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 |
|---|---|---|---|---|---|---|
RecordPOrder_MeetJoin_isLattice d T of POrder d T := {
meet : T -> T -> T;
join : T -> T -> T;
meetP : forall x y z, (x <= meet y z) = (x <= y) && (x <= z);
joinP : forall x y z, (join x y <= z) = (x <= z) && (y <= z);
}.
HB.builders Context d T of POrder_MeetJoin_isLattice d T.
HB.instance Definition _ := @POr... | HB.factory | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
RecordLattice_Meet_isDistrLattice d T of Lattice d T := {
meetUl : @left_distributive T T meet join;
}.
HB.builders Context d T of Lattice_Meet_isDistrLattice d T.
Let meetUr : right_distributive (@meet _ T) (@join _ T).
Proof. by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed.
Let joinIl : left_distributive (@joi... | HB.factory | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
RecordBDistrLattice_hasSectionalComplement d T
of BDistrLattice d T := {
diff : T -> T -> T;
diffKI : forall x y, y `&` diff x y = \bot;
joinIB : forall x y, (x `&` y) `|` diff x y = x;
}. | HB.factory | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
Buildd T :=
(BDistrLattice_hasSectionalComplement.Build d T) (only parsing). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Build | |
hasRelativeComplementd T :=
(BDistrLattice_hasSectionalComplement d T) (only parsing).
HB.builders Context d T of BDistrLattice_hasSectionalComplement d 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",
"From mathcomp Require Export preorder"
] | order/order.v | hasRelativeComplement | |
rcomplx y z := (x `&` y) `|` diff (y `|` x) z.
Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.
Proof. by rewrite meetUr joinIKC meetUl diffKI joinx0 meetKU. Qed.
Fact rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.
Proof. by rewrite joinCA joinIB joinA meetUK joinC. Qed.
HB.inst... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | rcompl | |
rcomplx y z := (y `|` x) `&` codiff (x `&` y) z.
Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.
Proof. by rewrite meetCA meetUB meetA joinIK. Qed.
Fact rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.
Proof. by rewrite joinIr meetUKC joinIl codiffKU meetx1 joinKI. Qed.
HB.instan... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | rcompl | |
Buildd T := (CBDistrLattice_hasComplement.Build d T) (only parsing). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Build | |
hasComplementd T := (CBDistrLattice_hasComplement d T) (only parsing).
HB.builders Context d T of CBDistrLattice_hasComplement d T.
HB.instance Definition _ := @CDistrLattice_hasDualSectionalComplement.Build d T
(fun x y => rcompl x \top y) (fun _ _ => erefl).
Fact complEcodiff (x : T) : compl x = codiff (\bot : T) x... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | hasComplement | |
diffx y := x `&` compl 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",
"From mathcomp Require Export preorder"
] | order/order.v | diff | |
codiffx y := x `|` compl 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",
"From mathcomp Require Export preorder"
] | order/order.v | codiff | |
rcomplx y z := (x `&` y) `|` diff (y `|` x) z.
Fact diffKI x y : y `&` diff x y = \bot.
Proof. by rewrite meetCA meetxC meetx0. Qed.
Fact joinIB x y : (x `&` y) `|` diff x y = x.
Proof. by rewrite -meetUr joinxC meetx1. Qed.
HB.instance Definition _ :=
@BDistrLattice_hasSectionalComplement.Build d T diff diffKI joinI... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | rcompl | |
RecordLattice_isTotal d T of Lattice d T := {
le_total : total (<=%O : rel T)
}.
HB.builders Context d T of Lattice_isTotal d T.
Fact meetUl : @left_distributive T T meet join.
Proof.
pose leP x y := lcomparable_leP (le_total x y); move=> x y z; apply/esym.
by case: (leP x y) (leP x z) (leP y z) => [|/ltW] xy [|/ltW]... | HB.factory | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
meet:= @min _ 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",
"From mathcomp Require Export preorder"
] | order/order.v | meet | |
join:= @max _ T.
Fact meetC : commutative meet.
Proof. by move=> x y; rewrite /meet; have [] := ltgtP. Qed.
Fact joinC : commutative join.
Proof. by move=> x y; rewrite /join; have [] := ltgtP. Qed.
Fact meetA : associative meet.
Proof.
move=> x y z; rewrite /meet /min !(fun_if, if_arg).
case: (leP z y) (leP y x) (leP ... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | join | |
Definition_ := POrder.on T'.
HB.instance Definition _ := POrder_isTotal.Build d T' le_total.
Implicit Types (x y z : T').
Fact meetE x y : meet x y = x `&` y. Proof. by rewrite meet_def. Qed.
