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 |
|---|---|---|---|---|---|---|
tnth_compl:= @tnth_compl. | 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 | tnth_compl | |
complEtprod:= @complEtprod. | 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 | complEtprod | |
Definition_ n (T : porderType disp) :=
POrder.copy (n.-tuple T) (n.-tupleprod T).
HB.instance Definition _ n (T : bPOrderType disp) :=
BPOrder.copy (n.-tuple T) (n.-tupleprod T).
HB.instance Definition _ n (T : tPOrderType disp) :=
TPOrder.copy (n.-tuple T) (n.-tupleprod T).
HB.instance Definition _ n (T : tbPOrd... | 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 | |
lexi_tuplePn T (t1 t2 : n.-tuple T) :
reflect (exists k : 'I_n.+1, forall i : 'I_n, (i <= k)%N ->
tnth t1 i <= tnth t2 i ?= iff (i != k :> nat)) (t1 <= t2).
Proof.
elim: n => [|n IHn] in t1 t2 *.
by rewrite tuple0 [t2]tuple0/= lexx; constructor; exists ord0 => -[].
case: (tupleP t1) (tupleP t2) => [... | 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 | lexi_tupleP | |
ltxi_tuplePn T (t1 t2 : n.-tuple T) :
reflect (exists k : 'I_n, forall i : 'I_n, (i <= k)%N ->
tnth t1 i <= tnth t2 i ?= iff (i != k :> nat)) (t1 < t2).
Proof.
elim: n => [|n IHn] in t1 t2 *.
by rewrite tuple0 [t2]tuple0/= ltxx; constructor => - [] [].
case: (tupleP t1) (tupleP t2) => [x1 {}t1] [x2 ... | 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 | ltxi_tupleP | |
ltxi_tuplePltn T (t1 t2 : n.-tuple T) : reflect
(exists2 k : 'I_n, forall i : 'I_n, (i < k)%N -> tnth t1 i = tnth t2 i
& tnth t1 k < tnth t2 k)
(t1 < t2).
Proof.
apply: (iffP (ltxi_tupleP _ _)) => [[k kP]|[k kP ltk12]].
exists k => [i i_lt|]; last by rewrite (lt_le... | 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 | ltxi_tuplePlt | |
Definition_ (n : nat) (T : bPOrderType disp) :=
POrder.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : tPOrderType disp) :=
POrder.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : tbPOrderType disp) :=
POrder.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : b... | 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 | |
lexi_tupleP:= @lexi_tupleP.
Arguments lexi_tupleP {disp disp' n T 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 | lexi_tupleP | |
ltxi_tupleP:= @ltxi_tupleP.
Arguments ltxi_tupleP {disp disp' n T 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 | ltxi_tupleP | |
ltxi_tuplePlt:= @ltxi_tuplePlt.
Arguments ltxi_tuplePlt {disp disp' n T 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 | ltxi_tuplePlt | |
Definition_ n (T : porderType disp) :=
POrder.copy (n.-tuple T) (n.-tuplelexi T).
HB.instance Definition _ n (T : bPOrderType disp) :=
BPOrder.copy (n.-tuple T) (n.-tuplelexi T).
HB.instance Definition _ n (T : tPOrderType disp) :=
TPOrder.copy (n.-tuple T) (n.-tuplelexi T).
HB.instance Definition _ n (T : tbPOrd... | 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 | |
setKUCB A : A :&: (A :|: B) = A.
Proof. by rewrite setUC setKU. 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 | setKUC | |
setKICB A : A :|: (A :&: B) = A.
Proof. by rewrite setIC setKI. Qed.
Fact le_anti : antisymmetric (fun A B => A \subset B).
Proof. by move=> A B ABA; apply/eqP; rewrite eqEsubset. Qed.
#[export]
HB.instance Definition _ := Preorder_isPOrder.Build disp {subset T} le_anti.
#[export]
HB.instance Definition _ := POrder_Me... | 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 | setKIC | |
setIDvA B : B :&: (A :\: B) = set0.
Proof.
apply/eqP; rewrite -subset0; apply/subsetP => x.
by rewrite !inE => /and3P[->].
Qed.
#[export]
HB.instance Definition _ :=
@BDistrLattice_hasSectionalComplement.Build disp {subset T}
(@setD _) setIDv (@setID _). | 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 | setIDv | |
setTDsymA : ~: A = setT :\: A.
