count",
"le_lt_trans",
"le_trans",
"path_sortedE",
"seq",
"sorted",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_filter_le x s :
sorted <=%O s -> [seq y <- s | y <= x] = take (count (<= x) s) s. | Proof.
elim: s => [//|y s IHs]/=; rewrite (path_sortedE le_trans) => /andP[le_y_s ss].
case: ifP => [|leyxF]; rewrite IHs//.
rewrite (@eq_in_count _ _ pred0) ?count_pred0/= ?take0// => z.
by move=> /(allP le_y_s) yz; apply: contraFF leyxF; apply: le_trans.
Qed. | Lemma | sorted_filter_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... | [
"allP",
"apply",
"count",
"count_pred0",
"eq_in_count",
"le_trans",
"path_sortedE",
"seq",
"sorted",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_count_le x x0 s i : sorted <=%O s ->
(i < count (<= x) s)%N -> nth x0 s i <= x. | Proof.
move=> ss iltc; rewrite -(nth_take _ iltc) -sorted_filter_le //.
by apply/(all_nthP _ (filter_all (<= x) _)); rewrite size_filter.
Qed. | Lemma | nth_count_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... | [
"all_nthP",
"apply",
"count",
"filter_all",
"nth",
"nth_take",
"size_filter",
"sorted",
"sorted_filter_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_count_lt x x0 s i : sorted <=%O s ->
(i < count (< x) s)%N -> nth x0 s i < x. | Proof.
move=> ss iltc; rewrite -(nth_take _ iltc) -sorted_filter_lt //.
by apply/(all_nthP _ (filter_all (< x) _)); rewrite size_filter.
Qed. | Lemma | nth_count_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... | [
"all_nthP",
"apply",
"count",
"filter_all",
"nth",
"nth_take",
"size_filter",
"sorted",
"sorted_filter_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_le_id s : sorted <=%O s -> sort <=%O s = s. | Proof. exact/sorted_sort/le_trans. Qed. | Lemma | sort_le_id | 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",
"sort",
"sorted",
"sorted_sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_lt_id s : sorted <%O s -> sort <%O s = s. | Proof. exact/sorted_sort/lt_trans. Qed. | Lemma | sort_lt_id | 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_trans",
"sort",
"sorted",
"sorted_sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_leNgt x y : x >=< y -> (x <= y) = ~~ (y < x). | Proof.
rewrite /comparable lt_le_def.
by case: (x <= y) => //=; case: (y <= x).
Qed. | Lemma | comparable_leNgt | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_ltNge x y : x >=< y -> (x < y) = ~~ (y <= x). | Proof.
rewrite /comparable lt_le_def.
by case: (x <= y) => //=; case: (y <= x).
Qed. | Lemma | comparable_ltNge | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_sym x y : (y >=< x) = (x >=< y). | Proof. by rewrite /comparable orbC. Qed. | Lemma | comparable_sym | 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 | |
comparablexx x : x >=< x. | Proof. by rewrite /comparable lexx. Qed. | Lemma | comparablexx | 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",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incomparable_eqF x y : (x >< y) -> (x == y) = false. | Proof. by apply: contraNF => /eqP ->; rewrite comparablexx. Qed. | Lemma | incomparable_eqF | 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",
"comparablexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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