Fact joinE x y : join x y = x `|` y. Proof. by rewrite join_def. Qed.
Fact meetC : commutative meet.
Proof. by move=> *; rewrite... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ := @POrder_Meet_isDistrLattice.Build d T
meet join meetC joinC meetA joinA joinKI meetKU le_def meetUl.
HB.instance Definition _ := DistrLattice_isTotal.Build d T le_total.
HB.end.
HB.factory Record LtOrder (d : disp_t) T of Choice T := {
le : rel T;
lt : rel T;
meet : T -> T -> T;
join : T -> T -... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Pcan:= isPOrder.Build disp (Choice.Pack (Choice.class T))
lt_def (@refl T disp' T' f) anti (@trans T disp' T' f). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Pcan | |
Canf' (f_can : cancel f f') := Pcan (can_pcan f_can). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Can | |
PCanIsPartial:= CancelPartial.Pcan. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | PCanIsPartial | |
CanIsPartial:= CancelPartial.Can.
#[export]
HB.instance Definition _ (disp : disp_t) (T : choiceType)
(disp' : disp_t) (T' : porderType disp') (f : T -> T')
(f' : T' -> option T) (f_can : pcancel f f') :=
Preorder_isPOrder.Build disp (pcan_type f_can) (CancelPartial.anti f_can).
#[export]
HB.instance Definition _... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | CanIsPartial | |
Definition_ :=
MonoTotal.Build disp (pcan_type f_can) (fun _ _ => erefl). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
PCanIsTotal: DistrLattice_isTotal _ (pcan_type f_can) :=
Total.on (pcan_type f_can). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | PCanIsTotal | |
Definition_ :=
MonoTotal.Build disp (can_type f_can) (fun _ _ => erefl). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
CanIsTotal: DistrLattice_isTotal _ (can_type f_can) :=
Total.on (can_type f_can). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | CanIsTotal | |
RecordIsoLattice disp T of POrder disp T := {
disp' : disp_t;
T' : latticeType disp';
f : T -> T';
f' : T' -> T;
f_can : cancel f f';
f'_can : cancel f' f;
f_mono : {mono f : x y / x <= y};
}.
HB.builders Context disp T of IsoLattice disp T. | HB.factory | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
meet(x y : T) := f' (meet (f x) (f 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",
"From mathcomp Require Export preorder"
] | order/order.v | meet | |
join(x y : T) := f' (join (f x) (f y)).
Fact meetC : commutative meet. Proof. by move=> x y; rewrite /meet meetC. Qed.
Fact joinC : commutative join. Proof. by move=> x y; rewrite /join joinC. Qed.
Fact meetA : associative meet.
Proof. by move=> y x z; rewrite /meet !f'_can meetA. Qed.
Fact joinA : associative join.
Pr... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | join | |
omorph_lt(d : disp_t) (T : porderType d) (d' : disp_t) (T' : porderType d')
(f : {omorphism T -> T'}) : injective f -> {homo f : x y / x < y}.
Proof. by move/inj_homo_lt; apply; apply: omorph_le. 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",
"From mathcomp Require Export preorder"
] | order/order.v | omorph_lt | |
meet_morphismd (T : latticeType d) d' (T' : latticeType d')
(f : T -> T') : Prop := {morph 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",
"From mathcomp Require Export preorder"
] | order/order.v | meet_morphism | |
join_morphismd (T : latticeType d) d' (T' : latticeType d')
(f : T -> T') : Prop := {morph f : x y / x `|` y}.
HB.mixin Record isMeetLatticeMorphism d (T : latticeType d)
d' (T' : latticeType d') (apply : T -> T') := {
omorphI_subproof : meet_morphism apply;
}.
HB.mixin Record isJoinLatticeMorphism d (T : latti... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | join_morphism | |
omorphI(f : {mlmorphism T -> T'}) : {morph f : x y / x `&` y}.
Proof. exact: omorphI_subproof. 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",
"From mathcomp Require Export preorder"
] | order/order.v | omorphI | |
omorphU(f : {jlmorphism T -> T'}) : {morph f : x y / x `|` y}.
Proof. exact: omorphU_subproof. 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",
"From mathcomp Require Export preorder"
] | order/order.v | omorphU | |
Definition_ := isMeetLatticeMorphism.Build d T d T idfun
idfun_is_meet_morphism.
Fact comp_is_meet_morphism : meet_morphism (f \o g).
Proof. by move=> x y; rewrite /= !omorphI. Qed.
#[export]
HB.instance Definition _ := isMeetLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_meet_morphism. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ := isJoinLatticeMorphism.Build d T d T idfun
idfun_is_join_morphism.