Proof. by rewrite setTD. Qed.
#[export]
HB.instance Definition _ :=
CBDistrLattice_hasComplement.Build disp {subset T} setTDsym. | 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 | setTDsym | |
meetEsubsetA B : A `&` B = A :&: B.
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 | meetEsubset | |
joinEsubsetA B : A `|` B = A :|: B.
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 | joinEsubset | |
botEsubset: \bot = set0 :> {subset 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 | botEsubset | |
topEsubset: \top = setT :> {subset 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 | topEsubset | |
subEsubsetA B : A `\` B = A :\: B.
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 | subEsubset | |
complEsubsetA : ~` A = ~: A.
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 | complEsubset | |
meetEsubset:= @meetEsubset. | 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 | meetEsubset | |
joinEsubset:= @joinEsubset. | 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 | joinEsubset | |
botEsubset:= @botEsubset. | 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 | botEsubset | |
topEsubset:= @topEsubset. | 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 | topEsubset | |
subEsubset:= @subEsubset. | 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 | subEsubset | |
complEsubset:= @complEsubset. | 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 | complEsubset | |
Definition_ (T : finType) :=
CTBDistrLattice.copy {set T} {subset 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 | |
mono_uniqued (T T' : finPOrderType d) (f g : T -> T') :
total (<=%O : rel T) -> (#|T'| <= #|T|)%N ->
{mono f : x y / x <= y} -> {mono g : x y / x <= y} ->
f =1 g.
Proof.
move=> le_total leT'T lef leg x0; move: {+}x0.
suff: finfun f = finfun g by move=> /ffunP + x => /(_ x); rewrite !ffunE.
apply: (can_inj fgr... | 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 | mono_unique | |
le_enum_valA : {mono @enum_val _ _ A : i j / i <= j}.
Proof.
apply: le_mono => i j le_ij.
rewrite /enum_val (set_nth_default (enum_default j)) -?cardE//.
apply: (sorted_ltn_nth lt_trans); rewrite -?topredE/= -?cardE//.
by rewrite lt_sorted_uniq_le enum_uniq/= sort_sorted.
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_enum_val | |
le_enum_rank_inx0 A (Ax0 : x0 \in A) :
{in A &, {mono enum_rank_in Ax0 : x y / x <= y}}.
Proof.
apply: can_mono_in (@in2W _ _ predT predT _ (@le_enum_val A)) => //.
exact/onW_can_in/enum_rankK_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 | le_enum_rank_in | |
le_enum_rank: {mono @enum_rank d T : i j / i <= j}.
Proof. exact: (can_mono (@enum_rankK _ _) (@le_enum_val predT)). 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_enum_rank | |
le_enum_val:= le_enum_val. | 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 | le_enum_val | |
le_enum_rank_in:= le_enum_rank_in. | 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 | le_enum_rank_in | |
le_enum_rank:= le_enum_rank. | 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 | le_enum_rank | |
card: #|{: T}| = \sum_i p_ i.
Proof.
rewrite card_tagged sumnE/= big_map big_enum.
by apply: eq_bigr => i _; rewrite card_ord.
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 | card | |
sig: ordsum -> T := enum_val \o cast_ord (esym card). | 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 | sig | |
rank: T -> ordsum := cast_ord card \o enum_rank. | 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 | rank | |
sigK: cancel sig rank.
Proof.
by move=> s; rewrite /sig/rank/= enum_valK cast_ord_comp cast_ord_id.
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 | sigK | |
sig_inj: injective sig. Proof. exact: can_inj sigK. 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 | sig_inj | |
rankK: cancel rank sig.
Proof.
by move=> p; rewrite /sig/rank/= cast_ord_comp cast_ord_id enum_rankK.
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 | rankK | |
rank_inj: injective rank. Proof. exact: can_inj rankK. 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 | rank_inj | |
sig1s : 'I_n := tag (sig 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 | sig1 | |
sig2s : 'I_(p_ (sig1 s)) := tagged (sig 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 | sig2 | |
Ranki (j : 'I_(p_ i)) := rank (Tagged _ j). | 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 | Rank | |
sigE12s : sig s = @Tagged _ (sig1 s) _ (sig2 s).
