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	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
to_dirr (B : {set gT}) (phi : 'CF(B)) (i : Iirr B) : dIirr B	:= ('[phi, 'chi_i] < 0, i).	Definition	to_dirr	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "Iirr", "dIirr", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
of_irr (B : {set gT}) (i : dIirr B) : Iirr B	:= i.2.	Definition	of_irr	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "Iirr", "dIirr", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
irr_constt_to_dirr (phi: 'CF(G)) i : phi \in 'Z[irr G] -> (i \in irr_constt phi) = (to_dirr phi i \in dirr_constt phi).	Proof. move=> Zphi; rewrite irr_consttE dirr_consttE cfdotZr rmorph_sign /=. by rewrite -real_normrEsign ?normr_gt0 ?Rreal_int // Cint_cfdot_vchar_irr. Qed.	Lemma	irr_constt_to_dirr	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "Cint_cfdot_vchar_irr", "Rreal_int", "cfdotZr", "dirr_constt", "dirr_consttE", "irr", "irr_constt", "irr_consttE", "normr_gt0", "real_normrEsign", "rmorph_sign", "to_dirr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
to_dirrK (phi: 'CF(G)) : cancel (to_dirr phi) (@of_irr G).	Proof. by []. Qed.	Lemma	to_dirrK	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "of_irr", "to_dirr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
of_irrK (phi: 'CF(G)) : {in dirr_constt phi, cancel (@of_irr G) (to_dirr phi)}.	Proof. case=> b i; rewrite dirr_consttE cfdotZr rmorph_sign /= /to_dirr mulr_sign. by rewrite fun_if oppr_gt0; case: b => [\|/ltW/le_gtF] ->. Qed.	Lemma	of_irrK	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "cfdotZr", "dirr_constt", "dirr_consttE", "le_gtF", "ltW", "mulr_sign", "of_irr", "oppr_gt0", "rmorph_sign", "to_dirr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfdot_todirrE (phi: 'CF(G)) i (phi_i := dchi (to_dirr phi i)) : '[phi, phi_i] : phi_i = '[phi, 'chi_i] : 'chi_i.	Proof. by rewrite cfdotZr rmorph_sign mulrC -scalerA signrZK. Qed.	Lemma	cfdot_todirrE	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "cfdotZr", "dchi", "mulrC", "rmorph_sign", "scalerA", "signrZK", "to_dirr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfun_sum_dconstt (phi : 'CF(G)) : phi \in 'Z[irr G] -> phi = \sum_(i in dirr_constt phi) '[phi, dchi i] *: dchi i.	Proof. move=> PiZ; rewrite [LHS]cfun_sum_constt. rewrite (reindex (to_dirr phi))=> [\| /=]. by exists (@of_irr _)=> //; apply: of_irrK . by apply: eq_big => i; rewrite ?irr_constt_to_dirr // cfdot_todirrE. Qed.	Lemma	cfun_sum_dconstt	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "apply", "cfdot_todirrE", "cfun_sum_constt", "dchi", "dirr_constt", "eq_big", "irr", "irr_constt_to_dirr", "of_irr", "of_irrK", "reindex", "to_dirr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cnorm_dconstt (phi : 'CF(G)) : phi \in 'Z[irr G] -> '[phi] = \sum_(i in dirr_constt phi) '[phi, dchi i] ^+ 2.	Proof. move=> PiZ; rewrite {1 2}(cfun_sum_dconstt PiZ). rewrite cfdot_suml; apply: eq_bigr=> i IiD. rewrite cfdot_sumr (bigD1 i) //= big1 ?addr0 => [j /andP [JiD IdJ]\|]. rewrite cfdotZr cfdotZl cfdot_dchi eq_sym (negPf IdJ) -natrB ?mulr0 //. by rewrite (negPf (contraNneq _ (dirr_constt_oppl JiD))) => // <-. rewrite...	Lemma	cnorm_dconstt	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "Cnat_dirr", "addr0", "apply", "aut_natr", "big1", "bigD1", "cfdotZl", "cfdotZr", "cfdot_dchi", "cfdot_suml", "cfdot_sumr", "cfun_sum_dconstt", "contraNneq", "dchi", "dirr_constt", "dirr_constt_oppl", "eq_bigr", "eq_sym", "eqxx", "irr", "mulr0", "mulr1", "natrB", "ndirr...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dirr_small_norm (phi : 'CF(G)) n : phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N -> [/\ #\|dirr_constt phi\| = n, dirr_constt phi :&: dirr_constt (- phi) = set0 & phi = \sum_(i in dirr_constt phi) dchi i].	Proof. move=> PiZ Pln; rewrite ltnNge -leC_nat => Nl4. suffices Fd i: i \in dirr_constt phi -> '[phi, dchi i] = 1. split; last 2 [by apply/setP=> u; rewrite !inE cfdotNl oppr_gt0 lt_asym]. apply/eqP; rewrite -eqC_nat -sumr_const -Pln (cnorm_dconstt PiZ). by apply/eqP/eq_bigr=> i Hi; rewrite Fd // expr1n. re...	Lemma	dirr_small_norm	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "Cnat_dirr", "apply", "bigD1", "cfdotNl", "cfun_sum_dconstt", "cnorm_dconstt", "contraNeq", "dchi", "dirr_constt", "dirr_consttE", "eqC_nat", "eq_bigr", "eq_sym", "expr1n", "exprn_ge0", "inE", "irr", "last", "leC_nat", "leq_sqr", "ler_nat", "ler_wpDr", "ltC_nat", "lt_as...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfdot_sum_dchi (phi1 phi2 : 'CF(G)) : '[\sum_(i in dirr_constt phi1) dchi i, \sum_(i in dirr_constt phi2) dchi i] = #\|dirr_constt phi1 :&: dirr_constt phi2\|%:R - #\|dirr_constt phi1 :&: dirr_constt (- phi2)\|%:R.	Proof. rewrite addrC (big_setID (dirr_constt (- phi2))) /= cfdotDl; congr (_ + _). rewrite cfdot_suml -sumr_const -sumrN; apply: eq_bigr => i /setIP[p1i p2i]. rewrite cfdot_sumr (bigD1 (ndirr i)) -?dirr_constt_oppr //= dchi_ndirrE. rewrite cfdotNr cfnorm_dchi big1 ?addr0 // => j /andP[p2j i'j]. rewrite cfdot_dc...	Lemma	cfdot_sum_dchi	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "add0r", "addr0", "addrC", "apply", "big1", "bigD1", "big_setID", "cfdotDl", "cfdotNr", "cfdot_dchi", "cfdot_suml", "cfdot_sumr", "cfnorm_dchi", "contraNneq", "contraTneq", "dchi", "dchi_ndirrE", "dirr_constt", "dirr_constt_oppl", "dirr_constt_oppr", "eq_bigr", "eq_sym", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfdot_dirr_eq1 : {in dirr G &, forall phi psi, ('[phi, psi] == 1) = (phi == psi)}.	Proof. move=> _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]]. rewrite eq_signed_irr cfdotZl cfdotZr rmorph_sign cfdot_irr mulrA -signr_addb. rewrite pmulrn -rmorphMsign (eqr_int _ _ 1) -negb_add. by case: (b1 (+) b2) (i1 == i2) => [] []. Qed.	Lemma	cfdot_dirr_eq1	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "cfdotZl", "cfdotZr", "cfdot_irr", "dirr", "dirrP", "eq_signed_irr", "eqr_int", "mulrA", "negb_add", "pmulrn", "rmorphMsign", "rmorph_sign", "signr_addb" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfdot_add_dirr_eq1 : {in dirr G & &, forall phi1 phi2 psi, '[phi1 + phi2, psi] = 1 -> psi = phi1 \/ psi = phi2}.	Proof. move=> _ _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]] /dirrP[c [j ->]] /eqP. rewrite cfdotDl !cfdotZl !cfdotZr !rmorph_sign !cfdot_irr !mulrA -!signr_addb. rewrite 2!{1}signrE !mulrBl !mul1r -!natrM addrCA -subr_eq0 -!addrA. rewrite -!opprD addrA subr_eq0 -mulrSr -!natrD eqr_nat => eq_phi_psi. apply/pred2P; rewrite...	Lemma	cfdot_add_dirr_eq1	group_representation	group_representation/vcharacter.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...	[ "addrA", "addrCA", "apply", "cfdotDl", "cfdotZl", "cfdotZr", "cfdot_irr", "dirr", "dirrP", "eq_signed_irr", "eq_sym", "eqr_nat", "mul1r", "mulrA", "mulrBl", "mulrSr", "natrD", "natrM", "negb_add", "opprD", "pred2P", "rmorph_sign", "signrE", "signr_addb", "subr_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lel_xor_gt (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set	:= \| LelNotGt of x <= y : lel_xor_gt x y x x y y x x y y true false \| GtlNotLe of y < x : lel_xor_gt x y y y x x y y x x false true.	Variant	lel_xor_gt	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
