phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
ltNleif x y C : x <= y ?= iff ~~ C -> (x < y) = C.	Proof. by move=> /lt_leif; rewrite negbK. Qed.	Lemma	ltNleif	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lt_leif" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif_anti C1 C2 x y : (x < y ?<= if C1) && (y < x ?<= if C2) = C1 && C2 && (x == y).	Proof. by case: C1 C2 => [][]; rewrite lte_anti. Qed.	Lemma	lteif_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",...	[ "lte_anti" ]	lteif	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteifN C x y : x < y ?<= if ~~ C -> ~~ (y < x ?<= if C).	Proof. by case: C => /=; case: comparableP. Qed.	Lemma	lteifN	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minEle x y : min x y = if x <= y then x else y.	Proof. by case: comparableP. Qed.	Lemma	minEle	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]	min and max	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxEle x y : max x y = if x <= y then y else x.	Proof. by case: comparableP. Qed.	Lemma	maxEle	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minEgt x y : x >=< y -> min x y = if x > y then y else x.	Proof. by case: comparableP. Qed.	Lemma	comparable_minEgt	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxEgt x y : x >=< y -> max x y = if x > y then x else y.	Proof. by case: comparableP. Qed.	Lemma	comparable_maxEgt	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minEge x y : x >=< y -> min x y = if x >= y then y else x.	Proof. by case: comparableP. Qed.	Lemma	comparable_minEge	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxEge x y : x >=< y -> max x y = if x >= y then x else y.	Proof. by case: comparableP. Qed.	Lemma	comparable_maxEge	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_l x y : x <= y -> min x y = x.	Proof. by case: comparableP. Qed.	Lemma	min_l	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_r x y : y <= x -> min x y = y.	Proof. by case: comparableP. Qed.	Lemma	min_r	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_l x y : y <= x -> max x y = x.	Proof. by case: comparableP. Qed.	Lemma	max_l	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_r x y : x <= y -> max x y = y.	Proof. by case: comparableP. Qed.	Lemma	max_r	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_minl x y : (min x y == x) = (x <= y).	Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.	Lemma	eq_minl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "eqxx", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_maxr x y : (max x y == y) = (x <= y).	Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.	Lemma	eq_maxr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "eqxx", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_idPl x y : reflect (min x y = x) (x <= y).	Proof. by rewrite -eq_minl; apply/eqP. Qed.	Lemma	min_idPl	order	order/order.v	[ "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_minl", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_idPr x y : reflect (max x y = y) (x <= y).	Proof. by rewrite -eq_maxr; apply/eqP. Qed.	Lemma	max_idPr	order	order/order.v	[ "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_maxr", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minC : min x y = min y x.	Proof. by case: comparableP cmp_xy. Qed.	Lemma	comparable_minC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxC : max x y = max y x.	Proof. by case: comparableP cmp_xy. Qed.	Lemma	comparable_maxC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_eq_minr : (min x y == y) = (y <= x).	Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.	Lemma	comparable_eq_minr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "eqxx", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_eq_maxl : (max x y == x) = (y <= x).	Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.	Lemma	comparable_eq_maxl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "eqxx", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_min_idPr : reflect (min x y = y) (y <= x).	Proof. by rewrite -comparable_eq_minr; apply/eqP. Qed.	Lemma	comparable_min_idPr	order	order/order.v	[ "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", "comparable_eq_minr", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_max_idPl : reflect (max x y = x) (y <= x).	Proof. by rewrite -comparable_eq_maxl; apply/eqP. Qed.	Lemma	comparable_max_idPl	order	order/order.v	[ "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", "comparable_eq_maxl", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_lteifNE C : x >=< y -> (x < y ?<= if ~~ C) = ~~ (y < x ?<= if C).	Proof. by case: C => /=; case: comparableP. Qed.	Lemma	comparable_lteifNE	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
P	:= comparableP.	Let	P	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_max_minl : max (min x y) z = min (max x z) (max y z).	Proof. move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/=. move: (P x y) (P x z) (P y z). move=> [xy\|xy\|xy\|<-] [xz\|xz\|xz\|<-] [yz\|yz\|yz\|//->]//= _; rewrite ?ltxx//. - by have := lt_trans xy (lt_trans yz xz); rewrite ltxx. - by have := lt_trans xy (lt_trans xz yz); rewrite ltxx. Qed.	Lemma	comparable_max_minl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lt_trans", "ltxx", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_le_min2 : x <= z -> y <= w -> Order.min x y <= Order.min z w.	Proof. move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw. - exact: le_trans xy yw. - exact: le_trans (ltW xy) xz. Qed.	Lemma	comparable_le_min2	order	order/order.v	[ "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_trans", "ltW", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_le_max2 : x <= z -> y <= w -> Order.max x y <= Order.max z w.	Proof. move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw. - exact: le_trans yw (ltW zw). - exact: le_trans xz zw. Qed.	Lemma	comparable_le_max2	order	order/order.v	[ "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_trans", "ltW", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minAC x y z : x >=< y -> x >=< z -> y >=< z -> min (min x y) z = min (min x z) y.	Proof. move=> xy xz yz; rewrite -comparable_minA// [min y z]comparable_minC//. by rewrite comparable_minA// 1?comparable_sym. Qed.	