Proof. by rewrite /sig1 /sig2; case: sig. 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 | sigE12 | |
rankEp : rank p = @Rank (tag p) (tagged p). Proof. by case: p. 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 | rankE | |
sig2Ks : Rank (sig2 s) = s. Proof. by rewrite -rankE sigK. 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 | sig2K | |
Rank1Ki0 (k : 'I_(p_ i0)) : sig1 (Rank k) = i0.
Proof. by rewrite /sig1 /Rank/= rankK/=. 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 | Rank1K | |
Rank2Ki0 (k : 'I_(p_ i0)) :
sig2 (Rank k) = cast_ord (congr1 p_ (esym (Rank1K k))) k.
Proof. by apply: val_inj; rewrite /sig2/sig1/Rank/= rankK. Qed.
#[local] Hint Resolve sigK rankK : 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",
"From mathcomp Require Export preorder"
] | order/order.v | Rank2K | |
rank_bij: bijective rank. Proof. by exists sig. 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 | rank_bij | |
sig_bij: bijective sig. Proof. by exists rank. 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 | sig_bij | |
rank_bij_on: {on [pred _ | true], bijective rank}.
Proof. exact/onW_bij/rank_bij. 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 | rank_bij_on | |
sig_bij_on: {on [pred _ | true], bijective sig}.
Proof. exact/onW_bij/sig_bij. 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 | sig_bij_on | |
le_sig: {mono sig : i j / i <= j}.
Proof. by move=> i j; rewrite /sig/= le_enum_val//; apply: le_total. 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_sig | |
le_sig1: {homo sig1 : i j / i <= j}.
Proof. by move=> i j; rewrite /sig1/= -le_sig leEsig/=; case: leP. 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_sig1 | |
le_rank: {mono rank : p q / p <= q}.
Proof. exact: can_mono le_sig. 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_rank | |
le_Ranki : {mono @Rank i : j k / j <= k}.
Proof. by move=> j k; rewrite /Rank le_rank/= leEsig/= tagged_asE 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",
"From mathcomp Require Export preorder"
] | order/order.v | le_Rank | |
lt_sig: {mono sig : i j / i < j}.
Proof. by move=> i j; rewrite !ltNge le_sig. 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_sig | |
lt_rank: {mono rank : p q / p < q}.
Proof. by move=> p q; rewrite !ltNge le_rank. 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_rank | |
lt_Ranki : {mono @Rank i : j k / j < k}.
Proof. by move=> j k; rewrite !ltNge le_Rank. 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_Rank | |
eq_Ranki i' (j : 'I_(p_ i)) (j': 'I_(p_ i')) :
(Rank j == Rank j' :> nat) = (i == i') && (j == j' :> nat).
Proof.
rewrite val_eqE /Rank -(can_eq sigK) !rankK.
case: (i =P i') => ii' /=; last by case: eqVneq => // -[].
by case: _ / ii' in j' *; rewrite eq_Tagged.
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 | eq_Rank | |
rankEsump : rank p = \sum_(i < n | (i < tag p)%N) p_ i + tagged p :> nat.
Proof.
pose sum p := \sum_(i < n | (i < tag p)%N) p_ i + tagged p.
rewrite -/(sum _); have sumlt : forall p, (sum p < \sum_i p_ i)%N.
rewrite /sum => -[/= i j].
rewrite [ltnRHS](bigID [pred i' : 'I__ | (i' < i)%N])/= ltn_add2l.
by rewrite (... | 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 | rankEsum | |
RankEsumi j : @Rank i j = \sum_(k < n | (k < i)%N) p_ k + j :> nat.
Proof. by rewrite /Rank rankEsum/=. 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 | RankEsum | |
rects : s = \sum_(i < n | (i < sig1 s)%N) p_ i + sig2 s :> nat.
Proof. by rewrite -[s]sigK rankEsum /= sigK. 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 | rect | |
eqRank(i0 j : nat) (li0 : (i0 < n)%N) (lj : (j < p_ (Ordinal li0))%N) :
(\sum_(i < n | (i < i0)%N) p_ i) + j = Rank (Ordinal lj) :> nat.
Proof. by rewrite RankEsum. 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 | eqRank | |
RecordisDuallyPreorder (d : disp_t) T of Equality T := {
le : rel T;
lt : rel T;
lt_def : forall x y, lt x y = (le x y) && ~~ (le y x);
gt_def : forall x y, lt y x = (le y x) && ~~ (le x y);
le_refl : reflexive le;
ge_refl : reflexive (fun x y => le y x);
le_trans : transitive ... | 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"
] | order/preorder.v | Record | |
comparable: rel T := fun (x y : T) => (x <= y) || (y <= x).