ltl_xor_ge (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set	:= \| LtlNotGe of x < y : ltl_xor_ge x y x x y y x x y y false true \| GelNotLt of y <= x : ltl_xor_ge x y y y x x y y x x true false.	Variant	ltl_xor_ge	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
comparel (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> Set	:= \| ComparelLt of x < y : comparel x y x x y y x x y y false false false true false true \| ComparelGt of y < x : comparel x y y y x x y y x x false false true false true false \| ComparelEq of x = y : comparel x y x x x x x x x x true true true true false false.	Variant	comparel	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
incomparel (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> Set	:= \| InComparelLt of x < y : incomparel x y x x y y x x y y false false false true false true true true \| InComparelGt of y < x : incomparel x y y y x x y y x x false false true false true false true true \| InComparel of x >< y : incomparel x y x y y x (meet y x) (meet x y) (join y x) (join x y) ...	Variant	incomparel	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
"x `&` y"	:= (meet x y) : order_scope.	Notation	x `&` y	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",...	[ "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x `\|` y"	:= (join x y) : order_scope.	Notation	x `\|` y	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i <- r \| P ) F"	:= (\big[@join _ _ / \bot]_(i <- r \| P%B) F%O) : order_scope.	Notation	\join_ ( i <- r \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i <- r ) F"	:= (\big[@join _ _ / \bot]_(i <- r) F%O) : order_scope.	Notation	\join_ ( i <- r ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i \| P ) F"	:= (\big[@join _ _ / \bot]_(i \| P%B) F%O) : order_scope.	Notation	\join_ ( i \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ i F"	:= (\big[@join _ _ / \bot]_i F%O) : order_scope.	Notation	\join_ i F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i : I \| P ) F"	:= (\big[@join _ _ / \bot]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\join_ ( i : I \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i : I ) F"	:= (\big[@join _ _ / \bot]_(i : I) F%O) (only parsing) : order_scope.	Notation	\join_ ( i : I ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( m <= i < n \| P ) F"	:= (\big[@join _ _ / \bot]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\join_ ( m <= i < n \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( m <= i < n ) F"	:= (\big[@join _ _ / \bot]_(m <= i < n) F%O) : order_scope.	Notation	\join_ ( m <= i < n ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i < n \| P ) F"	:= (\big[@join _ _ / \bot]_(i < n \| P%B) F%O) : order_scope.	Notation	\join_ ( i < n \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i < n ) F"	:= (\big[@join _ _ / \bot]_(i < n) F%O) : order_scope.	Notation	\join_ ( i < n ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i 'in' A \| P ) F"	:= (\big[@join _ _ / \bot]_(i in A \| P%B) F%O) : order_scope.	Notation	\join_ ( i 'in' A \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join_ ( i 'in' A ) F"	:= (\big[@join _ _ / \bot]_(i in A) F%O) : order_scope.	Notation	\join_ ( i 'in' A ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i <- r \| P ) F"	:= (\big[meet / \top]_(i <- r \| P%B) F%O) : order_scope.	Notation	\meet_ ( i <- r \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i <- r ) F"	:= (\big[meet / \top]_(i <- r) F%O) : order_scope.	Notation	\meet_ ( i <- r ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i \| P ) F"	:= (\big[meet / \top]_(i \| P%B) F%O) : order_scope.	Notation	\meet_ ( i \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ i F"	:= (\big[meet / \top]_i F%O) : order_scope.	Notation	\meet_ i F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i : I \| P ) F"	:= (\big[meet / \top]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\meet_ ( i : I \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i : I ) F"	:= (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.	Notation	\meet_ ( i : I ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( m <= i < n \| P ) F"	:= (\big[meet / \top]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\meet_ ( m <= i < n \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( m <= i < n ) F"	:= (\big[meet / \top]_(m <= i < n) F%O) : order_scope.	Notation	\meet_ ( m <= i < n ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i < n \| P ) F"	:= (\big[meet / \top]_(i < n \| P%B) F%O) : order_scope.	Notation	\meet_ ( i < n \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i < n ) F"	:= (\big[meet / \top]_(i < n) F%O) : order_scope.	Notation	\meet_ ( i < n ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i 'in' A \| P ) F"	:= (\big[meet / \top]_(i in A \| P%B) F%O) : order_scope.	Notation	\meet_ ( i 'in' A \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet_ ( i 'in' A ) F"	:= (\big[meet / \top]_(i in A) F%O) : order_scope.	Notation	\meet_ ( i 'in' A ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x `\` y"	:= (diff x y) : order_scope.	Notation	x `\` y	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"~` A"	:= (compl A) : order_scope.	Notation	~` A	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dual_meet	:= (@meet (dual_display _) _).	Notation	dual_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",...	[ "dual_display", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dual_join	:= (@join (dual_display _) _).	