phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
leBUK y x : y <= x -> (x `\` y) `\|` y = x.	Proof. by move=> leyx; rewrite joinC leBKU. Qed.	Lemma	leBUK	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinC", "leBKU" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBLR x y z : (x `\` y <= z) = (x <= y `\|` z).	Proof. apply/idP/idP; first by move=> /join_r <-; rewrite joinA diffKU joinAC leUr. by rewrite -{1}[x](joinIB y) => /(leU2r_le (diffIK _ _)). Qed.	Lemma	leBLR	order	order/order.v	[ "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", "diffIK", "diffKU", "joinA", "joinAC", "joinIB", "join_r", "leU2r_le", "leUr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffUx x y z : (x `\|` y) `\` z = (x `\` z) `\|` (y `\` z).	Proof. apply/eqP; rewrite eq_le leUx !leBl ?leUr ?leUl ?andbT //. by rewrite leBLR joinA diffKU joinAC diffKU joinAC -joinA leUr. Qed.	Lemma	diffUx	order	order/order.v	[ "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", "diffKU", "eq_le", "joinA", "joinAC", "leBLR", "leBl", "leUl", "leUr", "leUx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diff_eq0 x y : (x `\` y == \bot) = (x <= y).	Proof. by rewrite -lex0 leBLR joinx0. Qed.	Lemma	diff_eq0	order	order/order.v	[ "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", "joinx0", "leBLR", "lex0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinxB x y z : x `\|` (y `\` z) = ((x `\|` y) `\` z) `\|` (x `&` z).	Proof. by rewrite diffUx joinAC joinBI. Qed.	Lemma	joinxB	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffUx", "joinAC", "joinBI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinBx x y z : (y `\` z) `\|` x = ((y `\|` x) `\` z) `\|` (z `&` x).	Proof. by rewrite ![_ `\|` x]joinC ![_ `&` x]meetC joinxB. Qed.	Lemma	joinBx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinC", "joinxB", "meetC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBr z x y : x <= y -> z `\` y <= z `\` x.	Proof. by move=> lexy; rewrite leBLR joinxB meet_r ?leBUK ?leUr ?lexUl. Qed.	Lemma	leBr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinxB", "leBLR", "leBUK", "leUr", "lexUl", "meet_r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leB2 x y z t : x <= z -> t <= y -> x `\` y <= z `\` t.	Proof. by move=> /(@leBl t) ? /(@leBr x) /le_trans ->. Qed.	Lemma	leB2	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leBl", "leBr", "le_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_eq0E_diff z x y : x <= z -> (x `&` y == \bot) = (x <= z `\` y).	Proof. move=> xz; apply/idP/idP; last by move=> /meet_r <-; rewrite -meetA meetBI. by move=> /eqP xIy_eq0; rewrite -[x](joinIB y) xIy_eq0 join0x leBl. Qed.	Lemma	meet_eq0E_diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "join0x", "joinIB", "last", "leBl", "meetA", "meetBI", "meet_r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBRL x y z : (x <= z `\` y) = (x <= z) && (x `&` y == \bot).	Proof. apply/idP/idP => [xyz\|]; first by rewrite (@meet_eq0E_diff z) // (le_trans xyz). by move=> /andP [?]; rewrite -meet_eq0E_diff. Qed.	Lemma	leBRL	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "le_trans", "meet_eq0E_diff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_diff x y z : (x `\` y == z) = (z <= x <= y `\|` z) && (z `&` y == \bot).	Proof. by rewrite eq_le leBLR leBRL andbCA andbA. Qed.	Lemma	eq_diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "eq_le", "leBLR", "leBRL" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffxU x y z : z `\` (x `\|` y) = (z `\` x) `&` (z `\` y).	Proof. apply/eqP; rewrite eq_le lexI !leBr ?leUl ?leUr //=. rewrite leBRL leIx2 ?leBx //= meetUr meetAC diffIK -meetA diffIK. by rewrite meet0x meetx0 joinx0. Qed.	Lemma	diffxU	order	order/order.v	[ "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", "diffIK", "eq_le", "joinx0", "leBRL", "leBr", "leBx", "leIx2", "leUl", "leUr", "lexI", "meet0x", "meetA", "meetAC", "meetUr", "meetx0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffx0 x : x `\` \bot = x.	Proof. by apply/eqP; rewrite eq_diff join0x meetx0 lexx eqxx. Qed.	Lemma	diffx0	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "eq_diff", "eqxx", "join0x", "lexx", "meetx0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diff0x x : \bot `\` x = \bot.	Proof. by apply/eqP; rewrite eq_diff joinx0 meet0x lexx eqxx le0x. Qed.	Lemma	diff0x	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "eq_diff", "eqxx", "joinx0", "le0x", "lexx", "meet0x" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffIx x y z : (x `&` y) `\` z = (x `\` z) `&` (y `\` z).	Proof. apply/eqP; rewrite eq_diff joinIr ?leI2 ?diffKU ?leUr ?leBx //=. by rewrite -meetA diffIK meetx0. Qed.	Lemma	diffIx	order	order/order.v	[ "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", "diffIK", "diffKU", "eq_diff", "joinIr", "leBx", "leI2", "leUr", "meetA", "meetx0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetxB x y z : x `&` (y `\` z) = (x `&` y) `\` z.	Proof. by rewrite diffIx -{1}[x](joinBI z) meetUl meetIB joinx0. Qed.	Lemma	meetxB	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffIx", "joinBI", "joinx0", "meetIB", "meetUl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetBx x y z : (x `\` y) `&` z = (x `&` z) `\` y.	Proof. by rewrite ![_ `&` z]meetC meetxB. Qed.	Lemma	meetBx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meetC", "meetxB" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffxI x y z : x `\` (y `&` z) = (x `\` y) `\|` (x `\` z).	Proof. apply/eqP; rewrite eq_diff leUx !leBx //= joinIl joinA joinCA !diffKU. rewrite joinCA -joinA [_ `\|` x]joinC ![x `\|` _]join_l //. by rewrite -joinIl leUr /= meetUl {1}[_ `&` z]meetC ?meetBI joinx0. Qed.	Lemma	diffxI	order	order/order.v	[ "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", "diffKU", "eq_diff", "joinA", "joinC", "joinCA", "joinIl", "join_l", "joinx0", "leBx", "leUr", "leUx", "meetBI", "meetC", "meetUl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffBx x y z : (x `\` y) `\` z = x `\` (y `\|` z).	Proof. apply/eqP; rewrite eq_diff leBr ?leUl //=. by rewrite diffxU joinIr diffKU -joinIr meet_l ?leUr //= -meetA diffIK meetx0. Qed.	Lemma	diffBx	order	order/order.v	[ "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", "diffIK", "diffKU", "diffxU", "eq_diff", "joinIr", "leBr", "leUl", "leUr", "meetA", "meet_l", "meetx0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffxB x y z : x `\` (y `\` z) = (x `\` y) `\|` (x `&` z).	Proof. rewrite -[y in RHS](joinIB z) diffxU joinIl diffxI -joinA joinBI join_r //. by rewrite joinBx meetKU meetA meetAC diffIK meet0x joinx0 meet_r. Qed.	Lemma	diffxB	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffIK", "diffxI", "diffxU", "joinA", "joinBI", "joinBx", "joinIB", "joinIl", "join_r", "joinx0", "meet0x", "meetA", "meetAC", "meetKU", "meet_r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinBK x y : (y `\|` x) `\` x = (y `\` x).	