Lemma	comparable_minAC	order	order/order.v	[ "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_minA", "comparable_minC", "comparable_sym", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxAC x y z : x >=< y -> x >=< z -> y >=< z -> max (max x y) z = max (max x z) y.	Proof. move=> xy xz yz; rewrite -comparable_maxA// [max y z]comparable_maxC//. by rewrite comparable_maxA// 1?comparable_sym. Qed.	Lemma	comparable_maxAC	order	order/order.v	[ "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_maxA", "comparable_maxC", "comparable_sym", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minCA x y z : x >=< y -> x >=< z -> y >=< z -> min x (min y z) = min y (min x z).	Proof. move=> xy xz yz; rewrite comparable_minA// [min x y]comparable_minC//. by rewrite -comparable_minA// 1?comparable_sym. Qed.	Lemma	comparable_minCA	order	order/order.v	[ "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_minA", "comparable_minC", "comparable_sym", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxCA x y z : x >=< y -> x >=< z -> y >=< z -> max x (max y z) = max y (max x z).	Proof. move=> xy xz yz; rewrite comparable_maxA// [max x y]comparable_maxC//. by rewrite -comparable_maxA// 1?comparable_sym. Qed.	Lemma	comparable_maxCA	order	order/order.v	[ "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_maxA", "comparable_maxC", "comparable_sym", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_minACA x y z t : x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t -> min (min x y) (min z t) = min (min x z) (min y t).	Proof. move=> xy xz xt yz yt zt; rewrite comparable_minA// ?comparable_minl//. rewrite [min _ z]comparable_minAC// -comparable_minA// ?comparable_minl//. by rewrite inE comparable_sym. Qed.	Lemma	comparable_minACA	order	order/order.v	[ "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_minA", "comparable_minAC", "comparable_minl", "comparable_sym", "inE", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_maxACA x y z t : x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t -> max (max x y) (max z t) = max (max x z) (max y t).	Proof. move=> xy xz xt yz yt zt; rewrite comparable_maxA// ?comparable_maxl//. rewrite [max _ z]comparable_maxAC// -comparable_maxA// ?comparable_maxl//. by rewrite inE comparable_sym. Qed.	Lemma	comparable_maxACA	order	order/order.v	[ "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_maxA", "comparable_maxAC", "comparable_maxl", "comparable_sym", "inE", "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable_min_maxr x y z : x >=< y -> x >=< z -> y >=< z -> min x (max y z) = max (min x y) (min x z).	Proof. move=> xy xz yz; rewrite ![min x _]comparable_minC// ?comparable_maxr//. by rewrite comparable_min_maxl// 1?comparable_sym. Qed.	Lemma	comparable_min_maxr	order	order/order.v	[ "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_maxr", "comparable_minC", "comparable_min_maxl", "comparable_sym", "max", "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mono_in_leif (A : {pred T}) (f : T -> T) C : {in A &, {mono f : x y / x <= y}} -> {in A &, forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C)}.	Proof. by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.	Lemma	mono_in_leif	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eq_le", "leif" ]	monotonicity	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mono_leif (f : T -> T) C : {mono f : x y / x <= y} -> forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C).	Proof. by move=> mf x y; rewrite /leif !eq_le !mf. Qed.	Lemma	mono_leif	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eq_le", "leif" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nmono_in_leif (A : {pred T}) (f : T -> T) C : {in A &, {mono f : x y /~ x <= y}} -> {in A &, forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C)}.	Proof. by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.	Lemma	nmono_in_leif	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eq_le", "leif" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nmono_leif (f : T -> T) C : {mono f : x y /~ x <= y} -> forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C).	Proof. by move=> mf x y; rewrite /leif !eq_le !mf. Qed.	Lemma	nmono_leif	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eq_le", "leif" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_le : x0 <= x -> (forall i, P i -> f i <= x) -> \big[max/x0]_(i <- r \| P i) f i <= x.	Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite maxEle; case: ifPn. Qed.	Lemma	bigmax_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",...	[ "big_ind", "max", "maxEle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_bigmin : x <= x0 -> (forall i, P i -> x <= f i) -> x <= \big[min/x0]_(i <- r \| P i) f i.	Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite minEle; case: ifPn. Qed.	Lemma	le_bigmin	order	order/order.v	[ "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_ind", "min", "minEle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_leT b x y : (~~ b -> x < y) -> (y <= x -> b).	Proof. by case: comparableP; case: b. Qed.	Lemma	contra_leT	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_ltT b x y : (~~ b -> x <= y) -> (y < x -> b).	Proof. by case: comparableP; case: b. Qed.	Lemma	contra_ltT	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_leN b x y : (b -> x < y) -> (y <= x -> ~~ b).	Proof. by case: comparableP; case: b. Qed.	Lemma	contra_leN	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_ltN b x y : (b -> x <= y) -> (y < x -> ~~ b).	Proof. by case: comparableP; case: b. Qed.	Lemma	contra_ltN	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_le_not P x y : (P -> x < y) -> (y <= x -> ~ P).	Proof. by case: comparableP => // _ PF _ /PF. Qed.	Lemma	contra_le_not	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_lt_not P x y : (P -> x <= y) -> (y < x -> ~ P).	Proof. by case: comparableP => // _ PF _ /PF. Qed.	Lemma	contra_lt_not	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_leF b x y : (b -> x < y) -> (y <= x -> b = false).	Proof. by case: comparableP; case: b => // _ /implyP. Qed.	Lemma	contra_leF	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_ltF b x y : (b -> x <= y) -> (y < x -> b = false).	