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 |
|---|---|---|---|---|---|---|
homo_ltn_lt: (forall i, f i < f i.+1) -> {homo f : i j / i < j}.
Proof. by apply: homo_ltn; apply: 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
|
homo_ltn_lt
| |
nondecnP: (forall i, f i <= f i.+1) -> {homo f : i j / i <= j}.
Proof. by apply: homo_leq => //; 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
|
nondecnP
| |
nhomo_ltn_lt: (forall i, f i > f i.+1) -> {homo f : i j /~ i < j}.
Proof.
move=> f_dec; apply: homo_sym.
by apply: homo_ltn f_dec => ? ? ? ? /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
|
nhomo_ltn_lt
| |
nonincnP: (forall i, f i >= f i.+1) -> {homo f : i j /~ i <= j}.
Proof.
move=> /= f_dec; apply: homo_sym.
by apply: homo_leq f_dec => //= ? ? ? ? /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
|
nonincnP
| |
dvd:= (@le dvd_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
|
dvd
| |
sdvd:= (@lt dvd_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
|
sdvd
| |
nat0:= (@top dvd_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
|
nat0
| |
nat1:= (@bottom dvd_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
|
nat1
| |
t:= nat.
#[export]
HB.instance Definition _ := Choice.copy t nat.
|
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
|
t
| |
Definition_ := @Le_isPreorder.Build
dvd_display t dvdn dvdnn dvdn_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
| |
Definition_ := @hasBottom.Build _ t 1 dvd1n.
#[export]
HB.instance Definition _ := @hasTop.Build _ t 0 dvdn0.
Import DvdSyntax.
|
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
| |
dvdE: dvd = dvdn :> rel 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
|
dvdE
| |
nat1E: nat1 = 1%N :> 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
|
nat1E
| |
nat0E: nat0 = 0%N :> 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
|
nat0E
| |
natdvd:= t.
|
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
|
natdvd
| |
dvdEnat:= dvdE.
|
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
|
dvdEnat
| |
nat1E:= nat1E.
|
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
|
nat1E
| |
nat0E:= nat0E.
|
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
|
nat0E
| |
Definition_ :=
[SubChoice_isSubPreorder of 'I_n by <: with ord_display].
|
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
| |
leEord: (le : rel 'I_n) = leq. 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
|
leEord
| |
ltEord: (lt : rel 'I_n) = (fun m n => m < n)%N. 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
|
ltEord
| |
botEord: \bot = ord0. 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
|
botEord
| |
topEord: \top = ord_max. 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
|
topEord
| |
leEord:= leEord.
|
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
|
leEord
| |
ltEord:= ltEord.
|
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
|
ltEord
| |
botEord:= botEord.
|
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
|
botEord
| |
topEord:= topEord.
|
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
|
topEord
| |
leEbool: le = (leq : rel bool). 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
|
leEbool
| |
ltEboolx y : (x < y) = (x < y)%N. 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
|
ltEbool
| |
leEbool:= leEbool.
|
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
|
leEbool
| |
ltEbool:= ltEbool.
|
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
|
ltEbool
| |
prod_display(displ dispr : disp_t) : disp_t :=
Disp (prod_display_unit (d1 displ) (d1 dispr))
(prod_display_unit (d2 displ) (d2 dispr)).
Fact seqprod_display (disp : disp_t) : disp_t. Proof. exact: disp. Qed.
|
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
|
prod_display
| |
type(disp : disp_t) (T T' : Type) := T * T'.
|
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
|
type
| |
type_(disp1 disp2 : disp_t) (T : preorderType disp1) (T' : preorderType disp2) :=
type (prod_display disp1 disp2) T T'.
|
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
|
type_
| |
Definition_ := Preorder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := Preorder.on T2'.
|
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
| |
lex y := (x.1 <= y.1) && (x.2 <= y.2).
|
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
| |
ltx y := (x.1 < y.1) && (x.2 <= y.2) || (x.1 <= y.1) && (x.2 < y.2).
#[export]
HB.instance Definition _ := @isDuallyPreorder.Build disp3 (T1 * T2) le lt
(@lt_def _ _ T1' T2') (@lt_def _ _ T1^d T2^d)
(@refl _ _ T1' T2') (@refl _ _ T1^d T2^d)
(@trans _ _ T1' T2') (@trans _ _ T1^d T2^d).
|
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
| |
leEprodx y : (x <= y) = (x.1 <= y.1) && (x.2 <= y.2). 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
|
leEprod
| |
ltEprodx y :
(x < y) = (x.1 < y.1) && (x.2 <= y.2) || (x.1 <= y.1) && (x.2 < y.2).