Proof. by rewrite diffUx diffxx joinx0. Qed.	Lemma	joinBK	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffUx", "diffxx", "joinx0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinBKC x y : (x `\|` y) `\` x = (y `\` x).	Proof. by rewrite diffUx diffxx join0x. Qed.	Lemma	joinBKC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffUx", "diffxx", "join0x" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disj_le x y : x `&` y == \bot -> (x <= y) = (x == \bot).	Proof. by rewrite [x == \bot]eq_sym -eq_meetl => /eqP ->. Qed.	Lemma	disj_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",...	[ "bot", "eq_meetl", "eq_sym" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disj_leC x y : y `&` x == \bot -> (x <= y) = (x == \bot).	Proof. by rewrite meetC => /disj_le. Qed.	Lemma	disj_leC	order	order/order.v	[ "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", "disj_le", "meetC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disj_diffl x y : x `&` y == \bot -> x `\` y = x.	Proof. by move=> dxy; apply/eqP; rewrite eq_diff dxy lexx leUr. Qed.	Lemma	disj_diffl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "bot", "eq_diff", "leUr", "lexx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disj_diffr x y : x `&` y == \bot -> y `\` x = y.	Proof. by rewrite meetC => /disj_diffl. Qed.	Lemma	disj_diffr	order	order/order.v	[ "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", "disj_diffl", "meetC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0B x y : x < y -> \bot < y `\` x.	Proof. by move=> ?; rewrite lt_leAnge le0x leBLR joinx0 /= lt_geF. Qed.	Lemma	lt0B	order	order/order.v	[ "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", "joinx0", "le0x", "leBLR", "lt_geF", "lt_leAnge" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffErcompl x y : codiff x y = rcompl x \top y.	Proof. exact: codiffErcompl. Qed.	Lemma	codiffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff", "rcompl", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEdiff x : ~` x = \top `\` x.	Proof. exact: complEdiff. Qed.	Lemma	complEdiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complE	:= complEdiff.	Notation	complE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEcodiff x : ~` x = codiff \bot x.	Proof. exact: complEcodiff. Qed.	Lemma	complEcodiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "codiff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complErcompl x : ~` x = rcompl \bot \top x.	Proof. by rewrite complEdiff diffErcompl. Qed.	Lemma	complErcompl	order	order/order.v	[ "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", "complEdiff", "diffErcompl", "rcompl", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diff1x x : \top `\` x = ~` x.	Proof. exact/esym/complEdiff. Qed.	Lemma	diff1x	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffE x y : x `\` y = x `&` ~` y.	Proof. by rewrite complEdiff meetxB meetx1. Qed.	Lemma	diffE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "meetx1", "meetxB" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complK : involutive (@compl _ L).	Proof. by move=> x; rewrite !complEdiff diffxB diffxx meet1x join0x. Qed.	Lemma	complK	order	order/order.v	[ "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", "complEdiff", "diffxB", "diffxx", "join0x", "meet1x" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl_inj : injective (@compl _ L).	Proof. exact/inv_inj/complK. Qed.	Lemma	compl_inj	order	order/order.v	[ "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", "complK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disj_leC x y : (x `&` y == \bot) = (x <= ~` y).	Proof. by rewrite -diff_eq0 diffE complK. Qed.	Lemma	disj_leC	order	order/order.v	[ "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", "complK", "diffE", "diff_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leCx x y : (~` x <= y) = (~` y <= x).	Proof. by rewrite !complEdiff !leBLR joinC. Qed.	Lemma	leCx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "joinC", "leBLR" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexC x y : (x <= ~` y) = (y <= ~` x).	Proof. by rewrite -[x in LHS]complK leCx complK. Qed.	Lemma	lexC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complK", "leCx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leC x y : (~` x <= ~` y) = (y <= x).	Proof. by rewrite leCx complK. Qed.	Lemma	leC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complK", "leCx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complU x y : ~` (x `\|` y) = ~` x `&` ~` y.	Proof. by rewrite !complEdiff diffxU. Qed.	Lemma	complU	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "diffxU" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complI x y : ~` (x `&` y) = ~` x `\|` ~` y.	Proof. by rewrite !complEdiff diffxI. Qed.	Lemma	complI	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "diffxI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinxC x : x `\|` ~` x = \top.	Proof. by rewrite complEdiff diffKU joinx1. Qed.	Lemma	joinxC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complEdiff", "diffKU", "joinx1", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinCx x : ~` x `\|` x = \top.	Proof. by rewrite joinC joinxC. Qed.	Lemma	joinCx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinC", "joinxC", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetxC x : x `&` ~` x = \bot.	Proof. by rewrite complEdiff diffKI. Qed.	Lemma	meetxC	order	order/order.v	[ "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", "complEdiff", "diffKI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetCx x : ~` x `&` x = \bot.	Proof. by rewrite meetC meetxC. Qed.	Lemma	meetCx	order	order/order.v	[ "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", "meetC", "meetxC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl1 : ~` \top = \bot :> L.	Proof. by rewrite complEdiff diffxx. Qed.	Lemma	compl1	order	order/order.v	[ "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", "complEdiff", "diffxx", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl0 : ~` \bot = \top :> L.	Proof. by rewrite -compl1 complK. Qed.	Lemma	compl0	order	order/order.v	[ "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", "compl1", "complK", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complB x y : ~` (x `\` y) = ~` x `\|` y.	Proof. by rewrite diffE complI complK. Qed.	Lemma	complB	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "complI", "complK", "diffE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leBC x y : x `\` y <= ~` y.	Proof. by rewrite leBLR joinxC lex1. Qed.	Lemma	leBC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinxC", "leBLR", "lex1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl_joins (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) : ~` (\join_(j <- r \| P j) F j) = \meet_(j <- r \| P j) ~` F j.	