Proof. by case: comparableP; case: b => // _ /implyP. Qed.	Lemma	contra_ltF	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_le_leq x y m n : ((n < m)%N -> y < x) -> (x <= y -> (m <= n)%N).	Proof. by case: comparableP; case: ltngtP. Qed.	Lemma	contra_le_leq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "ltngtP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_le_ltn x y m n : ((n <= m)%N -> y < x) -> (x <= y -> (m < n)%N).	Proof. by case: comparableP; case: ltngtP. Qed.	Lemma	contra_le_ltn	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "ltngtP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_lt_leq x y m n : ((n < m)%N -> y <= x) -> (x < y -> (m <= n)%N).	Proof. by case: comparableP; case: ltngtP. Qed.	Lemma	contra_lt_leq	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "ltngtP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_lt_ltn x y m n : ((n <= m)%N -> y <= x) -> (x < y -> (m < n)%N).	Proof. by case: comparableP; case: ltngtP. Qed.	Lemma	contra_lt_ltn	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "comparableP", "ltngtP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leT_anti	:= @le_anti _ T.	Let	leT_anti	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "le_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge_antiT : antisymmetric (>=%O : rel T).	Proof. by move=> ? ? /le_anti. Qed.	Let	ge_antiT	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "le_anti", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltW_homo : {homo f : x y / x < y} -> {homo f : x y / x <= y}.	Proof. exact: homoW. Qed.	Lemma	ltW_homo	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "homoW" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltW_nhomo : {homo f : x y /~ x < y} -> {homo f : x y /~ x <= y}.	Proof. by apply: homoW=> // x y; rewrite eq_sym. Qed.	Lemma	ltW_nhomo	order	order/order.v	[ "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_sym", "homoW" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inj_homo_lt : injective f -> {homo f : x y / x <= y} -> {homo f : x y / x < y}.	Proof. exact: inj_homo. Qed.	Lemma	inj_homo_lt	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "inj_homo" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inj_nhomo_lt : injective f -> {homo f : x y /~ x <= y} -> {homo f : x y /~ x < y}.	Proof. by apply: inj_homo=> // x y; rewrite eq_sym. Qed.	Lemma	inj_nhomo_lt	order	order/order.v	[ "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_sym", "inj_homo" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inc_inj : {mono f : x y / x <= y} -> injective f.	Proof. exact: mono_inj. Qed.	Lemma	inc_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",...	[ "mono_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dec_inj : {mono f : x y /~ x <= y} -> injective f.	Proof. exact: mono_inj. Qed.	Lemma	dec_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",...	[ "mono_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltW_homo_in : {in D & D', {homo f : x y / x < y}} -> {in D & D', {homo f : x y / x <= y}}.	Proof. exact: homoW_in. Qed.	Lemma	ltW_homo_in	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "homoW_in" ]	Monotony in D D'	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltW_nhomo_in : {in D & D', {homo f : x y /~ x < y}} -> {in D & D', {homo f : x y /~ x <= y}}.	Proof. by apply: homoW_in=> // x y; rewrite eq_sym. Qed.	Lemma	ltW_nhomo_in	order	order/order.v	[ "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_sym", "homoW_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inj_homo_lt_in : {in D & D', injective f} -> {in D & D', {homo f : x y / x <= y}} -> {in D & D', {homo f : x y / x < y}}.	Proof. exact: inj_homo_in. Qed.	Lemma	inj_homo_lt_in	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "inj_homo_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inj_nhomo_lt_in : {in D & D', injective f} -> {in D & D', {homo f : x y /~ x <= y}} -> {in D & D', {homo f : x y /~ x < y}}.	Proof. by apply: inj_homo_in=> // x y; rewrite eq_sym. Qed.	Lemma	inj_nhomo_lt_in	order	order/order.v	[ "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_sym", "inj_homo_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inc_inj_in : {in D &, {mono f : x y / x <= y}} -> {in D &, injective f}.	Proof. exact: mono_inj_in. Qed.	Lemma	inc_inj_in	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "mono_inj_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dec_inj_in : {in D &, {mono f : x y /~ x <= y}} -> {in D &, injective f}.	Proof. exact: mono_inj_in. Qed.	Lemma	dec_inj_in	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "mono_inj_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lex0 x : (x <= \bot) = (x == \bot).	Proof. by rewrite le_eqVlt ltx0 orbF. Qed.	Lemma	lex0	order	order/order.v	[ "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", "le_eqVlt", "ltx0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0x x : (\bot < x) = (x != \bot).	Proof. by rewrite lt_def le0x andbT. Qed.	Lemma	lt0x	order	order/order.v	[ "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", "le0x", "lt_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq0_xor_gt0 x : bool -> bool -> Set	:= Eq0NotPOs : x = \bot -> eq0_xor_gt0 x true false \| POsNotEq0 : \bot < x -> eq0_xor_gt0 x false true.	Variant	eq0_xor_gt0	order	order/order.v	[ "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
