phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
lt_Taggedr x (u : T' (tag x)) : (x < Tagged T' u) = (tagged x < u).	Proof. by case: x => [t v]/= in u *; rewrite ltEsig/= lexx/= tagged_asE. Qed.	Lemma	lt_Taggedr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "lexx", "ltEsig", "tagged_asE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0x (x : {t : T & T' t}) : Tagged T' (\bot : T' \bot) <= x.	Proof. by rewrite leEsig /= !le0x implybT. Qed.	Fact	le0x	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "bot", "leEsig" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
botEsig : \bot = Tagged T' (\bot : T' \bot).	Proof. by []. Qed.	Lemma	botEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "bot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lex1 (x : {t : T & T' t}) : x <= Tagged T' (\top : T' \top).	Proof. rewrite leEsig /=; case: comparableP (lex1 (tag x)) => //=. by case: x => //= x px x0; rewrite x0 in px *; rewrite tagged_asE lex1. Qed.	Fact	lex1	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "comparableP", "leEsig", "tagged_asE", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
topEsig : \top = Tagged T' (\top : T' \top).	Proof. by []. Qed.	Lemma	topEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
total : total (<=%O : rel {t : T & T' t}).	Proof. move=> x y; rewrite !leEsig; case: (ltgtP (tag x) (tag y)) => //=. case: x y => [x x'] [y y']/= eqxy; elim: _ /eqxy in y' *. by rewrite !tagged_asE le_total. Qed.	Fact	total	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "T'", "leEsig", "le_total", "ltgtP", "rel", "tagged_asE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEsig	:= @leEsig.	Definition	leEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltEsig	:= @ltEsig.	Definition	ltEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_Taggedl	:= @le_Taggedl.	Definition	le_Taggedl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_Taggedl	:= @lt_Taggedl.	Definition	lt_Taggedl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_Taggedr	:= @le_Taggedr.	Definition	le_Taggedr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_Taggedr	:= @lt_Taggedr.	Definition	lt_Taggedr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
topEsig	:= @topEsig.	Definition	topEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
botEsig	:= @botEsig.	Definition	botEsig	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
anti : antisymmetric (@le disp1 disp2 disp2 T1 T2).	Proof. by rewrite /le => -[x x'] [y y'] /=; case: comparableP => //= -> /le_anti->. Qed.	Fact	anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "le", "le_anti" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
total : total (<=%O : rel (T1 * T2)).	Proof. by move=> x y; rewrite !leEprodlexi; case: ltgtP => //= _; exact: le_total. Qed.	Fact	total	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leEprodlexi", "le_total", "ltgtP", "rel" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodlexi T1 T2	:= T1 *lexi[lexi_display disp1 disp2] T2.	Let	prodlexi	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lexi_display" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq	:= (type disp').	Notation	seq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
anti : antisymmetric (@le disp disp' T).	Proof. by elim=> [\|x s ihs] [\|y s'] //=; rewrite andbACA => /andP[/le_anti-> /ihs->]. Qed.	Fact	anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "le", "le_anti" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet s1 s2	:= match s1, s2 with \| x1 :: s1', x2 :: s2' => (x1 `&` x2) :: meet s1' s2' \| _, _ => [::] end.	Fixpoint	meet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexI s1 s2 s3 : (s1 <= meet s2 s3) = (s1 <= s2) && (s1 <= s3).	Proof. elim: s1 s2 s3 => [\|x s1 IHs1] [\|y s2] [\|z s3] //=; first by rewrite andbF. by rewrite leEseq lexI IHs1 andbACA. Qed.	Fact	lexI	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leEseq", "meet", "s1", "s2", "s3" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEseq s1 s2 : s1 `&` s2 = [seq x.1 `&` x.2 \| x <- zip s1 s2].	Proof. by elim: s1 s2 => [\|x s1 ihs1] [\|y s2]//=; rewrite -ihs1. Qed.	Lemma	meetEseq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2", "seq", "zip" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_cons x1 s1 x2 s2 : (x1 :: s1 : seq T) `&` (x2 :: s2) = (x1 `&` x2) :: s1 `&` s2.	Proof. by []. Qed.	Lemma	meet_cons	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join s1 s2	:= match s1, s2 with \| [::], _ => s2 \| _, [::] => s1 \| x1 :: s1', x2 :: s2' => (x1 `\|` x2) :: join s1' s2' end.	Fixpoint	join	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leUx s1 s2 s3 : (join s1 s2 <= s3) = (s1 <= s3) && (s2 <= s3).	Proof. elim : s1 s2 s3 => [\|x s1 IHs1] [\|y s2] [\|z s3] //=; first by rewrite andbT. by rewrite leEseq leUx IHs1 andbACA. Qed.	