Proof. by elim/big_rec2: _=> [\|i x y ? <-]; rewrite ?compl0 ?complU. Qed.	Lemma	compl_joins	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "big_rec2", "compl0", "complU", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl_meets (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) : ~` (\meet_(j <- r \| P j) F j) = \join_(j <- r \| P j) ~` F j.	Proof. by elim/big_rec2: _=> [\|i x y ? <-]; rewrite ?compl1 ?complI. Qed.	Lemma	compl_meets	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "big_rec2", "compl1", "complI", "seq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_anti : antisymmetric (fun x y => @le d T y x).	Proof. by move=> x y; rewrite andbC; apply: le_anti. Qed.	Let	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",...	[ "apply", "le", "le_anti" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_le_def x y : lt x y = le x y && ~~ le y x.	Proof. rewrite lt_def andbC; case /boolP: (le x y) => //= xy. have [->\|/negP xyE /=] := eqVneq y x; first by rewrite le_refl. by apply/esym/negP => yx; apply/xyE/eqP/le_anti; rewrite yx. Qed.	Let	lt_le_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", "eqVneq", "le", "le_anti", "le_refl", "lt", "lt_def" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_anti : antisymmetric le.	Proof. move=> x y; rewrite !le_def [y == _]eq_sym. have [//\|neq_xy/=] := eqVneq x y => /andP[xy yx]. by have := lt_trans xy yx; rewrite lt_irr. Qed.	Let	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",...	[ "eqVneq", "eq_sym", "le", "le_def", "lt_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetxx : idempotent_op meet.	Proof. by move=> x; apply/eqP; rewrite -leEmeet. Qed.	Fact	meetxx	order	order/order.v	[ "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", "idempotent_op", "leEmeet", "meet" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexI x y z : (x <= meet y z) = (x <= y) && (x <= z).	Proof. rewrite !leEmeet; apply/eqP/andP => [<-\|[/eqP<- /eqP<-]]. split; apply/eqP; last by rewrite meetA -meetA meetxx. by rewrite -!meetA (meetC z) (meetA y) meetxx. by rewrite -!meetA (meetC z) -meetA (meetA y) !meetxx. Qed.	Fact	lexI	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "last", "leEmeet", "meet", "meetA", "meetC", "meetxx", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinxx : idempotent_op join.	Proof. by move=> x; apply/eqP; rewrite -leEjoin. Qed.	Fact	joinxx	order	order/order.v	[ "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", "idempotent_op", "join", "leEjoin" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leUx x y z : (join x y <= z) = (x <= z) && (y <= z).	rewrite !leEjoin; apply/eqP/andP => [<-\|[/eqP<- /eqP<-]]. split; apply/eqP; last by rewrite joinA -joinA joinxx. by rewrite -joinA (joinC _ x) (joinA x) joinxx. by rewrite -!joinA (joinC y) -joinA (joinA x) !joinxx. Qed.	Fact	leUx	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "join", "joinA", "joinC", "joinxx", "last", "leEjoin", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEjoin x y : (y <= x) = (join x y == x).	Proof. rewrite leEmeet; apply/eqP/eqP => <-. by rewrite meetC meetKU. by rewrite joinC joinKI. Qed.	Fact	leEjoin	order	order/order.v	[ "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", "join", "joinC", "joinKI", "leEmeet", "meetC", "meetKU" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUr : right_distributive (@meet _ T) (@join _ T).	Proof. by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed.	Let	meetUr	order	order/order.v	[ "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", "meetC", "meetUl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinIl : left_distributive (@join _ T) (@meet _ T).	Proof. by move=> x y z; rewrite meetUr joinIK meetUl -joinA meetUKC. Qed.	Let	joinIl	order	order/order.v	[ "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", "joinA", "joinIK", "meet", "meetUKC", "meetUl", "meetUr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_refl : reflexive le.	Proof. by move=> x; rewrite le_def meetxx. Qed.	Fact	le_refl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "le", "le_def", "meetxx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_anti : antisymmetric le.	Proof. by move=> x y; rewrite !le_def meetC => /andP [] /eqP {2}<- /eqP ->. Qed.	Fact	le_anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "le", "le_def", "meetC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_trans : transitive le.	Proof. move=> y x z; rewrite !le_def => /eqP lexy /eqP leyz; apply/eqP. by rewrite -[in LHS]lexy -meetA leyz lexy. Qed.	Fact	le_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", "le", "le_def", "meetA" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_le_def x y : lt x y = (le x y) && ~~ (le y x).	Proof. rewrite lt_def andbC; case/boolP: (le x y) => //= xy. congr negb; apply/eqP/idP => [->\|yx]; first exact/le_refl. by apply/le_anti/andP; split. Qed.	Fact	lt_le_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", "le", "le_anti", "le_refl", "lt", "lt_def", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build d T	:= (BDistrLattice_hasSectionalComplement.Build d T) (only parsing).	Notation	Build	order	order/order.v	[ "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
hasRelativeComplement d T	:= (BDistrLattice_hasSectionalComplement d T) (only parsing).	Notation	hasRelativeComplement	order	order/order.v	[ "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
rcompl x y z	:= (x `&` y) `\|` diff (y `\|` x) z.	Definition	rcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPmeet x y z : ((x `&` y) `\|` z) `&` rcompl x y z = x `&` y.	Proof. by rewrite meetUr joinIKC meetUl diffKI joinx0 meetKU. Qed.	Fact	rcomplPmeet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "diffKI", "joinIKC", "joinx0", "meetKU", "meetUl", "meetUr", "rcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPjoin x y z : ((y `\|` x) `&` z) `\|` rcompl x y z = y `\|` x.	Proof. by rewrite joinCA joinIB joinA meetUK joinC. Qed.	Fact	rcomplPjoin	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinA", "joinC", "joinCA", "joinIB", "meetUK", "rcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffErcompl x y : diff x y = rcompl \bot x y.	Proof. by rewrite /rcompl meet0x join0x joinx0. Qed.	Fact	diffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "diff", "join0x", "joinx0", "meet0x", "rcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcompl x y z	:= (y `\|` x) `&` codiff (x `&` y) z.	Definition	rcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPmeet x y z : ((x `&` y) `\|` z) `&` rcompl x y z = x `&` y.	Proof. by rewrite meetCA meetUB meetA joinIK. Qed.	Fact	rcomplPmeet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinIK", "meetA", "meetCA", "rcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcomplPjoin x y z : ((y `\|` x) `&` z) `\|` rcompl x y z = y `\|` x.	Proof. by rewrite joinIr meetUKC joinIl codiffKU meetx1 joinKI. Qed.	Fact	rcomplPjoin	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "joinIl", "joinIr", "joinKI", "meetUKC", "meetx1", "rcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffErcompl x y : codiff x y = rcompl x \top y.	Proof. by rewrite /rcompl join1x meet1x meetx1. Qed.	Fact	codiffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff", "join1x", "meet1x", "meetx1", "rcompl", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build d T	:= (CBDistrLattice_hasComplement.Build d T) (only parsing).	Notation	Build	order	order/order.v	[ "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