posxP x : eq0_xor_gt0 x (x == \bot) (\bot < x).	Proof. by rewrite lt0x; have [] := eqVneq; constructor; rewrite ?lt0x. Qed.	Lemma	posxP	order	order/order.v	[ "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", "eq0_xor_gt0", "eqVneq", "lt0x" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le1x x : (\top <= x) = (x == \top).	Proof. exact: (@lex0 _ T^d). Qed.	Lemma	le1x	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lex0", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltx1 x : (x < \top) = (x != \top).	Proof. exact: (@lt0x _ T^d). Qed.	Lemma	ltx1	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lt0x", "top" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lexI x y z : (x <= y `&` z) = (x <= y) && (x <= z).	Proof. exact: lexI. Qed.	Lemma	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",...	[]	interaction with order	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIr x y : y `&` x <= x.	Proof. by have:= le_refl (meet y x); rewrite lexI => /andP []. Qed.	Lemma	leIr	order	order/order.v	[ "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_refl", "lexI", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIl x y : x `&` y <= x.	Proof. by have:= le_refl (meet x y); rewrite lexI => /andP []. Qed.	Lemma	leIl	order	order/order.v	[ "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_refl", "lexI", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIxl x y z : y <= x -> y `&` z <= x.	Proof. exact/le_trans/leIl. Qed.	Lemma	leIxl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leIl", "le_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIxr x y z : z <= x -> y `&` z <= x.	Proof. exact/le_trans/leIr. Qed.	Lemma	leIxr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leIr", "le_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIx2 x y z : (y <= x) \|\| (z <= x) -> y `&` z <= x.	Proof. by case/orP => [/leIxl\|/leIxr]. Qed.	Lemma	leIx2	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leIxl", "leIxr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEmeet x y : (x <= y) = (x `&` y == x).	Proof. by rewrite eq_le lexI leIl lexx. Qed.	Lemma	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",...	[ "eq_le", "leIl", "lexI", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_meetl x y : (x `&` y == x) = (x <= y).	Proof. by apply/esym/leEmeet. Qed.	Lemma	eq_meetl	order	order/order.v	[ "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", "leEmeet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_meetr x y : (x `&` y == y) = (y <= x).	Proof. by rewrite eq_le lexI leIr lexx andbT. Qed.	Lemma	eq_meetr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "eq_le", "leIr", "lexI", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_idPl {x y} : reflect (x `&` y = x) (x <= y).	Proof. by rewrite -eq_meetl; apply/eqP. Qed.	Lemma	meet_idPl	order	order/order.v	[ "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_meetl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_idPr {x y} : reflect (y `&` x = x) (x <= y).	Proof. by rewrite -eq_meetr; apply/eqP. Qed.	Lemma	meet_idPr	order	order/order.v	[ "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_meetr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_l x y : x <= y -> x `&` y = x.	Proof. exact/meet_idPl. Qed.	Lemma	meet_l	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meet_idPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_r x y : y <= x -> x `&` y = y.	Proof. exact/meet_idPr. Qed.	Lemma	meet_r	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meet_idPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIidl x y : (x <= x `&` y) = (x <= y).	Proof. by rewrite lexI lexx. Qed.	Lemma	leIidl	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lexI", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leIidr x y : (x <= y `&` x) = (x <= y).	Proof. by rewrite lexI lexx andbT. Qed.	Lemma	leIidr	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "lexI", "lexx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leI2 x y z t : x <= z -> y <= t -> x `&` y <= z `&` t.	Proof. by move=> xz yt; rewrite lexI !leIx2 ?xz ?yt ?orbT //. Qed.	Lemma	leI2	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "leIx2", "lexI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetC : commutative (@meet _ L).	Proof. by move=> x y; apply: le_anti; rewrite !lexI !leIr !leIl. Qed.	Lemma	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",...	[ "apply", "leIl", "leIr", "le_anti", "lexI", "meet" ]	algebraic properties	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetA : associative (@meet _ L).	Proof. move=> x y z; apply: le_anti. rewrite !lexI leIr leIl /= andbT -andbA. rewrite ![_ `&` (_ `&` _) <= _]leIxr ?(leIr, leIl) //=. by rewrite leIxl ?leIl // leIxl // leIr. Qed.	Lemma	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", "leIl", "leIr", "leIxl", "leIxr", "le_anti", "lexI", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetxx : idempotent_op (@meet _ L).	Proof. by move=> x; apply/eqP; rewrite -leEmeet. Qed.	Lemma	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
meetAC : right_commutative (@meet _ L).	Proof. by move=> x y z; rewrite -!meetA [X in _ `&` X]meetC. Qed.	Lemma	meetAC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meet", "meetA", "meetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetCA : left_commutative (@meet _ L).	Proof. by move=> x y z; rewrite !meetA [X in X `&` _]meetC. Qed.	Lemma	meetCA	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meet", "meetA", "meetC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetACA : interchange (@meet _ L) (@meet _ L).	Proof. by move=> x y z t; rewrite !meetA [X in X `&` _]meetAC. Qed.	Lemma	meetACA	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meet", "meetA", "meetAC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKI y x : x `&` (x `&` y) = x `&` y.	Proof. by rewrite meetA meetxx. Qed.	Lemma	meetKI	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meetA", "meetxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetIK y x : (x `&` y) `&` y = x `&` y.	Proof. by rewrite -meetA meetxx. Qed.	Lemma	meetIK	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meetA", "meetxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetKIC y x : x `&` (y `&` x) = x `&` y.	Proof. by rewrite meetC meetIK meetC. Qed.	Lemma	meetKIC	order	order/order.v	[ "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", "meetIK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meetIKC y x : y `&` x `&` y = x `&` y.	Proof. by rewrite meetAC meetC meetxx. Qed.	Lemma	meetIKC	order	order/order.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "preorder", "Order", "BPreorderTheory", "PreCancelPartial", "OrderMorphismTheory", "NatOrder",...	[ "meetAC", "meetC", "meetxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet0x : left_zero \bot (@meet _ L).	Proof. by move=> x; apply/eqP; rewrite -leEmeet. Qed.	Lemma	meet0x	order	order/order.v	[ "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", "leEmeet", "meet" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> /lt_leif; rewrite negbK. Qed.