Fact comp_is_join_morphism : join_morphism (f \o g).
Proof. by move=> x y; rewrite /= !omorphU. Qed.
#[export]
HB.instance Definition _ := isJoinLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_join_morphism. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
RecordisBLatticeMorphism d (T : bLatticeType d)
d' (T' : bLatticeType d') (apply : T -> T') := {
omorph0_subproof : apply \bot = \bot;
}.
HB.mixin Record isTLatticeMorphism d (T : tLatticeType d)
d' (T' : tLatticeType d') (apply : T -> T') := {
omorph1_subproof : apply \top = \top;
}.
HB.structure Definitio... | HB.mixin | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
omorph0: f \bot = \bot.
Proof. exact: omorph0_subproof. 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",
"From mathcomp Require Export preorder"
] | order/order.v | omorph0 | |
Definition_ := isBLatticeMorphism.Build d T d T idfun
idfun_is_bottom_morphism.
Fact comp_is_bottom_morphism : (f \o g) \bot = \bot.
Proof. by rewrite /= !omorph0. Qed.
#[export]
HB.instance Definition _ := isBLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_bottom_morphism. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
omorph1: f \top = \top.
Proof. exact: omorph1_subproof. 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",
"From mathcomp Require Export preorder"
] | order/order.v | omorph1 | |
Definition_ := isTLatticeMorphism.Build d T d T idfun
idfun_is_top_morphism.
Fact comp_is_top_morphism : (f \o g) \top = \top.
Proof. by rewrite /= !omorph1. Qed.
#[export]
HB.instance Definition _ := isTLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_top_morphism. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
meet_closed:= {in S &, forall u v, u `&` v \in S}. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | meet_closed | |
join_closed:= {in S &, forall u v, u `|` v \in S}. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | join_closed | |
RecordisMeetLatticeClosed d (T : latticeType d) (S : {pred T}) := {
opredI : meet_closed S;
}.
HB.mixin Record isJoinLatticeClosed d (T : latticeType d) (S : {pred T}) := {
opredU : join_closed S;
}.
HB.mixin Record isBLatticeClosed d (T : bLatticeType d) (S : {pred T}) := {
opred0 : \bot \in S;
}.
HB.mixin Recor... | HB.mixin | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Record | |
DefinitionMeetLatticeClosed d T :=
{S of isMeetLatticeClosed d T S}.
#[short(type="joinLatticeClosed")]
HB.structure Definition JoinLatticeClosed d T :=
{S of isJoinLatticeClosed d T S}.
#[short(type="latticeClosed")]
HB.structure Definition LatticeClosed d T :=
{S of @MeetLatticeClosed d T S & @JoinLatticeClosed... | HB.structure | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
opredI(S : meetLatticeClosed T) : {in S &, forall u v, u `&` v \in S}.
Proof. exact: opredI. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opredI | |
opredU(S : joinLatticeClosed T) : {in S &, forall u v, u `|` v \in S}.
Proof. exact: opredU. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opredU | |
opred0(S : bLatticeClosed T) : \bot \in S.
Proof. exact: opred0. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opred0 | |
opred_joins(S : bJoinLatticeClosed T) I r (P : pred I) F :
(forall i, P i -> F i \in S) -> \join_(i <- r | P i) F i \in S.
Proof. by move=> FS; elim/big_ind: _; [exact: opred0 | exact: opredU |]. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opred_joins | |
opred1(S : tLatticeClosed T) : \top \in S.
Proof. exact: opred1. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opred1 | |
opred_meets(S : tMeetLatticeClosed T) I r (P : pred I) F :
(forall i, P i -> F i \in S) -> \meet_(i <- r | P i) F i \in S.
Proof. by move=> FS; elim/big_ind: _; [exact: opred1 | exact: opredI |]. 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",
"From mathcomp Require Export preorder"
] | order/order.v | opred_meets | |
DefinitionSubPOrder d (T : porderType d) S d' :=
{ U of SubEquality T S U & POrder d' U & isSubPreorder d T S d' U }.
HB.factory Record SubChoice_isSubPOrder d (T : porderType d) S (d' : disp_t) U
of SubChoice T S U := {}.
HB.builders Context d T S d' U of SubChoice_isSubPOrder d T S d' U.
HB.instance Definition ... | HB.structure | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
joinUKIy x : meetU x (joinU x y) = x.
Proof. by apply: val_inj; rewrite !SubK joinKI. Qed.
Let meetUKU y x : joinU x (meetU x y) = x.