hasComplement d T	:= (CBDistrLattice_hasComplement d T) (only parsing).	Notation	hasComplement	order	order/order.v	[ "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
complEcodiff (x : T) : compl x = codiff (\bot : T) x.	Proof. by rewrite complEdiff diffErcompl. Qed.	Fact	complEcodiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "codiff", "compl", "complEdiff", "diffErcompl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEdiff (x : T) : compl x = (\top : T) `\` x.	Proof. by rewrite complEcodiff codiffErcompl. Qed.	Fact	complEdiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiffErcompl", "compl", "complEcodiff", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diff x y	:= x `&` compl y.	Definition	diff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "compl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiff x y	:= x `\|` compl y.	Definition	codiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "compl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffKI x y : y `&` diff x y = \bot.	Proof. by rewrite meetCA meetxC meetx0. Qed.	Fact	diffKI	order	order/order.v	[ "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", "diff", "meetCA", "meetx0", "meetxC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinIB x y : (x `&` y) `\|` diff x y = x.	Proof. by rewrite -meetUr joinxC meetx1. Qed.	Fact	joinIB	order	order/order.v	[ "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", "joinxC", "meetUr", "meetx1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codiffErcompl x y : codiff x y = rcompl x \top y.	Proof. by rewrite /rcompl /diff join1x meetx1 meet1x. Qed.	Fact	codiffErcompl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "codiff", "diff", "join1x", "meet1x", "meetx1", "rcompl", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEdiff x : compl x = diff \top x.	Proof. exact/esym/meet1x. Qed.	Fact	complEdiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "compl", "diff", "meet1x", "top" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complEcodiff x : compl x = codiff \bot x.	Proof. exact/esym/join0x. Qed.	Fact	complEcodiff	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "bot", "codiff", "compl", "join0x" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetUl : @left_distributive T T meet join.	Proof. pose leP x y := lcomparable_leP (le_total x y); move=> x y z; apply/esym. by case: (leP x y) (leP x z) (leP y z) => [\|/ltW] xy [\|/ltW] xz [\|/ltW] yz; (apply/join_idPl \|\| apply/join_idPr) => //; apply: le_trans xy. Qed.	Fact	meetUl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "apply", "join", "join_idPl", "join_idPr", "lcomparable_leP", "leP", "le_total", "le_trans", "ltW", "meet" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparableT x y : x >=< y	:= le_total x y.	Let	comparableT	order	order/order.v	[ "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_total" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltgtP 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. exact: comparable_ltgtP. Qed.	Fact	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",...	[ "comparable_ltgtP", "compare", "max", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leP x y : le_xor_gt x y (min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).	Proof. exact: comparable_leP. Qed.	Fact	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_leP", "le_xor_gt", "max", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet	:= @min _ T.	Definition	meet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
join	:= @max _ T.	Definition	join	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "max" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetC : commutative meet.	Proof. by move=> x y; rewrite /meet; have [] := ltgtP. Qed.	Fact	meetC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "ltgtP", "meet" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinC : commutative join.	Proof. by move=> x y; rewrite /join; have [] := ltgtP. Qed.	Fact	joinC	order	order/order.v	[ "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", "ltgtP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetA : associative meet.	Proof. move=> x y z; rewrite /meet /min !(fun_if, if_arg). case: (leP z y) (leP y x) (leP z x) => [] zy [] yx [] zx//=. by have := le_lt_trans (le_trans zy yx) zx; rewrite ltxx. by apply/eqP; rewrite eq_le zx ltW// (lt_trans yx). Qed.	Fact	meetA	order	order/order.v	[ "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", "leP", "le_lt_trans", "le_trans", "ltW", "lt_trans", "ltxx", "meet", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinA : associative join.	Proof. move=> x y z; rewrite /meet /min !(fun_if, if_arg). case: (leP z y) (leP y x) (leP z x) => [] zy [] yx [] zx//=. by have := le_lt_trans (le_trans zy yx) zx; rewrite ltxx. by apply/eqP; rewrite eq_le zx ltW// (lt_trans yx). Qed.	Fact	joinA	order	order/order.v	[ "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", "join", "leP", "le_lt_trans", "le_trans", "ltW", "lt_trans", "ltxx", "meet", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinKI y x : meet x (join x y) = x.	Proof. rewrite /meet /join /min /max !(fun_if, if_arg). by have []// := ltgtP x y; rewrite ltxx. Qed.	Fact	joinKI	order	order/order.v	[ "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", "ltgtP", "ltxx", "max", "meet", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKU y x : join x (meet x y) = x.	Proof. rewrite /meet /join /min /max !(fun_if, if_arg). by have []// := ltgtP x y; rewrite ltxx. Qed.	Fact	meetKU	order	order/order.v	[ "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", "ltgtP", "ltxx", "max", "meet", "min" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEmeet x y : (x <= y) = (meet x y == x).	Proof. by rewrite /meet; case: leP => ?; rewrite ?eqxx ?lt_eqF. Qed.	Fact	leEmeet	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eqxx", "leP", "lt_eqF", "meet" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBUK y x : y <= x -> (x `\` y) `|` y = x.