Lemma

ltNleif

order

order/order.v

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

[ "lt_leif" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lteif_anti C1 C2 x y : (x < y ?<= if C1) && (y < x ?<= if C2) = C1 && C2 && (x == y).

Proof. by case: C1 C2 => [][]; rewrite lte_anti. Qed.

Lemma

lteif_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",...

[ "lte_anti" ]

lteif

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lteifN C x y : x < y ?<= if ~~ C -> ~~ (y < x ?<= if C).

Proof. by case: C => /=; case: comparableP. Qed.

Lemma

lteifN

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minEle x y : min x y = if x <= y then x else y.

Proof. by case: comparableP. Qed.

Lemma

minEle

order

order/order.v

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

[ "comparableP", "min" ]

min and max

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxEle x y : max x y = if x <= y then y else x.

Proof. by case: comparableP. Qed.

Lemma

maxEle

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minEgt x y : x >=< y -> min x y = if x > y then y else x.

Proof. by case: comparableP. Qed.

Lemma

comparable_minEgt

order

order/order.v

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

[ "comparableP", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxEgt x y : x >=< y -> max x y = if x > y then x else y.

Proof. by case: comparableP. Qed.

Lemma

comparable_maxEgt

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minEge x y : x >=< y -> min x y = if x >= y then y else x.

Proof. by case: comparableP. Qed.

Lemma

comparable_minEge

order

order/order.v

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

[ "comparableP", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxEge x y : x >=< y -> max x y = if x >= y then x else y.

Proof. by case: comparableP. Qed.

Lemma

comparable_maxEge

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

min_l x y : x <= y -> min x y = x.

Proof. by case: comparableP. Qed.

Lemma

min_l

order

order/order.v

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

[ "comparableP", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

min_r x y : y <= x -> min x y = y.

Proof. by case: comparableP. Qed.

Lemma

min_r

order

order/order.v

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

[ "comparableP", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_l x y : y <= x -> max x y = x.

Proof. by case: comparableP. Qed.

Lemma

max_l

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_r x y : x <= y -> max x y = y.

Proof. by case: comparableP. Qed.

Lemma

max_r

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_minl x y : (min x y == x) = (x <= y).

Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.

Lemma

eq_minl

order

order/order.v

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

[ "comparableP", "eqxx", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_maxr x y : (max x y == y) = (x <= y).

Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.

Lemma

eq_maxr

order

order/order.v

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

[ "comparableP", "eqxx", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

min_idPl x y : reflect (min x y = x) (x <= y).

Proof. by rewrite -eq_minl; apply/eqP. Qed.

Lemma

min_idPl

order

order/order.v

[ "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_minl", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_idPr x y : reflect (max x y = y) (x <= y).

Proof. by rewrite -eq_maxr; apply/eqP. Qed.

Lemma

max_idPr

order

order/order.v

[ "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_maxr", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minC : min x y = min y x.

Proof. by case: comparableP cmp_xy. Qed.

Lemma

comparable_minC

order

order/order.v

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

[ "comparableP", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxC : max x y = max y x.

Proof. by case: comparableP cmp_xy. Qed.

Lemma

comparable_maxC

order

order/order.v

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

[ "comparableP", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_eq_minr : (min x y == y) = (y <= x).

Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.

Lemma

comparable_eq_minr

order

order/order.v

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

[ "comparableP", "eqxx", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_eq_maxl : (max x y == x) = (y <= x).

Proof. by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.

Lemma

comparable_eq_maxl

order

order/order.v

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

[ "comparableP", "eqxx", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_min_idPr : reflect (min x y = y) (y <= x).

Proof. by rewrite -comparable_eq_minr; apply/eqP. Qed.

Lemma

comparable_min_idPr

order

order/order.v

[ "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", "comparable_eq_minr", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_max_idPl : reflect (max x y = x) (y <= x).

Proof. by rewrite -comparable_eq_maxl; apply/eqP. Qed.

Lemma

comparable_max_idPl

order

order/order.v

[ "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", "comparable_eq_maxl", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_lteifNE C : x >=< y -> (x < y ?<= if ~~ C) = ~~ (y < x ?<= if C).

Proof. by case: C => /=; case: comparableP. Qed.

Lemma

comparable_lteifNE

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

P

:= comparableP.

Let

P

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_max_minl : max (min x y) z = min (max x z) (max y z).

Proof. move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/=. move: (P x y) (P x z) (P y z). move=> [xy|xy|xy|<-] [xz|xz|xz|<-] [yz|yz|yz|//->]//= _; rewrite ?ltxx//. - by have := lt_trans xy (lt_trans yz xz); rewrite ltxx. - by have := lt_trans xy (lt_trans xz yz); rewrite ltxx. Qed.

Lemma

comparable_max_minl

order

order/order.v

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

[ "lt_trans", "ltxx", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_le_min2 : x <= z -> y <= w -> Order.min x y <= Order.min z w.

Proof. move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw. - exact: le_trans xy yw. - exact: le_trans (ltW xy) xz. Qed.

Lemma

comparable_le_min2

order

order/order.v

[ "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_trans", "ltW", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_le_max2 : x <= z -> y <= w -> Order.max x y <= Order.max z w.

Proof. move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw. - exact: le_trans yw (ltW zw). - exact: le_trans xz zw. Qed.

Lemma

comparable_le_max2

order

order/order.v

[ "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_trans", "ltW", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minAC x y z : x >=< y -> x >=< z -> y >=< z -> min (min x y) z = min (min x z) y.

Proof. move=> xy xz yz; rewrite -comparable_minA// [min y z]comparable_minC//. by rewrite comparable_minA// 1?comparable_sym. Qed.

Lemma

comparable_minAC

order

order/order.v

[ "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_minA", "comparable_minC", "comparable_sym", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxAC x y z : x >=< y -> x >=< z -> y >=< z -> max (max x y) z = max (max x z) y.

Proof. move=> xy xz yz; rewrite -comparable_maxA// [max y z]comparable_maxC//. by rewrite comparable_maxA// 1?comparable_sym. Qed.

