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 |
|---|---|---|---|---|---|---|---|---|---|---|
meetx0 : right_zero \bot (@meet _ L). | Proof. by move=> x; rewrite meetC meet0x. Qed. | Lemma | meetx0 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"meet",
"meet0x",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetx1 : right_id \top (@meet _ L). | Proof. by move=> x; apply/eqP; rewrite -leEmeet. Qed. | Lemma | meetx1 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"leEmeet",
"meet",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meet1x : left_id \top (@meet _ L). | Proof. by move=> x; apply/eqP; rewrite meetC meetx1. Qed. | Lemma | meet1x | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"meet",
"meetC",
"meetx1",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meet_eq1 x y : (x `&` y == \top) = (x == \top) && (y == \top). | Proof.
apply/idP/idP; last by move=> /andP[/eqP-> /eqP->]; rewrite meetx1.
by move=> /eqP xIy1; rewrite -!le1x -xIy1 leIl leIr.
Qed. | Lemma | meet_eq1 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"last",
"le1x",
"leIl",
"leIr",
"meetx1",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_inf_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) :
x \in r -> P x -> \meet_(i <- r | P i) F i <= F x. | Proof. by move=> xr Px; rewrite (big_rem x) ?Px //= leIl. Qed. | Lemma | meets_inf_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"Px",
"big_rem",
"leIl",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_max_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) (u : L) :
x \in r -> P x -> F x <= u -> \meet_(x <- r | P x) F x <= u. | Proof. by move=> ? ?; apply/le_trans/meets_inf_seq. Qed. | Lemma | meets_max_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"le_trans",
"meets_inf_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_inf I (j : I) (P : {pred I}) (F : I -> L) :
P j -> \meet_(i | P i) F i <= F j. | Proof. exact: meets_inf_seq. Qed. | Lemma | meets_inf | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_inf_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_max I (j : I) (u : L) (P : {pred I}) (F : I -> L) :
P j -> F j <= u -> \meet_(i | P i) F i <= u. | Proof. exact: meets_max_seq. Qed. | Lemma | meets_max | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_max_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_ge J (r : seq J) (P : {pred J}) (F : J -> L) (u : L) :
(forall x : J, P x -> u <= F x) -> u <= \meet_(x <- r | P x) F x. | Proof. by move=> leFm; elim/big_rec: _ => // i x Px xu; rewrite lexI leFm. Qed. | Lemma | meets_ge | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"Px",
"big_rec",
"lexI",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetsP_seq T (r : seq T) (P : {pred T}) (F : T -> L) (l : L) :
reflect (forall x : T, x \in r -> P x -> l <= F x)
(l <= \meet_(x <- r | P x) F x). | Proof.
apply: (iffP idP) => leFm => [x xr Px|].
exact/(le_trans leFm)/meets_inf_seq.
by rewrite big_seq_cond meets_ge// => x /andP[/leFm].
Qed. | Lemma | meetsP_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"Px",
"apply",
"big_seq_cond",
"le_trans",
"meets_ge",
"meets_inf_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetsP I (l : L) (P : {pred I}) (F : I -> L) :
reflect (forall i : I, P i -> l <= F i) (l <= \meet_(i | P i) F i). | Proof. by apply: (iffP (meetsP_seq _ _ _ _)) => H ? ?; apply: H. Qed. | Lemma | meetsP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"meetsP_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_meets I (A B : {set I}) (F : I -> L) :
A \subset B -> \meet_(i in B) F i <= \meet_(i in A) F i. | Proof. by move=> /subsetP AB; apply/meetsP => i iA; apply/meets_inf/AB. Qed. | Lemma | le_meets | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"meetsP",
"meets_inf",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_setU I (A B : {set I}) (F : I -> L) :
\meet_(i in (A :|: B)) F i = \meet_(i in A) F i `&` \meet_(i in B) F i. | Proof.
rewrite -!big_enum; have /= <- := @big_cat _ _ meet.
apply/eq_big_idem; first exact: meetxx.
by move=> ?; rewrite mem_cat !fintype.mem_enum inE.