Proof. by move=> leyx; rewrite joinC leBKU. Qed.

Lemma

leBUK

order

order/order.v

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

[ "joinC", "leBKU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBLR x y z : (x `\` y <= z) = (x <= y `|` z).

Proof. apply/idP/idP; first by move=> /join_r <-; rewrite joinA diffKU joinAC leUr. by rewrite -{1}[x](joinIB y) => /(leU2r_le (diffIK _ _)). Qed.

Lemma

leBLR

order

order/order.v

[ "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", "diffIK", "diffKU", "joinA", "joinAC", "joinIB", "join_r", "leU2r_le", "leUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffUx x y z : (x `|` y) `\` z = (x `\` z) `|` (y `\` z).

Proof. apply/eqP; rewrite eq_le leUx !leBl ?leUr ?leUl ?andbT //. by rewrite leBLR joinA diffKU joinAC diffKU joinAC -joinA leUr. Qed.

Lemma

diffUx

order

order/order.v

[ "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", "diffKU", "eq_le", "joinA", "joinAC", "leBLR", "leBl", "leUl", "leUr", "leUx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diff_eq0 x y : (x `\` y == \bot) = (x <= y).

Proof. by rewrite -lex0 leBLR joinx0. Qed.

Lemma

diff_eq0

order

order/order.v

[ "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", "joinx0", "leBLR", "lex0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinxB x y z : x `|` (y `\` z) = ((x `|` y) `\` z) `|` (x `&` z).

Proof. by rewrite diffUx joinAC joinBI. Qed.

Lemma

joinxB

order

order/order.v

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

[ "diffUx", "joinAC", "joinBI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinBx x y z : (y `\` z) `|` x = ((y `|` x) `\` z) `|` (z `&` x).

Proof. by rewrite ![_ `|` x]joinC ![_ `&` x]meetC joinxB. Qed.

Lemma

joinBx

order

order/order.v

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

[ "joinC", "joinxB", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBr z x y : x <= y -> z `\` y <= z `\` x.

Proof. by move=> lexy; rewrite leBLR joinxB meet_r ?leBUK ?leUr ?lexUl. Qed.

Lemma

leBr

order

order/order.v

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

[ "joinxB", "leBLR", "leBUK", "leUr", "lexUl", "meet_r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leB2 x y z t : x <= z -> t <= y -> x `\` y <= z `\` t.

Proof. by move=> /(@leBl t) ? /(@leBr x) /le_trans ->. Qed.

Lemma

leB2

order

order/order.v

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

[ "leBl", "leBr", "le_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_eq0E_diff z x y : x <= z -> (x `&` y == \bot) = (x <= z `\` y).

Proof. move=> xz; apply/idP/idP; last by move=> /meet_r <-; rewrite -meetA meetBI. by move=> /eqP xIy_eq0; rewrite -[x](joinIB y) xIy_eq0 join0x leBl. Qed.

Lemma

meet_eq0E_diff

order

order/order.v

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

[ "apply", "bot", "join0x", "joinIB", "last", "leBl", "meetA", "meetBI", "meet_r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBRL x y z : (x <= z `\` y) = (x <= z) && (x `&` y == \bot).

Proof. apply/idP/idP => [xyz|]; first by rewrite (@meet_eq0E_diff z) // (le_trans xyz). by move=> /andP [?]; rewrite -meet_eq0E_diff. Qed.

Lemma

leBRL

order

order/order.v

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

[ "apply", "bot", "le_trans", "meet_eq0E_diff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_diff x y z : (x `\` y == z) = (z <= x <= y `|` z) && (z `&` y == \bot).

Proof. by rewrite eq_le leBLR leBRL andbCA andbA. Qed.

Lemma

eq_diff

order

order/order.v

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

[ "bot", "eq_le", "leBLR", "leBRL" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffxU x y z : z `\` (x `|` y) = (z `\` x) `&` (z `\` y).

Proof. apply/eqP; rewrite eq_le lexI !leBr ?leUl ?leUr //=. rewrite leBRL leIx2 ?leBx //= meetUr meetAC diffIK -meetA diffIK. by rewrite meet0x meetx0 joinx0. Qed.

Lemma

diffxU

order

order/order.v

[ "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", "diffIK", "eq_le", "joinx0", "leBRL", "leBr", "leBx", "leIx2", "leUl", "leUr", "lexI", "meet0x", "meetA", "meetAC", "meetUr", "meetx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffx0 x : x `\` \bot = x.

Proof. by apply/eqP; rewrite eq_diff join0x meetx0 lexx eqxx. Qed.

Lemma

diffx0

order

order/order.v

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

[ "apply", "bot", "eq_diff", "eqxx", "join0x", "lexx", "meetx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diff0x x : \bot `\` x = \bot.

Proof. by apply/eqP; rewrite eq_diff joinx0 meet0x lexx eqxx le0x. Qed.

Lemma

diff0x

order

order/order.v

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

[ "apply", "bot", "eq_diff", "eqxx", "joinx0", "le0x", "lexx", "meet0x" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffIx x y z : (x `&` y) `\` z = (x `\` z) `&` (y `\` z).

Proof. apply/eqP; rewrite eq_diff joinIr ?leI2 ?diffKU ?leUr ?leBx //=. by rewrite -meetA diffIK meetx0. Qed.

Lemma

diffIx

order

order/order.v

[ "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", "diffIK", "diffKU", "eq_diff", "joinIr", "leBx", "leI2", "leUr", "meetA", "meetx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetxB x y z : x `&` (y `\` z) = (x `&` y) `\` z.

Proof. by rewrite diffIx -{1}[x](joinBI z) meetUl meetIB joinx0. Qed.

Lemma

meetxB

order

order/order.v

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

[ "diffIx", "joinBI", "joinx0", "meetIB", "meetUl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetBx x y z : (x `\` y) `&` z = (x `&` z) `\` y.

Proof. by rewrite ![_ `&` z]meetC meetxB. Qed.

Lemma

meetBx

order

order/order.v

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

[ "meetC", "meetxB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffxI x y z : x `\` (y `&` z) = (x `\` y) `|` (x `\` z).

Proof. apply/eqP; rewrite eq_diff leUx !leBx //= joinIl joinA joinCA !diffKU. rewrite joinCA -joinA [_ `|` x]joinC ![x `|` _]join_l //. by rewrite -joinIl leUr /= meetUl {1}[_ `&` z]meetC ?meetBI joinx0. Qed.

Lemma

diffxI

order

order/order.v

[ "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", "diffKU", "eq_diff", "joinA", "joinC", "joinCA", "joinIl", "join_l", "joinx0", "leBx", "leUr", "leUx", "meetBI", "meetC", "meetUl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffBx x y z : (x `\` y) `\` z = x `\` (y `|` z).

Proof. apply/eqP; rewrite eq_diff leBr ?leUl //=. by rewrite diffxU joinIr diffKU -joinIr meet_l ?leUr //= -meetA diffIK meetx0. Qed.