Proof.
rewrite lt_leAnge !leEprod negb_and andb_orr andbAC -lt_leAnge -andbA.
by rewrite -lt_leAnge.
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
|
ltEprod
| |
le_pair(x1 y1 : T1) (x2 y2 : T2) :
(x1, x2) <= (y1, y2) :> T1 * T2 = (x1 <= y1) && (x2 <= y2).
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
|
le_pair
| |
lt_pair(x1 y1 : T1) (x2 y2 : T2) :
(x1, x2) < (y1, y2) :> T1 * T2
= (x1 < y1) && (x2 <= y2) || (x1 <= y1) && (x2 < y2).
Proof. exact/ltEprod. 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_pair
| |
Definition_ := BPreorder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := BPreorder.on T2'.
#[export]
HB.instance Definition _ :=
@hasBottom.Build disp3 (T1 * T2) (\bot, \bot) (@le0x _ _ T1' T2').
|
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
| |
botEprod: \bot = (\bot, \bot) :> T1 * T2. 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
|
botEprod
| |
Definition_ :=
@hasTop.Build disp3 (T1 * T2) (\top, \top) (@le0x _ _ T1^d T2^d).
|
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
| |
topEprod: \top = (\top, \top) :> T1 * T2. 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
|
topEprod
| |
Definition_ (disp1 disp2 disp3 : disp_t)
(T1 : tbPreorderType disp1) (T2 : tbPreorderType disp2) :=
Preorder.on (type disp3 T1 T2).
|
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
| |
Definition_ (T1 : finPreorderType disp1)
(T2 : finPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finBPreorderType disp1)
(T2 : finBPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finTPreorderType disp1)
(T2 : finTPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finTBPreorderType disp1)
(T2 : finTBPreorderType disp2) := Preorder.on (type disp3 T1 T2).
|
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
| |
leEprod:= @leEprod.
|
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
|
leEprod
| |
ltEprod:= @ltEprod.
|
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
|
ltEprod
| |
le_pair:= @le_pair.
|
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_pair
| |
lt_pair:= @lt_pair.
|
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_pair
| |
botEprod:= @botEprod.
|
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
|
botEprod
| |
topEprod:= @topEprod.
|
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
|
topEprod
| |
Definition_ (T : preorderType disp1) (T' : preorderType disp2) :=
Preorder.copy (T * T')%type (T *p T').
HB.instance Definition _ (T1 : bPreorderType disp1) (T2 : bPreorderType disp2) :=
BPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _ (T1 : tPreorderType disp1) (T2 : tPreorderType disp2) :=
TPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _ (T1 : tbPreorderType disp1) (T2 : tbPreorderType disp2) :=
TBPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _
(T1 : finPreorderType disp1) (T2 : finPreorderType disp2) :=
FinPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _
(T1 : finBPreorderType disp1) (T2 : finBPreorderType disp2) :=
FinBPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _
(T1 : finTPreorderType disp1) (T2 : finTPreorderType disp2) :=
FinTPreorder.copy (T1 * T2)%type (prod T1 T2).
HB.instance Definition _
(T1 : finTBPreorderType disp1) (T2 : finTBPreorderType disp2) :=
FinTBPreorder.copy (T1 * T2)%type (prod T1 T2).
|
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
| |
type(disp : disp_t) (T T' : Type) := T * T'.
|
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
|
type
| |
type_(disp1 disp2 : disp_t) (T : preorderType disp1) (T' : preorderType disp2) :=
type (lexi_display disp1 disp2) T T'.
|
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
|
type_
| |
Definition_ := Preorder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := Preorder.on T2'.
|
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
| |
lex y := (x.1 <= y.1) && ((x.1 >= y.1) ==> (x.2 <= y.2)).
|
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
| |
ltx y := (x.1 <= y.1) && ((x.1 >= y.1) ==> (x.2 < y.2)).
#[export]
HB.instance Definition _ := @isDuallyPreorder.Build disp3 (T1 * T2) le lt
(@lt_le_def _ _ T1' T2') (@lt_le_def _ _ T1^d T2^d)
(@refl _ _ T1' T2') (@refl _ _ T1^d T2^d)
(@trans _ _ T1' T2') (@trans _ _ T1^d T2^d).
|
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
| |
leEprodlexix y :
(x <= y) = (x.1 <= y.1) && ((x.1 >= y.1) ==> (x.2 <= y.2)).
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
|
leEprodlexi
| |
ltEprodlexix y :
(x < y) = (x.1 <= y.1) && ((x.1 >= y.1) ==> (x.2 < y.2)).