Local Notation "x >=< y" := (comparable x y) : order_scope.
Local Notation "x >< y" := (~~ (x >=< y)) : order_scope. | 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 | comparable | |
ge: simpl_rel T := [rel x y | 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"
] | order/preorder.v | ge | |
gt: simpl_rel T := [rel x y | 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"
] | order/preorder.v | gt | |
leif(x y : T) C : Prop := ((x <= y) * ((x == y) = C))%type. | 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 | leif | |
le_of_leifx y C (le_xy : @leif x y C) := le_xy.1 : le 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 | le_of_leif | |
lteif(x y : T) C := if C then x <= y else 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 | lteif | |
le_xor_gt(x y : T) :
T -> T -> T -> T -> bool -> bool -> Set :=
| LeNotGt of x <= y : le_xor_gt x y x x y y true false
| GtNotLe of y < x : le_xor_gt x y y y x x false true. | Variant | order | [
"From HB Require Import structures",
"From mathcomp Require Import 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_xor_gt | |
lt_xor_ge(x y : T) :
T -> T -> T -> T -> bool -> bool -> Set :=
| LtNotGe of x < y : lt_xor_ge x y x x y y false true
| GeNotLt of y <= x : lt_xor_ge x y y y x x true false. | Variant | order | [
"From HB Require Import structures",
"From mathcomp Require Import 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_xor_ge | |
min(x y : T) := if x < y then x else 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 | min | |
max(x y : T) := if x < y then y else 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"
] | order/preorder.v | max | |
compare(x y : T) :
T -> T -> T -> T ->
bool -> bool -> bool -> bool -> bool -> bool -> Set :=
| CompareLt of x < y : compare x y
x x y y false false false true false true
| CompareGt of y < x : compare x y
y y x x false false true false true false
| CompareEq of x = y : compare x y
x x x x true ... | Variant | order | [
"From HB Require Import structures",
"From mathcomp Require Import 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 | compare | |
incompare(x y : T) :
T -> T -> T -> T ->
bool -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> Set :=
| InCompareLt of x < y : incompare x y
x x y y false false false true false true true true
| InCompareGt of y < x : incompare x y
y y x x false false true false true false true true
| InCom... | Variant | order | [
"From HB Require Import structures",
"From mathcomp Require Import 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 | incompare | |
arg_min{I : finType} := @extremum T I le. | 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 | arg_min | |
arg_max{I : finType} := @extremum T I ge. | 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 | arg_max | |
min_funf g x := min (f x) (g 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"
] | order/preorder.v | min_fun | |
max_funf g x := max (f x) (g 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"
] | order/preorder.v | max_fun | |
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 | |
nondecreasing:= nondecreasing. | 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 | nondecreasing | |
min:= min. | 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 | min | |
max:= max. | 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 | max | |
leLHS:= (X in (X <= _)%O)%pattern. | 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 | leLHS | |
leRHS:= (X in (_ <= X)%O)%pattern. | 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 | leRHS | |
ltLHS:= (X in (X < _)%O)%pattern. | 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 | ltLHS | |
ltRHS:= (X in (_ < X)%O)%pattern. | 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 | ltRHS | |
le_of_leif: leif >-> is_true. | Coercion | order | [
"From HB Require Import structures",
"From mathcomp Require Import 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_of_leif | |
DefinitionFinPreorder d := { T of Finite T & Preorder d T }.
#[short(type="finBPreorderType")]
HB.structure Definition FinBPreorder d := { T of FinPreorder d T & hasBottom d T }.
#[short(type="finTPreorderType")]
HB.structure Definition FinTPreorder d := { T of FinPreorder d T & hasTop d T }.
#[short(type="finTBPreorde... | 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"
] | order/preorder.v | Definition | |
dualT : Type := 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 | dual | |
dual_display(d : disp_t) := {| d1 := d2 d; d2 := d1 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 | dual_display | |
dual_le:= (@le (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_le | |
dual_lt:= (@lt (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_lt | |
dual_comparable:= (@comparable (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_comparable | |
dual_ge:= (@ge (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_ge | |
dual_gt:= (@gt (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_gt |
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