Lemma

diffBx

order

order/order.v

[ "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", "diffIK", "diffKU", "diffxU", "eq_diff", "joinIr", "leBr", "leUl", "leUr", "meetA", "meet_l", "meetx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffxB x y z : x `\` (y `\` z) = (x `\` y) `|` (x `&` z).

Proof. rewrite -[y in RHS](joinIB z) diffxU joinIl diffxI -joinA joinBI join_r //. by rewrite joinBx meetKU meetA meetAC diffIK meet0x joinx0 meet_r. Qed.

Lemma

diffxB

order

order/order.v

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

[ "diffIK", "diffxI", "diffxU", "joinA", "joinBI", "joinBx", "joinIB", "joinIl", "join_r", "joinx0", "meet0x", "meetA", "meetAC", "meetKU", "meet_r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinBK x y : (y `|` x) `\` x = (y `\` x).

Proof. by rewrite diffUx diffxx joinx0. Qed.

Lemma

joinBK

order

order/order.v

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

[ "diffUx", "diffxx", "joinx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinBKC x y : (x `|` y) `\` x = (y `\` x).

Proof. by rewrite diffUx diffxx join0x. Qed.

Lemma

joinBKC

order

order/order.v

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

[ "diffUx", "diffxx", "join0x" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disj_le x y : x `&` y == \bot -> (x <= y) = (x == \bot).

Proof. by rewrite [x == \bot]eq_sym -eq_meetl => /eqP ->. Qed.

Lemma

disj_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",...

[ "bot", "eq_meetl", "eq_sym" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disj_leC x y : y `&` x == \bot -> (x <= y) = (x == \bot).

Proof. by rewrite meetC => /disj_le. Qed.

Lemma

disj_leC

order

order/order.v

[ "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", "disj_le", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disj_diffl x y : x `&` y == \bot -> x `\` y = x.

Proof. by move=> dxy; apply/eqP; rewrite eq_diff dxy lexx leUr. Qed.

Lemma

disj_diffl

order

order/order.v

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

[ "apply", "bot", "eq_diff", "leUr", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disj_diffr x y : x `&` y == \bot -> y `\` x = y.

Proof. by rewrite meetC => /disj_diffl. Qed.

Lemma

disj_diffr

order

order/order.v

[ "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", "disj_diffl", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0B x y : x < y -> \bot < y `\` x.

Proof. by move=> ?; rewrite lt_leAnge le0x leBLR joinx0 /= lt_geF. Qed.

Lemma

lt0B

order

order/order.v

[ "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", "joinx0", "le0x", "leBLR", "lt_geF", "lt_leAnge" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffErcompl x y : codiff x y = rcompl x \top y.

Proof. exact: codiffErcompl. Qed.

Lemma

codiffErcompl

order

order/order.v

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

[ "codiff", "rcompl", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEdiff x : ~` x = \top `\` x.

Proof. exact: complEdiff. Qed.

Lemma

complEdiff

order

order/order.v

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

[ "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complE

:= complEdiff.

Notation

complE

order

order/order.v

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

[ "complEdiff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEcodiff x : ~` x = codiff \bot x.

Proof. exact: complEcodiff. Qed.

Lemma

complEcodiff

order

order/order.v

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

[ "bot", "codiff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complErcompl x : ~` x = rcompl \bot \top x.

Proof. by rewrite complEdiff diffErcompl. Qed.

Lemma

complErcompl

order

order/order.v

[ "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", "complEdiff", "diffErcompl", "rcompl", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diff1x x : \top `\` x = ~` x.

Proof. exact/esym/complEdiff. Qed.

Lemma

diff1x

order

order/order.v

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

[ "complEdiff", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffE x y : x `\` y = x `&` ~` y.

Proof. by rewrite complEdiff meetxB meetx1. Qed.

Lemma

diffE

order

order/order.v

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

[ "complEdiff", "meetx1", "meetxB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complK : involutive (@compl _ L).

Proof. by move=> x; rewrite !complEdiff diffxB diffxx meet1x join0x. Qed.

Lemma

complK

order

order/order.v

[ "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", "complEdiff", "diffxB", "diffxx", "join0x", "meet1x" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl_inj : injective (@compl _ L).

Proof. exact/inv_inj/complK. Qed.

Lemma

compl_inj

order

order/order.v

[ "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", "complK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disj_leC x y : (x `&` y == \bot) = (x <= ~` y).

Proof. by rewrite -diff_eq0 diffE complK. Qed.

Lemma

disj_leC

order

order/order.v

[ "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", "complK", "diffE", "diff_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leCx x y : (~` x <= y) = (~` y <= x).

Proof. by rewrite !complEdiff !leBLR joinC. Qed.

Lemma

leCx

order

order/order.v

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

[ "complEdiff", "joinC", "leBLR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexC x y : (x <= ~` y) = (y <= ~` x).

Proof. by rewrite -[x in LHS]complK leCx complK. Qed.

Lemma

lexC

order

order/order.v

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

[ "complK", "leCx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leC x y : (~` x <= ~` y) = (y <= x).

Proof. by rewrite leCx complK. Qed.

Lemma

leC

order

order/order.v

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

[ "complK", "leCx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complU x y : ~` (x `|` y) = ~` x `&` ~` y.

Proof. by rewrite !complEdiff diffxU. Qed.

Lemma

complU

order

order/order.v

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

[ "complEdiff", "diffxU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complI x y : ~` (x `&` y) = ~` x `|` ~` y.

Proof. by rewrite !complEdiff diffxI. Qed.

Lemma

complI

order

order/order.v

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

[ "complEdiff", "diffxI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinxC x : x `|` ~` x = \top.

Proof. by rewrite complEdiff diffKU joinx1. Qed.

Lemma

joinxC

order

order/order.v

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

[ "complEdiff", "diffKU", "joinx1", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinCx x : ~` x `|` x = \top.

Proof. by rewrite joinC joinxC. Qed.

Lemma

joinCx

order

order/order.v

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

[ "joinC", "joinxC", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetxC x : x `&` ~` x = \bot.

Proof. by rewrite complEdiff diffKI. Qed.

Lemma

meetxC

order

order/order.v

[ "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", "complEdiff", "diffKI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetCx x : ~` x `&` x = \bot.

Proof. by rewrite meetC meetxC. Qed.

Lemma

meetCx

order

order/order.v

[ "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", "meetC", "meetxC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl1 : ~` \top = \bot :> L.

Proof. by rewrite complEdiff diffxx. Qed.

Lemma

compl1

order

order/order.v

[ "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", "complEdiff", "diffxx", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl0 : ~` \bot = \top :> L.