Qed. | Lemma | meets_setU | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"big_cat",
"big_enum",
"eq_big_idem",
"inE",
"meet",
"meetxx",
"mem_cat",
"mem_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_seq I (r : seq I) (F : I -> L) :
\meet_(i <- r) F i = \meet_(i in r) F i. | Proof.
by rewrite -big_enum; apply/eq_big_idem => ?; rewrite /= ?meetxx ?fintype.mem_enum.
Qed. | Lemma | meets_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"big_enum",
"eq_big_idem",
"meetxx",
"mem_enum",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leUx x y z : (x `|` y <= z) = (x <= z) && (y <= z). | Proof. exact: leUx. Qed. | Lemma | leUx | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [] | interaction with order | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
leUr x y : x <= y `|` x. | Proof. exact: (@leIr _ L^d). Qed. | Lemma | leUr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leUl x y : x <= x `|` y. | Proof. exact: (@leIl _ L^d). Qed. | Lemma | leUl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lexUl x y z : x <= y -> x <= y `|` z. | Proof. exact: (@leIxl _ L^d). Qed. | Lemma | lexUl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIxl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lexUr x y z : x <= z -> x <= y `|` z. | Proof. exact: (@leIxr _ L^d). Qed. | Lemma | lexUr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lexU2 x y z : (x <= y) || (x <= z) -> x <= y `|` z. | Proof. exact: (@leIx2 _ L^d). Qed. | Lemma | lexU2 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIx2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leEjoin x y : (x <= y) = (x `|` y == y). | Proof. by rewrite [LHS](@leEmeet _ L^d) meetC. Qed. | Lemma | leEjoin | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leEmeet",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_joinl x y : (x `|` y == x) = (y <= x). | Proof. exact: (@eq_meetl _ L^d). Qed. | Lemma | eq_joinl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"eq_meetl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_joinr x y : (x `|` y == y) = (x <= y). | Proof. exact: (@eq_meetr _ L^d). Qed. | Lemma | eq_joinr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"eq_meetr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join_idPl {x y} : reflect (y `|` x = y) (x <= y). | Proof. exact: (@meet_idPl _ L^d). Qed. | Lemma | join_idPl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meet_idPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join_idPr {x y} : reflect (x `|` y = y) (x <= y). | Proof. exact: (@meet_idPr _ L^d). Qed. | Lemma | join_idPr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meet_idPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join_l x y : y <= x -> x `|` y = x. | Proof. exact/join_idPl. Qed. | Lemma | join_l | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join_idPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join_r x y : x <= y -> x `|` y = y. | Proof. exact/join_idPr. Qed. | Lemma | join_r | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join_idPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leUidl x y : (x `|` y <= y) = (x <= y). | Proof. exact: (@leIidr _ L^d). Qed. | Lemma | leUidl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIidr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leUidr x y : (y `|` x <= y) = (x <= y). | Proof. exact: (@leIidl _ L^d). Qed. | Lemma | leUidr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leIidl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leU2 x y z t : x <= z -> y <= t -> x `|` y <= z `|` t. | Proof. exact: (@leI2 _ L^d). Qed. | Lemma | leU2 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leI2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinC : commutative (@join _ L). | Proof. exact: (@meetC _ L^d). Qed. | Lemma | joinC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetC"
] | algebraic properties | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
joinA : associative (@join _ L). | Proof. exact: (@meetA _ L^d). Qed. | Lemma | joinA | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinxx : idempotent_op (@join _ L). | Proof. exact: (@meetxx _ L^d). Qed. | Lemma | joinxx | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"idempotent_op",
"join",
"meetxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinAC : right_commutative (@join _ L). | Proof. exact: (@meetAC _ L^d). Qed. | Lemma | joinAC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetAC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinCA : left_commutative (@join _ L). | Proof. exact: (@meetCA _ L^d). Qed. | Lemma | joinCA | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinACA : interchange (@join _ L) (@join _ L). | Proof. exact: (@meetACA _ L^d). Qed. | Lemma | joinACA | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetACA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinKU y x : x `|` (x `|` y) = x `|` y. | Proof. exact: (@meetKI _ L^d). Qed. | Lemma | joinKU | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetKI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinUK y x : (x `|` y) `|` y = x `|` y. | Proof. exact: (@meetIK _ L^d). Qed. | Lemma | joinUK | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetIK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinKUC y x : x `|` (y `|` x) = x `|` y. | Proof. exact: (@meetKIC _ L^d). Qed. | Lemma | joinKUC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetKIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinUKC y x : y `|` x `|` y = x `|` y. | Proof. exact: (@meetIKC _ L^d). Qed. | Lemma | joinUKC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetIKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinx0 : right_id \bot (@join _ L). | Proof. exact: (@meetx1 _ L^d). Qed. | Lemma | joinx0 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"join",
"meetx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join0x : left_id \bot (@join _ L). | Proof. exact: (@meet1x _ L^d). Qed. | Lemma | join0x | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"join",
"meet1x"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join_eq0 x y : (x `|` y == \bot) = (x == \bot) && (y == \bot). | Proof. exact: (@meet_eq1 _ L^d). Qed. | Lemma | join_eq0 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"meet_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_sup_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) :
x \in r -> P x -> F x <= \join_(i <- r | P i) F i. | Proof. exact: (@meets_inf_seq _ L^d). Qed. | Lemma | joins_sup_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_inf_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_min_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) (l : L) :
x \in r -> P x -> l <= F x -> l <= \join_(x <- r | P x) F x. | Proof. exact: (@meets_max_seq _ L^d). Qed. | Lemma | joins_min_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_max_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_sup I (j : I) (P : {pred I}) (F : I -> L) :
P j -> F j <= \join_(i | P i) F i. | Proof. exact: (@meets_inf _ L^d). Qed. | Lemma | joins_sup | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_inf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_min I (j : I) (l : L) (P : {pred I}) (F : I -> L) :
P j -> l <= F j -> l <= \join_(i | P i) F i. | Proof. exact: (@meets_max _ L^d). Qed. | Lemma | joins_min | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_le J (r : seq J) (P : {pred J}) (F : J -> L) (u : L) :
(forall x : J, P x -> F x <= u) -> \join_(x <- r | P x) F x <= u. | Proof. exact: (@meets_ge _ L^d). Qed. | Lemma | joins_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_ge",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinsP_seq T (r : seq T) (P : {pred T}) (F : T -> L) (u : L) :
reflect (forall x : T, x \in r -> P x -> F x <= u)
(\join_(x <- r | P x) F x <= u). | Proof. exact: (@meetsP_seq _ L^d). Qed. | Lemma | joinsP_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetsP_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinsP I (u : L) (P : {pred I}) (F : I -> L) :
reflect (forall i : I, P i -> F i <= u) (\join_(i | P i) F i <= u). | Proof. exact: (@meetsP _ L^d). Qed. | Lemma | joinsP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetsP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_joins I (A B : {set I}) (F : I -> L) :
A \subset B -> \join_(i in A) F i <= \join_(i in B) F i. | Proof. exact: (@le_meets _ L^d). Qed. | Lemma | le_joins | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_meets"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_setU I (A B : {set I}) (F : I -> L) :
\join_(i in (A :|: B)) F i = \join_(i in A) F i `|` \join_(i in B) F i. | Proof. exact: (@meets_setU _ L^d). Qed. | Lemma | joins_setU | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_seq I (r : seq I) (F : I -> L) :
\join_(i <- r) F i = \join_(i in r) F i. | Proof. exact: (@meets_seq _ L^d). Qed. | Lemma | joins_seq | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meets_seq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinx1 : right_zero \top (@join _ L). | Proof. exact: (@meetx0 _ L^d). Qed. | Lemma | joinx1 | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meetx0",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