Fact	leUx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "join", "leEseq", "s1", "s2", "s3" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEseq s1 s2 : s1 `\|` s2 = match s1, s2 with \| [::], _ => s2 \| _, [::] => s1 \| x1 :: s1', x2 :: s2' => (x1 `\|` x2) :: ((s1' : seq _) `\|` s2') end.	Proof. by case: s1. Qed.	Lemma	joinEseq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join_cons x1 s1 x2 s2 : (x1 :: s1 : seq T) `\|` (x2 :: s2) = (x1 `\|` x2) :: s1 `\|` s2.	Proof. by []. Qed.	Lemma	join_cons	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : left_distributive (@meet T) (@join T).	Proof. by elim=> [\|? ? ih] [\|? ?] [\|? ?] //=; rewrite meetUl ih. Qed.	Fact	meetUl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "join", "meet" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEseq	:= @meetEseq.	Definition	meetEseq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_cons	:= @meet_cons.	Definition	meet_cons	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEseq	:= @joinEseq.	Definition	joinEseq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
anti: antisymmetric (@le disp disp' T).	Proof. move=> x y /andP []; elim: x y => [\|x sx ih] [\|y sy] //=. by case: comparableP => //= -> lesxsy /(ih _ lesxsy) ->. Qed.	Fact	anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "le" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neqhead_lexiE (x y : T) s1 s2 : x != y -> (x :: s1 <= y :: s2 :> seq _) = (x < y).	Proof. by rewrite lexi_cons; case: comparableP. Qed.	Lemma	neqhead_lexiE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "lexi_cons", "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neqhead_ltxiE (x y : T) s1 s2 : x != y -> (x :: s1 < y :: s2 :> seq _) = (x < y).	Proof. by rewrite ltxi_cons; case: (comparableP x y). Qed.	Lemma	neqhead_ltxiE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "ltxi_cons", "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
total : total (<=%O : rel (seq T)).	Proof. by elim=> [\|x1 s1 ihs1] [\|x2 s2]//=; rewrite !lexi_cons; case: ltgtP => /=. Qed.	Fact	total	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lexi_cons", "ltgtP", "rel", "s1", "s2", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neqhead_lexiE	:= @neqhead_lexiE.	Definition	neqhead_lexiE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
neqhead_ltxiE	:= @neqhead_ltxiE.	Definition	neqhead_ltxiE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"n .-tuple"	:= (type disp' n) : type_scope.	Notation	n .-tuple	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet t1 t2 : n.-tuple T	:= [tuple tnth t1 i `&` tnth t2 i \| i < n].	Definition	meet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexI t1 t2 t3 : (t1 <= meet t2 t3) = (t1 <= t2) && (t1 <= t3).	Proof. rewrite !leEtprod; apply/forallP/andP => [H\|[Ht12 Ht13] i]; last first. by rewrite tnth_mktuple lexI (forallP Ht12) (forallP Ht13). by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple lexI => /andP[]. Qed.	Fact	lexI	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "forallP", "last", "leEtprod", "meet", "split", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_meet t1 t2 i : tnth (t1 `&` t2) i = tnth t1 i `&` tnth t2 i.	Proof. exact: tnth_mktuple. Qed.	Lemma	tnth_meet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEtprod t1 t2 : t1 `&` t2 = [tuple tnth t1 i `&` tnth t2 i \| i < n].	Proof. by []. Qed.	Lemma	meetEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join t1 t2 : n.-tuple T	:= [tuple tnth t1 i `\|` tnth t2 i \| i < n].	Definition	join	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leUx t1 t2 t3 : (join t1 t2 <= t3) = (t1 <= t3) && (t2 <= t3).	Proof. rewrite !leEtprod; apply/forallP/andP => [H\|[Ht13 Ht23] i]; last first. by rewrite tnth_mktuple leUx (forallP Ht13) (forallP Ht23). by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple leUx => /andP[]. Qed.	Fact	leUx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "forallP", "join", "last", "leEtprod", "split", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_join t1 t2 i : tnth (t1 `\|` t2) i = tnth t1 i `\|` tnth t2 i.	Proof. exact: tnth_mktuple. Qed.	Lemma	tnth_join	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEtprod t1 t2 : t1 `\|` t2 = [tuple tnth t1 i `\|` tnth t2 i \| i < n].	Proof. by []. Qed.	Lemma	joinEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : left_distributive (@meet n T) (@join n T).	Proof. by move=> t1 t2 t3; apply: eq_from_tnth => i; rewrite !tnth_mktuple meetUl. Qed.	Fact	meetUl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "eq_from_tnth", "join", "meet", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcompl t1 t2 t3	:= [tuple rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) \| i < n].	Definition	rcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPmeet x y z : ((x `&` y) `\|` z) `&` rcompl x y z = x `&` y.	Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPmeet. Qed.	Fact	rcomplPmeet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "eq_from_tnth", "rcompl", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPjoin x y z : ((y `\|` x) `&` z) `\|` rcompl x y z = y `\|` x.	Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPjoin. Qed.	Fact	rcomplPjoin	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "eq_from_tnth", "rcompl", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_rcompl t1 t2 t3 i : tnth (Order.rcompl t1 t2 t3) i = Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i).	Proof. exact: tnth_mktuple. Qed.	Lemma	tnth_rcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "rcompl", "tnth", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplEtprod t1 t2 t3 : Order.rcompl t1 t2 t3 = [tuple Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) \| i < n].	Proof. by []. Qed.	Lemma	rcomplEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "rcompl", "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diff t1 t2 : n.-tuple T	:= [tuple tnth t1 i `\` tnth t2 i \| i < n].	Definition	diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffErcompl t1 t2 : diff t1 t2 = rcompl \bot t1 t2.	Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple diffErcompl. Qed.	Fact	diffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "diff", "eq_from_tnth", "rcompl", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_diff t1 t2 i : tnth (diff t1 t2) i = tnth t1 i `\` tnth t2 i.	Proof. exact: tnth_mktuple. Qed.	Lemma	tnth_diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diff", "tnth", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffEtprod t1 t2 : t1 `\` t2 = [tuple tnth t1 i `\` tnth t2 i \| i < n].	Proof. by []. Qed.	Lemma	diffEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiff t1 t2 : n.-tuple T	:= [tuple Order.codiff (tnth t1 i) (tnth t2 i) \| i < n].	Definition	codiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffErcompl t1 t2 : codiff t1 t2 = rcompl t1 \top t2.	Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple codiffErcompl. Qed.	Fact	codiffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "codiff", "eq_from_tnth", "rcompl", "tnth_mktuple", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_codiff t1 t2 i : tnth (Order.codiff t1 t2) i = Order.codiff (tnth t1 i) (tnth t2 i).	Proof. exact: tnth_mktuple. Qed.	Lemma	tnth_codiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff", "tnth", "tnth_mktuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffEtprod t1 t2 : Order.codiff t1 t2 = [tuple Order.codiff (tnth t1 i) (tnth t2 i) \| i < n].	Proof. by []. Qed.	Lemma	codiffEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff", "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl t : n.-tuple T	:= map_tuple compl t.	Definition	compl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "map_tuple", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEdiff t : compl t = (\top : n.-tuple T) `\` t.	Proof. by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEdiff. Qed.	Fact	complEdiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "compl", "eq_from_tnth", "tnth_map", "tnth_mktuple", "top", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEcodiff t : compl t = codiff (\bot : n.-tuple T) t.	Proof. by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEcodiff. Qed.	Fact	complEcodiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "codiff", "compl", "eq_from_tnth", "tnth_map", "tnth_mktuple", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_compl t i : tnth (~` t) i = ~` tnth t i.	Proof. by rewrite tnth_map. Qed.	Lemma	tnth_compl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "tnth", "tnth_map" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEtprod t : ~` t = map_tuple Order.compl t.	Proof. by []. Qed.	Lemma	complEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "compl", "map_tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_meet	:= @tnth_meet.	Definition	tnth_meet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEtprod	:= @meetEtprod.	Definition	meetEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_join	:= @tnth_join.	Definition	tnth_join	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEtprod	:= @joinEtprod.	Definition	joinEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_rcompl	:= @tnth_rcompl.	Definition	tnth_rcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplEtprod	:= @rcomplEtprod.	Definition	rcomplEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_diff	:= @tnth_diff.	Definition	tnth_diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffEtprod	:= @diffEtprod.	Definition	diffEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_codiff	:= @tnth_codiff.	