Proof. by apply: val_inj; rewrite !SubK meetKU. Qed.
Let le_meetU x y : (x <= y) = (meetU x y == x).
Proof. by rewrite -le_val -(inj_eq val_inj) SubK leEmeet. Qed.
HB.instance Definition... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | joinUKI | |
totalU: total (<=%O : rel U).
Proof. by move=> x y; rewrite -!le_val le_total. Qed.
HB.instance Definition _ := Lattice_isTotal.Build d' U totalU.
HB.end.
HB.factory Record SubPOrder_isSubOrder d (T : orderType d) S d' U
of @SubPOrder d T S d' U := {}.
HB.builders Context d T S d' U of SubPOrder_isSubOrder d T S d'... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | totalU | |
Definition_ :=
SubPOrder_isSubOrder.Build disp T P disp (sub_type sT). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ :=
Preorder_isPOrder.Build nat_display nat anti_leq.
#[export]
HB.instance Definition _ :=
POrder_isTotal.Build nat_display nat leq_total. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
incn_inP: {in D, forall i, i.+1 \in D -> f i < f i.+1} ->
{in D &, {mono f : i j / i <= j}}.
Proof. by move=> f_inc; apply/le_mono_in/homo_ltn_lt_in. 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",
"From mathcomp Require Export preorder"
] | order/order.v | incn_inP | |
decn_inP: {in D, forall i, i.+1 \in D -> f i > f i.+1} ->
{in D &, {mono f : i j /~ i <= j}}.
Proof. by move=> f_dec; apply/le_nmono_in/nhomo_ltn_lt_in. 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",
"From mathcomp Require Export preorder"
] | order/order.v | decn_inP | |
incnP: (forall i, f i < f i.+1) -> {mono f : i j / i <= j}.
Proof. by move=> f_inc; apply/le_mono/homo_ltn_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",
"From mathcomp Require Export preorder"
] | order/order.v | incnP | |
decnP: (forall i, f i > f i.+1) -> {mono f : i j /~ i <= j}.
Proof. by move=> f_dec; apply/le_nmono/nhomo_ltn_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",
"From mathcomp Require Export preorder"
] | order/order.v | decnP | |
gcd:= (@meet 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",
"From mathcomp Require Export preorder"
] | order/order.v | gcd | |
lcm:= (@join 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",
"From mathcomp Require Export preorder"
] | order/order.v | lcm | |
lcmnnn : lcmn n n = n.
Proof. by case: n => // n; rewrite /lcmn gcdnn mulnK. 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",
"From mathcomp Require Export preorder"
] | order/order.v | lcmnn | |
le_defm n : m %| n = (gcdn m n == m)%N.
Proof. by apply/gcdn_idPl/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",
"From mathcomp Require Export preorder"
] | order/order.v | le_def | |
joinKIn m : gcdn m (lcmn m n) = m.
Proof. by rewrite (gcdn_idPl _)// dvdn_lcml. 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",
"From mathcomp Require Export preorder"
] | order/order.v | joinKI | |
meetKUn m : lcmn m (gcdn m n) = m.
Proof. by rewrite (lcmn_idPl _)// dvdn_gcdl. 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",
"From mathcomp Require Export preorder"
] | order/order.v | meetKU | |
meetUl: left_distributive gcdn lcmn.
Proof.
move=> [|m'] [|n'] [|p'] //=; rewrite ?lcmnn ?lcm0n ?lcmn0 ?gcd0n ?gcdn0//.
- by rewrite gcdnC meetKU.
- by rewrite lcmnC gcdnC meetKU.
apply: eqn_from_log; rewrite ?(gcdn_gt0, lcmn_gt0)//= => p.
by rewrite !(logn_gcd, logn_lcm) ?(gcdn_gt0, lcmn_gt0)// minn_maxl.
Qed.
Fact dv... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | meetUl | |
sdvdE(m n : t) : m %<| n = (n != m) && (m %| n).