Proof. by rewrite -compl1 complK. Qed.

Lemma

compl0

order

order/order.v

[ "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", "compl1", "complK", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complB x y : ~` (x `\` y) = ~` x `|` y.

Proof. by rewrite diffE complI complK. Qed.

Lemma

complB

order

order/order.v

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

[ "complI", "complK", "diffE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leBC x y : x `\` y <= ~` y.

Proof. by rewrite leBLR joinxC lex1. Qed.

Lemma

leBC

order

order/order.v

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

[ "joinxC", "leBLR", "lex1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl_joins (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) : ~` (\join_(j <- r | P j) F j) = \meet_(j <- r | P j) ~` F j.

Proof. by elim/big_rec2: _=> [|i x y ? <-]; rewrite ?compl0 ?complU. Qed.

Lemma

compl_joins

order

order/order.v

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

[ "big_rec2", "compl0", "complU", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl_meets (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) : ~` (\meet_(j <- r | P j) F j) = \join_(j <- r | P j) ~` F j.

Proof. by elim/big_rec2: _=> [|i x y ? <-]; rewrite ?compl1 ?complI. Qed.

Lemma

compl_meets

order

order/order.v

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

[ "big_rec2", "compl1", "complI", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge_anti : antisymmetric (fun x y => @le d T y x).

Proof. by move=> x y; rewrite andbC; apply: le_anti. Qed.

Let

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",...

[ "apply", "le", "le_anti" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_le_def x y : lt x y = le x y && ~~ le y x.

Proof. rewrite lt_def andbC; case /boolP: (le x y) => //= xy. have [->|/negP xyE /=] := eqVneq y x; first by rewrite le_refl. by apply/esym/negP => yx; apply/xyE/eqP/le_anti; rewrite yx. Qed.

Let

lt_le_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", "eqVneq", "le", "le_anti", "le_refl", "lt", "lt_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_anti : antisymmetric le.

Proof. move=> x y; rewrite !le_def [y == _]eq_sym. have [//|neq_xy/=] := eqVneq x y => /andP[xy yx]. by have := lt_trans xy yx; rewrite lt_irr. Qed.

Let

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",...

[ "eqVneq", "eq_sym", "le", "le_def", "lt_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetxx : idempotent_op meet.

Proof. by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Fact

meetxx

order

order/order.v

[ "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", "idempotent_op", "leEmeet", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lexI x y z : (x <= meet y z) = (x <= y) && (x <= z).

Proof. rewrite !leEmeet; apply/eqP/andP => [<-|[/eqP<- /eqP<-]]. split; apply/eqP; last by rewrite meetA -meetA meetxx. by rewrite -!meetA (meetC z) (meetA y) meetxx. by rewrite -!meetA (meetC z) -meetA (meetA y) !meetxx. Qed.

Fact

lexI

order

order/order.v

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

[ "apply", "last", "leEmeet", "meet", "meetA", "meetC", "meetxx", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinxx : idempotent_op join.

Proof. by move=> x; apply/eqP; rewrite -leEjoin. Qed.

Fact

joinxx

order

order/order.v

[ "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", "idempotent_op", "join", "leEjoin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leUx x y z : (join x y <= z) = (x <= z) && (y <= z).

rewrite !leEjoin; apply/eqP/andP => [<-|[/eqP<- /eqP<-]]. split; apply/eqP; last by rewrite joinA -joinA joinxx. by rewrite -joinA (joinC _ x) (joinA x) joinxx. by rewrite -!joinA (joinC y) -joinA (joinA x) !joinxx. Qed.

Fact

leUx

order

order/order.v

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

[ "apply", "join", "joinA", "joinC", "joinxx", "last", "leEjoin", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEjoin x y : (y <= x) = (join x y == x).

Proof. rewrite leEmeet; apply/eqP/eqP => <-. by rewrite meetC meetKU. by rewrite joinC joinKI. Qed.

Fact

leEjoin

order

order/order.v

[ "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", "join", "joinC", "joinKI", "leEmeet", "meetC", "meetKU" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetUr : right_distributive (@meet _ T) (@join _ T).

Proof. by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed.

Let

meetUr

order

order/order.v

[ "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", "meetC", "meetUl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinIl : left_distributive (@join _ T) (@meet _ T).

Proof. by move=> x y z; rewrite meetUr joinIK meetUl -joinA meetUKC. Qed.

Let

joinIl

order

order/order.v

[ "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", "joinA", "joinIK", "meet", "meetUKC", "meetUl", "meetUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_refl : reflexive le.

Proof. by move=> x; rewrite le_def meetxx. Qed.

Fact

le_refl

order

order/order.v

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

[ "le", "le_def", "meetxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_anti : antisymmetric le.

Proof. by move=> x y; rewrite !le_def meetC => /andP [] /eqP {2}<- /eqP ->. Qed.

Fact

le_anti

order

order/order.v

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

[ "le", "le_def", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_trans : transitive le.

Proof. move=> y x z; rewrite !le_def => /eqP lexy /eqP leyz; apply/eqP. by rewrite -[in LHS]lexy -meetA leyz lexy. Qed.

Fact

le_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", "le", "le_def", "meetA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_le_def x y : lt x y = (le x y) && ~~ (le y x).

Proof. rewrite lt_def andbC; case/boolP: (le x y) => //= xy. congr negb; apply/eqP/idP => [->|yx]; first exact/le_refl. by apply/le_anti/andP; split. Qed.

Fact

lt_le_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", "le", "le_anti", "le_refl", "lt", "lt_def", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build d T

:= (BDistrLattice_hasSectionalComplement.Build d T) (only parsing).

Notation

Build

order

order/order.v

[ "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

hasRelativeComplement d T

:= (BDistrLattice_hasSectionalComplement d T) (only parsing).

Notation

hasRelativeComplement

order

order/order.v

[ "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

rcompl x y z

:= (x `&` y) `|` diff (y `|` x) z.

Definition

rcompl

order

order/order.v

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

[ "diff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite meetUr joinIKC meetUl diffKI joinx0 meetKU. Qed.

Fact

rcomplPmeet

order

order/order.v

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

[ "diffKI", "joinIKC", "joinx0", "meetKU", "meetUl", "meetUr", "rcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite joinCA joinIB joinA meetUK joinC. Qed.

Fact

rcomplPjoin

order

order/order.v

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

[ "joinA", "joinC", "joinCA", "joinIB", "meetUK", "rcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffErcompl x y : diff x y = rcompl \bot x y.