Definition	tnth_codiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffEtprod	:= @codiffEtprod.	Definition	codiffEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tnth_compl	:= @tnth_compl.	Definition	tnth_compl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEtprod	:= @complEtprod.	Definition	complEtprod	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexi_tupleP n 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) => [x1 {}t1] [x2 {}t2]. rewrite [_ <= _]lexi_cons; apply: (iffP idP) => [\|[k leif_xt12]]. case: comparableP => //= [ltx12 _\|-> /IHn[k kP]]. exists ord0 => i; rewrite leqn0 =...	Lemma	lexi_tupleP	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "comparableP", "eqxx", "leifP", "leqn0", "lexi_cons", "lexx", "lift", "lt_geF", "ltnS", "nat", "ord0", "ord_inj", "tnth", "tnth0", "tnthS", "tuple", "tuple0", "tupleP", "unliftP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxi_tupleP n 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 {}t2]. rewrite [_ < _]ltxi_cons; apply: (iffP idP) => [\|[k leif_xt12]]. case: (comparableP x1 x2) => //= [ltx12 _\|-> /IHn[k kP]]. exists ord0 => i; rewrite leqn0 => ...	Lemma	ltxi_tupleP	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "comparableP", "eqxx", "leifP", "leqn0", "lexx", "lift", "lt_geF", "ltnS", "ltxi_cons", "ltxx", "nat", "ord0", "ord_inj", "tnth", "tnth0", "tnthS", "tuple", "tuple0", "tupleP", "unliftP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxi_tuplePlt n 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_leif (kP _ _)) ?eqxx ?leqnn. by have /eqTleif->// := kP i (ltnW i_lt); rewrite ltn_eqF. by exists k => i; case: ltngtP => //= [/kP-> _\|/ord_inj-> _]; apply/leifP. Qed.	Lemma	ltxi_tuplePlt	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "eqTleif", "eqxx", "last", "leifP", "leqnn", "lt_leif", "ltnW", "ltn_eqF", "ltngtP", "ltxi_tupleP", "ord_inj", "tnth", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexi_tupleP	:= @lexi_tupleP.	Definition	lexi_tupleP	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxi_tupleP	:= @ltxi_tupleP.	Definition	ltxi_tupleP	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltxi_tuplePlt	:= @ltxi_tuplePlt.	Definition	ltxi_tuplePlt	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'subset' T }"	:= (type disp T).	Notation	{ 'subset' T }	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setKUC B A : A :&: (A :\|: B) = A.	Proof. by rewrite setUC setKU. Qed.	Lemma	setKUC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "setKU", "setUC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setKIC B A : A :\|: (A :&: B) = A.	Proof. by rewrite setIC setKI. Qed.	Lemma	setKIC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "setIC", "setKI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_anti : antisymmetric (fun A B => A \subset B).	Proof. by move=> A B ABA; apply/eqP; rewrite eqEsubset. Qed.	Fact	le_anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "eqEsubset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIDv A B : B :&: (A :\: B) = set0.	Proof. apply/eqP; rewrite -subset0; apply/subsetP => x. by rewrite !inE => /and3P[->]. Qed.	Lemma	setIDv	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "inE", "set0", "subset0", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setTDsym A : ~: A = setT :\: A.	Proof. by rewrite setTD. Qed.	Lemma	setTDsym	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "setT", "setTD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEsubset A B : A `&` B = A :&: B.	Proof. by []. Qed.	Lemma	meetEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEsubset A B : A `\|` B = A :\|: B.	Proof. by []. Qed.	Lemma	joinEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
botEsubset : \bot = set0 :> {subset T}.	Proof. by []. Qed.	Lemma	botEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "set0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
topEsubset : \top = setT :> {subset T}.	Proof. by []. Qed.	Lemma	topEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "setT", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subEsubset A B : A `\` B = A :\: B.	Proof. by []. Qed.	Lemma	subEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEsubset A : ~` A = ~: A.	Proof. by []. Qed.	Lemma	complEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEsubset	:= @meetEsubset.	Definition	meetEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinEsubset	:= @joinEsubset.	Definition	joinEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
botEsubset	:= @botEsubset.	Definition	botEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
topEsubset	:= @topEsubset.	Definition	topEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subEsubset	:= @subEsubset.	Definition	subEsubset	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_Taggedr x (u : T' (tag x)) : (x < Tagged T' u) = (tagged x < u).

