fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
dual_leif:= (@leif (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_leif
| |
dual_lteif:= (@lteif (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_lteif
| |
dual_max:= (@max (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_max
| |
dual_min:= (@min (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_min
| |
dual_bottom:= (@bottom (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_bottom
| |
dual_top:= (@top (dual_display _) _).
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
dual_top
| |
Definition_ (T : eqType) := Equality.on T^d.
HB.instance Definition _ (T : choiceType) := Choice.on T^d.
HB.instance Definition _ (T : countType) := Countable.on T^d.
HB.instance Definition _ (T : finType) := Finite.on T^d.
HB.instance Definition _ (d : disp_t) (T : preorderType d) :=
isDuallyPreorder.Build (dual_display d) T^d
gt_def lt_def ge_refl le_refl ge_trans le_trans.
|
HB.instance
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
Definition
| |
leEdual(d : disp_t) (T : preorderType d) (x y : T) :
(x <=^d y :> T^d) = (y <= x).
Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
leEdual
| |
ltEdual(d : disp_t) (T : preorderType d) (x y : T) :
(x <^d y :> T^d) = (y < x).
Proof. by []. Qed.
HB.instance Definition _ d (T : tPreorderType d) :=
hasBottom.Build (dual_display d) T^d lex1.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ltEdual
| |
botEduald (T : tPreorderType d) : (dual_bottom : T^d) = \top :> T.
Proof. by []. Qed.
HB.instance Definition _ d (T : bPreorderType d) :=
hasTop.Build (dual_display d) T^d le0x.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
botEdual
| |
topEduald (T : bPreorderType d) : (dual_top : T^d) = \bot :> T.
Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
topEdual
| |
nondecreasingdisp' (T' : preorderType disp') (f : T -> T') : Prop :=
{homo f : x y / x <= y}.
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
nondecreasing
| |
geEx y : ge x y = (y <= x). Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
geE
| |
gtEx y : gt x y = (y < x). Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
gtE
| |
lexx(x : T) : x <= x.
Proof. exact: le_refl. Qed.
Hint Resolve lexx : core.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lexx
| |
le_refl: reflexive le := lexx.
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_refl
| |
ge_refl: reflexive ge := lexx.
Hint Resolve le_refl : core.
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ge_refl
| |
le_trans: transitive (<=%O : rel T).
Proof. exact: le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_trans
| |
ge_trans: transitive (>=%O : rel T).
Proof. by move=> ? ? ? ? /le_trans; apply. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ge_trans
| |
le_le_transx y z t : z <= x -> y <= t -> x <= y -> z <= t.
Proof. by move=> + /(le_trans _)/[apply]; apply: le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_le_trans
| |
lt_le_defx y: (x < y) = (x <= y) && ~~ (y <= x).
Proof. exact: lt_def. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_le_def
| |
ltxxx: x < x = false.
Proof. by rewrite lt_le_def andbN. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ltxx
| |
lt_irreflexive: irreflexive lt := ltxx.
Hint Resolve lt_irreflexive : core.
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_irreflexive
| |
ltexx:= (lexx, ltxx).
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ltexx
| |
lt_eqFx y: x < y -> x == y = false.
Proof. by apply: contraTF => /eqP ->; rewrite ltxx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_eqF
| |
gt_eqFx y : y < x -> x == y = false.
Proof. by move=> /lt_eqF; rewrite eq_sym. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
gt_eqF
| |
ltWx y: x < y -> x <= y.
Proof. by rewrite lt_le_def => /andP[]. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ltW
| |
lt_le_transy 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_le_trans
| |
lt_trans: transitive (<%O : rel T).
Proof. by move=> y x z le1 /ltW le2; apply/(@lt_le_trans y). Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_trans
| |
le_lt_transy 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_lt_trans
| |
lt_nsymx y : x < y -> y < x -> False.
Proof. by move=> xy /(lt_trans xy); rewrite ltxx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_nsym
| |
lt_asymx y : x < y < x = false.
Proof. by apply/negP => /andP []; apply: lt_nsym. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_asym
| |
le_gtFx y: x <= y -> y < x = false.
Proof.
by move=> le_xy; apply/negP => /lt_le_trans /(_ le_xy); rewrite ltxx.
Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_gtF
| |
lt_geFx y : x < y -> y <= x = false.