Notation	dual_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",...	[ "dual_display", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x `&^d` y"	:= (dual_meet x y) : order_scope.	Notation	x `&^d` y	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",...	[ "dual_meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x `\|^d` y"	:= (dual_join x y) : order_scope.	Notation	x `\|^d` y	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",...	[ "dual_join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bot"	:= dual_bottom.	Notation	\bot	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",...	[ "dual_bottom" ]	notations later. Do not remove them.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\top"	:= dual_top.	Notation	\top	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",...	[ "dual_top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join	:= dual_join.	Notation	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",...	[ "dual_join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet	:= dual_meet.	Notation	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",...	[ "dual_meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i <- r \| P ) F"	:= (\big[join / \bot]_(i <- r \| P%B) F%O) : order_scope.	Notation	\join^d_ ( i <- r \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i <- r ) F"	:= (\big[join / \bot]_(i <- r) F%O) : order_scope.	Notation	\join^d_ ( i <- r ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i \| P ) F"	:= (\big[join / \bot]_(i \| P%B) F%O) : order_scope.	Notation	\join^d_ ( i \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ i F"	:= (\big[join / \bot]_i F%O) : order_scope.	Notation	\join^d_ i F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i : I \| P ) F"	:= (\big[join / \bot]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\join^d_ ( i : I \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i : I ) F"	:= (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.	Notation	\join^d_ ( i : I ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( m <= i < n \| P ) F"	:= (\big[join / \bot]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\join^d_ ( m <= i < n \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( m <= i < n ) F"	:= (\big[join / \bot]_(m <= i < n) F%O) : order_scope.	Notation	\join^d_ ( m <= i < n ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i < n \| P ) F"	:= (\big[join / \bot]_(i < n \| P%B) F%O) : order_scope.	Notation	\join^d_ ( i < n \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i < n ) F"	:= (\big[join / \bot]_(i < n) F%O) : order_scope.	Notation	\join^d_ ( i < n ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i 'in' A \| P ) F"	:= (\big[join / \bot]_(i in A \| P%B) F%O) : order_scope.	Notation	\join^d_ ( i 'in' A \| P ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\join^d_ ( i 'in' A ) F"	:= (\big[join / \bot]_(i in A) F%O) : order_scope.	Notation	\join^d_ ( i 'in' A ) F	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", "join" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i <- r \| P ) F"	:= (\big[meet / \top]_(i <- r \| P%B) F%O) : order_scope.	Notation	\meet^d_ ( i <- r \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i <- r ) F"	:= (\big[meet / \top]_(i <- r) F%O) : order_scope.	Notation	\meet^d_ ( i <- r ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i \| P ) F"	:= (\big[meet / \top]_(i \| P%B) F%O) : order_scope.	Notation	\meet^d_ ( i \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ i F"	:= (\big[meet / \top]_i F%O) : order_scope.	Notation	\meet^d_ i F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i : I \| P ) F"	:= (\big[meet / \top]_(i : I \| P%B) F%O) (only parsing) : order_scope.	Notation	\meet^d_ ( i : I \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i : I ) F"	:= (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.	Notation	\meet^d_ ( i : I ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( m <= i < n \| P ) F"	:= (\big[meet / \top]_(m <= i < n \| P%B) F%O) : order_scope.	Notation	\meet^d_ ( m <= i < n \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( m <= i < n ) F"	:= (\big[meet / \top]_(m <= i < n) F%O) : order_scope.	Notation	\meet^d_ ( m <= i < n ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i < n \| P ) F"	:= (\big[meet / \top]_(i < n \| P%B) F%O) : order_scope.	Notation	\meet^d_ ( i < n \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i < n ) F"	:= (\big[meet / \top]_(i < n) F%O) : order_scope.	Notation	\meet^d_ ( i < n ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i 'in' A \| P ) F"	:= (\big[meet / \top]_(i in A \| P%B) F%O) : order_scope.	Notation	\meet^d_ ( i 'in' A \| P ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\meet^d_ ( i 'in' A ) F"	:= (\big[meet / \top]_(i in A) F%O) : order_scope.	Notation	\meet^d_ ( i 'in' A ) F	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",...	[ "meet", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetEdual d (T : joinSemilatticeType d) (x y : T) : ((x : T^d) `&^d` y) = (x `\|` y).	Proof. by []. Qed.	Lemma	meetEdual	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
joinEdual d (T : meetSemilatticeType d) (x y : T) : ((x : T^d) `\|^d` y) = (x `&` y).	Proof. by []. Qed.	Lemma	joinEdual	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_anti: antisymmetric (<=%O : rel T).	Proof. exact: le_anti. Qed.	Lemma	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",...	[ "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_anti: antisymmetric (>=%O : rel T).	Proof. by move=> x y /le_anti. Qed.	Lemma	ge_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_anti", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_le x y: (x == y) = (x <= y <= x).	Proof. by apply/eqP/idP => [->\|/le_anti]; rewrite ?lexx. Qed.	Lemma	eq_le	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", "le_anti", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_def x y : (x < y) = (y != x) && (x <= y).	Proof. rewrite andbC lt_le_def; case/boolP: (x <= y) => //= xy. congr negb; apply/idP/eqP => [yx\|->]; last exact/lexx. by apply/le_anti; rewrite yx. Qed.	Lemma	lt_def	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", "last", "le_anti", "lexx", "lt_le_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_neqAle x y: (x < y) = (x != y) && (x <= y).	