Proof. by case: x => [t v]/= in u *; rewrite ltEsig/= lexx/= tagged_asE. Qed.

Lemma

lt_Taggedr

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "lexx", "ltEsig", "tagged_asE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le0x (x : {t : T & T' t}) : Tagged T' (\bot : T' \bot) <= x.

Proof. by rewrite leEsig /= !le0x implybT. Qed.

Fact

le0x

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "bot", "leEsig" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

botEsig : \bot = Tagged T' (\bot : T' \bot).

Proof. by []. Qed.

Lemma

botEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "bot" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lex1 (x : {t : T & T' t}) : x <= Tagged T' (\top : T' \top).

Proof. rewrite leEsig /=; case: comparableP (lex1 (tag x)) => //=. by case: x => //= x px x0; rewrite x0 in px *; rewrite tagged_asE lex1. Qed.

Fact

lex1

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "comparableP", "leEsig", "tagged_asE", "top" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

topEsig : \top = Tagged T' (\top : T' \top).

Proof. by []. Qed.

Lemma

topEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "top" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

total : total (<=%O : rel {t : T & T' t}).

Proof. move=> x y; rewrite !leEsig; case: (ltgtP (tag x) (tag y)) => //=. case: x y => [x x'] [y y']/= eqxy; elim: _ /eqxy in y' *. by rewrite !tagged_asE le_total. Qed.

Fact

total

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "T'", "leEsig", "le_total", "ltgtP", "rel", "tagged_asE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEsig

:= @leEsig.

Definition

leEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltEsig

:= @ltEsig.

Definition

ltEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_Taggedl

:= @le_Taggedl.

Definition

le_Taggedl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_Taggedl

:= @lt_Taggedl.

Definition

lt_Taggedl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_Taggedr

:= @le_Taggedr.

Definition

le_Taggedr

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_Taggedr

:= @lt_Taggedr.

Definition

lt_Taggedr

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

topEsig

:= @topEsig.

Definition

topEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

botEsig

:= @botEsig.

Definition

botEsig

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

anti : antisymmetric (@le disp1 disp2 disp2 T1 T2).

Proof. by rewrite /le => -[x x'] [y y'] /=; case: comparableP => //= -> /le_anti->. Qed.

Fact

anti

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "comparableP", "le", "le_anti" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

total : total (<=%O : rel (T1 * T2)).

Proof. by move=> x y; rewrite !leEprodlexi; case: ltgtP => //= _; exact: le_total. Qed.

Fact

total

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "leEprodlexi", "le_total", "ltgtP", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodlexi T1 T2

:= T1 *lexi[lexi_display disp1 disp2] T2.

Let

prodlexi

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "lexi_display" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq

:= (type disp').

Notation

seq

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "type" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

anti : antisymmetric (@le disp disp' T).

Proof. by elim=> [|x s ihs] [|y s'] //=; rewrite andbACA => /andP[/le_anti-> /ihs->]. Qed.

Fact

anti

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "le", "le_anti" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet s1 s2

:= match s1, s2 with | x1 :: s1', x2 :: s2' => (x1 `&` x2) :: meet s1' s2' | _, _ => [::] end.

Fixpoint

meet

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexI s1 s2 s3 : (s1 <= meet s2 s3) = (s1 <= s2) && (s1 <= s3).

Proof. elim: s1 s2 s3 => [|x s1 IHs1] [|y s2] [|z s3] //=; first by rewrite andbF. by rewrite leEseq lexI IHs1 andbACA. Qed.

Fact

lexI

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "leEseq", "meet", "s1", "s2", "s3" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEseq s1 s2 : s1 `&` s2 = [seq x.1 `&` x.2 | x <- zip s1 s2].

Proof. by elim: s1 s2 => [|x s1 ihs1] [|y s2]//=; rewrite -ihs1. Qed.

Lemma

meetEseq

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2", "seq", "zip" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_cons x1 s1 x2 s2 : (x1 :: s1 : seq T) `&` (x2 :: s2) = (x1 `&` x2) :: s1 `&` s2.

Proof. by []. Qed.

Lemma

meet_cons

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join s1 s2

:= match s1, s2 with | [::], _ => s2 | _, [::] => s1 | x1 :: s1', x2 :: s2' => (x1 `|` x2) :: join s1' s2' end.