Proof. by apply: contraTF => /le_gtF ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_geF
| |
lt_gtFx y hxy := le_gtF (@ltW x y hxy).
|
Definition
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_gtF
| |
lt_leAngex y : (x < y) = (x <= y) && ~~ (y <= x).
Proof. exact: lt_le_def. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_leAnge
| |
lt_le_asymx y : x < y <= x = false.
Proof. by apply/negP; move=> /andP[] xy /(lt_le_trans xy); rewrite ltxx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_le_asym
| |
le_lt_asymx y : x <= y < x = false.
Proof. by rewrite andbC lt_le_asym. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_lt_asym
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_leP
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_geP
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_ltP
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_gtP
| |
le_path_minx s : path <=%O x s -> all (>= x) s.
Proof. exact/order_path_min/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_path_min
| |
lt_path_minx s : path <%O x s -> all (> x) s.
Proof. exact/order_path_min/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_path_min
| |
le_path_sortedEx s : path <=%O x s = all (>= x) s && sorted <=%O s.
Proof. exact/path_sortedE/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_path_sortedE
| |
lt_path_sortedEx s : path <%O x s = all (> x) s && sorted <%O s.
Proof. exact/path_sortedE/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_path_sortedE
| |
le_sorted_pairwises : sorted <=%O s = pairwise <=%O s.
Proof. exact/sorted_pairwise/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_sorted_pairwise
| |
lt_sorted_pairwises : sorted <%O s = pairwise <%O s.
Proof. exact/sorted_pairwise/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_pairwise
| |
le_path_pairwisex s : path <=%O x s = pairwise <=%O (x :: s).
Proof. exact/path_pairwise/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_path_pairwise
| |
lt_path_pairwisex s : path <%O x s = pairwise <%O (x :: s).
Proof. exact/path_pairwise/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_path_pairwise
| |
lt_sorted_is_uniq_les : 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_is_uniq_le
| |
le_sorted_maskm s : sorted <=%O s -> sorted <=%O (mask m s).
Proof. exact/sorted_mask/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_sorted_mask
| |
lt_sorted_maskm s : sorted <%O s -> sorted <%O (mask m s).
Proof. exact/sorted_mask/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_mask
| |
le_sorted_filtera s : sorted <=%O s -> sorted <=%O (filter a s).
Proof. exact/sorted_filter/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_sorted_filter
| |
lt_sorted_filtera s : sorted <%O s -> sorted <%O (filter a s).
Proof. exact/sorted_filter/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_filter
| |
le_path_maskx m s : path <=%O x s -> path <=%O x (mask m s).
Proof. exact/path_mask/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_path_mask
| |
lt_path_maskx m s : path <%O x s -> path <%O x (mask m s).
Proof. exact/path_mask/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_path_mask
| |
le_path_filterx a s : path <=%O x s -> path <=%O x (filter a s).
Proof. exact/path_filter/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_path_filter
| |
lt_path_filterx a s : path <%O x s -> path <%O x (filter a s).
Proof. exact/path_filter/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_path_filter
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_sorted_ltn_nth
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_sorted_leq_nth
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_leq_nth
| |
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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_ltn_nth
| |
subseq_le_pathx s1 s2 : subseq s1 s2 -> path <=%O x s2 -> path <=%O x s1.
Proof. exact/subseq_path/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
subseq_le_path
| |
subseq_lt_pathx s1 s2 : subseq s1 s2 -> path <%O x s2 -> path <%O x s1.
Proof. exact/subseq_path/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
subseq_lt_path
| |
subseq_le_sorteds1 s2 : subseq s1 s2 -> sorted <=%O s2 -> sorted <=%O s1.
Proof. exact/subseq_sorted/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
subseq_le_sorted
| |
subseq_lt_sorteds1 s2 : subseq s1 s2 -> sorted <%O s2 -> sorted <%O s1.
Proof. exact/subseq_sorted/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
subseq_lt_sorted
| |
lt_sorted_uniqs : sorted <%O s -> uniq s.
Proof. exact/sorted_uniq/ltxx/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_uniq
| |
lt_sorted_eqs1 s2 :
sorted <%O s1 -> sorted <%O s2 -> s1 =i s2 -> s1 = s2.