Proof. by rewrite lt_def eq_sym. Qed.	Lemma	lt_neqAle	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",...	[ "eq_sym", "lt_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_eqVlt x y: (x <= y) = (x == y) \|\| (x < y).	Proof. by rewrite lt_neqAle; case: eqP => //= ->; rewrite lexx. Qed.	Lemma	le_eqVlt	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",...	[ "lexx", "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lte_anti	:= (=^~ eq_le, @lt_asym disp T, @lt_le_asym disp T, @le_lt_asym disp T).	Definition	lte_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",...	[ "eq_le", "le_lt_asym", "lt_asym", "lt_le_asym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_geP {x y} : reflect (forall z, (z <= x) = (z <= y)) (x == y).	Proof. by apply: (iffP idP) => [/eqP->//\|/[dup]] /[!eq_le] -> <-; rewrite !lexx. Qed.	Lemma	eq_geP	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_le", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_leP {x y} : reflect (forall z, (x <= z) = (y <= z)) (x == y).	Proof. by apply: (iffP idP) => [/eqP->//\|/[dup]] /[!eq_le] <- ->; rewrite !lexx. Qed.	Lemma	eq_leP	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_le", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_sorted_uniq_le s : sorted <%O s = uniq s && sorted <=%O s.	Proof. rewrite le_sorted_pairwise lt_sorted_pairwise uniq_pairwise -pairwise_relI. by apply/eq_pairwise => ? ?; rewrite lt_neqAle. Qed.	Lemma	lt_sorted_uniq_le	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_pairwise", "le_sorted_pairwise", "lt_neqAle", "lt_sorted_pairwise", "pairwise_relI", "sorted", "uniq", "uniq_pairwise" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_sorted_eq s1 s2 : sorted <=%O s1 -> sorted <=%O s2 -> perm_eq s1 s2 -> s1 = s2.	Proof. exact/sorted_eq/le_anti/le_trans. Qed.	Lemma	le_sorted_eq	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_anti", "le_trans", "perm_eq", "s1", "s2", "sorted", "sorted_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
count_lt_le_mem x s : (count (< x) s < count (<= x) s)%N = (x \in s).	Proof. have := count_predUI (pred1 x) (< x) s. have -> : count (predI (pred1 x) (< x)) s = 0%N. rewrite (@eq_count _ _ pred0) ?count_pred0 // => y /=. by rewrite lt_neqAle; case: eqP => //= ->; rewrite eqxx. have /eq_count-> : [predU1 x & < x] =1 (<= x) by move=> y /=; rewrite le_eqVlt. by rewrite addn0 => ->; rewr...	Lemma	count_lt_le_mem	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",...	[ "add1n", "addn0", "count", "count_pred0", "count_predUI", "eq_count", "eqxx", "has_count", "has_pred1", "le_eqVlt", "leq_add2r", "lt_neqAle", "pred1", "predU1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_ltgtP x y : x >=< y -> compare x y (min y x) (min x y) (max y x) (max x y) (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).	Proof. rewrite /min /max />=<%O !le_eqVlt [y == x]eq_sym. have := (eqVneq x y, (boolP (x < y), boolP (y < x))). move=> [[->//\|neq_xy /=] [[] xy [] //=]] ; do ?by rewrite ?ltxx; constructor. by rewrite ltxx in xy. by rewrite le_gtF // ltW. Qed.	Lemma	comparable_ltgtP	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",...	[ "compare", "eqVneq", "eq_sym", "le_eqVlt", "le_gtF", "ltW", "ltxx", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_leP x y : x >=< y -> le_xor_gt x y (min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).	Proof. by move=> /comparable_ltgtP [?\|?\|->]; constructor; rewrite // ltW. Qed.	Lemma	comparable_leP	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",...	[ "comparable_ltgtP", "le_xor_gt", "ltW", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_ltP x y : x >=< y -> lt_xor_ge x y (min y x) (min x y) (max y x) (max x y) (y <= x) (x < y).	Proof. by move=> /comparable_ltgtP [?\|?\|->]; constructor; rewrite // ltW. Qed.	Lemma	comparable_ltP	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",...	[ "comparable_ltgtP", "ltW", "lt_xor_ge", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparableP x y : incompare x y (min y x) (min x y) (max y x) (max x y) (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y) (y >=< x) (x >=< y).	Proof. rewrite ![y >=< _]comparable_sym; have [c_xy\|i_xy] := boolP (x >=< y). by case: (comparable_ltgtP c_xy) => ?; constructor. by rewrite /min /max ?incomparable_eqF ?incomparable_leF; rewrite ?incomparable_ltF// 1?comparable_sym //; constructor. Qed.	Lemma	comparableP	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",...	[ "comparable_ltgtP", "comparable_sym", "incomparable_eqF", "incomparable_leF", "incomparable_ltF", "incompare", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leifP x y C : reflect (x <= y ?= iff C) (if C then x == y else x < y).	Proof. rewrite /leif le_eqVlt; apply: (iffP idP)=> [\|[]]. by case: C => [/eqP->\|lxy]; rewrite ?eqxx // lxy lt_eqF. by move=> /orP[/eqP->\|lxy] <-; rewrite ?eqxx // lt_eqF. Qed.	Lemma	leifP	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", "eqxx", "le_eqVlt", "leif", "lt_eqF" ]	leif	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leif_trans x1 x2 x3 C12 C23 : x1 <= x2 ?= iff C12 -> x2 <= x3 ?= iff C23 -> x1 <= x3 ?= iff C12 && C23.	Proof. move=> ltx12 ltx23; apply/leifP; rewrite -ltx12. case eqx12: (x1 == x2). by rewrite (eqP eqx12) lt_neqAle !ltx23 andbT; case C23. by rewrite (@lt_le_trans _ _ x2) ?ltx23 // lt_neqAle eqx12 ltx12. Qed.	Lemma	leif_trans	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", "leifP", "lt_le_trans", "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leif_le x y : x <= y -> x <= y ?= iff (x >= y).	Proof. by move=> lexy; split=> //; rewrite eq_le lexy. Qed.	Lemma	leif_le	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",...	[ "eq_le", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leif_eq x y : x <= y -> x <= y ?= iff (x == y).	Proof. by []. Qed.	Lemma	leif_eq	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
ge_leif x y C : x <= y ?= iff C -> (y <= x) = C.	Proof. by case=> le_xy; rewrite eq_le le_xy. Qed.	Lemma	ge_leif	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",...	[ "eq_le" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_leif x y C : x <= y ?= iff C -> (x < y) = ~~ C.	Proof. by move=> le_xy; rewrite lt_neqAle !le_xy andbT. Qed.	Lemma	lt_leif	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",...	[ "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