Fixpoint

join

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leUx s1 s2 s3 : (join s1 s2 <= s3) = (s1 <= s3) && (s2 <= s3).

Proof. elim : s1 s2 s3 => [|x s1 IHs1] [|y s2] [|z s3] //=; first by rewrite andbT. by rewrite leEseq leUx IHs1 andbACA. Qed.

Fact

leUx

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "join", "leEseq", "s1", "s2", "s3" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Proof. by case: s1. Qed.

Lemma

joinEseq

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join_cons x1 s1 x2 s2 : (x1 :: s1 : seq T) `|` (x2 :: s2) = (x1 `|` x2) :: s1 `|` s2.

Proof. by []. Qed.

Lemma

join_cons

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetUl : left_distributive (@meet T) (@join T).

Proof. by elim=> [|? ? ih] [|? ?] [|? ?] //=; rewrite meetUl ih. Qed.

Fact

meetUl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "join", "meet" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEseq

:= @meetEseq.

Definition

meetEseq

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_cons

:= @meet_cons.

Definition

meet_cons

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinEseq

:= @joinEseq.

Definition

joinEseq

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

anti: antisymmetric (@le disp disp' T).

Proof. move=> x y /andP []; elim: x y => [|x sx ih] [|y sy] //=. by case: comparableP => //= -> lesxsy /(ih _ lesxsy) ->. Qed.

Fact

anti

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "comparableP", "le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neqhead_lexiE (x y : T) s1 s2 : x != y -> (x :: s1 <= y :: s2 :> seq _) = (x < y).

Proof. by rewrite lexi_cons; case: comparableP. Qed.

Lemma

neqhead_lexiE

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "comparableP", "lexi_cons", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neqhead_ltxiE (x y : T) s1 s2 : x != y -> (x :: s1 < y :: s2 :> seq _) = (x < y).

Proof. by rewrite ltxi_cons; case: (comparableP x y). Qed.

Lemma

neqhead_ltxiE

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "comparableP", "ltxi_cons", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

total : total (<=%O : rel (seq T)).

Proof. by elim=> [|x1 s1 ihs1] [|x2 s2]//=; rewrite !lexi_cons; case: ltgtP => /=. Qed.

Fact

total

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "lexi_cons", "ltgtP", "rel", "s1", "s2", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neqhead_lexiE

:= @neqhead_lexiE.

Definition

neqhead_lexiE

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

neqhead_ltxiE

:= @neqhead_ltxiE.

Definition

neqhead_ltxiE

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"n .-tuple"

:= (type disp' n) : type_scope.

Notation

n .-tuple

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "type" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet t1 t2 : n.-tuple T

:= [tuple tnth t1 i `&` tnth t2 i | i < n].

Definition

meet

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexI t1 t2 t3 : (t1 <= meet t2 t3) = (t1 <= t2) && (t1 <= t3).

Proof. rewrite !leEtprod; apply/forallP/andP => [H|[Ht12 Ht13] i]; last first. by rewrite tnth_mktuple lexI (forallP Ht12) (forallP Ht13). by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple lexI => /andP[]. Qed.

Fact

lexI

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "forallP", "last", "leEtprod", "meet", "split", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_meet t1 t2 i : tnth (t1 `&` t2) i = tnth t1 i `&` tnth t2 i.

Proof. exact: tnth_mktuple. Qed.

Lemma

tnth_meet

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEtprod t1 t2 : t1 `&` t2 = [tuple tnth t1 i `&` tnth t2 i | i < n].

Proof. by []. Qed.

Lemma

meetEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join t1 t2 : n.-tuple T

:= [tuple tnth t1 i `|` tnth t2 i | i < n].

Definition

join

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leUx t1 t2 t3 : (join t1 t2 <= t3) = (t1 <= t3) && (t2 <= t3).

Proof. rewrite !leEtprod; apply/forallP/andP => [H|[Ht13 Ht23] i]; last first. by rewrite tnth_mktuple leUx (forallP Ht13) (forallP Ht23). by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple leUx => /andP[]. Qed.

Fact

leUx

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "forallP", "join", "last", "leEtprod", "split", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_join t1 t2 i : tnth (t1 `|` t2) i = tnth t1 i `|` tnth t2 i.

Proof. exact: tnth_mktuple. Qed.