Lemma

comparable_maxAC

order

order/order.v

[ "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_maxA", "comparable_maxC", "comparable_sym", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minCA x y z : x >=< y -> x >=< z -> y >=< z -> min x (min y z) = min y (min x z).

Proof. move=> xy xz yz; rewrite comparable_minA// [min x y]comparable_minC//. by rewrite -comparable_minA// 1?comparable_sym. Qed.

Lemma

comparable_minCA

order

order/order.v

[ "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_minA", "comparable_minC", "comparable_sym", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxCA x y z : x >=< y -> x >=< z -> y >=< z -> max x (max y z) = max y (max x z).

Proof. move=> xy xz yz; rewrite comparable_maxA// [max x y]comparable_maxC//. by rewrite -comparable_maxA// 1?comparable_sym. Qed.

Lemma

comparable_maxCA

order

order/order.v

[ "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_maxA", "comparable_maxC", "comparable_sym", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_minACA x y z t : x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t -> min (min x y) (min z t) = min (min x z) (min y t).

Proof. move=> xy xz xt yz yt zt; rewrite comparable_minA// ?comparable_minl//. rewrite [min _ z]comparable_minAC// -comparable_minA// ?comparable_minl//. by rewrite inE comparable_sym. Qed.

Lemma

comparable_minACA

order

order/order.v

[ "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_minA", "comparable_minAC", "comparable_minl", "comparable_sym", "inE", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_maxACA x y z t : x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t -> max (max x y) (max z t) = max (max x z) (max y t).

Proof. move=> xy xz xt yz yt zt; rewrite comparable_maxA// ?comparable_maxl//. rewrite [max _ z]comparable_maxAC// -comparable_maxA// ?comparable_maxl//. by rewrite inE comparable_sym. Qed.

Lemma

comparable_maxACA

order

order/order.v

[ "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_maxA", "comparable_maxAC", "comparable_maxl", "comparable_sym", "inE", "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comparable_min_maxr x y z : x >=< y -> x >=< z -> y >=< z -> min x (max y z) = max (min x y) (min x z).

Proof. move=> xy xz yz; rewrite ![min x _]comparable_minC// ?comparable_maxr//. by rewrite comparable_min_maxl// 1?comparable_sym. Qed.

Lemma

comparable_min_maxr

order

order/order.v

[ "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_maxr", "comparable_minC", "comparable_min_maxl", "comparable_sym", "max", "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mono_in_leif (A : {pred T}) (f : T -> T) C : {in A &, {mono f : x y / x <= y}} -> {in A &, forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C)}.

Proof. by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.

Lemma

mono_in_leif

order

order/order.v

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

[ "eq_le", "leif" ]

monotonicity

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mono_leif (f : T -> T) C : {mono f : x y / x <= y} -> forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C).

Proof. by move=> mf x y; rewrite /leif !eq_le !mf. Qed.

Lemma

mono_leif

order

order/order.v

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

[ "eq_le", "leif" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nmono_in_leif (A : {pred T}) (f : T -> T) C : {in A &, {mono f : x y /~ x <= y}} -> {in A &, forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C)}.

Proof. by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.

Lemma

nmono_in_leif

order

order/order.v

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

[ "eq_le", "leif" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nmono_leif (f : T -> T) C : {mono f : x y /~ x <= y} -> forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C).

Proof. by move=> mf x y; rewrite /leif !eq_le !mf. Qed.

Lemma

nmono_leif

order

order/order.v

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

[ "eq_le", "leif" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_le : x0 <= x -> (forall i, P i -> f i <= x) -> \big[max/x0]_(i <- r | P i) f i <= x.

Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite maxEle; case: ifPn. Qed.

Lemma

bigmax_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",...

[ "big_ind", "max", "maxEle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_bigmin : x <= x0 -> (forall i, P i -> x <= f i) -> x <= \big[min/x0]_(i <- r | P i) f i.

Proof. by move=> ? ?; elim/big_ind: _ => // *; rewrite minEle; case: ifPn. Qed.

Lemma

le_bigmin

order

order/order.v

[ "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_ind", "min", "minEle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_leT b x y : (~~ b -> x < y) -> (y <= x -> b).

Proof. by case: comparableP; case: b. Qed.

Lemma

contra_leT

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_ltT b x y : (~~ b -> x <= y) -> (y < x -> b).

Proof. by case: comparableP; case: b. Qed.

Lemma

contra_ltT

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_leN b x y : (b -> x < y) -> (y <= x -> ~~ b).

Proof. by case: comparableP; case: b. Qed.

Lemma

contra_leN

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_ltN b x y : (b -> x <= y) -> (y < x -> ~~ b).

Proof. by case: comparableP; case: b. Qed.

Lemma

contra_ltN

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_le_not P x y : (P -> x < y) -> (y <= x -> ~ P).

Proof. by case: comparableP => // _ PF _ /PF. Qed.

Lemma

contra_le_not

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_lt_not P x y : (P -> x <= y) -> (y < x -> ~ P).

Proof. by case: comparableP => // _ PF _ /PF. Qed.

Lemma

contra_lt_not

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_leF b x y : (b -> x < y) -> (y <= x -> b = false).

Proof. by case: comparableP; case: b => // _ /implyP. Qed.

Lemma

contra_leF

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_ltF b x y : (b -> x <= y) -> (y < x -> b = false).

Proof. by case: comparableP; case: b => // _ /implyP. Qed.

Lemma

contra_ltF

order

order/order.v

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

[ "comparableP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_le_leq x y m n : ((n < m)%N -> y < x) -> (x <= y -> (m <= n)%N).