to_dirr (B : {set gT}) (phi : 'CF(B)) (i : Iirr B) : dIirr B

:= ('[phi, 'chi_i] < 0, i).

Definition

to_dirr

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "Iirr", "dIirr", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

of_irr (B : {set gT}) (i : dIirr B) : Iirr B

:= i.2.

Definition

of_irr

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "Iirr", "dIirr", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

irr_constt_to_dirr (phi: 'CF(G)) i : phi \in 'Z[irr G] -> (i \in irr_constt phi) = (to_dirr phi i \in dirr_constt phi).

Proof. move=> Zphi; rewrite irr_consttE dirr_consttE cfdotZr rmorph_sign /=. by rewrite -real_normrEsign ?normr_gt0 ?Rreal_int // Cint_cfdot_vchar_irr. Qed.

Lemma

irr_constt_to_dirr

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "Cint_cfdot_vchar_irr", "Rreal_int", "cfdotZr", "dirr_constt", "dirr_consttE", "irr", "irr_constt", "irr_consttE", "normr_gt0", "real_normrEsign", "rmorph_sign", "to_dirr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

to_dirrK (phi: 'CF(G)) : cancel (to_dirr phi) (@of_irr G).

Proof. by []. Qed.

Lemma

to_dirrK

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "of_irr", "to_dirr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

of_irrK (phi: 'CF(G)) : {in dirr_constt phi, cancel (@of_irr G) (to_dirr phi)}.

Proof. case=> b i; rewrite dirr_consttE cfdotZr rmorph_sign /= /to_dirr mulr_sign. by rewrite fun_if oppr_gt0; case: b => [|/ltW/le_gtF] ->. Qed.

Lemma

of_irrK

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "cfdotZr", "dirr_constt", "dirr_consttE", "le_gtF", "ltW", "mulr_sign", "of_irr", "oppr_gt0", "rmorph_sign", "to_dirr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfdot_todirrE (phi: 'CF(G)) i (phi_i := dchi (to_dirr phi i)) : '[phi, phi_i] *: phi_i = '[phi, 'chi_i] *: 'chi_i.

Proof. by rewrite cfdotZr rmorph_sign mulrC -scalerA signrZK. Qed.

Lemma

cfdot_todirrE

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "cfdotZr", "dchi", "mulrC", "rmorph_sign", "scalerA", "signrZK", "to_dirr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfun_sum_dconstt (phi : 'CF(G)) : phi \in 'Z[irr G] -> phi = \sum_(i in dirr_constt phi) '[phi, dchi i] *: dchi i.

Proof. move=> PiZ; rewrite [LHS]cfun_sum_constt. rewrite (reindex (to_dirr phi))=> [| /=]. by exists (@of_irr _)=> //; apply: of_irrK . by apply: eq_big => i; rewrite ?irr_constt_to_dirr // cfdot_todirrE. Qed.

Lemma

cfun_sum_dconstt

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "apply", "cfdot_todirrE", "cfun_sum_constt", "dchi", "dirr_constt", "eq_big", "irr", "irr_constt_to_dirr", "of_irr", "of_irrK", "reindex", "to_dirr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cnorm_dconstt (phi : 'CF(G)) : phi \in 'Z[irr G] -> '[phi] = \sum_(i in dirr_constt phi) '[phi, dchi i] ^+ 2.

Proof. move=> PiZ; rewrite {1 2}(cfun_sum_dconstt PiZ). rewrite cfdot_suml; apply: eq_bigr=> i IiD. rewrite cfdot_sumr (bigD1 i) //= big1 ?addr0 => [j /andP [JiD IdJ]|]. rewrite cfdotZr cfdotZl cfdot_dchi eq_sym (negPf IdJ) -natrB ?mulr0 //. by rewrite (negPf (contraNneq _ (dirr_constt_oppl JiD))) => // <-. rewrite...

Lemma

cnorm_dconstt

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "Cnat_dirr", "addr0", "apply", "aut_natr", "big1", "bigD1", "cfdotZl", "cfdotZr", "cfdot_dchi", "cfdot_suml", "cfdot_sumr", "cfun_sum_dconstt", "contraNneq", "dchi", "dirr_constt", "dirr_constt_oppl", "eq_bigr", "eq_sym", "eqxx", "irr", "mulr0", "mulr1", "natrB", "ndirr...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dirr_small_norm (phi : 'CF(G)) n : phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N -> [/\ #|dirr_constt phi| = n, dirr_constt phi :&: dirr_constt (- phi) = set0 & phi = \sum_(i in dirr_constt phi) dchi i].

Proof. move=> PiZ Pln; rewrite ltnNge -leC_nat => Nl4. suffices Fd i: i \in dirr_constt phi -> '[phi, dchi i] = 1. split; last 2 [by apply/setP=> u; rewrite !inE cfdotNl oppr_gt0 lt_asym]. apply/eqP; rewrite -eqC_nat -sumr_const -Pln (cnorm_dconstt PiZ). by apply/eqP/eq_bigr=> i Hi; rewrite Fd // expr1n. re...

Lemma

dirr_small_norm

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "Cnat_dirr", "apply", "bigD1", "cfdotNl", "cfun_sum_dconstt", "cnorm_dconstt", "contraNeq", "dchi", "dirr_constt", "dirr_consttE", "eqC_nat", "eq_bigr", "eq_sym", "expr1n", "exprn_ge0", "inE", "irr", "last", "leC_nat", "leq_sqr", "ler_nat", "ler_wpDr", "ltC_nat", "lt_as...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfdot_sum_dchi (phi1 phi2 : 'CF(G)) : '[\sum_(i in dirr_constt phi1) dchi i, \sum_(i in dirr_constt phi2) dchi i] = #|dirr_constt phi1 :&: dirr_constt phi2|%:R - #|dirr_constt phi1 :&: dirr_constt (- phi2)|%:R.

Proof. rewrite addrC (big_setID (dirr_constt (- phi2))) /= cfdotDl; congr (_ + _). rewrite cfdot_suml -sumr_const -sumrN; apply: eq_bigr => i /setIP[p1i p2i]. rewrite cfdot_sumr (bigD1 (ndirr i)) -?dirr_constt_oppr //= dchi_ndirrE. rewrite cfdotNr cfnorm_dchi big1 ?addr0 // => j /andP[p2j i'j]. rewrite cfdot_dc...