Lemma

tnth_join

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinEtprod t1 t2 : t1 `|` t2 = [tuple tnth t1 i `|` tnth t2 i | i < n].

Proof. by []. Qed.

Lemma

joinEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetUl : left_distributive (@meet n T) (@join n T).

Proof. by move=> t1 t2 t3; apply: eq_from_tnth => i; rewrite !tnth_mktuple meetUl. Qed.

Fact

meetUl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "eq_from_tnth", "join", "meet", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcompl t1 t2 t3

:= [tuple rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) | i < n].

Definition

rcompl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPmeet. Qed.

Fact

rcomplPmeet

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "eq_from_tnth", "rcompl", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPjoin. Qed.

Fact

rcomplPjoin

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "eq_from_tnth", "rcompl", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_rcompl t1 t2 t3 i : tnth (Order.rcompl t1 t2 t3) i = Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i).

Proof. exact: tnth_mktuple. Qed.

Lemma

tnth_rcompl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "rcompl", "tnth", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplEtprod t1 t2 t3 : Order.rcompl t1 t2 t3 = [tuple Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) | i < n].

Proof. by []. Qed.

Lemma

rcomplEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "rcompl", "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diff t1 t2 : n.-tuple T

:= [tuple tnth t1 i `\` tnth t2 i | i < n].

Definition

diff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffErcompl t1 t2 : diff t1 t2 = rcompl \bot t1 t2.

Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple diffErcompl. Qed.

Fact

diffErcompl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "bot", "diff", "eq_from_tnth", "rcompl", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_diff t1 t2 i : tnth (diff t1 t2) i = tnth t1 i `\` tnth t2 i.

Proof. exact: tnth_mktuple. Qed.

Lemma

tnth_diff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "diff", "tnth", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffEtprod t1 t2 : t1 `\` t2 = [tuple tnth t1 i `\` tnth t2 i | i < n].

Proof. by []. Qed.

Lemma

diffEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiff t1 t2 : n.-tuple T

:= [tuple Order.codiff (tnth t1 i) (tnth t2 i) | i < n].

Definition

codiff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffErcompl t1 t2 : codiff t1 t2 = rcompl t1 \top t2.

Proof. by apply: eq_from_tnth => i; rewrite !tnth_mktuple codiffErcompl. Qed.

Fact

codiffErcompl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "codiff", "eq_from_tnth", "rcompl", "tnth_mktuple", "top" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_codiff t1 t2 i : tnth (Order.codiff t1 t2) i = Order.codiff (tnth t1 i) (tnth t2 i).

Proof. exact: tnth_mktuple. Qed.

Lemma

tnth_codiff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "codiff", "tnth", "tnth_mktuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffEtprod t1 t2 : Order.codiff t1 t2 = [tuple Order.codiff (tnth t1 i) (tnth t2 i) | i < n].

Proof. by []. Qed.

Lemma

codiffEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "codiff", "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl t : n.-tuple T

:= map_tuple compl t.

Definition

compl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "map_tuple", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEdiff t : compl t = (\top : n.-tuple T) `\` t.

Proof. by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEdiff. Qed.

Fact

complEdiff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "compl", "eq_from_tnth", "tnth_map", "tnth_mktuple", "top", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEcodiff t : compl t = codiff (\bot : n.-tuple T) t.

Proof. by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEcodiff. Qed.

Fact

complEcodiff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "bot", "codiff", "compl", "eq_from_tnth", "tnth_map", "tnth_mktuple", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_compl t i : tnth (~` t) i = ~` tnth t i.

Proof. by rewrite tnth_map. Qed.

Lemma

tnth_compl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "tnth", "tnth_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEtprod t : ~` t = map_tuple Order.compl t.

Proof. by []. Qed.

Lemma

complEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "compl", "map_tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_meet

:= @tnth_meet.

Definition

tnth_meet

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEtprod

:= @meetEtprod.

Definition

meetEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_join

:= @tnth_join.

Definition

tnth_join

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinEtprod

:= @joinEtprod.

Definition

joinEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_rcompl

:= @tnth_rcompl.

Definition

tnth_rcompl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcomplEtprod

:= @rcomplEtprod.

Definition

rcomplEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_diff

:= @tnth_diff.

Definition

tnth_diff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffEtprod

:= @diffEtprod.

Definition

diffEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_codiff

:= @tnth_codiff.