Proof. by case: comparableP; case: ltngtP. Qed.

Lemma

contra_le_leq

order

order/order.v

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

[ "comparableP", "ltngtP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_le_ltn x y m n : ((n <= m)%N -> y < x) -> (x <= y -> (m < n)%N).

Proof. by case: comparableP; case: ltngtP. Qed.

Lemma

contra_le_ltn

order

order/order.v

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

[ "comparableP", "ltngtP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_lt_leq x y m n : ((n < m)%N -> y <= x) -> (x < y -> (m <= n)%N).

Proof. by case: comparableP; case: ltngtP. Qed.

Lemma

contra_lt_leq

order

order/order.v

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

[ "comparableP", "ltngtP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_lt_ltn x y m n : ((n <= m)%N -> y <= x) -> (x < y -> (m < n)%N).

Proof. by case: comparableP; case: ltngtP. Qed.

Lemma

contra_lt_ltn

order

order/order.v

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

[ "comparableP", "ltngtP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leT_anti

:= @le_anti _ T.

Let

leT_anti

order

order/order.v

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

[ "le_anti" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> ? ? /le_anti. Qed.

Let

ge_antiT

order

order/order.v

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

[ "le_anti", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltW_homo : {homo f : x y / x < y} -> {homo f : x y / x <= y}.

Proof. exact: homoW. Qed.

Lemma

ltW_homo

order

order/order.v

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

[ "homoW" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltW_nhomo : {homo f : x y /~ x < y} -> {homo f : x y /~ x <= y}.

Proof. by apply: homoW=> // x y; rewrite eq_sym. Qed.

Lemma

ltW_nhomo

order

order/order.v

[ "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_sym", "homoW" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inj_homo_lt : injective f -> {homo f : x y / x <= y} -> {homo f : x y / x < y}.

Proof. exact: inj_homo. Qed.

Lemma

inj_homo_lt

order

order/order.v

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

[ "inj_homo" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inj_nhomo_lt : injective f -> {homo f : x y /~ x <= y} -> {homo f : x y /~ x < y}.

Proof. by apply: inj_homo=> // x y; rewrite eq_sym. Qed.

Lemma

inj_nhomo_lt

order

order/order.v

[ "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_sym", "inj_homo" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inc_inj : {mono f : x y / x <= y} -> injective f.

Proof. exact: mono_inj. Qed.

Lemma

inc_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",...

[ "mono_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dec_inj : {mono f : x y /~ x <= y} -> injective f.

Proof. exact: mono_inj. Qed.

Lemma

dec_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",...

[ "mono_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltW_homo_in : {in D & D', {homo f : x y / x < y}} -> {in D & D', {homo f : x y / x <= y}}.

Proof. exact: homoW_in. Qed.

Lemma

ltW_homo_in

order

order/order.v

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

[ "homoW_in" ]

Monotony in D D'

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltW_nhomo_in : {in D & D', {homo f : x y /~ x < y}} -> {in D & D', {homo f : x y /~ x <= y}}.

Proof. by apply: homoW_in=> // x y; rewrite eq_sym. Qed.

Lemma

ltW_nhomo_in

order

order/order.v

[ "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_sym", "homoW_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inj_homo_lt_in : {in D & D', injective f} -> {in D & D', {homo f : x y / x <= y}} -> {in D & D', {homo f : x y / x < y}}.

Proof. exact: inj_homo_in. Qed.

Lemma

inj_homo_lt_in

order

order/order.v

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

[ "inj_homo_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inj_nhomo_lt_in : {in D & D', injective f} -> {in D & D', {homo f : x y /~ x <= y}} -> {in D & D', {homo f : x y /~ x < y}}.

Proof. by apply: inj_homo_in=> // x y; rewrite eq_sym. Qed.

Lemma

inj_nhomo_lt_in

order

order/order.v

[ "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_sym", "inj_homo_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inc_inj_in : {in D &, {mono f : x y / x <= y}} -> {in D &, injective f}.

Proof. exact: mono_inj_in. Qed.

Lemma

inc_inj_in

order

order/order.v

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

[ "mono_inj_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dec_inj_in : {in D &, {mono f : x y /~ x <= y}} -> {in D &, injective f}.

Proof. exact: mono_inj_in. Qed.

Lemma

dec_inj_in

order

order/order.v

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

[ "mono_inj_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lex0 x : (x <= \bot) = (x == \bot).

Proof. by rewrite le_eqVlt ltx0 orbF. Qed.

Lemma

lex0

order

order/order.v

[ "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", "le_eqVlt", "ltx0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0x x : (\bot < x) = (x != \bot).

Proof. by rewrite lt_def le0x andbT. Qed.

Lemma

lt0x

order

order/order.v

[ "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", "le0x", "lt_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq0_xor_gt0 x : bool -> bool -> Set

:= Eq0NotPOs : x = \bot -> eq0_xor_gt0 x true false | POsNotEq0 : \bot < x -> eq0_xor_gt0 x false true.

Variant

eq0_xor_gt0

order

order/order.v

[ "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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

posxP x : eq0_xor_gt0 x (x == \bot) (\bot < x).

Proof. by rewrite lt0x; have [] := eqVneq; constructor; rewrite ?lt0x. Qed.

Lemma

posxP

order

order/order.v

[ "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", "eq0_xor_gt0", "eqVneq", "lt0x" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le1x x : (\top <= x) = (x == \top).

Proof. exact: (@lex0 _ T^d). Qed.

Lemma

le1x

order

order/order.v

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

[ "lex0", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltx1 x : (x < \top) = (x != \top).

Proof. exact: (@lt0x _ T^d). Qed.