join1x : left_zero \top (@join _ L). | Proof. exact: (@meet0x _ L^d). Qed. | Lemma | join1x | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meet0x",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetUK x y : (x `&` y) `|` y = y. | Proof. exact/join_idPr/leIr. Qed. | Lemma | meetUK | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join_idPr",
"leIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetUKC x y : (y `&` x) `|` y = y. | Proof. by rewrite meetC meetUK. Qed. | Lemma | meetUKC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetC",
"meetUK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetKUC y x : x `|` (y `&` x) = x. | Proof. by rewrite joinC meetUK. Qed. | Lemma | meetKUC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinC",
"meetUK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetKU y x : x `|` (x `&` y) = x. | Proof. by rewrite meetC meetKUC. Qed. | Lemma | meetKU | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"meetC",
"meetKUC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinIK x y : (x `|` y) `&` y = y. | Proof. exact/meet_idPr/leUr. Qed. | Lemma | joinIK | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"leUr",
"meet_idPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinIKC x y : (y `|` x) `&` y = y. | Proof. by rewrite joinC joinIK. Qed. | Lemma | joinIKC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinC",
"joinIK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinKIC y x : x `&` (y `|` x) = x. | Proof. by rewrite meetC joinIK. Qed. | Lemma | joinKIC | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinIK",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinKI y x : x `&` (x `|` y) = x. | Proof. by rewrite joinC joinKIC. Qed. | Lemma | joinKI | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinC",
"joinKIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcomparableP x y : incomparel x y
(min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y)
(y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y) (y >=< x) (x >=< y). | Proof.
by case: (comparableP x) => [hxy|hxy|hxy|->]; do 1?have hxy' := ltW hxy;
rewrite ?(meetxx, joinxx);
rewrite ?(meet_l hxy', meet_r hxy', join_l hxy', join_r hxy');
constructor.
Qed. | Lemma | lcomparableP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparableP",
"incomparel",
"join_l",
"join_r",
"joinxx",
"ltW",
"max",
"meet_l",
"meet_r",
"meetxx",
"min"
] | comparison predicates | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lcomparable_ltgtP x y : x >=< y ->
comparel x y (min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y)
(y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y). | Proof. by case: (lcomparableP x) => // *; constructor. Qed. | Lemma | lcomparable_ltgtP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparel",
"lcomparableP",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcomparable_leP x y : x >=< y ->
lel_xor_gt x y (min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y) (x <= y) (y < x). | Proof. by move/lcomparable_ltgtP => [/ltW xy|xy|->]; constructor. Qed. | Lemma | lcomparable_leP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"lcomparable_ltgtP",
"lel_xor_gt",
"ltW",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcomparable_ltP x y : x >=< y ->
ltl_xor_ge x y (min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y) (y <= x) (x < y). | Proof. by move=> /lcomparable_ltgtP [xy|/ltW xy|->]; constructor. Qed. | Lemma | lcomparable_ltP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"lcomparable_ltgtP",
"ltW",
"ltl_xor_ge",
"max",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetUl : left_distributive (@meet _ L) (@join _ L). | Proof. exact: meetUl. Qed. | Lemma | meetUl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meetUr : right_distributive (@meet _ L) (@join _ L). | Proof. by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed. | Lemma | meetUr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meet",
"meetC",
"meetUl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinIl : left_distributive (@join _ L) (@meet _ L). | Proof. exact: joinIl. Qed. | Lemma | joinIl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"meet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joinIr : right_distributive (@join _ L) (@meet _ L). | Proof. by move=> x y z; rewrite ![x `|` _]joinC joinIl. Qed. | Lemma | joinIr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"join",
"joinC",
"joinIl",
"meet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leU2l_le y t x z : x `&` t = \bot -> x `|` y <= z `|` t -> x <= z. | Proof.
by move=> xIt0 /(leI2 (lexx x)); rewrite joinKI meetUr xIt0 joinx0 leIidl.