Definition

tnth_codiff

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffEtprod

:= @codiffEtprod.

Definition

codiffEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tnth_compl

:= @tnth_compl.

Definition

tnth_compl

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEtprod

:= @complEtprod.

Definition

complEtprod

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexi_tupleP n 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) => [x1 {}t1] [x2 {}t2]. rewrite [_ <= _]lexi_cons; apply: (iffP idP) => [|[k leif_xt12]]. case: comparableP => //= [ltx12 _|-> /IHn[k kP]]. exists ord0 => i; rewrite leqn0 =...

Lemma

lexi_tupleP

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "comparableP", "eqxx", "leifP", "leqn0", "lexi_cons", "lexx", "lift", "lt_geF", "ltnS", "nat", "ord0", "ord_inj", "tnth", "tnth0", "tnthS", "tuple", "tuple0", "tupleP", "unliftP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltxi_tupleP n 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 {}t2]. rewrite [_ < _]ltxi_cons; apply: (iffP idP) => [|[k leif_xt12]]. case: (comparableP x1 x2) => //= [ltx12 _|-> /IHn[k kP]]. exists ord0 => i; rewrite leqn0 => ...

Lemma

ltxi_tupleP

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "comparableP", "eqxx", "leifP", "leqn0", "lexx", "lift", "lt_geF", "ltnS", "ltxi_cons", "ltxx", "nat", "ord0", "ord_inj", "tnth", "tnth0", "tnthS", "tuple", "tuple0", "tupleP", "unliftP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltxi_tuplePlt n 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_leif (kP _ _)) ?eqxx ?leqnn. by have /eqTleif->// := kP i (ltnW i_lt); rewrite ltn_eqF. by exists k => i; case: ltngtP => //= [/kP-> _|/ord_inj-> _]; apply/leifP. Qed.

Lemma

ltxi_tuplePlt

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "eqTleif", "eqxx", "last", "leifP", "leqnn", "lt_leif", "ltnW", "ltn_eqF", "ltngtP", "ltxi_tupleP", "ord_inj", "tnth", "tuple" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexi_tupleP

:= @lexi_tupleP.

Definition

lexi_tupleP

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltxi_tupleP

:= @ltxi_tupleP.

Definition

ltxi_tupleP

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltxi_tuplePlt

:= @ltxi_tuplePlt.

Definition

ltxi_tuplePlt

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'subset' T }"

:= (type disp T).

Notation

{ 'subset' T }

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "type" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setKUC B A : A :&: (A :|: B) = A.

Proof. by rewrite setUC setKU. Qed.

Lemma

setKUC

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "setKU", "setUC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setKIC B A : A :|: (A :&: B) = A.

Proof. by rewrite setIC setKI. Qed.

Lemma

setKIC

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "setIC", "setKI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_anti : antisymmetric (fun A B => A \subset B).

Proof. by move=> A B ABA; apply/eqP; rewrite eqEsubset. Qed.

Fact

le_anti

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "eqEsubset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIDv A B : B :&: (A :\: B) = set0.

Proof. apply/eqP; rewrite -subset0; apply/subsetP => x. by rewrite !inE => /and3P[->]. Qed.

Lemma

setIDv

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "apply", "inE", "set0", "subset0", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setTDsym A : ~: A = setT :\: A.

Proof. by rewrite setTD. Qed.

Lemma

setTDsym

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "setT", "setTD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEsubset A B : A `&` B = A :&: B.

Proof. by []. Qed.

Lemma

meetEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinEsubset A B : A `|` B = A :|: B.

Proof. by []. Qed.

Lemma

joinEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

botEsubset : \bot = set0 :> {subset T}.

Proof. by []. Qed.

Lemma

botEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "bot", "set0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

topEsubset : \top = setT :> {subset T}.

Proof. by []. Qed.

Lemma

topEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[ "setT", "top" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subEsubset A B : A `\` B = A :\: B.

Proof. by []. Qed.

Lemma

subEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEsubset A : ~` A = ~: A.

Proof. by []. Qed.

Lemma

complEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEsubset

:= @meetEsubset.

Definition

meetEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinEsubset

:= @joinEsubset.

Definition

joinEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

botEsubset

:= @botEsubset.

Definition

botEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

topEsubset

:= @topEsubset.

Definition

topEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subEsubset

:= @subEsubset.

Definition

subEsubset

order

order/order.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d