Lemma

cfdot_sum_dchi

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "add0r", "addr0", "addrC", "apply", "big1", "bigD1", "big_setID", "cfdotDl", "cfdotNr", "cfdot_dchi", "cfdot_suml", "cfdot_sumr", "cfnorm_dchi", "contraNneq", "contraTneq", "dchi", "dchi_ndirrE", "dirr_constt", "dirr_constt_oppl", "dirr_constt_oppr", "eq_bigr", "eq_sym", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfdot_dirr_eq1 : {in dirr G &, forall phi psi, ('[phi, psi] == 1) = (phi == psi)}.

Proof. move=> _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]]. rewrite eq_signed_irr cfdotZl cfdotZr rmorph_sign cfdot_irr mulrA -signr_addb. rewrite pmulrn -rmorphMsign (eqr_int _ _ 1) -negb_add. by case: (b1 (+) b2) (i1 == i2) => [] []. Qed.

Lemma

cfdot_dirr_eq1

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "cfdotZl", "cfdotZr", "cfdot_irr", "dirr", "dirrP", "eq_signed_irr", "eqr_int", "mulrA", "negb_add", "pmulrn", "rmorphMsign", "rmorph_sign", "signr_addb" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfdot_add_dirr_eq1 : {in dirr G & &, forall phi1 phi2 psi, '[phi1 + phi2, psi] = 1 -> psi = phi1 \/ psi = phi2}.

Proof. move=> _ _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]] /dirrP[c [j ->]] /eqP. rewrite cfdotDl !cfdotZl !cfdotZr !rmorph_sign !cfdot_irr !mulrA -!signr_addb. rewrite 2!{1}signrE !mulrBl !mul1r -!natrM addrCA -subr_eq0 -!addrA. rewrite -!opprD addrA subr_eq0 -mulrSr -!natrD eqr_nat => eq_phi_psi. apply/pred2P; rewrite...

Lemma

cfdot_add_dirr_eq1

group_representation

group_representation/vcharacter.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "order", "ssralg", "poly", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "...

[ "addrA", "addrCA", "apply", "cfdotDl", "cfdotZl", "cfdotZr", "cfdot_irr", "dirr", "dirrP", "eq_signed_irr", "eq_sym", "eqr_nat", "mul1r", "mulrA", "mulrBl", "mulrSr", "natrD", "natrM", "negb_add", "opprD", "pred2P", "rmorph_sign", "signrE", "signr_addb", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lel_xor_gt (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set

:= | LelNotGt of x <= y : lel_xor_gt x y x x y y x x y y true false | GtlNotLe of y < x : lel_xor_gt x y y y x x y y x x false true.

Variant

lel_xor_gt

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

ltl_xor_ge (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set

:= | LtlNotGe of x < y : ltl_xor_ge x y x x y y x x y y false true | GelNotLt of y <= x : ltl_xor_ge x y y y x x y y x x true false.

Variant

ltl_xor_ge

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

comparel (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> Set

:= | ComparelLt of x < y : comparel x y x x y y x x y y false false false true false true | ComparelGt of y < x : comparel x y y y x x y y x x false false true false true false | ComparelEq of x = y : comparel x y x x x x x x x x true true true true false false.

Variant

comparel

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

incomparel (x y : T) : T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> Set

:= | InComparelLt of x < y : incomparel x y x x y y x x y y false false false true false true true true | InComparelGt of y < x : incomparel x y y y x x y y x x false false true false true false true true | InComparel of x >< y : incomparel x y x y y x (meet y x) (meet x y) (join y x) (join x y) ...

Variant

incomparel

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

"x `&` y"

:= (meet x y) : order_scope.

Notation

x `&` y

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",...

[ "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x `|` y"

:= (join x y) : order_scope.

Notation

x `|` y

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i <- r | P ) F"

:= (\big[@join _ _ / \bot]_(i <- r | P%B) F%O) : order_scope.

Notation

\join_ ( i <- r | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i <- r ) F"

:= (\big[@join _ _ / \bot]_(i <- r) F%O) : order_scope.

Notation

\join_ ( i <- r ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i | P ) F"

:= (\big[@join _ _ / \bot]_(i | P%B) F%O) : order_scope.

Notation

\join_ ( i | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ i F"

:= (\big[@join _ _ / \bot]_i F%O) : order_scope.

Notation

\join_ i F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i : I | P ) F"

:= (\big[@join _ _ / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\join_ ( i : I | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i : I ) F"

:= (\big[@join _ _ / \bot]_(i : I) F%O) (only parsing) : order_scope.

Notation

\join_ ( i : I ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( m <= i < n | P ) F"

:= (\big[@join _ _ / \bot]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\join_ ( m <= i < n | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( m <= i < n ) F"

:= (\big[@join _ _ / \bot]_(m <= i < n) F%O) : order_scope.

Notation

\join_ ( m <= i < n ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i < n | P ) F"

:= (\big[@join _ _ / \bot]_(i < n | P%B) F%O) : order_scope.

Notation

\join_ ( i < n | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i < n ) F"

:= (\big[@join _ _ / \bot]_(i < n) F%O) : order_scope.

Notation

\join_ ( i < n ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i 'in' A | P ) F"

:= (\big[@join _ _ / \bot]_(i in A | P%B) F%O) : order_scope.

Notation

\join_ ( i 'in' A | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join_ ( i 'in' A ) F"

:= (\big[@join _ _ / \bot]_(i in A) F%O) : order_scope.

Notation

\join_ ( i 'in' A ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i <- r | P ) F"

:= (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.

Notation

\meet_ ( i <- r | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i <- r ) F"

:= (\big[meet / \top]_(i <- r) F%O) : order_scope.

Notation

\meet_ ( i <- r ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i | P ) F"

:= (\big[meet / \top]_(i | P%B) F%O) : order_scope.

Notation

\meet_ ( i | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ i F"

:= (\big[meet / \top]_i F%O) : order_scope.

Notation

\meet_ i F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i : I | P ) F"

:= (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\meet_ ( i : I | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i : I ) F"

:= (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.