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",
"From mathcomp Require Export preorder"
] | order/order.v | sdvdE | |
gcdE: gcd = gcdn :> (t -> t -> 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",
"From mathcomp Require Export preorder"
] | order/order.v | gcdE | |
lcmE: lcm = lcmn :> (t -> t -> 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",
"From mathcomp Require Export preorder"
] | order/order.v | lcmE | |
sdvdEnat:= sdvdE. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | sdvdEnat | |
gcdEnat:= gcdE. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | gcdEnat | |
lcmEnat:= lcmE. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | lcmEnat | |
Definition_ (n : nat) :=
[SubChoice_isSubOrder 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
andEbool: meet = andb. 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",
"From mathcomp Require Export preorder"
] | order/order.v | andEbool | |
orEbool: meet = andb. 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",
"From mathcomp Require Export preorder"
] | order/order.v | orEbool | |
subEboolx y : x `\` y = 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",
"From mathcomp Require Export preorder"
] | order/order.v | subEbool | |
complEbool: compl = negb. 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",
"From mathcomp Require Export preorder"
] | order/order.v | complEbool | |
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",
"From mathcomp Require Export preorder"
] | order/order.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",
"From mathcomp Require Export preorder"
] | order/order.v | ltEbool | |
andEbool:= andEbool. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | andEbool | |
orEbool:= orEbool. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | orEbool | |
subEbool:= subEbool. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | subEbool | |
complEbool:= complEbool. | 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",
"From mathcomp Require Export preorder"
] | order/order.v | complEbool | |
meetlexi:= (@meet (lexi_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",
"From mathcomp Require Export preorder"
] | order/order.v | meetlexi | |
joinlexi:= (@join (lexi_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",
"From mathcomp Require Export preorder"
] | order/order.v | joinlexi | |
meetlexi:= (@meet (seqlexi_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",
"From mathcomp Require Export preorder"
] | order/order.v | meetlexi | |
joinlexi:= (@join (seqlexi_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",
"From mathcomp Require Export preorder"
] | order/order.v | joinlexi | |
Definition_ := POrder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := POrder.on T2'.
#[export]
HB.instance Definition _ :=
Preorder_isDuallyPOrder.Build disp3 (T1 * T2)
(@anti _ _ T1' T2') (@anti _ _ 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
ltEprodx y : (x < y) = [&& x != y, x.1 <= y.1 & x.2 <= y.2].
Proof. by rewrite lt_neqAle. 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",
"From mathcomp Require Export preorder"
] | order/order.v | ltEprod | |
lt_pair(x1 y1 : T1) (x2 y2 : T2) : (x1, x2) < (y1, y2) :> T1 * T2 =
[&& (x1 != y1) || (x2 != y2), x1 <= y1 & x2 <= y2].
Proof. by rewrite ltEprod negb_and. 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",
"From mathcomp Require Export preorder"
] | order/order.v | lt_pair | |
Definition_ := MeetSemilattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := MeetSemilattice.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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
meetx y := (x.1 `&` y.1, x.2 `&` y.2).
#[export]
HB.instance Definition _ :=
@POrder_isMeetSemilattice.Build disp3 (T1 * T2) meet (@lexI _ _ T1' T2'). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | meet | |
meetEprodx 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",
"From mathcomp Require Export preorder"
] | order/order.v | meetEprod | |
joinx y := (x.1 `|` y.1, x.2 `|` y.2).
#[export]
HB.instance Definition _ :=
@POrder_isJoinSemilattice.Build disp3 (T1 * T2) join
(fun x y z => @lexI _ _ T1^d T2^d z 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",
"From mathcomp Require Export preorder"
] | order/order.v | join | |
joinEprodx 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",
"From mathcomp Require Export preorder"
] | order/order.v | joinEprod | |
Definition_ (disp1 disp2 disp3 : disp_t)
(T1 : bPOrderType disp1) (T2 : bPOrderType disp2) :=
POrder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (disp1 disp2 disp3 : disp_t)
(T1 : tPOrderType disp1) (T2 : tPOrderType disp2) :=
POrder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (disp1... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ := DistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := DistrLattice.on T2'.
#[export]
HB.instance Definition _ := Lattice_isDistributive.Build disp3 (T1 * T2)
(@meetUl _ _ T1' T2') (@meetUl _ _ 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ (disp1 disp2 disp3 : disp_t)
(T1 : bDistrLatticeType disp1) (T2 : bDistrLatticeType disp2) :=
POrder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (disp1 disp2 disp3 : disp_t)
(T1 : tDistrLatticeType disp1) (T2 : tDistrLatticeType disp2) :=
POrder.on (type disp3 T1 T2).
#[export]
HB.inst... | 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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
Definition_ := CDistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := CDistrLattice.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",
"From mathcomp Require Export preorder"
] | order/order.v | Definition | |
rcomplx y z := (rcompl x.1 y.1 z.1, rcompl x.2 y.2 z.2).
#[export]
HB.instance Definition _ :=
@DistrLattice_hasRelativeComplement.Build disp3 (T1 * T2)
rcompl (@rcomplPmeet _ _ T1' T2')
(fun x y => @rcomplPmeet _ _ T1^d T2^d y x). | 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",
"From mathcomp Require Export preorder"
] | order/order.v | rcompl |
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