Qed. | Lemma | leU2l_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"joinKI",
"joinx0",
"leI2",
"leIidl",
"lexx",
"meetUr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leU2r_le y t x z : x `&` t = \bot -> y `|` x <= t `|` z -> x <= z. | Proof. by rewrite joinC [_ `|` z]joinC => /leU2l_le H /H. Qed. | Lemma | leU2r_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"joinC",
"leU2l_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
disjoint_lexUl z x y : x `&` z = \bot -> (x <= y `|` z) = (x <= y). | Proof.
move=> xz0; apply/idP/idP=> xy; last by rewrite lexU2 ?xy.
by apply: (@leU2l_le x z); rewrite ?joinxx.
Qed. | Lemma | disjoint_lexUl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"bot",
"joinxx",
"last",
"leU2l_le",
"lexU2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
disjoint_lexUr z x y : x `&` z = \bot -> (x <= z `|` y) = (x <= y). | Proof. by move=> xz0; rewrite joinC; rewrite disjoint_lexUl. Qed. | Lemma | disjoint_lexUr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bot",
"disjoint_lexUl",
"joinC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leU2E x y z t : x `&` t = \bot -> y `&` z = \bot ->
(x `|` y <= z `|` t) = (x <= z) && (y <= t). | Proof.
move=> dxt dyz; apply/idP/andP; last by case=> ? ?; exact: leU2.
by move=> lexyzt; rewrite (leU2l_le _ lexyzt) // (leU2r_le _ lexyzt).
Qed. | Lemma | leU2E | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"bot",
"last",
"leU2",
"leU2l_le",
"leU2r_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
joins_disjoint (I : finType) (d : L) (P : {pred I}) (F : I -> L) :
(forall i : I, P i -> d `&` F i = \bot) -> d `&` \join_(i | P i) F i = \bot. | Proof.
move=> d_Fi_disj; have : \big[andb/true]_(i | P i) (d `&` F i == \bot).
rewrite big_all_cond; apply/allP => i _ /=.
by apply/implyP => /d_Fi_disj ->.
elim/big_rec2: _ => [|i y]; first by rewrite meetx0.
case; rewrite (andbF, andbT) // => Pi /(_ isT) dy /eqP dFi.
by rewrite meetUr dy dFi joinxx.
Qed. | Lemma | joins_disjoint | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"allP",
"apply",
"big_all_cond",
"big_rec2",
"bot",
"joinxx",
"meetUr",
"meetx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leI2l_le y t x z : y `|` z = \top -> x `&` y <= z `&` t -> x <= z. | Proof. by rewrite joinC; exact: (@leU2l_le _ L^d). Qed. | Lemma | leI2l_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinC",
"leU2l_le",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leI2r_le y t x z : y `|` z = \top -> y `&` x <= t `&` z -> x <= z. | Proof. by rewrite joinC; exact: (@leU2r_le _ L^d). Qed. | Lemma | leI2r_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joinC",
"leU2r_le",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cover_leIxl z x y : z `|` y = \top -> (x `&` z <= y) = (x <= y). | Proof. by rewrite joinC; exact: (@disjoint_lexUl _ L^d). Qed. | Lemma | cover_leIxl | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"disjoint_lexUl",
"joinC",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cover_leIxr z x y : z `|` y = \top -> (z `&` x <= y) = (x <= y). | Proof. by rewrite joinC; exact: (@disjoint_lexUr _ L^d). Qed. | Lemma | cover_leIxr | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"disjoint_lexUr",
"joinC",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leI2E x y z t : x `|` t = \top -> y `|` z = \top ->
(x `&` y <= z `&` t) = (x <= z) && (y <= t). | Proof. by move=> ? ?; apply: (@leU2E _ L^d); rewrite meetC. Qed. | Lemma | leI2E | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"leU2E",
"meetC",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
meets_total (I : finType) (d : L) (P : {pred I}) (F : I -> L) :
(forall i : I, P i -> d `|` F i = \top) -> d `|` \meet_(i | P i) F i = \top. | Proof. exact: (@joins_disjoint _ L^d). Qed. | Lemma | meets_total | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"joins_disjoint",
"top"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_total : total (<=%O : rel T) | := le_total. | Definition | le_total | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"rel",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_total : total (>=%O : rel T). | Proof. by move=> ? ?; apply: le_total. Qed. | Lemma | ge_total | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"apply",
"le_total",