Notation

\meet_ ( i : I ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( m <= i < n | P ) F"

:= (\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\meet_ ( m <= i < n | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( m <= i < n ) F"

:= (\big[meet / \top]_(m <= i < n) F%O) : order_scope.

Notation

\meet_ ( m <= i < n ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i < n | P ) F"

:= (\big[meet / \top]_(i < n | P%B) F%O) : order_scope.

Notation

\meet_ ( i < n | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i < n ) F"

:= (\big[meet / \top]_(i < n) F%O) : order_scope.

Notation

\meet_ ( i < n ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i 'in' A | P ) F"

:= (\big[meet / \top]_(i in A | P%B) F%O) : order_scope.

Notation

\meet_ ( i 'in' A | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet_ ( i 'in' A ) F"

:= (\big[meet / \top]_(i in A) F%O) : order_scope.

Notation

\meet_ ( i 'in' A ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x `\` y"

:= (diff x y) : order_scope.

Notation

x `\` y

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"~` A"

:= (compl A) : order_scope.

Notation

~` A

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dual_meet

:= (@meet (dual_display _) _).

Notation

dual_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",...

[ "dual_display", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dual_join

:= (@join (dual_display _) _).

Notation

dual_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",...

[ "dual_display", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x `&^d` y"

:= (dual_meet x y) : order_scope.

Notation

x `&^d` y

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",...

[ "dual_meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x `|^d` y"

:= (dual_join x y) : order_scope.

Notation

x `|^d` y

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",...

[ "dual_join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bot"

:= dual_bottom.

Notation

\bot

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",...

[ "dual_bottom" ]

notations later. Do not remove them.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\top"

:= dual_top.

Notation

\top

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",...

[ "dual_top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join

:= dual_join.

Notation

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",...

[ "dual_join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet

:= dual_meet.

Notation

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",...

[ "dual_meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i <- r | P ) F"

:= (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.

Notation

\join^d_ ( i <- r | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i <- r ) F"

:= (\big[join / \bot]_(i <- r) F%O) : order_scope.

Notation

\join^d_ ( i <- r ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i | P ) F"

:= (\big[join / \bot]_(i | P%B) F%O) : order_scope.

Notation

\join^d_ ( i | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ i F"

:= (\big[join / \bot]_i F%O) : order_scope.

Notation

\join^d_ i F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i : I | P ) F"

:= (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\join^d_ ( i : I | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i : I ) F"

:= (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.

Notation

\join^d_ ( i : I ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( m <= i < n | P ) F"

:= (\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\join^d_ ( m <= i < n | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( m <= i < n ) F"

:= (\big[join / \bot]_(m <= i < n) F%O) : order_scope.

Notation

\join^d_ ( m <= i < n ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i < n | P ) F"

:= (\big[join / \bot]_(i < n | P%B) F%O) : order_scope.

Notation

\join^d_ ( i < n | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i < n ) F"

:= (\big[join / \bot]_(i < n) F%O) : order_scope.

Notation

\join^d_ ( i < n ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i 'in' A | P ) F"

:= (\big[join / \bot]_(i in A | P%B) F%O) : order_scope.

Notation

\join^d_ ( i 'in' A | P ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\join^d_ ( i 'in' A ) F"

:= (\big[join / \bot]_(i in A) F%O) : order_scope.

Notation

\join^d_ ( i 'in' A ) F

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", "join" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i <- r | P ) F"

:= (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.

Notation

\meet^d_ ( i <- r | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i <- r ) F"

:= (\big[meet / \top]_(i <- r) F%O) : order_scope.

Notation

\meet^d_ ( i <- r ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i | P ) F"

:= (\big[meet / \top]_(i | P%B) F%O) : order_scope.

Notation

\meet^d_ ( i | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ i F"

:= (\big[meet / \top]_i F%O) : order_scope.

Notation

\meet^d_ i F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i : I | P ) F"

:= (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.

Notation

\meet^d_ ( i : I | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i : I ) F"

:= (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.

Notation

\meet^d_ ( i : I ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( m <= i < n | P ) F"

:= (\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.

Notation

\meet^d_ ( m <= i < n | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( m <= i < n ) F"

:= (\big[meet / \top]_(m <= i < n) F%O) : order_scope.

Notation

\meet^d_ ( m <= i < n ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i < n | P ) F"

:= (\big[meet / \top]_(i < n | P%B) F%O) : order_scope.

Notation

\meet^d_ ( i < n | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i < n ) F"

:= (\big[meet / \top]_(i < n) F%O) : order_scope.

Notation

\meet^d_ ( i < n ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i 'in' A | P ) F"

:= (\big[meet / \top]_(i in A | P%B) F%O) : order_scope.

Notation

\meet^d_ ( i 'in' A | P ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\meet^d_ ( i 'in' A ) F"

:= (\big[meet / \top]_(i in A) F%O) : order_scope.

Notation

\meet^d_ ( i 'in' A ) F

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",...

[ "meet", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetEdual d (T : joinSemilatticeType d) (x y : T) : ((x : T^d) `&^d` y) = (x `|` y).

Proof. by []. Qed.

Lemma

meetEdual

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

joinEdual d (T : meetSemilatticeType d) (x y : T) : ((x : T^d) `|^d` y) = (x `&` y).

Proof. by []. Qed.

Lemma

joinEdual

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_anti: antisymmetric (<=%O : rel T).

Proof. exact: le_anti. Qed.

Lemma

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",...

[ "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge_anti: antisymmetric (>=%O : rel T).

Proof. by move=> x y /le_anti. Qed.

Lemma

ge_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_anti", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_le x y: (x == y) = (x <= y <= x).

Proof. by apply/eqP/idP => [->|/le_anti]; rewrite ?lexx. Qed.

Lemma

eq_le

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", "le_anti", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_def x y : (x < y) = (y != x) && (x <= y).

Proof. rewrite andbC lt_le_def; case/boolP: (x <= y) => //= xy. congr negb; apply/idP/eqP => [yx|->]; last exact/lexx. by apply/le_anti; rewrite yx. Qed.

Lemma

lt_def

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", "last", "le_anti", "lexx", "lt_le_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_neqAle x y: (x < y) = (x != y) && (x <= y).

Proof. by rewrite lt_def eq_sym. Qed.

Lemma

lt_neqAle

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",...

[ "eq_sym", "lt_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_eqVlt x y: (x <= y) = (x == y) || (x < y).

Proof. by rewrite lt_neqAle; case: eqP => //= ->; rewrite lexx. Qed.