Proof. by rewrite /rcompl meet0x join0x joinx0. Qed.

Fact

diffErcompl

order

order/order.v

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

[ "bot", "diff", "join0x", "joinx0", "meet0x", "rcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcompl x y z

:= (y `|` x) `&` codiff (x `&` y) z.

Definition

rcompl

order

order/order.v

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

[ "codiff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite meetCA meetUB meetA joinIK. Qed.

Fact

rcomplPmeet

order

order/order.v

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

[ "joinIK", "meetA", "meetCA", "rcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite joinIr meetUKC joinIl codiffKU meetx1 joinKI. Qed.

Fact

rcomplPjoin

order

order/order.v

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

[ "joinIl", "joinIr", "joinKI", "meetUKC", "meetx1", "rcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffErcompl x y : codiff x y = rcompl x \top y.

Proof. by rewrite /rcompl join1x meet1x meetx1. Qed.

Fact

codiffErcompl

order

order/order.v

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

[ "codiff", "join1x", "meet1x", "meetx1", "rcompl", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build d T

:= (CBDistrLattice_hasComplement.Build d T) (only parsing).

Notation

Build

order

order/order.v

[ "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

hasComplement d T

:= (CBDistrLattice_hasComplement d T) (only parsing).

Notation

hasComplement

order

order/order.v

[ "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

complEcodiff (x : T) : compl x = codiff (\bot : T) x.

Proof. by rewrite complEdiff diffErcompl. Qed.

Fact

complEcodiff

order

order/order.v

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

[ "bot", "codiff", "compl", "complEdiff", "diffErcompl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEdiff (x : T) : compl x = (\top : T) `\` x.

Proof. by rewrite complEcodiff codiffErcompl. Qed.

Fact

complEdiff

order

order/order.v

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

[ "codiffErcompl", "compl", "complEcodiff", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diff x y

:= x `&` compl y.

Definition

diff

order

order/order.v

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

[ "compl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiff x y

:= x `|` compl y.

Definition

codiff

order

order/order.v

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

[ "compl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffKI x y : y `&` diff x y = \bot.

Proof. by rewrite meetCA meetxC meetx0. Qed.

Fact

diffKI

order

order/order.v

[ "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", "diff", "meetCA", "meetx0", "meetxC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinIB x y : (x `&` y) `|` diff x y = x.

Proof. by rewrite -meetUr joinxC meetx1. Qed.

Fact

joinIB

order

order/order.v

[ "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", "joinxC", "meetUr", "meetx1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codiffErcompl x y : codiff x y = rcompl x \top y.

Proof. by rewrite /rcompl /diff join1x meetx1 meet1x. Qed.

Fact

codiffErcompl

order

order/order.v

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

[ "codiff", "diff", "join1x", "meet1x", "meetx1", "rcompl", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEdiff x : compl x = diff \top x.

Proof. exact/esym/meet1x. Qed.

Fact

complEdiff

order

order/order.v

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

[ "compl", "diff", "meet1x", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complEcodiff x : compl x = codiff \bot x.

Proof. exact/esym/join0x. Qed.

Fact

complEcodiff

order

order/order.v

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

[ "bot", "codiff", "compl", "join0x" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetUl : @left_distributive T T meet join.

Proof. pose leP x y := lcomparable_leP (le_total x y); move=> x y z; apply/esym. by case: (leP x y) (leP x z) (leP y z) => [|/ltW] xy [|/ltW] xz [|/ltW] yz; (apply/join_idPl || apply/join_idPr) => //; apply: le_trans xy. Qed.

Fact

meetUl

order

order/order.v

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

[ "apply", "join", "join_idPl", "join_idPr", "lcomparable_leP", "leP", "le_total", "le_trans", "ltW", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparableT x y : x >=< y

:= le_total x y.

Let

comparableT

order

order/order.v

[ "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_total" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltgtP 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. exact: comparable_ltgtP. Qed.

Fact

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",...

[ "comparable_ltgtP", "compare", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leP x y : le_xor_gt x y (min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).

Proof. exact: comparable_leP. Qed.

Fact

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_leP", "le_xor_gt", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet

:= @min _ T.

Definition

meet

order

order/order.v

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

[ "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

join

:= @max _ T.

Definition

join

order

order/order.v

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

[ "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetC : commutative meet.

Proof. by move=> x y; rewrite /meet; have [] := ltgtP. Qed.

Fact

meetC

order

order/order.v

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

[ "ltgtP", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinC : commutative join.

Proof. by move=> x y; rewrite /join; have [] := ltgtP. Qed.

Fact

joinC

order

order/order.v

[ "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", "ltgtP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetA : associative meet.

Proof. move=> x y z; rewrite /meet /min !(fun_if, if_arg). case: (leP z y) (leP y x) (leP z x) => [] zy [] yx [] zx//=. by have := le_lt_trans (le_trans zy yx) zx; rewrite ltxx. by apply/eqP; rewrite eq_le zx ltW// (lt_trans yx). Qed.

Fact

meetA

order

order/order.v

[ "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", "leP", "le_lt_trans", "le_trans", "ltW", "lt_trans", "ltxx", "meet", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinA : associative join.

Proof. move=> x y z; rewrite /meet /min !(fun_if, if_arg). case: (leP z y) (leP y x) (leP z x) => [] zy [] yx [] zx//=. by have := le_lt_trans (le_trans zy yx) zx; rewrite ltxx. by apply/eqP; rewrite eq_le zx ltW// (lt_trans yx). Qed.

Fact

joinA

order

order/order.v

[ "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", "join", "leP", "le_lt_trans", "le_trans", "ltW", "lt_trans", "ltxx", "meet", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinKI y x : meet x (join x y) = x.

Proof. rewrite /meet /join /min /max !(fun_if, if_arg). by have []// := ltgtP x y; rewrite ltxx. Qed.

Fact

joinKI

order

order/order.v

[ "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", "ltgtP", "ltxx", "max", "meet", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetKU y x : join x (meet x y) = x.

Proof. rewrite /meet /join /min /max !(fun_if, if_arg). by have []// := ltgtP x y; rewrite ltxx. Qed.

Fact

meetKU

order

order/order.v

[ "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", "ltgtP", "ltxx", "max", "meet", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEmeet x y : (x <= y) = (meet x y == x).

Proof. by rewrite /meet; case: leP => ?; rewrite ?eqxx ?lt_eqF. Qed.

Fact

leEmeet

order

order/order.v

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

[ "eqxx", "leP", "lt_eqF", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d