Lemma

ltx1

order

order/order.v

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

[ "lt0x", "top" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. exact: lexI. Qed.

Lemma

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",...

[]

interaction with order

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIr x y : y `&` x <= x.

Proof. by have:= le_refl (meet y x); rewrite lexI => /andP []. Qed.

Lemma

leIr

order

order/order.v

[ "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_refl", "lexI", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIl x y : x `&` y <= x.

Proof. by have:= le_refl (meet x y); rewrite lexI => /andP []. Qed.

Lemma

leIl

order

order/order.v

[ "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_refl", "lexI", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIxl x y z : y <= x -> y `&` z <= x.

Proof. exact/le_trans/leIl. Qed.

Lemma

leIxl

order

order/order.v

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

[ "leIl", "le_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIxr x y z : z <= x -> y `&` z <= x.

Proof. exact/le_trans/leIr. Qed.

Lemma

leIxr

order

order/order.v

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

[ "leIr", "le_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIx2 x y z : (y <= x) || (z <= x) -> y `&` z <= x.

Proof. by case/orP => [/leIxl|/leIxr]. Qed.

Lemma

leIx2

order

order/order.v

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

[ "leIxl", "leIxr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEmeet x y : (x <= y) = (x `&` y == x).

Proof. by rewrite eq_le lexI leIl lexx. Qed.

Lemma

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",...

[ "eq_le", "leIl", "lexI", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_meetl x y : (x `&` y == x) = (x <= y).

Proof. by apply/esym/leEmeet. Qed.

Lemma

eq_meetl

order

order/order.v

[ "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", "leEmeet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_meetr x y : (x `&` y == y) = (y <= x).

Proof. by rewrite eq_le lexI leIr lexx andbT. Qed.

Lemma

eq_meetr

order

order/order.v

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

[ "eq_le", "leIr", "lexI", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_idPl {x y} : reflect (x `&` y = x) (x <= y).

Proof. by rewrite -eq_meetl; apply/eqP. Qed.

Lemma

meet_idPl

order

order/order.v

[ "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_meetl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_idPr {x y} : reflect (y `&` x = x) (x <= y).

Proof. by rewrite -eq_meetr; apply/eqP. Qed.

Lemma

meet_idPr

order

order/order.v

[ "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_meetr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_l x y : x <= y -> x `&` y = x.

Proof. exact/meet_idPl. Qed.

Lemma

meet_l

order

order/order.v

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

[ "meet_idPl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_r x y : y <= x -> x `&` y = y.

Proof. exact/meet_idPr. Qed.

Lemma

meet_r

order

order/order.v

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

[ "meet_idPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIidl x y : (x <= x `&` y) = (x <= y).

Proof. by rewrite lexI lexx. Qed.

Lemma

leIidl

order

order/order.v

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

[ "lexI", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leIidr x y : (x <= y `&` x) = (x <= y).

Proof. by rewrite lexI lexx andbT. Qed.

Lemma

leIidr

order

order/order.v

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

[ "lexI", "lexx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leI2 x y z t : x <= z -> y <= t -> x `&` y <= z `&` t.

Proof. by move=> xz yt; rewrite lexI !leIx2 ?xz ?yt ?orbT //. Qed.

Lemma

leI2

order

order/order.v

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

[ "leIx2", "lexI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetC : commutative (@meet _ L).

Proof. by move=> x y; apply: le_anti; rewrite !lexI !leIr !leIl. Qed.

Lemma

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",...

[ "apply", "leIl", "leIr", "le_anti", "lexI", "meet" ]

algebraic properties

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetA : associative (@meet _ L).

Proof. move=> x y z; apply: le_anti. rewrite !lexI leIr leIl /= andbT -andbA. rewrite ![_ `&` (_ `&` _) <= _]leIxr ?(leIr, leIl) //=. by rewrite leIxl ?leIl // leIxl // leIr. Qed.

Lemma

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", "leIl", "leIr", "leIxl", "leIxr", "le_anti", "lexI", "meet" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetxx : idempotent_op (@meet _ L).

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

Lemma

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

meetAC : right_commutative (@meet _ L).

Proof. by move=> x y z; rewrite -!meetA [X in _ `&` X]meetC. Qed.

Lemma

meetAC

order

order/order.v

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

[ "meet", "meetA", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetCA : left_commutative (@meet _ L).

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

Lemma

meetCA

order

order/order.v

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

[ "meet", "meetA", "meetC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetACA : interchange (@meet _ L) (@meet _ L).

Proof. by move=> x y z t; rewrite !meetA [X in X `&` _]meetAC. Qed.

Lemma

meetACA

order

order/order.v

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

[ "meet", "meetA", "meetAC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetKI y x : x `&` (x `&` y) = x `&` y.

Proof. by rewrite meetA meetxx. Qed.

Lemma

meetKI

order

order/order.v

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

[ "meetA", "meetxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetIK y x : (x `&` y) `&` y = x `&` y.

Proof. by rewrite -meetA meetxx. Qed.

Lemma

meetIK

order

order/order.v

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

[ "meetA", "meetxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetKIC y x : x `&` (y `&` x) = x `&` y.

Proof. by rewrite meetC meetIK meetC. Qed.

Lemma

meetKIC

order

order/order.v

[ "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", "meetIK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meetIKC y x : y `&` x `&` y = x `&` y.

Proof. by rewrite meetAC meetC meetxx. Qed.

Lemma

meetIKC

order

order/order.v

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

[ "meetAC", "meetC", "meetxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet0x : left_zero \bot (@meet _ L).

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

Lemma

meet0x

order

order/order.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d