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
|
ltEprodlexi
| |
lexi_pair(x1 y1 : T1) (x2 y2 : T2) :
(x1, x2) <= (y1, y2) :> T1 * T2 = (x1 <= y1) && ((x1 >= y1) ==> (x2 <= y2)).
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
|
lexi_pair
| |
ltxi_pair(x1 y1 : T1) (x2 y2 : T2) :
(x1, x2) < (y1, y2) :> T1 * T2 = (x1 <= y1) && ((x1 >= y1) ==> (x2 < y2)).
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
|
ltxi_pair
| |
Definition_ := BPreorder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := BPreorder.on T2'.
#[export]
HB.instance Definition _ :=
@hasBottom.Build disp3 (T1 * T2) (\bot, \bot) (@le0x _ _ T1' T2').
|
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
| |
botEprodlexi: \bot = (\bot, \bot) :> T1 * T2. 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
|
botEprodlexi
| |
Definition_ :=
@hasTop.Build disp3 (T1 * T2) (\top, \top) (@le0x _ _ T1^d T2^d).
|
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
| |
topEprodlexi: \top = (\top, \top) :> T1 * T2. 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
|
topEprodlexi
| |
Definition_ (disp1 disp2 disp3 : disp_t)
(T1 : tbPreorderType disp1) (T2 : tbPreorderType disp2) :=
Preorder.on (type disp3 T1 T2).
|
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
| |
sub_prod_lexi(disp1 disp2 disp3 disp4 : disp_t)
(T1 : preorderType disp1) (T2 : preorderType disp2) :
subrel (<=%O : rel (T1 *prod[disp3] T2)) (<=%O : rel (type disp4 T1 T2)).
Proof.
case=> [x1 x2] [y1 y2]; rewrite leEprod leEprodlexi /= => /andP[] -> ->.
exact: implybT.
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
|
sub_prod_lexi
| |
Definition_ (T1 : bPreorderType disp1)
(T2 : bPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export, warning="-HB.no-new-instance"]
HB.instance Definition _ (T1 : tPreorderType disp1)
(T2 : tPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export, warning="-HB.no-new-instance"]
HB.instance Definition _ (T1 : tbPreorderType disp1)
(T2 : tbPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finPreorderType disp1)
(T2 : finPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finBPreorderType disp1)
(T2 : finBPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finTPreorderType disp1)
(T2 : finTPreorderType disp2) := Preorder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (T1 : finTBPreorderType disp1)
(T2 : finTBPreorderType disp2) := Preorder.on (type disp3 T1 T2).
|
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
| |
leEprodlexi:= @leEprodlexi.
|
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
|
leEprodlexi
| |
ltEprodlexi:= @ltEprodlexi.
|
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
|
ltEprodlexi
| |
lexi_pair:= @lexi_pair.
|
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
|
lexi_pair
| |
ltxi_pair:= @ltxi_pair.
|
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
|
ltxi_pair
| |
topEprodlexi:= @topEprodlexi.
|
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
|
topEprodlexi
| |
botEprodlexi:= @botEprodlexi.
|
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
|
botEprodlexi
| |
sub_prod_lexi:= @sub_prod_lexi.
|
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
|
sub_prod_lexi
| |
Definition_ (T1 : preorderType disp1) (T2 : preorderType disp2) :=
Preorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _ (T1 : bPreorderType disp1) (T2 : bPreorderType disp2) :=
BPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _ (T1 : tPreorderType disp1) (T2 : tPreorderType disp2) :=
TPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _ (T1 : tbPreorderType disp1) (T2 : tbPreorderType disp2) :=
TBPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _
(T1 : finPreorderType disp1) (T2 : finPreorderType disp2) :=
FinPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _
(T1 : finBPreorderType disp1) (T2 : finBPreorderType disp2) :=
FinBPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _
(T1 : finTPreorderType disp1) (T2 : finTPreorderType disp2) :=
FinTPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
HB.instance Definition _
(T1 : finTBPreorderType disp1) (T2 : finTBPreorderType disp2) :=
FinTBPreorder.copy (T1 * T2)%type (prodlexi T1 T2).
|
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
| |
type(disp : disp_t) T := seq T.
|
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
|
type
| |
type_(disp : disp_t) (T : preorderType disp) :=
type (seqprod_display disp) T.
Context {disp disp' : disp_t}.
Local Notation seq := (type disp').
#[export] HB.instance Definition _ (T : eqType) := Equality.on (seq T).
#[export] HB.instance Definition _ (T : choiceType) := Choice.on (seq T).