Proof. exact/irr_sorted_eq/ltxx/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_sorted_eq
| |
filter_lt_nthx0 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
filter_lt_nth
| |
count_lt_nthx0 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
count_lt_nth
| |
filter_le_nthx0 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
filter_le_nth
| |
count_le_nthx0 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
count_le_nth
| |
sorted_filter_ltx 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
sorted_filter_lt
| |
sorted_filter_lex 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
sorted_filter_le
| |
nth_count_lex 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
nth_count_le
| |
nth_count_ltx 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
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
nth_count_lt
| |
sort_le_ids : sorted <=%O s -> sort <=%O s = s.
Proof. exact/sorted_sort/le_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
sort_le_id
| |
sort_lt_ids : sorted <%O s -> sort <%O s = s.
Proof. exact/sorted_sort/lt_trans. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
sort_lt_id
| |
comparable_leNgtx y : x >=< y -> (x <= y) = ~~ (y < x).
Proof.
rewrite /comparable lt_le_def.
by case: (x <= y) => //=; case: (y <= x).
Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
comparable_leNgt
| |
comparable_ltNgex y : x >=< y -> (x < y) = ~~ (y <= x).
Proof.
rewrite /comparable lt_le_def.
by case: (x <= y) => //=; case: (y <= x).
Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
comparable_ltNge
| |
comparable_symx y : (y >=< x) = (x >=< y).
Proof. by rewrite /comparable orbC. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
comparable_sym
| |
comparablexxx : x >=< x.
Proof. by rewrite /comparable lexx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
comparablexx
| |
incomparable_eqFx y : (x >< y) -> (x == y) = false.
Proof. by apply: contraNF => /eqP ->; rewrite comparablexx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
incomparable_eqF
| |
incomparable_leFx y : (x >< y) -> (x <= y) = false.
Proof. by apply: contraNF; rewrite /comparable => ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
incomparable_leF
| |
incomparable_ltFx y : (x >< y) -> (x < y) = false.
Proof. by rewrite lt_le_def => /incomparable_leF ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
incomparable_ltF
| |
le_comparable(x y : T) : x <= y -> x >=< y.
Proof. by rewrite /comparable => ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
le_comparable
| |
lt_comparable(x y : T) : x < y -> x >=< y.
Proof. by rewrite /comparable => /ltW ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lt_comparable
| |
ge_comparable(x y : T) : y <= x -> x >=< y.
Proof. by rewrite /comparable orbC => ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
ge_comparable
| |
gt_comparable(x y : T) : y < x -> x >=< y.
Proof. by rewrite /comparable orbC => /ltW ->. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
gt_comparable
| |
leif_reflx C : reflect (x <= x ?= iff C) C.
Proof. by apply: (iffP idP) => [-> | <-] //; split; rewrite ?eqxx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
leif_refl
| |
eq_leifx y C : x <= y ?= iff C -> (x == y) = C.
Proof. by move=> []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
eq_leif
| |
eqTleifx y C : x <= y ?= iff C -> C -> x = y.
Proof. by move=> [] _ <- /eqP. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
eqTleif
| |
lteif_transx y z C1 C2 :
x < y ?<= if C1 -> y < z ?<= if C2 -> x < z ?<= if C1 && C2.
Proof.
case: C1 C2 => [][];
[exact: le_trans | exact: le_lt_trans | exact: lt_le_trans | exact: lt_trans].
Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteif_trans
| |
lteifxxx C : (x < x ?<= if C) = C.
Proof. by case: C; rewrite /= ltexx. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteifxx
| |
lteifNFx y C : y < x ?<= if ~~ C -> x < y ?<= if C = false.
Proof. by case: C => [/lt_geF|/le_gtF]. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteifNF
| |
lteifSx y C : x < y -> x < y ?<= if C.
Proof. by case: C => //= /ltW. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteifS
| |
lteifTx y : x < y ?<= if true = (x <= y). Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteifT
| |
lteifFx y : x < y ?<= if false = (x < y). Proof. by []. Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteifF
| |
lteif_orbx y : {morph lteif x y : p q / p || q}.
Proof.
case=> [][] /=.
- by rewrite orbb.
- by case/boolP: (x < y) => [/ltW -> //|_]; rewrite orbF.
- by case/boolP: (x < y) => [/ltW ->|].
- by rewrite orbb.
Qed.
|
Lemma
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
lteif_orb
|
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