"rel",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparableT x y : x >=< y. | Proof. exact: le_total. Qed. | Lemma | comparableT | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_le_sorted s : sorted <=%O (sort <=%O s). | Proof. exact: sort_sorted. Qed. | Lemma | sort_le_sorted | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"sort",
"sort_sorted",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort_lt_sorted s : sorted <%O (sort <=%O s) = uniq s. | Proof. by rewrite lt_sorted_uniq_le sort_uniq sort_le_sorted andbT. Qed. | Lemma | sort_lt_sorted | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"lt_sorted_uniq_le",
"sort",
"sort_le_sorted",
"sort_uniq",
"sorted",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_sort_leP s1 s2 : reflect (sort <=%O s1 = sort <=%O s2) (perm_eq s1 s2). | Proof. exact/perm_sortP/le_anti/le_trans/le_total. Qed. | Lemma | perm_sort_leP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_anti",
"le_total",
"le_trans",
"perm_eq",
"perm_sortP",
"s1",
"s2",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_sort_le p s : filter p (sort <=%O s) = sort <=%O (filter p s). | Proof. exact/filter_sort/le_trans/le_total. Qed. | Lemma | filter_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"filter",
"filter_sort",
"le_total",
"le_trans",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_sort_le s (m : bitseq) :
{m_s : bitseq | mask m_s (sort <=%O s) = sort <=%O (mask m s)}. | Proof. exact/mask_sort/le_trans/le_total. Qed. | Lemma | mask_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bitseq",
"le_total",
"le_trans",
"mask",
"mask_sort",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_mask_sort_le s (m : bitseq) :
sorted <=%O (mask m s) -> {m_s : bitseq | mask m_s (sort <=%O s) = mask m s}. | Proof. exact/sorted_mask_sort/le_trans/le_total. Qed. | Lemma | sorted_mask_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"bitseq",
"le_total",
"le_trans",
"mask",
"sort",
"sorted",
"sorted_mask_sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_sort_le : {homo sort <=%O : s1 s2 / @subseq T s1 s2}. | Proof. exact/subseq_sort/le_trans/le_total. Qed. | Lemma | subseq_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_total",
"le_trans",
"s1",
"s2",
"sort",
"subseq",
"subseq_sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_subseq_sort_le s1 s2 :
subseq s1 s2 -> sorted <=%O s1 -> subseq s1 (sort <=%O s2). | Proof. exact/sorted_subseq_sort/le_trans/le_total. Qed. | Lemma | sorted_subseq_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_total",
"le_trans",
"s1",
"s2",
"sort",
"sorted",
"sorted_subseq_sort",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem2_sort_le s x y : x <= y -> mem2 s x y -> mem2 (sort <=%O s) x y. | Proof. exact/mem2_sort/le_trans/le_total. Qed. | Lemma | mem2_sort_le | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"le_total",
"le_trans",
"mem2",
"mem2_sort",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leNgt x y : (x <= y) = ~~ (y < x). | Proof. exact: comparable_leNgt. Qed. | Lemma | leNgt | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparable_leNgt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltNge x y : (x < y) = ~~ (y <= x). | Proof. exact: comparable_ltNge. Qed. | Lemma | ltNge | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparable_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltgtP x y | := LatticeTheory.lcomparable_ltgtP (comparableT x y). | Definition | ltgtP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparableT",
"lcomparable_ltgtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leP x y | := LatticeTheory.lcomparable_leP (comparableT x y). | Definition | leP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparableT",
"lcomparable_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltP x y | := LatticeTheory.lcomparable_ltP (comparableT x y). | Definition | ltP | order | order/order.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"path",
"fintype",
"tuple",
"bigop",
"finset",
"div",
"prime",
"finfun",
"preorder",
"Order",
"BPreorderTheory",
"PreCancelPartial",
"OrderMorphismTheory",
"NatOrder",... | [
"comparableT",
"lcomparable_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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