#[export] HB.instance Definition _ (T : countType) := Countable.on (seq T).
|
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
|
type_
| |
les1 s2 := if s1 isn't x1 :: s1' then true else
if s2 isn't x2 :: s2' then false else
(x1 <= x2) && le s1' s2'.
Fact refl : reflexive le. Proof. by elim=> //= ? ? ?; rewrite !lexx. Qed.
Fact trans : transitive le.
Proof.
elim=> [|y ys ihs] [|x xs] [|z zs] //= /andP[xy xys] /andP[yz yzs].
by rewrite (le_trans xy)// ihs.
Qed.
#[export]
HB.instance Definition _ := isPreorder.Build disp' (seq T) (rrefl _) refl trans.
|
Fixpoint
|
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
| |
leEseqs1 s2 : s1 <= s2 = if s1 isn't x1 :: s1' then true else
if s2 isn't x2 :: s2' then false else
(x1 <= x2) && (s1' <= s2' :> seq _).
Proof. by case: s1. 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
|
leEseq
| |
le0ss : [::] <= s :> seq _. 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
|
le0s
| |
les0s : s <= [::] = (s == [::]). Proof. by rewrite leEseq. 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
|
les0
| |
le_consx1 s1 x2 s2 :
x1 :: s1 <= x2 :: s2 :> seq _ = (x1 <= x2) && (s1 <= s2).
Proof. by []. Qed.
#[export]
HB.instance Definition _ := hasBottom.Build _ (seq T) le0s.
|
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_cons
| |
botEseq: \bot = [::] :> seq 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
|
botEseq
| |
seqprod_with:= type.
|
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
|
seqprod_with
| |
seqprod:= type_.
|
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
|
seqprod
| |
leEseq:= @leEseq.
|
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
|
leEseq
| |
le0s:= @le0s.
|
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
|
le0s
| |
les0:= @les0.
|
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
|
les0
| |
le_cons:= @le_cons.
|
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_cons
| |
botEseq:= @botEseq.
|
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
|
botEseq
| |
seqprod:= (seqprod_with (seqprod_display disp)).
HB.instance Definition _ (T : preorderType disp) :=
Preorder.copy (seq T) (seqprod T).
HB.instance Definition _ (T : preorderType disp) :=
BPreorder.copy (seq T) (seqprod T).
|
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
|
seqprod
| |
type(disp : disp_t) T := seq T.
|
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
|
type
| |
type_(disp : disp_t) (T : preorderType disp) :=
type (seqlexi_display disp) T.
Context {disp disp' : disp_t}.
Local Notation seq := (type disp').
#[export] HB.instance Definition _ (T : eqType) := Equality.on (seq T).
#[export] HB.instance Definition _ (T : choiceType) := Choice.on (seq T).
#[export] HB.instance Definition _ (T : countType) := Countable.on (seq T).
|
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
|
type_
| |
les1 s2 := if s1 isn't x1 :: s1' then true else
if s2 isn't x2 :: s2' then false else
(x1 <= x2) && ((x1 >= x2) ==> le s1' s2').
|
Fixpoint
|
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
| |
lts1 s2 := if s2 isn't x2 :: s2' then false else
if s1 isn't x1 :: s1' then true else
(x1 <= x2) && ((x1 >= x2) ==> lt s1' s2').
Fact refl: reflexive le.
Proof. by elim => [|x s ih] //=; rewrite lexx. Qed.
Fact trans: transitive le.
Proof.
elim=> [|y sy ihs] [|x sx] [|z sz] //= /andP[] xy /implyP yx /andP[] yz /implyP zy /=.
rewrite (le_trans xy yz)/=; apply/implyP => zx.
apply/ihs; first exact/yx/(le_trans yz zx).
exact/zy/(le_trans zx xy).
Qed.
|
Fixpoint
|
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
| |
lt_le_defs1 s2 : lt s1 s2 = le s1 s2 && ~~ le s2 s1.
Proof.
elim: s1 s2 => [|x s1 ihs1] [|y s2]//=; rewrite ihs1.
by case: (x <= y); case (y <= x).
Qed.
#[export]
HB.instance Definition _ := isPreorder.Build disp' (seq T) lt_le_def refl trans.
|
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
| |
leEseqlexis1 s2 :
s1 <= s2 = if s1 isn't x1 :: s1' then true else
if s2 isn't x2 :: s2' then false else
(x1 <= x2) && ((x1 >= x2) ==> (s1' <= s2' :> seq T)).
Proof. by case: s1. 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
|
leEseqlexi
|
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