Lemma

le_eqVlt

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",...

[ "lexx", "lt_neqAle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lte_anti

:= (=^~ eq_le, @lt_asym disp T, @lt_le_asym disp T, @le_lt_asym disp T).

Definition

lte_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",...

[ "eq_le", "le_lt_asym", "lt_asym", "lt_le_asym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_geP {x y} : reflect (forall z, (z <= x) = (z <= y)) (x == y).

Proof. by apply: (iffP idP) => [/eqP->//|/[dup]] /[!eq_le] -> <-; rewrite !lexx. Qed.

Lemma

eq_geP

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_le", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_leP {x y} : reflect (forall z, (x <= z) = (y <= z)) (x == y).

Proof. by apply: (iffP idP) => [/eqP->//|/[dup]] /[!eq_le] <- ->; rewrite !lexx. Qed.

Lemma

eq_leP

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_le", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_sorted_uniq_le s : sorted <%O s = uniq s && sorted <=%O s.

Proof. rewrite le_sorted_pairwise lt_sorted_pairwise uniq_pairwise -pairwise_relI. by apply/eq_pairwise => ? ?; rewrite lt_neqAle. Qed.

Lemma

lt_sorted_uniq_le

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_pairwise", "le_sorted_pairwise", "lt_neqAle", "lt_sorted_pairwise", "pairwise_relI", "sorted", "uniq", "uniq_pairwise" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_sorted_eq s1 s2 : sorted <=%O s1 -> sorted <=%O s2 -> perm_eq s1 s2 -> s1 = s2.

Proof. exact/sorted_eq/le_anti/le_trans. Qed.

Lemma

le_sorted_eq

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_anti", "le_trans", "perm_eq", "s1", "s2", "sorted", "sorted_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

count_lt_le_mem x s : (count (< x) s < count (<= x) s)%N = (x \in s).

Proof. have := count_predUI (pred1 x) (< x) s. have -> : count (predI (pred1 x) (< x)) s = 0%N. rewrite (@eq_count _ _ pred0) ?count_pred0 // => y /=. by rewrite lt_neqAle; case: eqP => //= ->; rewrite eqxx. have /eq_count-> : [predU1 x & < x] =1 (<= x) by move=> y /=; rewrite le_eqVlt. by rewrite addn0 => ->; rewr...

Lemma

count_lt_le_mem

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",...

[ "add1n", "addn0", "count", "count_pred0", "count_predUI", "eq_count", "eqxx", "has_count", "has_pred1", "le_eqVlt", "leq_add2r", "lt_neqAle", "pred1", "predU1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_ltgtP x y : x >=< y -> compare x y (min y x) (min x y) (max y x) (max x y) (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).

Proof. rewrite /min /max />=<%O !le_eqVlt [y == x]eq_sym. have := (eqVneq x y, (boolP (x < y), boolP (y < x))). move=> [[->//|neq_xy /=] [[] xy [] //=]] ; do ?by rewrite ?ltxx; constructor. by rewrite ltxx in xy. by rewrite le_gtF // ltW. Qed.

Lemma

comparable_ltgtP

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",...

[ "compare", "eqVneq", "eq_sym", "le_eqVlt", "le_gtF", "ltW", "ltxx", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_leP x y : x >=< y -> le_xor_gt x y (min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).

Proof. by move=> /comparable_ltgtP [?|?|->]; constructor; rewrite // ltW. Qed.

Lemma

comparable_leP

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",...

[ "comparable_ltgtP", "le_xor_gt", "ltW", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_ltP x y : x >=< y -> lt_xor_ge x y (min y x) (min x y) (max y x) (max x y) (y <= x) (x < y).

Proof. by move=> /comparable_ltgtP [?|?|->]; constructor; rewrite // ltW. Qed.

Lemma

comparable_ltP

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",...

[ "comparable_ltgtP", "ltW", "lt_xor_ge", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparableP x y : incompare x y (min y x) (min x y) (max y x) (max x y) (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y) (y >=< x) (x >=< y).

Proof. rewrite ![y >=< _]comparable_sym; have [c_xy|i_xy] := boolP (x >=< y). by case: (comparable_ltgtP c_xy) => ?; constructor. by rewrite /min /max ?incomparable_eqF ?incomparable_leF; rewrite ?incomparable_ltF// 1?comparable_sym //; constructor. Qed.

Lemma

comparableP

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",...

[ "comparable_ltgtP", "comparable_sym", "incomparable_eqF", "incomparable_leF", "incomparable_ltF", "incompare", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leifP x y C : reflect (x <= y ?= iff C) (if C then x == y else x < y).

Proof. rewrite /leif le_eqVlt; apply: (iffP idP)=> [|[]]. by case: C => [/eqP->|lxy]; rewrite ?eqxx // lxy lt_eqF. by move=> /orP[/eqP->|lxy] <-; rewrite ?eqxx // lt_eqF. Qed.

Lemma

leifP

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", "eqxx", "le_eqVlt", "leif", "lt_eqF" ]

leif

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leif_trans x1 x2 x3 C12 C23 : x1 <= x2 ?= iff C12 -> x2 <= x3 ?= iff C23 -> x1 <= x3 ?= iff C12 && C23.

Proof. move=> ltx12 ltx23; apply/leifP; rewrite -ltx12. case eqx12: (x1 == x2). by rewrite (eqP eqx12) lt_neqAle !ltx23 andbT; case C23. by rewrite (@lt_le_trans _ _ x2) ?ltx23 // lt_neqAle eqx12 ltx12. Qed.

Lemma

leif_trans

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", "leifP", "lt_le_trans", "lt_neqAle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leif_le x y : x <= y -> x <= y ?= iff (x >= y).

Proof. by move=> lexy; split=> //; rewrite eq_le lexy. Qed.

Lemma

leif_le

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",...

[ "eq_le", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leif_eq x y : x <= y -> x <= y ?= iff (x == y).

Proof. by []. Qed.

Lemma

leif_eq

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

ge_leif x y C : x <= y ?= iff C -> (y <= x) = C.

Proof. by case=> le_xy; rewrite eq_le le_xy. Qed.

Lemma

ge_leif

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",...

[ "eq_le" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_leif x y C : x <= y ?= iff C -> (x < y) = ~~ C.

Proof. by move=> le_xy; rewrite lt_neqAle !le_xy andbT. Qed.

Lemma

lt_leif

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",...

[ "lt_neqAle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d